Discussions on Dai-Freed Anomalies

Dissertation in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE WITH A MAJOR IN PHYSICS

Department of Physics and Astronomy

Huaiyu Li

[15 Oct 2019]

Contents

0.1 Introduction to anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2 Pati-Salam model and Spin(10) Grand Unified Theory(GUT) . . . . . . . 50.3 Dai-Freed anomalies and η-invariants for discrete symmetry . . . . . . . . 6

1 Mathematical and Physical Introductions 71.1 Spin structures and spin bundles . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Tangent spaces and tangent bundle . . . . . . . . . . . . . . . . . . 71.1.2 Clifford algebra and spin groups . . . . . . . . . . . . . . . . . . . 91.1.3 Spin/Pin structures and examples in low dimensions . . . . . . . . 161.1.4 Dirac Operator in Curved and Unorientable Space . . . . . . . . . 21

1.2 Exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.1 Cech cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . 231.2.2 Long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Dai-Freed theorem and Spin bordism 302.1 APS index theorem and Dai-Freed theorem . . . . . . . . . . . . . . . . . 30

2.1.1 Dirac operators revisited . . . . . . . . . . . . . . . . . . . . . . . . 322.1.2 Dai-Freed theorem and η-invariant . . . . . . . . . . . . . . . . . . 34

2.2 Anomalies with the Dai-Freed theorem and Spin bordism . . . . . . . . . 352.3 AHSS and some continuous gauge groups . . . . . . . . . . . . . . . . . . 38

2.3.1 Atiyah-Hirzebruch spectral sequence . . . . . . . . . . . . . . . . . 382.3.2 U(1) and Steenrod square . . . . . . . . . . . . . . . . . . . . . . . 402.3.3 SU(2) and SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Discrete Symmetry Anomalies and Particle Implementations 463.1 An easy way out for twisted Z4 . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 η-Invariant for Pin+(4) . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Calculation of BZn, twisted or not . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 BZn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2 SpinZ2n and SpinZ4 . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.3 Some comments on Z4 and 16 . . . . . . . . . . . . . . . . . . . . . 52

4 Summary 544.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Possible further developments and acknowledgement . . . . . . . . . . . . 54

4.2.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1

Abstract

In both field theories of high energy physics and field theories of condensed mattertheories anomalies have been imposing constraints and bringing up new theories. Whilethe perturbational and local anomalies from triangle diagrams are well-developed, wefollow and review two works leading to non-perturbational global anomalies both relatedwith Atiyah-Patogi-Sieger (APS) index theorem. In the context of topological fieldtheories the APS index theorem imposes an η-invariant which adds a global anomalouse−iπη/2 to the path integral from each fermions on the boundary. In the case of Pin+(4)this will require a multiple of 16 Majorana fermions for the global anomaly to cancel.On the other hand with Dai-Freed theorem, global ’t Hooft anomalies concerned toG symmetry group on 4-dimensional spin manifold which is also classifying space BGare studied. The existence of anomalies are related to the triviality of spin bordismgroup ΩSpin

5 (BG), where examples with SU(2), GSM and Zn are studied with Atiyah-Hirzebruch spectral sequence method and representation of Zn rings. Further there areextensions of spin groups from Pin, SpinZ4 and Spinc, where the anomaly cancellationconstraint of 16 fermions per generation from SpinZ4(4) is related to that of Pin+(4)in topological superconductor. A possible candidate of Z4 charge named B − L chargeor X charge from several commutative breaking patterns of Spin(10) GUT is brieflydiscussed with Pati-Salam model and R⊗ C⊗H⊗O.

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Svensk sammanfattning

Anomalier i fysik ar symmetrier som inte kan beskrivas ordentligt. En symmetri kan iallmanhet vara tydlig och valdefinierad men nar man byter till ett annat system, kommerden inte langre vara med i det nya systemets sprak. I regel betyder det att en typ avsymmetri i det verkliga livet som ar beskriven enligt sunt fornuft, exempelvis en varldi en spegel eller hur tittare ser ett fotbollsspel fran olika sittplatser, verkar overtradasi kvantteorier eller statistisk teori. Forutom att uppge att symmetrin ar bruten, vi kanlosa problemet med anomalier genom att tillagga olika slags krav till systemet for attsymmetrin annu kan bli kvar. De forekommer i kvantteorier nar man forsoker att ”gau-ge” en teori sa att forutsagelser inte kommer att skilja sig fran irrelevanta forandringar,liksom en katt som ar dod fran den ena sidan men livande fran den andra sidan. Enskenbar lojlig losning ar att inte tillata en halva katten att existera. I kvantteorier bety-der detta att man forutsager nya typer av partiklar eller nya slags fysikaliska faser medexotiska och fantastiska egenskaper. Ett fall dar anomalier inte ar sallsynta ar nar manandrar mangder av dimensioner dar en anomali betyder krav for saker att bli flata elleratt forekomma i den fjarde dimensionen. I stangteorin eller inom topologiska faltteorindar dimension ar ett nummer, kommer den satta granser till teorin som maste definie-ras, medan anomalier i var 3D+-tid kan betyda att en ny dimension kan existera medvissa villkor. Processen kan ge oss feedback till var varld genom unikheten av dess pro-cess genom att tvinga atminstone en ny typ av partiklar, s.k. hogerhant neutrino, sommanniskor har sokt under flera decennier att existera. Det kommer ocksa att utnyttjaandra krav pa supersymmetri och mork materia-partiklar som manniskor annu forsokeratt definiera. Ytterligare kommer det ocksa hjalpa oss for att forsta unikhet av var varlddar materia ar 3D och tiden ar 1D.

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Introduction

0.1 Introduction to anomalies

Quantum anomaly is initially to describe a local gauge symmetry failing to preservein a quantum theory, bringing up constraints to the theory in order to reach anomalycancellation. The first and classic case of anomaly from triangle diagrams for abeliangauge groups was indicating the gauge symmetries to be broken in path integrals. Suchanomalies are however not shown explicitly or phenomenologically in a Lagrangian andperturbational theory since it is cancelled within the theory after imposing constraints.Beyond the local symmetries, anomalies were also calculated for global and non-abeliansymmetries. A most famous case was the Witten’s SU(2) anomaly[1] for left-handedWeyl fermions, giving out the condition that doublets of left-handed fermion doubletsmust be even, or number of Dirac fermions must be even. In the mathematical view theexistence of anomalies means the lagrangian and path integral theory to be ill-defined.To the interest of this paper such problems may show up as fixing the partition functionfor its correspondence to path integral and generality among theories.

The traditional way of dealing with anomalies is to treat them as anomaly polyno-mials and a classic example of Atiyah-Singer index theorem for Dirac operators. Fu-jikawa’s method that derived anomalies through path integrals was successful to findcorrespondence between the physical predictive method and topological construction.The discussion upon types of anomalies is then led to whether the intentionally gaugedtransformations can be continuously deformed to identity, where local ones can butglobal ones cannot. The Witten’s anomaly for instance, shows it as the spectral flow ofDirac operator’s eigenvalues that changes signs and therefore affects the signs of actionand partition function. Compactification of d-dimensional flat space to moduli spaceand further d-spheres means that global anomalies are in principle related with homo-topy groups, where maps from the gauge transformations to spatial transformations arenon-trivial. Meanwhile a similar method recognized as mapping torus was introducedfor such constructions, in which case the gauging process describes a non-contractiblecircle on the mapping torus. Anomalies for such systems are studied as specific topolog-ical invariants that can be mod Z2 invariant for Witten’s anomalies, Dirac index whereAtiyah-Singer index theorem applies and η-invariant for Atiyah-Patodi-Singer (APS) in-dex theorem conditions, which is the center of this paper’s studies. Dai-Freed theoremis a generalization of such discussions, relating the anomalies to the APS index of Diracoperators on a manifold with an extra dimensions and the original manifold its boundarywith its influence upon the partition function. Readers with backgrounds may find abovecontents peculiar from not mentioning regularization, since in the interest of this papercancellation of anomalies can be cancelled with Dai-Freed theorem with proper mapping

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tori constructions. Dai-Freed theorem also makes global and local anomalies studied ina similar way, where global anomalies correspond to the boundary of mapping tori andthe mapping tori equivalence for local anomalies appear by adding two dimensions.

0.2 Pati-Salam model and Spin(10) Grand UnifiedTheory(GUT)

Pati-Salam model was introduced as a model of particles where lepton can be understoodas a fourth quark flavor, and a first attempt of grand unification. Up to expansions ofstandard model its gauge group is SU(2)L× SU(2)R × SU(4), where SU(4) breaks intoSU(3)× U(1) and further breaks into the standard model with right-handed neutrinos.In comparison to SU(5) Grand Unification Theory Pati-Salam model does not describe asimple gauge group, but on the contrary presents comprehensive representations betweenfermions and anti-fermions (including right-handed neutrinos), as well as isospin andweak isospin. Another advantage of SU(5) is that it shows the Z6 symmetry of standardmodel comprehensively.SU(5) GUT and Pati-Salam can however, both be included in a more general SO(10)

GUT gauge group, as the nowadays particle physicists acknowledge. Beyond whichhowever, there seems to be a right-handed neutrino that orders the number of fermionsto be 16 or 24. While certain branching rules can explain that, a more natural explanationto it is a group with spinor representations, aka Spin(10) group. The Spin(10) grouphas a ΛC5 exterior algebra, which contains 32 Weyl spinors or 16 Dirac spinors pergeneration. The exterior algebra has a ladder operator reduction to SU(5) and standardmodel, as well as a reduction to Spin(4)×Spin(6) which is another way to describe Pati-Salam model. Readers may want to refer to (add the paper here) for more informationabout the SU(5) and ladder operator formulation, while for the interest of our paper theSpin(4) × Spin(6) Pati-Salam theory allows a (Spin(4) × Z4)/Z2 structure group thatlives in Standard Model and is better defined than the so-called X charge from breakingSO(10) into SU(5).

Aside from the Pati-Salam attempts to grand unification, that Spin(10) belonging tothe representations of Clifford algebra Cl(10) also leads to algebraic models for describingthe representations. One specific case is R ⊗ C ⊗ H ⊗ O, where H is quaternions andO is octonions. In this model C ⊗ O leads to color charge and C ⊗ H leads to (weak-)isospin and chirality. A recent introduction is [2]. This representation theory alsoleads to a Z4 symmetry from B − L charge in standard model. Both Cl(10) modelsare indicating Clifford algebras to be a bigger scenario of particle physics, and with Xcharge or B−L charge the Z4 symmetry is explained before the symmetry breaking. Thisglobal symmetry which is proven to be anomaly-free is however, not guaranteed from thelepton number breaking interactions (for instance, double beta decay). Explanations tothe violation are Higgs doublets/triplets[3, 4] or with dark matter[5]. In the discussion ofthis paper such Z4 global symmetry in 4 dimensions will lead to Z16 in fermion numbers,which will serve as constraints for the extensions to standard model. An original targetis to show this relation between Z4 charge and the mod 16 fermion number in standard

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model.

0.3 Dai-Freed anomalies and η-invariants for discrete symmetry

With APS index theorem anomalies can be spotted with η-invariants, while with Dai-Freed theorem the η-invariants in one more dimensions can describe anomalies. However,in studies started with [6] and more developed by [7], [8] and [9], the construction itselfcan be ill-defined if we don’t require a mapping tori construction. in [7] this is calledDai-Freed anomalies. If the traditional anomalies mean one η-invariant, this type of nowanomalies means a group of η-invariants in the construction.

The group of η-invariants only affect when not being embedded trivially in U(1) forthe phase of partition function. One way of studying it is through its cause, whichare different spin manifolds as the boundary of d + 1-dimensional manifolds endowedwith gauge group G. We call the group of such boundary manifolds spin bordism groupΩSpinn+1 (BG), where BG is the ∞-dimensional classifying space of gauge group G. With

commutative diagrams of related principle bundles the spin bordism group is isomorphicto the homotopy from the manifold X to BG, which means different gauge bundlesexisting and thus a topologically distinct Y manifold where the e−2iπηY determines theanomaly. Such anomalies are unique when we don’t allow topology changes, when theyare allowed they reduce to traditional types of anomalies.

Leading the search into ΩSpin∗ (BG) groups mean we can leave the discussions totally

to algebraic topology. A method is spectral sequences [10] for calculating out out spinbordism. The spin bordisms are generalized homology theories, while the spectral se-quences are the process of generalizing under infinite (co)homology deductions. Adamsspectral sequence is the general method, with one variant called Atiyah-Hirzebruch spec-tral sequence which converges fast in low dimensions can be used for ΩSpin

n (BG) wheren ≤ 10. We will go through some simple cases, which will show some uniqueness to our4D world to be free from most of anomaly variants.

With discrete symmetry Zn however, discussions might not be that direct. RelevantBGs are lens spaces, where the η-invariants are already discussed in [11] with the tra-ditional calculations of anomaly polynomials nearly discussed again. However, based onrelated studies we can directly discuss on such η-invariants in specific representations.The constraints from Dai-Freed anomalies will restrict amount of fermions and chargesin a particle theory assuming Zn symmetry.

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1 Mathematical and Physical Introductions

1.1 Spin structures and spin bundles

This section is focused on an attempt to introduce spin structures and Dirac algebra froma natural ground-up way through tangent bundles and Clifford algebra of a vector bundle.Due to the finite background of the author the section is not targeted to discuss upona general Pin structure on a general space analytically without defects. Readers withinterest may also want to refer to [12] and its reference lists as well-defined introductions.Chapter 33-40 of Srednicki [13] provide a well grounded description of cases included inthis section in oriented Lorentz spacetime with Clifford algebra a “black box” and fiberbundles not a necessity.

1.1.1 Tangent spaces and tangent bundle

On a differential manifold M , given a point x we can draw infinite amount of linesthrough x. Of all lines choose certain N different lines that cannot linearly define eachothers but together can linearly describe all the lines, we can arrange them in a linearspace of vectors so that they are perpendicular to each other under the covariant productsof the coordinate system. Choose for instance, the coordinates at and around (does notmatter in our discussion even if it is global) x as x = xµ, to write the set of vectors asX = eµxµ, xµ ∈ x. For a specific line we can then parametrize it as c(t) and introducethe basis of vectors, to get a vector for the specific line in the very basis. Moreover, fora differentiable function f(c(t)) on a curve that the line belongs to, we first notice itsindependence from choices of X or x. A tangent vector for the generic function at t = t0is defined as follows:

X[f(c(t0))] = Xµ ∂f(c(t))

∂xµ

∣∣∣∣t=t0

=dxµ(c(t))

dt

∂f(c(t))

∂xµ

∣∣∣∣t=t0

=∂f(c(t))

∂t

∣∣∣∣t=t0

(1.1)

The point is, we split the part of vector coordinates from that of manifold coordinates,and following the convention mark them as X = Xµeµ, where eµ = ∂

∂xµ . The space iswell-known to be called a tangent space TxM .

It is not implicit that tangent spaces are not restricted to a choice of coordinates. Fordifferent coordinates we can define a whole bunch of tangent spaces locally on x. Tobetter describe the method we introduce fiber bundles and tangent bundles for tangentvectors. The manifold M is called the base space and the surjection from TxM to x iscalled the projection π. Groups of possible tangent vectors at x is called the fiber groupF and TxM is the fiber at x. We will come to notice that different choices of coordinatescan result π−1 to act differently on x, say x ∈ Ui ∩ Uj where Ui is an open covering

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of M , a specific vector v ∈ TxUi in TxUj will go through a transition function tij thattransforms vector basis for the two covers. The group of transition functions is a Liegroup called the structure group G and the transformed fiber is GF . We call the spacesof all the fibers on M the total space E. Finally, the well-acknowledged notation of afiber bundle is as follows:

F : Eπ−→M also written as F → E

π−→M (1.2)

Besides we also have for Ui a local map φi as a local trivialization so that π(φi(p, f)) = pand tij = φ−1

i φj . The bundle is trivial if local trivializations can be brought to a samefunction and thus transition functions being identities. The fiber bundle of tangentvectors is called tangent bundle, noted as TM . For out interest and convenience, we willthen use a basis of tangent vectors in its own space, signed with eµa where µ is the localcoordinates on M and a is the coordinate of vectors that are orthonormal in Euclideanmetrics. That is to say, such relations exist:

eµaeνbgµν = δab, δabeµae

νb = gµν (1.3)

We may still need to specify the relation of indices ijk and µνσ for vector bundles andcoordinates to avoid further confusions (also to Euclidean basis). We have for instance,a joint section Ui ∩ Uj , where coordinates are xµ on Ui and yµ on Uj . We have localtrivializations for vector spaces V = V mu∂xµ = V µ∂yµ written as, for a point p on Mand u on TM where π(u) = p:

φ−1i (u) = (p, V µ), φ−1

j (u) = (p, V µ) (1.4)

The general denotation about fiber bundle is (E, π,M,F,G) and considering local coor-dinates and coverings, (E, π,M,F,G, Ui, φi) for a specific covering map (also knownas a coordinate bundle). Such notations are in conformity with [14] and materials withdifferent orders of terms like [15] can be easily guessed out.

And naturally group of transition functions is that of the coordinates transitions. Wewill see the further uses in the next section. Before moving on, some specific cases ofbundles are needed throughout the chapters as basic concepts.

Principal bundles and Frames

Principal bundles are a special case with fiber bundles, where G = F so the fiber isthe same as structure group. The notation is further shortened to P (M,G) or simplyG bundle over M . Apart from the left transition GF we can then define how righttransition FG acts within a principal bundle. For any a ∈ G and u ∈ π−1(p), if theconventional local trivialization gives φ−1(u) = (p, gi) in the case of principal bundle,then we define the action of a on u as:

φ−1i (ua) = (p, gia) ⇔ ua = φ1(p, gia) (1.5)

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For any local trivialization ijk . . . as a property of principal bundle and permits a yetsimpler denotation P ×G→ P or (u, a) 7→ ua. Of the properties of principal bundle wemay need that it is transitive with π, which is:

π(ua) = π(u) (1.6)

In definition of spin structure and Clifford algebra, we will need such property for as-sociated bundle. Principal bundles have a canonical local trivialization si(p) = φi(p, e)which transits directly as si(p) = sj(p)tij(p).

For vector bundles on a manifold we define frame bundles. Locally over Ui a frame orlinear n-frame is defined as linearly independent sections1 of the vector bundle. A mostsimple case is the frame given by

∂∂xµ

over our tangent bundle, where the orthonormal

basis which we call an orthonormal frame is a solid case. The group for transformationsbetween linear n-frames is naturally general linear group GL(n,R) when considering nreal dimensions of the manifold. Noting the group G to act right of the frame eβ = eαG

αβ

and the transition function is G−1, we have a principal bundle over M . With notationsthat the set of frames being L(M) the frame bundle on a n-manifold is the principalbundle L(M,GL(n,R)).

The concept of frame bundles can be further extended to a K−vector bundle whereK = R, C or H2. We name it L(E) as the frame bundle of vector bundle E, and thestructure group related follows as GL(k,K) for K-vector bundle of rank k. Specifically,note L(E) =

⋃m∈M Lm for fiber m, when taking basis to be orthonormal Om we obtain

the orthonormal frame bundle O(E). Narrowing down to tangent bundle it is O(M),and an oriented case imposes the order over O(M) referred to O+(M) which reduces itto the SO group thereafter.

So far what we need about fiber bundles and tangent bundles are explained, moreadvanced contents included will be specified in the following chapters.

1.1.2 Clifford algebra and spin groups

We may instantly notice that for a tangent bundle on a manifold, rotation group thatpreserves the direction should naturally be a subgroup of G. For instance, we can con-struct the tangent bundle we wrote above on a Minkovski spacetime with structure groupSO(1, 3) There may be invariants preserved by the tangent bundle from moving alongloops on the manifold or reversing some of the directions of coordinates, for instance.We may then need to define a structure and related bundle that we call a spin bundlefor describing it. Depending on types of the values preserved by the structure group itcan be Z2 spin which we are familiar with on an orientable manifold, a spin-c group ingeneral for a U(1) conserved value. Classically the reader might have know well about

1A continuous map s from base space to total space that satisfies π s = id, a global section in aprincipal bundle is allowed only if the bundle is trivial.

2Researches for instance [16] are also interested on O the octonion bundles, aka. G2 in SUSY andstring theory. Since attempts including [2] are relating standard model with octonions we note them tobe also applicable in frame structure.

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a spin existing from quantization of angular momentum or from a Lorentz group repre-sentation, yet for definitions of a generic manifold and expanded class of values, we mayhave to dig into Clifford algebra and spin structures. Such methods should both apply toa physical spinor and the (weak)isospin in particle physics and all Riemannian metrics.Beyond spin group and spin structures discrete symmetries can extend them to a Pinstructure, which does not require the manifold to be orientable. In this part, we concernupon the formulation of related Spin and Pin groups, with structures introduced withfiber bundle language in the next subsection. This section follows Chapter 5 in [15] and[17].

Clifford algebra and exterior algebra

We may consider first a quadratic form for a vector space. Consider on a vector space Vover commutative field K and a quadratic form q(v) for an element of v, which we calla quadratic space (V, q). We consider the tensor algebra

T (V ) ≡∞⊕k=0

(k⊗V )

And using the form v ⊗ v − q1T (V ) we can generate the two-sided ideal, marked byJq(V ). The Clifford algebra is the quotient algebra T (V )/Jq(V ), noted as Cl(V, q). Wehave a linear mapping j : V → Cl(V, q), which fulfills naturally:

j(v)2 − q(v) ∗ 1 = 0 (1.7)

In other words, V is a linear subspace of the Clifford algebra. Under certain polarizationsof vectors and the quadratic form, we obtain a bilinear form for the projection andanticommutation condition:

j(u), j(v) = 2g(u, v) = q(u+ v)− q(u)− q(v) (1.8)

Where g(u, v) is the bilinear metric for the vectors space, which also applies to the metricof tangent space introduced before. For Clifford algebra we need these properties for ourfurther introduction:

1. Universal property: For the linear mapping F : V → A where A is a unital asso-ciative algebra s.t. F (v)2− q(v)1 = 0, it will extend to the unique homomorphismF : Cl(V, q)→ A where F = F j.An important corollary is that, the Clifford algebra Cl(V, q) is unique up to amisomorphism. It means that algebras linear with the linear mapping to V directlyor under the unique extension are all isomorphic to Cl(V, q).

2. Parity automorphism, canonical anti-automorphism and Z2 grading: There existsan automorphism in a Clifford algebra named parity automorphism p : Cl(V, q)→Cl(V, q) that has p j(v) = −j(v) and thus p2 = id. The parity automorphismimplies a Z2 grading of the Clifford algebra, written as:

Cl(V, q) = Cl0(V, q)⊕ Cl1(V, q) (1.9)

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Where Clk(V, q) means the elements a of Cl(V, q) where p(a) = (−1)ka. We thensee this module 2 grading between even and odd. Specifically, the indices can besummed:

Cli(V, q) · Clj(V, q) ⊂ Cli+j(V, q) (1.10)

The Z2 grading will also make Clifford algebra called a “superalgebra”, whichreaders with supersymmetry backgrounds might be familiar with. Apart from theautomorphism there is also a canonical anti-automorphism that writes as t(a) ≡aT :

t(a · b) = t(b) · t(a)

It is apparent that t2 = id and t commutes with p. We will further explain whatit is within spinor part.

3. Sum of spaces: For the sum of quadratic spaces, we have the isomorphism:

Cl(V1 ⊕ V2, q1 ⊕ q2) ∼= Cl(V1, q1)⊗Cl(V2, q2) (1.11)

Where the Z2 graded tensor product generates the odd and even grades with suchrelation:

(A⊗B)0 = (A0 ⊗B0)⊕ (A1 ⊗B1), (A⊗B)1 = (A0 ⊗B1)⊕ (A0 ⊗B1) (1.12)

We may also consider an exterior algebra ΛV . It is defined as a specific case of Cl(V, q)with q = 0. It is given by an orthonormal basis ei mentioned in section 1.1 from therelation ei, ej = 2η(ei, ej) and the isomorphism Cl(V, q) ∼= ΛV :

ei1 · ei2 · · · · · eik → ei1 ∧ ei2 ∧ · · · ∧ eik (1.13)

From the Z grading of exterior algebra, we can write a Cl(V, q) =⋃iCli(V, q) and the

mapping written with exterior algebra under the inverse isomorphism c : ΛV → Cl(V, q):

Cli(V, q) =i⊕

k=0

c(ΛkV ) (1.14)

Specifically, the Clifford algebra we concern about is on Rn and Cn. While the latter isa complexification of the former, the former one can be introduced in a Minkovski spacewhere the quadratic form is, for n = r + s:

q(x) = x21 + x2

2 + · · ·+ x2r − x2

r+1 − · · · − x2r+s (1.15)

With the corresponding ”pseudo-orthogonal” Clr,s. Specifically, we note Cln or for s = 0and the ”dual” Cl∗n for r = 0. The complexification to complex space of a Clifford algebrais marked as Cl(VC, qC) and has the isomorphism Cl(VC, qC) ∼= Cl(V, q)⊗R C. We notethat for complex numbers unique and non-degenerate quadratic forms are always sumof squares (otherwise a minus will cause it to be just an extra i). Thus we can write forthe Z-grading a general Cl(n) = Clcn = Cl(Cn, q) with the isomorphism:

Clcn∼= Clr,n−r ⊗R C (1.16)

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The isomorphism indicates the different Clr,s with identical r + s being just differentrealization of Cl(r + s). Before digging into explicit cases with spin groups and gammamatrices, we have some relations for computation of higher indices:

Cl0,n+2∼= Cln,0 ⊗ Cl0,2

Clr+1,s+1∼= Clr,s ⊗ Cl1,1

Cl0r+1,s∼= Cls,r

(1.17)

Which all survive exchanging indices. We may note in addition that for Clifford algebraon real spaces, it is apparent that a Z8 periodic isomorphism exists, noted as:

Cl0,n+8∼= Cl0,n ⊗ Cl0,8 (1.18)

We may now move to introduction of a spin group and show how the transitionsalternate properties of spinors in different dimensions.

Spin and Pin from Clifford algebra

From the above Clifford algebra constructions, we can consider a Lie group structurefrom the invertible elements of Clifford algebra. The set of invertible elements are thosewith nonzero q(v) and therefore v−1 = v/q(v). With such “nonzero” terms marked byCl(V, q)∗ that has the group structure, and a subsequent Lie group that is called theClifford group:

Γ(V, q) =a ∈ Cl(V, q)∗|∀v ∈ V,p(a)va−1 ∈ V

(1.19)

With definition it has a twisted adjoint representation to an automorphism on V , notedas Ad : Γ(V, q)→ Aut(V ) and acts on algebra as:

Ad(a)v = p(a)va−1 (1.20)

Now we show it to act on the vector space as reflections. That is, given an invertiblevector v ∈ V , for any w ∈ V , we have:

Ad(v)w = −vwv−1

= −vwvq(v)

=v2w − 2η(v, w)v

q(v)= w − 2

η(v, w)

qv

(1.21)

Which means a reflection on the direction of v. The kernel of Ad is shown below to bewith the same structure as identities in Cl(V, q), which is exactly non-zero terms K∗.For a to be the kernel, we have p(a)va−1 = v or equivalently p(a)v = va. Splitting toeven and odd grades we have further:

a0v = va0, a1v = −va1

12

Now consider the q-orthogonal basis ei, suppose the reflection to be upon v = e1,we can write a0 = p0 + e1p1 where p is generated by components orthogonal to e1 andwithout doubt p0 even/p1 odd under parity, we act it on v:

(p0 + e1p1)e1 = e1p0 − e21p1 = e1(p0 − e1p1)

Which means, for it to be a kernel, e21p1 = 0. Since e2

1 = a(e1) we have p1 = 0. Itis apparent that during this calculation the term with different parity from a is zero.Therefore in a1 = p1 + e1p0 we have p0 = 0. Since it can be applied to all the vectors,we have then the conclusion that a contains no vector term and therefore only K∗ ones.In addition, we have a relation between the reflection and parity, shown by:

Ad(p(a))v = p(p(a))vp(a−1) = −p(Ad(a)v) = Ad(a)v

Which gives the relation Ad(p(a)) = Ad(a). Therefore we find that for a ∈ ker(Ad),

p(a) ∈ ker(Ad) as well. Further, for any a acted with a−1 on the left a−1p(a) is also a

kernel of Ad, which means a−1p(a) ∈ K∗. We then have p(a) = ka, k ∈ K∗ and furtherk2 = 1 explicitly. Parity then is shown to be acting linearly on an element of Γ(V, q)and with definite value. We now show how these properties lead to rotation groups andspinor groups.

By a natural guess from the fact that the kernel of Ad being the K∗ elements, we canrelate Γ(V, q) with a certain group through Ad. Now calculate the inner product:

2η(Ad(a)v1, Ad(a)v2) =

Adv, Adw

= p(a)v, wp(a−1)

= 2η(v, w)

With an orthogonal group for vectors, we obtain the reflection with respect to an element.That is, we have the fiber bundle structure:

K∗1Cl(V,q) → Γ(V, q)Ad−−→ O(V, q) (1.22)

Where since k = 1 for the even terms Γ0(V, q) = Γ(V, q) ∩ Cl0(V, q)∗, they correspondwith the special orthogonal group SO(V, q).Now we can finally start the definition of spin groups and pin groups. We define themas elements of Γ0 and Γ with norm 1 respectively, noted as follows:

Spin(V, q) =a ∈ Γ0(V, q)|N(a) = 1

(1.23)

Pin(V, q) = a ∈ Γ(V, q)|N(a) = 1 (1.24)

Where we define the norm in the kernel of Ad as N(a) = aa where a = t p(a). Anapparent relation here is N(v) = −q(v) in order to preserve a(v). The norm is preservedunder parity and algebraic multiplication. We now show it being the kernel, using

13

following deduction:

Ad(aa)v = p(aa)v(aa)−1

= t(a)(p(a)va−1)a−1

= t(a)t(p(a)va−1)a−1

= t(a)t(a)−1vaa−1

= v

For the Clifford algebras discussed above, impose the spin groups on them we will obtainthe spin groups for orthogonal groups of (r, s) type. We naturally realize the isomorphismamong the same n = r + s for Spin/Pin groups as well. In the next part we will show adifference from dimensions and a greater spin structure.

Spin and Pin groups in different dimensions

Now we note, the even and odd n = r + s have a difference in their isomorphism toclassifying spaces. We start with n = 1 and n = 2. For n = 1 there is one vector andthe structure of algebra is in C ∼= R ⊕ R. Since R ⊗R C = C ⊗ C, when considered onClc1 groups they are uniformly C ⊗ C. Now with n = 2 then we have a R2 group orquaternion group H for r + s = 2, which certainly both complexify to C2. We calculatethe general rules of the isomorphisms, from the following deduction:

Clcn+2∼= (Cln,0 ⊗ Cl0,2)⊗R C ∼= (Cln,0 ⊗R C)⊗C (Cl0,2 ⊗R C)

We have such relations for higher n:

Clcn+2∼= Clcn ⊗C C2, Clc2n

∼= C2n , Clc2n+1∼= C2n ⊗ C2n (1.25)

Which means the even ”amount” or even dimensional vector spaces are irreducible whileodd dimensional ones can be reduced to two irreducible subspaces. In correspondenceto Clc complexification, we define the Spinc group as the subgroup of Clcn = Cln ⊗ C,so that:

Spinc(n) ∼= (Spin(n)× U(1))/Z2 (1.26)

Where moving the Z2 grading to Spin we have Spinc(n) ∼= SO(n) × U(1). Considerthen Spin(n) ⊂ (Clcn)0 and Pin(n) ⊂ Clcn, together with (Clcn)0 ∼= Clcn−1, we say forn > 2, odd dimensions have only one irreducible Spin representations but two Pin rep-resentations, while even dimensions have one Pin but two Spin representations. Shortlywe will present some examples. Here we may also notice that Clc has the Z2 gradingother than that of Z8 for Clp,q. Though there being various results from Z8 discussedlater that lead to existence of a Pin− structure related to dimensionality, we may notea general Z2 division for a general Pin group, which we will right bring up.

Pina,b,c and Pin±

For a Pin group, there is a way to extend its definition and obtain the Pin groups we needin this paper. From the Z2 mapping of Pinp,q → Op,q, there are symmetries resulted

14

from the (anti-)automorphisms mentioned above. First we note on a real K it is alsopossible to write:

Spinp,q =a ∈ Γ0

p,q|N(a) = ±1

(1.27)

Pinp,q = a ∈ Γp,q|N(a) = ±1 (1.28)

Through which claim, they becometh double coverings of SO(p, q) and O(p, q) groupsrespectively. For a double cover of O(p, q) or O(n,C) groups, consider the orthogo-nal group O which contains such connected components: One O0 which is connectedto identity and its variants through parity transformation P , time reversal T and thecombination PT , it is well known that O is the union of these components:

O = (O0) ∪ (PO0) ∪ (TO0) ∪ (PTO0) (1.29)

Where apparently P , T and PT all square to 1. While we have 1, P, T, PT ∼= Z2⊗Z2

and thereafter O01, P, T, PT ∼= O0Z2⊗Z2. Now to impose another Z2 we can seekfrom squares of discrete symmetries. In accordance to the Spin0(p, q) group which isconnected, such isomorphism exists from the semidirect product to describe the Pina,b,c

group, where a, b, c are Z2 valued for the parity of squared symmetries:

Pina,b,c(p, q) ∼=Spin0(p, q) Ca,b,c

Z2(1.30)

The conventional notation for Pina,b,c are three ± for the sign of P 2, T 2 and (PT )2

respectively. Readers may instantly notice that whether PT = TP or PT = −TP isdirectly related with c = ±ab. First we consider a general Pinc(n) and matrix represen-tations of algebra/transformations. To illustrate it clearly we write3 (which also appliesfor real F):

A∗ = WAW−1, A = EATE−1, A∗ = CAC−1 (1.31)

So that we have Aut(Cl) = I,W,E,C, specifically for n = 2m so that such automor-phisms are for an irreducible representation of Cn algebra. From the fact that the parityautomorphism can be expressed by the volume product we can build E and C with fixedW forms in the matrix representation. In the complex case things are simpler whereby noting n = 2m where m can be even or odd and with theorem 2 and 3 of [17], Pingroups are proved to be either (+,+,+) or (−,−,−), summarized as (εi is the γ matrixrepresentation of basis ei):

1. For oddm, W = ε1ε2 . . . ε2m. Of all 2m fundamental basis ε1, . . . , εm are even underreversion and odd under conjugation while εm+1, . . . , ε2m the other way around.We note E = ε1 . . . εm and C = εm+1 . . . ε2m to fulfill the condition. Exchangingthe indices we conclude W 2−1 and non-trivial items in I,W,E,C anti-commute.Up to group isomorphism we note it as Aut+(Cn) = Q4/Z2 which is quaterniongroup and signature (−,−,−), namely Cliffordian group Pin−,−,−(n,C).

3Here in consistency with other materials, we use A for canonical anti-automorphism and A∗ fort p, which follow though in later chapters.

15

2. For even m, W = ε1ε2 . . . ε2m. Accordingly for the same parity under reversion andconjugation we note E = εm+1 . . . ε2m and C = ε1 . . . εm. The items in I,W,E,Cthen commute and are noted as Aut−(Cn) = Z2 ⊗ Z2. The pin group is callednon-Cliffordian group Pin+,+,+ with signature (+,+,+).

For field F = R with Clifford algebra Clp,q, similar analysis can be done with matrixrepresentations of related discrete symmetries. Despite the complexity of a Z8 groupto describe in detail, we can likewise divide through p − q mod 8 and discuss aboutpossible pin groups. The general technique is called Dabrawski groups, while for ourinterest there exists two types of pin groups. Consider p − q = 0, 1, 2, 3, 4 mod 8, byexchanging p and q or equally from Pinp,q to Pinq,p we note Pin+ and Pin− groupsrespectively. We may as well, note the usual chart for Pin± that we refer to are relatedto the spatial reflection R, or the twisted adjoint automorphism.

A more developed way in the context of mathematical physics and string of describingthe existence of Pin± groups is through Stiefel-Whitney groups, which we will discussabout in the following subsection. For readers confused with the automorphisms andeightfold groups the reflection Pin± group can be viewed simply as a Z2 covering ofSpin group and exists on an unorientable manifold as well. Since this legality of thetopic on anomalies is determined on the existence and isomorphism of spin/pin repre-sentations other than an exact form (which we however will give an example), we mayskip the discussion upon representations considering the information about the amountsof irreducible representations being 1 or 2 when existing is our furthest interest.

1.1.3 Spin/Pin structures and examples in low dimensions

In this subsection we discuss upon the existence of Spin, Spinc and Pin± structureson manifolds, together with the low dimensional cases mentioned in Witten[6] and ourinterest. Information about associated (vector) bundle and line bundle for the extensionto Spinc and Pin± are introduced, whereas its usage should also be in the spinor/Diracbundle in the next subsection.

Associated bundles

Since a principal bundle permits a right action from G, when taking account anothermanifold F where G can act on the left we have on P × F the action:

(u, f)→ (ug, g−1f) , u ∈ P, f ∈ F (1.32)

If we consider G to be acting trivially and identified, with the equivalence class (P×F )/Gthe associated bundle is noted as P ×G F , whose local trivialization follows that of P .We can consider G with a specific representation ρ, which will clarify the concept ofassociated vector bundle P ×ρ V :

P ×ρ V = P × V/ ∼, (u, v) ∼ (ug, ρ(g)−1v)

Readers may be aware that the frame bundle is a specific case of P ×ρ V where G isGL(k,K).

16

For a closed subgroup H ⊂ G of G, naturally there exists the associated bundleP ×G G/H which is isomorphic to the quotient P/H and usually called the H-bundleof P . We can well perform it on the frame bundle L(E) where we call the reductionto H ⊂ GL(k,K) an H-structure on E. It has a bijective morphism to the smoothsections of the H-bundle L(E) ⊗GL(k,K) (GL(k,K)/H). The existence of such groupscan be viewed as topological properties of the manifold or vector bundle e.g. specificStiefel-Whitney class(es) for H = Z2 and resultant (S)pin structures.

Generalities of Spin structures, Spinc structures and Pin structures

In the language of fiber bundle, consider manifold M on which the total space froma vector bundle is E. We consider an orthogonal frame over the total space O(E)and a principle bundle over M with structure group S(E). For the spin structure toexist (which is however, not necessary for a pin structure), the manifold4 should beorientable so that O(E) can be reduced to oriented frame O+(E). A spin structure isa pair (S(E),Λ), where Λ : S(E)→ O+(E) is the two-sheeted covering map on O+(E).Having a Spin(n) structure group and the spin structure then means that the vectorbundle is divided with a Z2 subgroup that tells the directions preserved with continuoustransitions. Stiefel-Whitney classes are used to describe the phenomena with grouptopology, which we will cover in next part.

In analogous to the associated bundles, the spin structure is considered as an H-structure. The homomorphism Spin(n)→ SO(n) has a Z2 fiber structure, which trans-lated to Λ we can express in the language of associated bundles O+(E) = S(E)×Spin(n)

SO(n). With that we can construct S(E) as isomorphism on E expressed as (S(E), φ)being the pair on E, under such deduction:

E ∼= O+(E)×SO(n) Rn ∼= S(E)×Spin(n) Rn (1.33)

A Spinc structure is defined similar to Spin structure, with Spinc bundle Sc(E) overM . In the case of a Spinc bundle, the bundle morphism Λ : Sc(E) → O+(E) shouldhave a U(1) subgroup. Noting a principal U(1) bundle over M as P , we can explain iteasily by showing a two-fold covering5.:

Sc(E)→ O+(E)×M P (1.34)

The existence of a Spinc structure to the level of fiber bundle is given by proposition5.4.16 of [15], which we give as follows: Consider a complex associated line bundleL := P ×σ C over M and an oriented Riemannian vector bundle E over M , the Spinc

structure is allowed iff there exists a choice of L so that E ⊕ L admits a spin structure.The proposition above is consistent to the definition of Spinc group. In next subsection

the condition is expressed with the language of Stiefel-Whitney classes. Now we turn tothe Pin structures (Pinc is however, off the concern of our topic).

4In the context should be E, yet for the tangent bundles it can be equivalently noted to M , whichis more common in papers by physicists.

5Here the cross product is the fiber product of two principle bundles G1 and G2, with structuregroup G1 ×G2

17

With Pin structures, we follow [18] which is consistent to Witten[6] and Tachikawa[8].From the fact that Clifford algebra can be divided into Cl+ and Cl− regarding whethereIeI = ±1 and equivalently a1, a2 = ±2 < a1, a2 >, which will affect the twistedadjoint when written as transposed form other than the reversed, which we write asfollows:

Ad(a)v =

avt(a) , a ∈ Cl−

p(a)vt(a) , a ∈ Cl+(1.35)

The reader might remember the previous eightfold discussions upon Pinabc, while heretaking the vector frame to be Euclidean and orthonormal we can simply divide Pin±

from Cl± for all dimensions. As a double cover to S(E) → O+(E), we can think ofthe O(n) and the natural extension to O(n + 1) by adding an item at An+1,n+1. Wecan further add m terms to make it An+m. From [8] there is a Z4 division upon addeddimensions6 which arises Pin± for adding 1 or 3 dimensions as well as SpinZ4 for adding2. We note that apart from O(n,m) our discussions only apply to O(n)± Im. Considerthen adding one real line bundle7 ε over vector space V or specifically tangent spaceTM , which will then transform with ±1 depending on the loop of the path (or we callholonomy). We say that for the case of determinant line bundles, the spin structure onV ⊕ε is the Pin− structure on V or TM . Now consider a Pin+, which has a Z4 structurein the domain of reflection, we have V ⊕ ε⊕ ε⊕ ε to represent its matrix extension ±I3.On a trivial group we may note also that ε has naturally Pin+ and for ε⊗ ε⊗ ε a naturalPin1. A typical case of Pin− structure is a real projective space RP2, which we willdiscuss soon as examples on low-dimensional manifolds.

One dimensional

Before discussion upon a specific dimension we take a look at spinors. Spinors areelements of Clifford modules called spin modules, where irreducible Clifford algebrarepresentations are behaving as endomorphisms into irreducible modules. We use theγ matrices representations in the spinor representations of spin/pin, where since thegamma matrices are invertible it is an automorphism over endomorphism. Inheritedfrom the last subsection, for a Clc2n+1 as well as (Clc2n+2)0, Pinr,s (r + s = 2n + 1) orSpinr,s (r+ s = 2n+ 2) there are two irreducible representations of basis dimension 2n,while for the other case there is only one irreducible representation of basis dimension2n.

We may also consider how reflection acts on spinors. For the reflection with regardsto vector wi, its action on spinors ψ can be represented as:

ψ → aγiwiψ |a| = 1 (1.36)

6For O(n) and we have to note (3.2) of [8] has not reversed Pin+ and Pin−, where readers may takethe Z4 equivalence for the difference between added dimensions e2 = ±1.

7Readers dissatisfied to the narrative may want to refer to tautological real line bundle or doublecover, to which materials from nLab will be of general help.

18

In which representation, a = ±1 or usually 1 for a Pin+ and a = ±i for Pin−. For theconventional spinors the reflection operation is not preserved by the representation, yetin the case of Pin it is preserved.

Now consider one dimension, where O(1) is simply Z2 and SO(1) is identity. Depend-ing on e2 = ±1 the basis Clifford algebra is (1,−1) with R⊕R or i with C. There existsSpin(1) = Z2 as just ±1 eigenvalues to the representation, and the Pin±(1) both exist,each as Z2⊗Z2 for Pin+ and Z4 for Pin−. Clc1 is then with basis of C⊕C ∼= H, leavingthe Spinc(1) to be simply U(1) structure. Just as what we will see in Spin(2) the twoirreducible representations differ just through ±1 or a π rotation on U(1), leaving it justas conjugation under a same representation.

Two dimensional and RP2

In two dimensional real space R2 the Clifford algebra is R(2) or H, with the transfor-mation group O(2) and only two gamma matrices. Taking them as σ1 and σ3 as Paulimatrices we obtain Cl2,0 and generator Σ = 1

2σ1σ3 = − i2σ2 for the need of 4π as the pe-

riod of θa as generation angles. The spinors with eigenvalues ±1 are real 2-dimensional.Despite in principal there are two irreducible representations, it is apparent that whenthey transform with σ1σ3 it is nothing more than a π rotation, which means they areactually the same (which only applies to 2 dimensions). With that we only constructone specific a for the Pin+, where the reflection on spinor ψ is ψ → γ ·wψ. With [6] wecan take similarly ψ → iγ ·wψ. An example of Pin+(2) is to consider on a Klein bottleparametrized by:

(x1, x2) ∼= (x1 + 1, x2) ∼= (x1, x2 + 1) ∼= (x1 ±1

2,−x2) (1.37)

With the Majorana spinors having:

ψ(x1, x2) ∼= ψ(x1 + 1, x2) ∼= ψ(x1, x2 + 1) ∼= γ2ψ(x1 ±1

2,−x2) (1.38)

Where with the introduction of γ2 we are already unable to differ the chirality as “cir-culating” through x2 is the same as reversing x2.

Now consider a real projective space RP2. With Stiefel-Whitney class method it isproved to not permit a Pin+ structure but only Pin−. The formation of RP2 has three8

reversions in its triangulation and two at each point. Describe them as two Mobiusbands where one tangent vector e1 squares as −1 around its boundary. Since we havetwo “directions” we will need it to be squared to −1 to interpret two reversions at asame point with one third for any non-contractible circulation. Thus we need R2 = −1for the reversion and a Pin− structure over RP2.

Three and four dimensional

In three dimensions the Spin group is a double cover of SO(3) well acknowledged asSU(2). There exists three γ matrices γi = σi with Pauli matrices to be the matrix

8See Example 3.11 of [14].

19

representation of group transformations. Spinors ψi transforming in Spin(3) have theγ3 to be the helicity operator on x3 direction as taught in every quantum mechanicscourses. With the helicity among the group transformation, there is only one irreduciblerepresentation of Spin(3) for spinors. Further, if we compare the possible bilinear formsin two and three dimensions, we obtain through the gamma matrices:

ψψ = εabψaψb, ε being the Levi-Civita tensor (1.39)

We get to claim that it is pseudoreal since the form invariant to group transformationis antisymmetric. We can as well use unit quaternion numbers q = a + bi + cj + dkand its conjugate q† = a − bi − cj − dk. Naturally we have q†q = 1, which shows theinvariance under the SU(2) transformation. From its anticommutation however, we candeduct it being antisymmetric in ψaψb, which requires it being the Levi-Civita tensor.With Spin(2) however, it is complex and no antisymmetric, therefore being real, notedas habψ

aψb which is however zero. On the other hand the fact that even dimensions canbe divided to two irreducible representations means it can be written as habψ

aψb, whichis not in the same Fermi statistics.

For the Pin+(3) consider naturally the mapping SU(2) → U(2). The determinantsof a ∈ Pin+(3) is divided with the reflection γ · w which has determinant −1. Withthe determinant change the two irreducible representations can be given by the sign ofreflection:

ψa−→ aψ, ψ

a−→ a det aψ (1.40)

Where we can see the line bundle for Pin(3) manifold being the determinant bundle.Besides, for the invariant bilinear forms we have9,

(ψ,ψ)→ a · a(ψ,ψ) = det a(ψ,ψ) (1.41)

For both ψ and ψ. The combination (ψ, ψ) is however invariant as the “off-diagonal”mass. That is, two massless complex fermion in 3 dimensions can have a bare massin their combination. When considering T symmetry for instance, this will lead tocancellation of T -anomalies in the 3-dimensional case. To proceed on we show somebasics that continue to four dimensions.

In four dimensions, the readers may already be familiar with some representationsof gamma matrices. From that Spin(4) algebra being subgroup of Cl0(4) and Cl(3),we can expect two irreducible representations each isomorphic to the Spin(3) irre-ducible representations in previous paragraphs. From the well-known deductions theγ5 = γ1γ2γ3γ4 is the chirality operator that reduces the representations into two ”chi-ralities” each being the left and right-handed Spin(3) Weyl fermion representations.Within those Weyl fermions the representation is simply pseudoreal with respect toeach SU(2) representation. What’s more, in general we can write for the group struc-ture Spin(4) ∼= SU(2)L × SU(2)R. In cases like Pati-Salam model[19] the isomorphismrelates to the generalization of left-handed and right-handed electroweak charges.

9For the case of Pin−(3) the difference between ψ and ψ need to be told since (det a)2 = ±1, herewe omit the difference for the Pin+(3).

20

For Pin(4), the Z2 over the division of irreducible representations are combining themas one10. For instance, with the R transformation in Pin+ ψ → γ ·wψ, the anticommuta-tion with γ5 gives three minus signs and one plus sign, leaving the total anticommutationand thus shifting the chirality of SU(2). Up to now we have enough information for thefollowing discussions upon spin, while further Clifford algebra related structures will bediscussed when mentioned.

1.1.4 Dirac Operator in Curved and Unorientable Space

Consider the conventional differential operator on the spinor11 representations and gammamatrices as the representation of Clifford algebra, such differential operators are well-acknowledged to be the Dirac operators. Coupled to a gauge field or a connection formthe interactions in quantum field theories are explained[14, 13]. We introduce shortlythe conventional notations of Dirac operator and its extension in an unorientable case,while more general cases are presented in [15] and quantum gravity texts.

A Dirac operator acts on the spinor bundle, or more precisely sections on the spinorbundle. The sections are noted as Γ(M,F )12 for F : E

π−→ M , where the mapping ofidentity πs = idM for s ∈ Γ. With the notation clear, the Dirac operators are mappingsΓ(M,Spin(M))→ Γ(M,Spin(M)) for spin bundles.

Consider now, the spinors that are invariant in the spin group. For the covariantderivative on the manifold which is given as Di = ∂i + wiab ·

14

[γa, γb

], combining the

coordinate change and the rotation from the change of metrics. The latter part arisingfrom the parallel transport is also called spin connection. Conventionally it is the Levi-Civita connection, whose properties we may only notice its existence for the case of nota gravitational anomaly.

Consider then with section and the splitting between the two irreducible representa-tions of Cl(n) when n is even, locally we are able to construct the spin structure and onlocal sections, we define the Clifford module E as following on local U and X ∈ TU :

EU → U, ∃c : TU → End(EU ), c(X)2 = g(X,X)idE (1.42)

So that if we consider a local vector bundle V , we obtain the isomorphism EU ∼= V (U)⊗Wwhere W is a homomorphism. The global extension appears when the spin or pinstructure is global on M . When the vector bundle V splits into V → V + ⊕ V −, theClifford module will also split from the isomorphism.

Now with the mapping Γ(M,S(M))→ Γ(M,TM ⊗S(M)) from the covariant deriva-tive already imposed, we need another c acted on Γ(M,TM ⊗S(M)) to make the wholeoperation Γ(M,S(M)) → Γ(M,S(M)). Historically we will use iγµ as the transitionand write i /∇ = iγµ∇µ as the Dirac operator.

Following the splitting between the sections and Clifford module, calling the section∆(M) it will also split into ∆±(M) in the two irreducible representations case. We can

10Readers may consider twisting the 4-manifold. As the left and right-handed spinors are divided, by”twisting” their chiralities transmit to each other, thus combine as one same.

11In this section all ”spin” when not explicitly describing an orientable spin can also mean a pin.12Conventionally it is noted as Γ, which is the same as that of Clifford group.

21

notice the fact that one gamma matrix in i /∇ anticommutes to γ the chirality operator,which splits the representation. In this type of spinors called canonical spinor bundle,the Dirac operators are transferring between two irreducible representations. Also sincethere only being one γ it anticommutes to the reflection R as well. Or more precisely, itanticommutes with every single reflection, including T the time symmetry.

In the case where there is only one irreducible representation, we will consider twocases: Odd dimensions and unorientable manifold. For 2n + 1 dimensions, the Diracoperator does not operate on the subbundles but the whole. For instance, betweenSpin(4) and Spin(3), we have, writing the gamma matrices in 4 Euclidean dimensionsas:

i /∇ψ = i

(0 iαµ

−iᵆ 0

)∇µ, αµ = (I2,−iσ) (1.43)

Which is a Gamma matrices representation of Spin(4). From this we can deduct theproperties of 4-dimensional Dirac operators in the discussions above. The spaces of

spinors are then connected in their representations. If we then add γ5 =

(I2 00 −I2

)to the gamma matrices we get Spin(5), where the transformation between irreduciblerepresentations is among the group transformations, thus Dirac operator only transformsin the one representation again.

In the case of non-orientable, since the orientation is not present locally anymore andthe fact that the Dirac operator we gave anticommutes with the reflection operator, itcertainly transforms between the two possible Pin± structures to construct. For in-stance, we consider the process of construction as follows: Starting with the orientablespin manifold M , splitting into two manifolds from the reflection and call the resultingmanifold the quotient M = M/Z2. With the spin structures S(E) extending to P (E)

and P ′(E) from whether R acts as +1 or −1 on the related S(E)→ M . In even dimen-sions, the R reflection anticommutes with the parity operator. The Dirac operator canbe simply defined as γ /D which is even under the reflection. This is however impossiblein odd dimensions, meaning that is is not defined as self-adjoint and the Dirac operatorbehaves as transformations between P and P ′, just like it does to S+ and S− as irre-ducible representations. With that we showed the difference it makes from the parity ofdimensions in analog to the discussions in formally introducing the Clifford algebra.

1.2 Exact sequence

Following the fiber bundle theory, an exact sequence can be introduced for a sequenceof groups that have relations of fibers and total/base spaces. An exact sequence can bewritten as follows13:

1→ A→ B → C → · · · → Y → Z → 1 (1.44)

13Here and in following discussions we note 1 to be the identity or trivial group, while in othermaterials it can be 0. We may also use 0 in cases like Stiefel-Whitney classes to make it convenient tonew readers only searching for concerned paragraphs.

22

Where each three neighbors are related to a fiber bundle (in some cases not fiber bundlebut Serre fibration, see following discussion), and the ones from start to end are (notnecessarily) trivial groups. For instance, 1 → A → B means a mapping from groupA to group B through a trivial group and Y → Z → 1 means mappings from Z to atrivial group has group structure Y . We will see examples of such mappings in followingparagraphs and sections. For readers’ notice we already have for spin/pin groups:

1→ K∗1Cl(V,q) → Γ(V, q)Ad−−→ O(V, q)→ 1 (1.45)

1→ K∗1Cl(V,q) → Γ0(V, q)Ad−−→ SO(V, q)→ 1 (1.46)

1→ Z2 → Spinr,s → SOr,s → 1 (1.47)

1→ Z2 → Pinr,s → Or,s → 1 (1.48)

As we see, the exact sequences are not showing the explicit map, but the group structuresof the mappings. The information however is enough for us to learn its topologicalproperties.

1.2.1 Cech cohomology groups

Now up to the uses of exact sequences in our interest, we will recap upon homologygroups and homotopy groups for manifolds and bring up important exact sequences forour use. We start with homology groups. Homology groups are, on a topological space,the group of maps from cyclic groups that has no boundaries to those boundaries thatare in the same ranking. To make it clear, on the topological space X which we chooseto be the differential manifold for our interest and avoiding discussions on definitions,we can draw “circles” along the manifold which denote a closed area that can be fromzero dimensional to what the manifold permits. It is apparent that we can joint such“circles” with weights and form a “chain”. For instance, we denote them as follows, firstthe “circles” which we call singular q−simplexes:

Σq =

∑i

pixi|(xi) ∈ ∆q

(1.49)

Where standard q−simplexes ∆q are:

∆q =

(x0, x1 . . . xq) ∈ Rq+1|∑

xi = 1, xi ≥ 0

(1.50)

And the group of chains are noted as follows, given the group of ”weights” P:

Sq(X,P) =

c =∑s∈Σq

pss|p ∈ P

(1.51)

Now we define the boundaries. A boundary of elements in Sq is the sum of all boundariesof its simplex components which are q − 1−singular simplexes denoted as:

δq(sq) =

q∑i=0

(−1)is(x0, . . . , xi, . . . , xq) (1.52)

23

With notion what xi is excluded, as we use later too. The notion of −1 in some con-texts makes it called ”oriented simplexes”, while the use of it makes sense out of thesummations. It is apparent that we have an important relation:

δq−1 δq = 0 (1.53)

Following the relation a boundary is apparently with a zero boundary, though thosewith zero boundaries are not necessarily boundaries of a higher simplex. We call thegroup of zero boundaries r−cycle group Zr(X,P), group of boundaries r−boundarygroup Br(X,P). Finally we have the r− homology group of equivalences from cycles toboundaries:

Hr(X,P) =Zr(X,P)

Br(X,P)) (1.54)

It is apparent that homeomorphic manifolds have isomorphic homology groups, so thatthey are treated the same in our discussions upon its general properties. We then definethe dual of homology related things that we call cohomology. First with respect tosingular simplexes, their duals are defined with integrals on the simplexes. Formally wefirst note a standard q−simplex as σq and a singular q−simplex as sq, with functionsq = f (σq), then we have: ∫

sq

ωq =

∫σq

f∗ωq (1.55)

Where ωq is a differential q−form on the manifold and f∗ is the pull-back of f . Withthe boundary δq (in some materials called ∂q) given, we then have the coboundary fromStroke’s theorem that writes as:∫

sq

dωq−1 =

∫σq

d(f∗ωq−1) =

∫δ(σq)

f∗ωq =

∫δ(sq)

ωq (1.56)

From these we have made clear what a ”dual” is in its integral form and that thecoboundary will add to the indices. In what we call de-Rham cohomology, it is definedthen by those being coboundaries and with zero coboundaries, denoted as:

Hr(X,P) =Zr(X,P)

Br(X,P)(1.57)

Conventionally de-Rham cohomology is for discussions of R group homology. What wewill need for a spin theory is the Cech cohomology group that is the dual of Z2 chainsand describes a Z2-valued function, which is specifically −1, 1 for describing a spin. Inspecific, we have a clearer description of Cech cohomology group that that of de-Rhamcohomology group. We write, for a Z2-valued function f(r0, r1, . . . , rn), we call it ar-cochain Cr and its coboundary as follows:

d(f) =∏i

f(r0, . . . , ri, . . . , r1) (1.58)

24

An apparent fact is that d2(f) = 1 since every element in the product appears twice.Therefore, we can explicitly write down:

Zr(X,Z2) = f ∈ Cr(X,Z2)|d(f) = 1 (1.59)

Br(X,Z2) = d(f)|f ∈ Cr−1(X,Z2) (1.60)

Hr(X,Z2) = Zr(X,Z2)Br(X,Z2

(1.61)

The Cech cohomology group sheds its importance by determining the existences of ori-entation and spin structures. The characteristic classes called Stiefel-Whitney classeswhich should be invariant under local coordinates transformations of the tangent bundleare related to Cech cohomology groups by following deductions. Say that we choose anorthonormal frame for the tangent bundle TM over M , written as eia for a local map-ping Ui, the transition function tij lies in an orthogonal group. For an m-dimensional oneit is a O(m) group member, which means for mappings we can define a Cech 1-cochainas:

f1(i, j) ≡ det(tij) = ±1 (1.62)

To avoid confusion the det here is not over indices ij. Now we show a member ofC1(M,Z2) is exactly a member of 1-cocycle. It is simply proved as:

df1(i, j, k) = f(ij)f(jk)f(ki) = det(tij) det(tjk) det(tki) = 1 (1.63)

Since they are in an O group. Moreover, we obtain the cohomology group by studyingthe transition of frames. Say a new frame eµa is obtained by transition eia = eiahi, sinceeia = tij eja we instantly obtain tij = hitijh

−1j . Now with noticing that hi can define

the Cech 0-cochain as f0(i) ≡ det(hi), we have an interesting result from the transition,noting it as f1:

f1(i, j) = f1(i, j)f0(i)f0(j) = f1(i, j)df0(i, j) (1.64)

Therefore it is transformed by a B1(X,Z2) member, we can claim that a transition offrames does not change the Cech cohomology. The first Cech cohomology group is trivialif and only if the manifold is orientable so that the transition function is in SO(m) groupand always has determinant 1. This is called the first Stiefel-Whitney class w1(M). Wehave then the claim that the necessary and sufficiency condition for a manifold M to beorientable is w1(M) = 1.

For a possible spin structure, its existence as a double cover of SO(m) rotation groupsmakes its transition function tij with ϕ(tij) = tij . When the cocycle condition tij tjk tki =I is met, we can say it being a kernel of ϕ. The Cech 2-cochain f(i, j, k) is defined asfollows:

tij tjk tki = f(i, j, k)I (1.65)

From which we can tell it being the trivial group for an existing Spin(m) structure,which means ∃Spin(m) on M ⇒ H2(M,Z2) = 1. Beyond spin structures, Pin andSpinc structures’ existence are determined by certain combinations of Stiefel-Whitneyclasses as they are shown in last section as Spin structures on certain line bundles withthe original (tangent) vector bundles. A more concrete description for the Spinc case is

25

in [20]. For a general case, a Stiefel-Whitney class w(E) for a structure E =⊕

i Li is anexample of total Chern classes c(E), where we have the Whitney sum formula and itsinfluence upon w(L1 ⊕ L2):

c(E) =∏i

c(Li), w1(L1 ⊕ L2) = w1(L1) + w1(L2),

w2(L1 ⊕ L2) = w2(L1) + w2(L2) + w1(L1)w1(L2)

(1.66)

With which we can show the existence of Pin structures and Spinc structures.For Pin± structures, we may note that their existence should be consistent in the

Stiefel-Whitney classes they own and as locally spin they should be presented by thesecond order, which means they are either w2 or w2 +w2

1. Consider first the line bundles⊕n ε, where we show through a simple chart:

n Structure w1(⊕

n ε) w2(⊕

n ε)

1 Pin+ w1(ε) w2(ε)2 SpinZ4 0 w2

1(ε)3 Pin− w1(ε) w2(ε) + w2

1(ε)4 Spin× Z2 0 0

Where from the combinations and fact that w2(ε) is trivial, we then have w2 = 0 forexistence of Pin+ and w2 + w2

1 = 0 for existence of Pin−. Further, the relation can beshown from considering combining determinant line bundle for 1 time Pin− and 3 timesPin+, where Lemma 1.7 of [18] has a more detailed description.

From the existence condition in 1.1.3, the existence of Spinc structure on E is equiv-alent to the existence of E ⊕ L where L is a certain complex line bundle. Since E isalready oriented, we will only need w2(E) +w2(L) = 0 or under mod 2 w2(E) = w2(L).The condition can be also viewed as our (and the author’s) one first case of long exactsequence, which we will start discussing about in next part to relate it with H3(E,Z),namely the third integral Stiefel-Whitney group related to the long exact sequence incohomology.

1.2.2 Long exact sequences

An exact sequence mentioned above with five terms (including ones) is called a shortexact sequence. Moreover, for the case of short exact sequence:

0→ Af−→ B

g−→ C → 0

We say that the exactness requires im(f) = ker(g), that is, f being injective and g beingsurjective. Such properties are doubtlessly essential to the fibration14 structure A : B →

14The exact sequence structure itself however, does not require the group homomorphism to have afiber bundle structure but Serre fibration is enough, which means only homotopy equivalence of fibersare required and extends it to discrete groups such as homology/homotopy groups. Materials including[21] and [22] are good references to the topic on mathematicians’ view.

26

C. The exactness of short spectral sequences indicates structure of continuing exactsequences called long exact sequences. Typically for Ak, Bk, Ck and accordinglyinduced fk and gk, when such sequences are always exact:

0→ Akfk−→ Bk gk−→ Ck → 0

We may (or may not) have a resulted long exact sequence, which can be written as:

· · · → Ak → Bk → Ck → Ak+1 → · · ·

Or similar one yet with k− 1 as the continuation. We may note this to be a weak claimas individual examples are shown in different manners. First example is in cohomologywhere we follow [23] and note 3.1 of [24] as an alternative for a short introduction.

Of cohomology and connecting homomorphism

Consider for vector spaces V k, the “coboundary” with regard to de-Rham cohomology ordifferential is written as linear mappings dk : V k → V k → V k+1. We have dk dk−1 = 0and the familiar sequence where the relation conducts:

· · · → V k−1 dk−1−−−→ V k dk−→ V k+1 dk+1−−−→ · · · (1.67)

If we call V = V k the cochain complex, the cohomology vector space is defined as:

Hk(V) :=ker(dk)

im(dk−1)(1.68)

With respect to the cohomology group above Hr(X,K), we note V to be K-valuedvectors (also for Z and Zn). Now consider on A and B with d and d′ respectively, wedefine the cochain map ϕ that commutes with differentials as follows:

Bk = ϕk(Ak), d′k ϕk = ϕk+1 dk (1.69)

So that we have the following commutative diagram:

· · · Ak−1 Ak Ak+1 · · ·

· · · Bk−1 Bk Bk+1 · · ·

dk−1

ϕk−1

dk

ϕk ϕk+1

d′k−1 d′k

Now return to the short exact sequence for cochain complexes 0 → A i−→ B j−→ C → 0,which means for each k dimensions to A,B,C, i, j the sequence is short exact. Forconvenience we emit labels for d and i, j to obtain such commutative diagram:

0 Ak+1 Bk+1 Ck+1 0

0 Ak Bk Ck 0

i j

d

i

d

j

d

27

Now we consider an element in Ck which is in the cohomology group Hk(C), marked asck. As an element of cocycle group d(ck) = 0, we have through the commutative diagramthat there exists elements in Bk marked by bk so that ck = j(bk). The commutativitythen gives such relation:

j d(bk) = d j(bk) = 0

From which we have ker(jk+1) ∼= ker(dk) and following ker(j) = im(i) we have thoseconcerned elements of Ak+1 from the injective ik+1 as ak+1 which fulfills15:

d(ak+1) ∼= i d(ak+1) = d d(bk) = 0

As similar things can be done to A and B, we fine the homomorphism from Hk(C) toHk+1(A) which is called the connecting homomorphism or Bockstein homomorphism,

d∗ : Hk(C)→ Hk+1(A) (1.70)

Now consider the lifting of i, j to cohomology groups i∗, j∗ so that we have exact sequence

Hk(A)i∗−→ Hk(B)

j∗−→ Hk(C). Reading off from d(b) = 0 in Hk(B) we can see elements ofim(j∗k) are all kernels of d∗k. We prove that ker(d∗k) ∈ im(j∗k) to show their equivalence.For trivial elements in Hk+1(A) say ak+1 = d(ak), from jk given d∗ck = ak+1 we noteb ∈ Bk with j(b) = c. with ik we then have for bk − ik(ak) it belonging to im(jk):

jk(bk − ik(ak)) = jk(b

k)− j i(a) = jk(bk) = ck

and cocycle:d(b− i(ak)) = d(b)− i d(ak) = d(bk)− i(ak) = 0

so that we have [c] = j∗[b − i(a)] and ker(d∗) ∈ im(j∗). With that and similarly withi∗k+1 we have the long exact sequence in cohomology:

· · · d∗−→ Hk(A)

i∗−→ Hk(B)j∗−→ Hk(C) d∗−→ Hk+1(A)

i∗−→ · · · (1.71)

We give for example, with the short exact sequence 0 → Z → Z r−→ Z2 → 0, the relatedclip of long exact sequence is the following:

· · · → H2(E,Z)r−→ H2(E,Z2)

β−→ H3(E,Z) (1.72)

Which we can interpret as, considering the claim w2(E) = w2(L), the line bundle overw2(L) lives in a kernel of β. Therefore the condition for the existence of Spinc structureis W3(E) = H3(E,Z) = 0, known as the third integral Stiefel-Whitney class.

Apart from the use in W3, discussions above also applies to the homology groups whichcan be viewed as dual or “co-” over cohomology just as how cohomology was introducedin this thesis. We write the exact sequence as follows, with notations read off naturally.For exact sequence 0→ A→ B → C → 0:

· · · → Hk(X,A)→ Hk(X,B)→ Hk(X,C)d−→ Hk−1(X,A) (1.73)

15For comprehensive understanding here ”[]” may be emitted when meaning the group.

28

We may use such duals without proof. Materials including [21] and [25] will give well-acknowledged introduction. So far with an example of long exact sequence, the othersequences used may be introduced without proof but only name. To explain the legal-ity above and in further discussions, we bring up the universal coefficient theorem ofcohomology as the short exact sequence (see also Chapter 3.1 of [25]):

0→ ExtR(Hn−1(C,R), G)→ Hn(C,G)h−→ HomR(Hn(C,R), G)→ 0 (1.74)

For instance, if we take R = G and prove that ExtR(Hn−1(C,R), G) = 0, there will bea bijection between Hn(C,G) and Hn(C,G) or Hn(C,G) = Hn(C,G). Conditions areto take Hn−1(C,R) to be free (e.g. Z+) or the extension group being trivial when theprojections are simple enough, for instance R = G = Z2 being the simple injection. Sofar it is safe to say Hn(X,Z2) = Hn(X,Z2).

29

2 Dai-Freed theorem and Spin bordism

2.1 APS index theorem and Dai-Freed theorem

For the Dirac operator or a general differential operator D on the section1 Γ(M,E) →Γ(M,F ), the reversed operator D† : Γ(M,F ) → Γ(M,E) is defined as the adjointoperator. The “index” of operators are defined as difference of numbers of zero modes(zero eigenvectors s so that Ds or D†s equals 0) between the two manifolds, given asindD = dim kerD − dim kerD† or simply νD = nD − nD† . From the studies of Atiyahand Singer[26] the index is a topological invariant under topological transformations ona compact manifold without the boundary. To extend the theory to boundaries a specificcase for Dirac operators is achieved with Atiyah-Patodi-Singer index theorem[27, 28].

Consider now the manifold X being the boundary of manifold Y , the Dirac index onY is given with two parts: the original “Dirac index” form on Y and the contributionfrom the boundary. Within Nakahara’s[14] frame of description, given the form of theDirac index of twisted spin complex as follows:

indD =

∫X×I

A(R)c(F)− 1

2

[η(i /∇(A1))− i /∇(A0))

]=

∫M×I

A(R)c(F)− ηX2

∣∣∣t=1

t=0

= −spectral flow

(2.1)

The meanings of notations are as follows: Consider figure 2.1 as a visualization of themanifolds up to our concern after necessary compactifications, the gauge potential Atcorresponds to the two ends considered as boundaries with t = 0 and t = 1 each witha different gauge bundle while t is the parameter for the extension of the gauge bundleon Y .2 The part of A(R)c(F) in the expression is the contribution from the continuoustransformation in the bulk of Y , which is the same form as the AS index were it beingwithout boundary. The terms are invariant polynomials result described as CHern-Simons forms, which can be derived through Fujikawa’s method. The other part η refersto the η-invariants on the boundaries X, which is defined as the difference between theamount of positive and negative eigenvalues for the Dirac operators coupled with thegauge field A.

Such expressions are generalized as the “spectral flow”, which corresponds to howmany times the eigenvalues pass zero. The expressions of η-invariants for instance, are

1Although formally it is defined on a section, for the concern of this paper the cases are either globalon a manifold or two manifold with different global index ”glued”.

2As readers may notice it might be causing problems from nontrivial homotopy groups on Y , whosecontribution is then contained in the index on Y .

30

X1

Y

X0

Figure 2.1: As in the figure, the boundary X0 and X1 are parametrized with t = 0and t = 1, in which case the gauges field A transforms from left to rightand Y = X × I. The resulted Dirac operators have one contribution fromtransforming through the bulk and one another from the difference of modesin boundaries of two sides.

written in “friendly and comprehensive” way as well as a regularized expression:

η =∑λ>0

1−∑λ<0

1

= lims→0

∑k

sgn(λk)|ηk|s

= lims→0+

∑k

sgn(λk) exp(−sλ2k)

(2.2)

Where the exact way does not matter as long as they meet with the expression we needto analyze. Now with the Dirac index, we redraw the manifold in two ways. For instance,the above one can be considered with the boundary X0∪X1 and the overline means thereversed orientation in order to preserve in Y . The readers might be more familiar withsuch constructions for the extending of gauge, shown as follows 2.2:

Y

X

Figure 2.2: This shows with a general (no need to be contractible to a point or orientationpreserving) manifold Y together with the definition of gauge field A0. Theright ”tube” is glued to Y with the side X matching orientation and behavesas the previous 2.1. The total manifold is then Y ⊕ (X × I).

The extension “glued” to the manifold Y is not a “manifold homeomorphism” if the

31

gauge field A0 on that has a nonzero net η-invariant and thus changes the index. What’smore, if extended to infinity the manifold is then without boundary yet non-compact.Now consider the fact that there being every ways to extend the manifold from gluingwhile the different summation of η will result to a new term in the Dirac index for thenew closed manifold. The anomalies we analyze in this paper are mainly described withthe changes of η-invariant.

2.1.1 Dirac operators revisited

For our concern the Dirac operators now has the form iγµ(∂µ + ωµ +Aµ). This will notaffect us writing it into D(A) and its Hermitian conjugate D†(A). Now we consider thedimensionality to be n and its extension to n+1 dimensions. Now consider the manifoldX with i /D and Y with X = ∂Y and new dimension parameterized by t, the constructionof new sets of Dirac operators can be written as:

i /DY = iγt

(Dt + i

(0 /DX

− /D†X 0

)), γt =

(Id 00 −Id

)(2.3)

Or with the new set of gamma matrices:

γt =

(Id 00 −Id

), γi =

(0 iαi−iα†i 0

)(2.4)

Where αi are Hermitian gamma matrices on X. Instead of sticking on the exact formof γ-matrices we note that they are understood as combining the conjugate eigenvaluesto maintain itself Hermitian. Further, in the case of [6] for the extension with not flatmetric but gY (t) = d2t+ ε2gX(t) for a certain s value in the transmission of t it is then:

i /DY = iγt

(Dt +

1

ε

(0 /DX

− /D†X 0

)), γt =

(Id 00 −Id

)(2.5)

This looks similar to the case of extension from odd to even, while the other one can beachieved through complex or quaternion group isomorphism always (See Section 2.2).Now consider a simple proof of Atiyah-Singer index theorem. As the set of eigenvectorsψ of i /D with eigenvalues λ, written as i /Dψn = λnψn. Suppose we have naturally

different amounts of zero modes between i /D and −i /D†. Other modes with nonzeroeigenvalues λ may arise from ε and ωi as the gravity part or Aµ transform into zero

modes. Consider then when /DY (t) creates a zero mode /DY (t)ψ = 0, its /DX and /D†X

parts both contribute to the zero as a · 0 ± a† · 0. Then we can say exchanging DX

and D†X with γ the chirality operator, we have one other zero mode from the reversedoperator. Then we can safely say that the Atiyah-Singer index is preserved, or µ+ − µ−is invariant under a diffeomorphism from gauge or gravity transformations.

Now consider the boundaries being present in our case, the range of t can be specifiedas [0, 1) under compactification.3 For the case of our previous Atiyah-Singer index

3This is just an example for the APS boundary since we only care about 0 side.

32

definition call the boundary n dimensional and we may think of the Dirac index onn dimensions. Before calculating we may tell a difference first between two types offermion eigenstates: The Dirac (or Weyl) fermions have the states exchanged by the

Hermitian conjugate, shown as typically

(ψ1

ψ2

)where ψ2 6= ψ†1 and not both zero; the

Majorana Fermions are however its own Hermitian conjugates, meaning them to be(ψψ†

). When this Majorana fermion has an eigenvalue, we obtain two states under the

same eigenvalue. The outcome related to our discussions is that for “index” we have inour discussions upon Dirac fermions, in the Majorana case we have twice the amount.

Back to the transition of states. When we do the gauge or gravitational transforma-tion along the “lines” of the eigenvalue, zero modes may appear or disappear when aline passes zero. The index on Y then receives a contribution from the edge modes ifthey will transform past zero. The transition of spectrum can be monotonic under outconstruction, which means from each zero mode generated one negative λ becomes onepositive λ under A0 → Agt . Therefore we obtain η = n+(λ) − n−(λ) as our previousη-invariant for the boundary whose change in Y is a “bordism” invariant. Moreover,the bordism invariant combined with the AS index will bring about the correction termwhich corresponds to the new Atiyah-Patodi-Singer index:

IndAPS = IndAS −1

2ηtotal (2.6)

Where the “total” preserves the orientation (if there exists any). The Atiyah-Singerindex can be presented by the gauge and gravitational Chern-Simons classes, while theAPS index is accompanied with the boundary term regularized by the massless limitand t = 0. According to [29] as a complementary approach, the alternative expressionhas the form:

η(i /DX)

2=

CS(g,A)

2π

∣∣∣∣t=0

mod Z (2.7)

Whereas the combined regularized form of the invariant (the index or half the “η-invariant” in [29] and some other materials) on the twisted spin complex (the spincomplex coupled with the gauge field) is given as:

Indreg =CSbulk

2π− η(i /DX)

2(2.8)

That is, the same as bulk gauge-gravitational index minus the surface.Further, if we consider the zero modes or ker(i /DY ), the contribution to η-invariant

from the whole spectral flow is then:

η = lims→0

∑k

sgn(λk)|ηk|s + dim ker(i /D)Y (2.9)

Which is the total amount of boundary correction to the index. The η-invariant in thecase where the contribution from Chern-Simon terms is zero will equal to the APS index.

33

Figure 2.3: The spectral flow shows the change of the eigenvalues with the gauge trans-formation in the t dimension. The monotonicity of flow is guaranteed bythe construction of the homotopy. When one line passes zero one eigenvaluechanges sign and subtracts two to η.

Or as in Nakahara or earlier expression:

Ind = −spectral flow =1

2

∫Y

dηcdtdt− 1

2ηX (2.10)

2.1.2 Dai-Freed theorem and η-invariant

When we consider the partition function of a quantum system, whose fermionic partconventionally is given as:

Zψ =

∫Dψe−

12

∫ddxψi /Dψ = deti /D =

∏i

λi (2.11)

The sign of concerned product is determined by the sign of eigenvalues of Dirac operatori /D. For real or pseudo-real fermions the pairs of conjugate sections will lead our Diracoperator to be written in a more ”friendly” form:

i /D =

0 λ1

−λ2 00 λ2

−λ2 0. . .

0 λk−λk 0

(2.12)

For the states to have the partition function written with the fermion pfaffian of theDirac operator. Going back to our case for DY to DX it is the pseudoreal fermions on

34

X and boundary of Y . The partition function can then be written as the combined formas:

ZY =∣∣Pf(i /DY )

∣∣ e− iπηY2 (2.13)

Where in our convention, for a system transforming from n(λ+) = n(λ−) we roughlyhave ηY = 2(n(λ−) mod 2) and the amount should double for Majorana fermions on Y .The Dai-Freed theorem [20] says that this is a topological invariant for the boundaryconditions given. Generally we follow the convention of [8] to write it as:

Z = ZboundaryZbulk = det(i /D−+X )eiπηY /2

= det

(1

2(1− γt)(γtγiDi)

1

2(1 + γt)

)eiπηY /2

(2.14)

The total partition function will be a C value while either boundary or bulk partitionfunctions are determinant or pfaffian line bundles. Consider if Zboundary = (−1)F de-pending on the signs that vary by the gauge which gives a Z2 group, then Zbulk = (−1)F

′

where F + F ′ = 0 mod 2 and thus cancel the contribution to the global anomaly on Xaccompanied with the mapping torus construction.

2.2 Anomalies with the Dai-Freed theorem and Spin bordism

Anomalies in quantum field theory can be described with index theorems. The existenceof anomalies means the regarding symmetry to be unable to be gauged in a quantumtheory. The chiral anomaly for instance, has the form[14]:

∂µjµm+1 = −2iA = −2i

∑i

ψ†i γm+1ψi (2.15)

Where A is the anomaly factor regarding to states not matching and cancelling underchiral operator γm+1. Specifically, the term appears as ei

∫dx∂µjµ , which indicates that

Re(jµ) has the period 2π for its contribution to partition function. Introducing the heatkernel regulator e−(λi/M)2 the integration form then falls into:∫

dx∂µjµ = −2i

∫dx lim

M→∞

∑i

ψ†i γm+1ψie

−(λi/M)2

= limM→∞

∑i

⟨ψi

∣∣∣γm+1e−(i /∇/M)2∣∣∣ψi⟩

=∑i

〈0, i|0, i〉i /∇+−∑i

〈0, i|0, i〉i /∇−

= indi /∇+

(2.16)

That is, the Dirac index of positive chiral operator i /∇ is the anomaly itself. If we thencount the boundary contribution the APS index is then the anomaly. While the de-duction above concerns only the chiral symmetry, similar methods extend with gauge

35

symmetry Aµ coupled to ∇ and making D, where the index i /D is to describe the re-garding gauge current.

With anomaly for a gauge field Aµ considered as the Dirac index on X, using theformer mapping torus construction we extend it to Y where X = ∂Y . With Dai-Freed theorem the invariant term

∣∣Pf(i /D)∣∣ e−iπηY /2 means that the anomaly in partition

function ZX =∣∣Pf(i /D)

∣∣ can be cancelled with the mapping tori by the APS invariantηY . Now consider the APS index theorem when Y being the boundary of Z and thegauge field on X continues. Following the APS index theorem it is:

Indi /DZ =CSZ(g,A)

2π− 1

2ηY (2.17)

Which we recover now though the way of anomaly. The form of CSZ to our concernis 2π

∫Z A(R)c(F), which is the index density on Z being integrated or the anomaly

polynomial∫Z IdimZ . Since the Indi /D should be integer, exponentiating will then give:

eiπηY = eiCSZ(g,A) = e2πi∫Z IZ (2.18)

If we take Majorana fermion states then it is eiπηY /2.4 From this approach we recoverthe anomaly on n dimensional manifold X associated with symmetry G by the anomalyunder the same symmetryG extended to n+2-dimensional manifold Z, while the mediumis the η-invariant on n + 1 dimensional manifold Y constructed with mapping torus.Therefore, by calculating the η-invariant on Y we can determine whether the anomaliesexist or are trivial in partition function.

Constructions above can cancel local anomalies apparently when the “mapping torus”construction is totally contractible to identity. For the following discussions we focuson the structure of G-bundles related to the gauge and for clarification take theη-invariant phase term as 2iπη. In the case of Spin × G structure local anomaliescorrespond to a trivial G bundle, while a “global” anomaly is a nontrivial gauge bundle.The non-triviality is determined by the homotopy group πn(G) when we compactifyRn ⊕ ∞ → Sn. In general, for a generic mapping torus related to the gauge bundletrivial or not, the construction for Dai-Freed theorem is valid and will cancel the anomalyvia the η-invariant or declare the vanishing of anomaly when e2πiηY = 1. Practically thegauge transformations including O(n) and SU(n) consist of parts connected to identityand parts that do not and thus may cause both global or local anomalies, while Zntransformations are not connected to identity at all thus will be related to the globalanomalies. No matter which anomalies we are referring to, for gauge anomalies to cancelthe mapping torus should be always providing a trivial e2πiηY . Such discussions concludethe cancellation of traditional gauge anomalies.

Beyond the “mapping torus” we constructed, it is natural to question whether theresult applies to any kinds of dimension extension with closed Y . The original indextheorem constructions only require the APS boundary conditions but no other con-straints, so does the Dai-Freed theorem. For instance, such situation arises when we

4The ambiguity here can be more clearly explained if we take 2η as our expression from the start.Our discussion here will only focus on whether η2 in the text is zero mod 2. Usually this is just markedby 2πiηY as in [7] and other physics materials.

36

considering the “gluing” of two Y1 and Y2 manifolds with both X being the boundary.If the anomalies are always cancelled, we will expect ηY1∪Y2 being trivial in the glued

surface Y1 ∪ Y2. On the mutual boundaries of Y1 and Y2 we only have to ensure thatthe gauge bundle collapses to Spin×G. In the case where ηY1∪Y2 is not ensured by thegauge group G, a new type of anomalies are located by the Dai-Freed construction whenthere are only “one” or several constructions cancelling the anomaly. Specifically if thetraditional anomalies don’t exist it will then impose conditions to G bundle in order forDai-Freed theorem to preserve. We may draw the picture as follows: In order to find out

X

YY

Figure 2.4: An illustration of the “reversed” direction for showing the case relevant toDai-Freed anomalies. in order for the orientation at X to be preserved wesee Y2 being “inside-out”. If the combined bulk η-invariant is not trivial inthe new partition function it then violated the condition on index theorem.

the group ηY belongs to, we bring on the bordism group. When adding one dimensionto Z our Y1 and Y2 can both be considered as one side of boundary, while Y1 ∪ Y2 beingthe total boundary. The bordism group ΩSpin(M) is defined as the group of inequivalentspin manifolds mapped to W . Inequivalent spin manifolds Y1 and Y2 both with a prin-ciple G-bundle will then be reflexed by a nontrivial ΩSpin

d (M) where d = dimY . Underthe condition that Z is anomaly free our manifold Y has its η-invariant only dependenton its types of mappings towards a certain manifold W that will permit a principleG-bundle. In our case we take W = BG the classifying space, defined as follows: Theclassifying space is an infinite-dimensional space with the principle G-bundle, under suchone-to-one connection between G and group An so that we have such chain:

B0 B1 B2 · · · B∞

A0 A1 A2 · · · A∞

G G G G (2.19)

For instance, in the case of G = SU(2) we have An = HPn the quaternion projectivespace and Bn = S4n+3. If we have a spin manifold Y with a principle G-bundle, there

37

will be a mapping f from X to BG. We can then use the homotopy class from X to BGnoted as [X,BG] to determine if there are topologically inequivalent principle G-bundles.We can draw such diagram for a more explicit picture of our discussions above.

P1(X,G)

BG1

X EG

BG2

P2(X,G)

f1

f2

f∗1

f∗2

(2.20)

If we show that ΩSpind+1 (BG) is not trivial, we will then expect distinct gauge bundles

coupled to the spin structure on d-dimensional manifolds X, shown by extending underthe APS construction into generic d+ 1-dimensional manifolds Y . Our η-invariant thenmaps ΩSpin

d+1 (BG) into U(1) group for the partition function like it does to traditionalanomalies. To determine if such anomalies exist or being trivial we only have to calculatethe Ω(BG) and the η invariant. Similarly, for Pin± or Spinc manifolds they correspondto ΩPin or ΩSpinc , which provides a probe for possible anomalies especially when ana-lytical approach and exact expressions are not present yet. Moreover, such anomaliescan always be cancelled by having certain amount of fermions with demanded charge inthe theory and vise versa.

2.3 AHSS and some continuous gauge groups

Spectral sequences are a way for calculating the Ω(X) groups as topological properties.We loosen the condition of fibration to take Serre fibration into our consideration. Moreprecisely, we take the property of homology for the fibration and related exact sequencesto construct. In this section we will introduce the general strategy and implementationsinto some important continuous gauge symmetries following section 2 and 3 of [7].

2.3.1 Atiyah-Hirzebruch spectral sequence

A formal expression for the AHSS in Chapter 11 of [10] is as follows: For a spectrum Enand X being the space where we compute the (co)homology, the “Ep,q2 ” page is givenexplicitly as:

E2p,q = Hp(X;Eq) (2.21)

In the case of our concern, Eq = ΩSpinq (pt) and X is the the base space when we consider

the Serre fibration pt→ X → X. In order to have ΩSpin(X) we take a sequence of spaces

38

which ends to the desired spin bordism space as a generalized homology:

F−1ΩSpinn (X) = 0, ΩSpin

n (X) = FnΩSpinn (X),

FkΩSpinn (X)

Fk−1ΩSpinn (X)

= E∞k,n−k (2.22)

Therefore we can result our desired spin bordism group as the “product” of all terms inE∞k,n−k or the result of ⊕ from non-trivial terms in the infinite page E∞ with bidegreessummed up to n. With the Serre fibration we calculate the E2 page and continue turningpages under the rules of cohomology, given by:

Hp,q(E∗,∗, dr) =ker dr : Erp,q → Erp−r,q+r−1

im dr : Erp+1,q−r+1 → Erp,q(2.23)

With the next “page” given as terms Er+1p,q = Hr

p,q(Ep,q, dr). Under the rule that d2r = 0

we are then expecting all trivial cohomology terms to vanish in r → ∞. We mayinstantly result to that all terms that are neither projected to or from a zero term onthe concerned page under the page’s differential survive the page-turning, while the fateof terms with lines to other ones are to be determined by the cohomology.

E0,5

E3,4

E1,3 E2,3 E3,3

E1,2 E4,2

E3,1

E0,0 E5,0

Table 2.1: An example for E2 page where all other terms are zero (which is rather com-mon with groups). All those terms that are not connected to anything surviveto the next E3 page. For the lines that exist we consider the cohomology ofd2 concerned: If d2 : E4,2 → E2,3 is a surjection for instance, it will lead tothe vanishing of E2,3 starting from next page. Consider also the differentialbeing homomorphism, then usually if only one element of E4,2 is projected to0 it will not survive the cohomology either to next page.

The table 2.1 explains how the spectral sequence works on the second page. Whilethe arrows are treated individually from the algebraic way, the third page in the casewhere all possible cohomology vanish will lead us to such third page in table 2.2:

39

E0,5

E3,4

E1,3 E3,3

E0,0

Table 2.2: The third page when all terms with connections in the second page vanish. Onthe third page d3 brings us the new line which is probably causing vanishingagain.

If we repeat the process on and on and suppose no contributions from rightwards, theonly three terms left are E3,4, E1,3 and E0,0. We can therefore say that if this corresponds

to BG, the non-vanishing spin bordism groups are ΩSpin0 (BG) = E0,0, ΩSpin

4 (BG) = E1,3

and ΩSpin7 (BG) = E3,4. We take some important examples in SU(2), U(1) and SU(N).

2.3.2 U(1) and Steenrod square

For calculation centered to pt → BG → BG, we will need first ΩSpin∗ (BG) to start our

sequence, which is given by the Pontrjagin-Thom isomorphism. From [30] and [31] thecharts for low dimensions (up to consideration to type II-B string theory, for instance)are summarized as table 2.3:

n 0 1 2 3 4 5 6 7 8 9 10

ΩSpinn (pt) Z Z2 Z2 0 Z 0 0 0 Z⊕ Z Z2 ⊕ Z2 Z2 ⊕ Z2 ⊕ Z2

Table 2.3: The spin bordism group of a point in conform with [7].

Now our task is to calculate Hn(BU(1),Z2) and Hn(BU(1),Z) groups to draw E2

page. We take the long exact sequence of homotopy as follows:

· · · → πn(F )→ πn(E)→ πn(B)→ πn−1(F )→ · · · (2.24)

Where from the fact that the only non-trivial homotopy group of U(1) is π1(U(1)) = Z,the only non-trivial homotopy group of BU(1) is π2(BU(1)) = Z. Such groups withonly one non-trivial homotopy group are called Eilenberg–MacLane spaces[32] E(G,n),where E(U(1), 2) = RP∞. The cohomology ring of E(U(1), 2) [33] gives that for ourconsideration Hn(BU(1),Z) = Z for n ∈ 2Z and 0 for odd n. In determining the Z2

homology, we take use of the universal coefficient theorem5 with form of short exactsequence for homology:

0→ Hn(C,Z)⊗G→ Hn(C,G)→ Tor(Hn−1(C,Z), G)→ 0 (2.25)

5See chapter 3.A of [25] for more explanation.

40

In the case where Hn−1(C) is either kZ6 or 0, both being torsion free, we are left withthe exact sequence:

Hn(C,Z)⊗G→ Hn(C,G)→ 0 (2.26)

The only way to keep the projections exact is to let Hn(C,G) ∼= G when Hn(C,Z 6= 0)and Hn(c,G) = 0 when Hn(C,Z = 0). In our U(1) case we can then say The Z2 grouphomology is just the same with Z2, leaving us with the following chart(for convenienceto the first chart we save the zeros):

8 Z⊕ Z 0 Z⊕ Z 0 Z⊕ Z 0 Z⊕ Z

7 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0

4 Z 0 Z 0 Z 0 Z

3 0 0 0 0 0 0 0

2 Z20 Z2

0 Z2 0 Z2

1 Z2 0 Z20 Z2

0 Z2

0 Z 0 Z 0 Z 0 Z

0 1 2 3 4 5 6

α

β

α′

β′

Table 2.4: The E2 page for BU(1) with potentially vanishing differentials in dimensionless than 5 noted.

Now we decide if the differentials rule terms out in the second page since up to ΩSpin5

nothing is affected by higher differentials. We separate our differentials into two parts:The first is mod 2 reduction happening to Hk(X,Z) → Hk(X,Z2), which correspondsto our previous “clip” Z r−→ Z2 and can be just imagined as “even or odd” beyondfancy expressions. The second that goes from Hn(X,Z2) to Hk−n(X,Z2) is the dualof Steenrod square Sq from Lemma 2.3.2 of [34], endowed with such properties to ourinterest:

6For convenience and formatting in this paper we may use kG to represent⊕

kG, which I note incase of causing confusion.

41

(i) Cartan formula: Sqi(α ^ β) =∑

j Sqi−j(α) ^ Sqj(β);

(ii) Sqi(α) = 0 if i > j = |α| the degree of α, for instance, Hj(X,Z2);

(iii) Sqi(α) = α2 if i = |α|;

(iv) Sq0(α) = α;

(v) Sq1 is the Blockstein homomorphism Hn(X,Z2) → Hn+1(X,Z2) associated withthe short exact sequence 0→ Z2 → Z4 → Z2 → 0.

And for Chern classes, namely Wu’s formula [35]:

Sq2i(cj) =

i∑k=0

(j − k − 1i− k

)ckci+j−k (2.27)

To transform from cohomology to homology we don’t take explicit form, but with thedual operation Sq∗. The degree matching relation is Kronecker pairing 〈a, b〉 with a thehomology and b cohomology. Being a dual means we have 〈Sq∗(a), b〉 = 〈a, Sq(b)〉 alwayssatisfying, and thus bijection between Sq∗(Hn+2) and Sq(Hn) or the same topologicalstructures.

To start with, let’s consider α and β in 2.4. In specific, for a CP∞ space there isno degree over 2 in cohomology ring due to the equivalence relation. We then havenontrivial Hn(BU(1),Z2) = Z2[x] with x of degree 2 for Hn(BU(1)) under reduction,and composed with the Cartan formula we obtain:

Sq2(H4(BU(1),Z2)) = Sq2(Z2[x]) = Z2[x2] (2.28)

Therefore β gives Z2[x2] as its image. Now with α being just Sq∗, its dual acts on im βas:

Sq2(x2) = 2 ∗ Sq0(x) ^ Sq2(x) + Sq1(x) ^ Sq1(x)

= 2 ∗ x3 + 0 = 0 mod 2(2.29)

Therefore we have im β = kerα, which is equivalent to say H22,1(E2,1, d2) = 0 or E2,1

vanishes. The same process on α′ and β′ causes E4,1 to vanish too. Now we consider α′

with E2,2 → E3,0 = 0. Since α′ is the dual of Sq∗ or equally an injection, we can say thecohomology for page turning at E2,2 is also zero. We then arrive at the spin bordism

groups up to ΩSpin5 : As the result there is no anomaly in 2 or 4 dimensions with the

n 0 1 2 3 4 5

ΩSpinn (BU(1)) Z Z2 Z⊕ Z2 0 Z⊕ Z 0

global U(1) gauge.

42

2.3.3 SU(2) and SU(n)

SU(2)

With BSU(2) = HP∞, we start with its homology groups which can be calculatedout from cellular homology, with each 4 dimensions forming a cell of quaternion. Thewell-known result is as follows:

Hn(HP∞,Z) =

Z n = 4k,

0 otherwise.

From the universal coefficient theorem for homology it results to similar case forHn(HP∞,Z2)and leads to the E2 page 2.5:

5 0 04 Z Z3 0 02 Z2 Z2

1 Z2 Z2

0 Z Z0 1 2 3 4 5

Table 2.5: E2, E3 and E4 pages of BSU(2), since the big division between nontrivialHn(BSU(2)) groups they pass on until page 4.

Only d4k will affect the pages, so up to dimension 4 only differential concerned isE4,1 → E0,4 or Z2 → Z. This is a trivial problem since we have a homomorphism froma cyclic group to noncyclic group, which can only be d4(Z2) = 0. Finally, we receiveΩSpin

5 (BSU(2)) = Z2, the Z2 anomaly for 4 dimensional Weyl fermions in [1].

SU(n)

With SU(n) where n > 2, its classifying space is Grassmannian Gr(n,C∞). Fromchapter 11.2.4 of [14] we have ci(G) ∈ H2i(BG) in general presented by Chern classesand in our case since c1 = i

2π trF and F ∈ SU(n) being traceless, at most we reach insuch cohomology ring of SU(n) [35]:

H∗(BSU(n),Z) = Z[c2, c3, · · · , cn]

Note that it stops at cn. For n = 2 it also applies with only c2 left in all cohomologygroups, leading to our homology groups above with Hn = Hn ensured with the universalcoefficient theorem and the mod 4 cycle. For n > 2 we are safe to write it towards thesixth (and certainly seventh) column under the universal coefficient theorem as table 2.6:From E6,0 to E4,1, it goes through a similar mod 2 reduction as BU(1). The relevantSteenrod square is Sq2(Z2(c2)). We then take i = 1 in Wu’s formula to achieve:

Sq2(c2) = c1 ^ c2 + c3

= c3

(2.30)

43

6

5

4 Z Z Z

3

2 Z2 Z2 Z2

1 Z2 Z2Z2

0 Z Z Z0 1 2 3 4 5 6 7

Table 2.6: The E2 page of BSU(n) up to the seventh column shared by all n > 3. Upto the fifth spin bordism group only the differential drawn is concerned if itvanishes. If not, we have the same anomaly as in SU(2).

For Z2 reduction it is simply an injection in Z2. The undrawn d2(E4,1) then implies that

the homology is 0 and E4,1 does not survive. As a result, ΩSpin5 (BSU(n)) = 0 for n > 2.

Upon Standard Model

The gauge group of standard model is well acknowledged as SU(3)× SU(2)×U(1). Asfor U(1) charge there are different ways of division by Zn charges where n ∈ 2, 3, 6. Wecan look into the parent theory of the standard model, with its 10+5 representation forfermions not including right-handed neutrinos embedded in SU(5). That is, as subgroupof SU(5), SM is free from Dai-Freed anomalies on a 4-dimensional spin manifold.

A problem about the standard model is that, taking at most Z6 charge from U(1)Y itstill fails to tell right-handed neutrinos from positrons if they exist. They both belongto (1,1)6 representation. In embeddings to SO(10) or greater gauge groups, the 16representations will include the right-handed neutrino. However, restricting to standardmodel, we can discuss upon the Dai-Freed anomaly and confirm that no gauge conditionis put onto the subgroup. The standard model still survives without any necessarycounterpart, which agrees with the current experimental results [36]. A comprehensivediagram for the 15 fermions in 10 + 5 representation is:

10 + 5 =

0 ub −ug ur gr−ub 0 ur ug dgug −ur 0 ub db−ur −ug −ub 0 e+

−dr −dg −db −e+ 0

+

drdgdbe−ν

Since this theory is free from anomalies for gauge symmetry, one may also ask if theyhave global anomalies with discrete symmetries T , C or P . One way of discussing itin general is through extending into unorientable manifolds where a R in Euclidean

44

metric (for instance CP in Lorentz spacetime) is part of the extended spin structure, orour previously introduced Pin± structures. On a Pin manifold the point spin bordismis given as (up to degree 8)[5]: The Z16 in ΩPin+

4 (pt) will cause the Z16 anomaly for

n 0 1 2 3 4 5 6 7 8

ΩPin+(pt) Z2 0 Z2 Z2 Z16 0 0 0 Z2 ⊕ Z32

ΩPin−(pt) Z2 Z2 Z8 0 0 0 Z16 0 2Z2

boundary fermions in unorientable 4-manifolds. The ΩPin±5 (BSU(n)) groups can be

easily read off from the chart to be both zero. That is, even there is a parity counterpartof things in real life, as long as they are described by standard model no new anomalieswill be found even looking into a hidden dimension. From the claims the standard modelis tested despite its other deficits.

45

3 Discrete Symmetry Anomalies andParticle Implementations

In this chapter we will apply the Dai-Freed anomaly methods into discrete symmetrieswithin a particle physics model. Main references will be Chapter 4 of [7] and [9] fordiscrete symmetry anomalies. The part is not targeted at explaining the topic fullybut only serving as an introduction of topics around Z4 symmetry in SpinZ4 structuresinspired Z16 constraints. Generally the anomalies considered are ’t Hooft anomalies witha global symmetry, which is caused by one single (Weyl) fermion in representation of Zneither as a gauge symmetry Spin − Zn or twisted symmetry SpinZ2n. The anomaliesare cancelled by imposing constraints on the system of fermions with discrete symmetrycharges per generation restrained.

3.1 An easy way out for twisted Z4

In specific for twisted Z4 symmetry inspiring SpinZ4 manifolds, it can be treated as an Z2

extension to Spin in parallel to Pin±. We have a Smith homomorphism [8] written withSpin(d; k) as notations for the extension of spin groups. It is shown as Spin(d+k) actingon Rd ⊕ Rk manifolds with ±Ik being the extra k dimensions. Since the constructionis mod 4 it is the previous Spin(

⊕n ε) with “exchanged” dimension numbers from line

bundles, or written as:

k Spin(d; k)

1 Pin−(d)2 SpinZ4(d)3 Pin+(d)4 Spin(d)× Z2

We note Spin[k](d) for Spin(d; k). Consider then the homomorphism from Rd ⊕ Rk toRd−1 ⊕Rk+1, the Smith homomorphism is its version into spin bordism groups, writtenas:

ΩSpin[k]

d (pt)→ ΩSpin[k+1]

d−1 (pt) (3.1)

Where ΩSpin[k]

d (pt) = ΩSpin[k](pt) ⊕ ΩSpin

d (pt). An apparent result is ΩSpin(BZ2) canbe read off easily in low dimensions from Pin− bordism groups. Another result is sinceΩSpin

4 (pt) = 0:

ΩSpinZ45 (pt) ∼= ΩPin+

4 (pt) = Z16 (3.2)

We get out main result for the chapter easily. To describe the Pin+ side we follow [6]and [37] with the correspondent Majorana fermions on the boundary of Pin+ manifold.

46

3.1.1 η-Invariant for Pin+(4)

To start with, consider two Pin+ structures on a 4-dimensional Pin+ manifold X, oneas P and one with the canonical line bundle ε. The latter is noted as P ′ = P ⊗ ε and thecanonical line bundle is defined by its holonomy being −1 under an orientation reversingloop and 1 if the orientation preserves. It also means the transition function P ′ will bethe version of P changing sign when the orientation is reversed.

When combining one Majorana fermion coupled to P and one Majorana fermioncoupled to P ′ we obtain one Majorana fermion in the orientable double cover X, whichalso has an orientation-reversing symmetry on the corresponding directions. In this caseX is free from parity anomalies and has η = 0 mod Z. We have one relation ready:

ηP = ηP ′ = 0 mod Z (3.3)

The second relation will be 8ηP = 8ηP ′ mod Z. We extend X under APS boundaryconditions towards a five dimensional manifold Z = X × [0, 1]. The odd dimensionalmanifold has the simple form I = −2η|10 for index, will from definition is integer. More-over, consider the extended spin structure to be P⊗V , it is a certain Pin(5). Since I stillhas its physical meaning as the number of eigenvalues passing through 0, the Kramerdoubling of the eigenvalues regarding to antilinear operation T = Kγ2γ5

1 will guaranteea degeneracy of zero modes and thus I = 0 mod 2. As the result we have ηP⊗V beinginvariant mod Z.

In order to obtain ηP out from ηP⊗V , one way is to take a trivial V which candecompose ηP⊗V as multipliers of ηP or ηP ′ . For P it is trivial to take V = R8 toobtain 8ηP . For P ′ the equivalent choice will be V = ε⊕8. We consider two topologicalproperties for the discussion, the Cech cohomology group or Stiefel-Whitney classeswi(V ) and de-Rham cohomology related Pontrjagin class p1 ∈ H4(V,Z). The formeris simple with w(V ) = w(ε)8, where the coefficients from w1 to w4 all consist of evennumbers thus are trivial. The latter can be put trivial as follows:

The characteristic class p1(ε⊕8) can be deformed into 2p1(ε⊕4) under the cohomologydeformation. From the fact that p1(ε⊕4) =

∑41 c1(ε)2 = 0 if we consider the complexifi-

cation into ε ⊗R C for the analytic calculation of characteristic classes. Since we reachto a torsion group which is zero, this Pontrjagin class is also zero and thus vanishes. Weare safe to say that 8ηP = 8ηP ′ mod Z holds for our discussion.

The condition for those two conditions to hold can be various, while the most relaxedone will be ηP and ηP ′ both being a multiple of 1/16. So for we proved that η-invariantfor Pin+(4) is a multiple of 1/16 and the general spin bordism group cannot be greateror smaller than Z16. In specific, RP4 has η-invariant to be 1/16, which illustrates onecase.

1Here K is complex conjugation and γ2, γ5 are real gamma matrices.

47

3.2 Calculation of BZn, twisted or not

In this section the bordism property being considered is formally Γ5Spin ⊆ Ω5

Spin, which

is a proper subgroup of Hom(ΩSpinn (BG), U(1)) and reflexes the term e−2iπη precisely.

There is also one claim that because of the Chinese remainder theorem, Z2kn∼= Z2k⊕Zn

for odd n. If we “assume” a SpinZn structure it will be (Spin × Zn)/Z2, which doesnot exist since there is no shared generator of Zn and Z2. Moreover, for the SpinZ2n

case, if n is odd in principle we will expect SpinZ2n ∼ Spin × Zn under spin bordism.The explanation is briefly given as follows: First we restrict ourselves on Spin with avanishing w1 so that the orientable condition is satisfied. In the level of w2 the “twist”for manifolds not permitting a Spin is written in [9] as ΩSpin(BZn; ξ), where ξ is areal vector bundle over BZn. The twisted bordism group [38] has the triple description(M, s, f), with f : M → BG being a continuous map and our spin structure s on thebundle TM ⊕ f∗(BG). If ξ allows a spin structure, the pullback bundle added is trivial

and we can identify ΩSpin(BG, ξ) ∼= ΩSpin(BG), or equally ΩSpinZ2n ∼= ΩSpin(BZn).Since BZn = L∞n the infinity-dimensional lens group, we have:

H2(BZn,Z2) = Tor(Zn,Z2) = Zgcd(n,2) (3.4)

So we have H2(BZn) = Z2 for even n and 0 for odd n. Since the second Stiefel-Whitney

class disappear iff n is odd, we have ΩSpinZ2n ∼= ΩSpin(BZn) iff n is odd. Further, we

have the principle that ΩSpinZ4 is not ΩSpin(BZ2).

3.2.1 BZnThe discrete symmetry group Zn can be represented by elements of integer rings andgroup elements as e2iπ/n or sections of U1/circle being identified. Moreover, for a repre-sentation we give RU(Zn) =

⊕n−1s=0 ρs · Z for the description where ρs = λs mod n. We

may start with the spectral sequence for a sneak peak:To start with we need cohomology groups of BZn. The lens space is given by L2k

n =S2k−1/λ(Zn), where we obviously see the Zn being a principle bundle. Following example2.43 of [25] and setting k →∞, we read off after applying universal coefficient theorem:

Hk(L∞n ,Z) =

Z k = 0,

Zn k is odd,

0 k is even and nonzero.

And for discrete homology:

Hk(L∞n ,Zm) =

Zm k = 0,

Zm ⊗ Zn k 6= 0.

It is practical to obtain maximum spin bordism groups through AHSS only for odd n,drawing down the second E2 pages and starting individual discussions through Steenrodsquares for Z2 differentials concerned. We display the E2 pages here as references:

48

7654 Z Zn Zn Zn Zn32 Z2

1 Z2

0 Z Zn Zn Zn Zn0 1 2 3 4 5 6 7

Table 3.1: The E2 page of BZn bordism for odd n, it is sparse with hardly any differen-tials and easy to discuss upon.

The ΩSpin5 terms are not affected at all by differentials. We can claim it to be a

maximal Zn2 or note as some extension noted as e(Zn,Zn), so that such sequence isexact:

0→ Zn → e(Zn,Zn)→ Zn → 0 (3.5)

While further discussion should be left to another analysis, the even case is a bit com-plicated:

7654 Z Zn Zn Zn Zn32 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

1 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2

0 Z Zn Zn Zn Zn0 1 2 3 4 5 6 7

Table 3.2: The E2 page of BZn bordism for odd n, the non-vanishing differentials are tobe determined by each Steenrod squares.

The richness of differentials in 3.2 means usually the discussion is nontrivial. ForZ2 terms as an example, the Steenrod squares act on Stiefel-Whitney classes given asH∗(BZn,Z2) = Z2[w1]. We use Chinese remainder theorem Z2km = Z2k ⊕ Zm for modd to restrict our discussion to a Z2k problem and an odd problem which we havealready dealt with. With the problem simplified we are free to step forward and analyzeH i(Zn,Z2) = wi. The Steenrod squares now act in an iterative way, starting withSq1(w) = w2 and Sq2(w) = 0:

Sq1(wn) = w ^ Sq1(wn−1) + wn+1 = nwn+1 (3.6)

49

And,Sq2(wn) = w ^ Sq2(wn−1) + w2 ^ Sq1(wn−1) + 0

=

n−1∑i=1

iwn+2

=n(n− 1)

2wn+2

(3.7)

So all differentials in the second page ending in 4k + 2 and 4k + 3 terms are not caus-ing vanishing in turning page. In Ek,5−k terms this will cause E4,1 and E3,2 to van-ish, and the fate of E5,0 can be understood as a Z2 reduction since it being a cyclicgroup. For instance, In the case of BZ2 it will vanish to be trivial. However, dictatingE4,2 → E1,4 is not obvious here, as Z2 → Zn can be nonzero and homomorphic becauseof cyclic condition. On the other hand, with Adams spectral sequence [9] the condition

is∣∣∣ΩSpin

5 (BZ2k)∣∣∣ ≤ 22k/4, which agrees with our result. Moreover, from Smith isomor-

phism the BZ2 case is ΩSpinn (BZ2) = ΩPin−

n−1 (pt) ⊕ ΩSpinn (pt), which agrees with both

when giving ΩSpin5 (Z2) = 0.

When studying discrete symmetry anomalies we want conditions they impose to aparticle theory. Z3 symmetry is an example when AHSS predicts a maximum mod9 anomaly, which agrees with the local gauge and gravitational anomaly cancellationcondition. We show it through writing the U(1) charge qi = 3si+ri, where ri ∈ −1, 0, 1.Following the conditions:

0 =∑

qi = 3∑

si +∑

ri (3.8)

The first shows ri = 0, with mod 3 part absorbed into si. Moreover,

0 =∑

q3i =

∑27s3

i + 27s2i qi + 9siq

2i + q3

i

=∑

q3i mod 9

=∑

qi mod 9

(3.9)

That is, we have the Z9 valued mapping from η-invariants to U(1), which requires theZ3 charges to be summing up to 9 in the whole theory. We present the general casefor BZn anomaly cancellation as follows, embedding the symmetry into U(k) resultingto a representation τ(a1, a2, . . . , ak) = ρa1 ⊕ ρa2 ⊕ · · · ⊕ ρak . Set the lens space asL(n; a1, a2, . . . , ak), where we also note k ∈ 2Z in order for the spin structure to exist.The η-invariant reads [11]:

ηs(L(n, a1, a2, . . . , ak), R) =1

n

∑λ 6=1

tr(R(λ))λ

12

(a1+a2+···+an)

(1− a1)(1− a2) · · · (1− ak)·∑j

c1(Hj)[CP1]F(λj)

(3.10)Which takes distinct forms in different cases2. Take k = 2 for instance, it suffices todescribe the ΓSpin5 (BZn) situation. The spin manifolds are X(n; a1, a2) = S(H ⊗ H ⊕

2Up till finishing the author is unable to determine the exact form, other than following the formsin e.g. [9]

50

1)/τ(a1, a2). Results of ΩSpin5 (BZ2) = ΓSpin5 (BZ2) are:

ΓSpin5 (BZn) ∼= In/In ∩RU0(Zn)4

where In =⊕j≥0

(ρ1 − ρ−1)(ρ0 − ρ1)2jρj−1 · Z

and RU0(Zn) = (ρ1 − ρ0) ·RU(Zn)

∼=

0 n = 2,

Zn ⊕ Zn/4 n = 2m,m > 1

Z3n ⊕ Zn/3 n = 3m

Zn ⊕ Zn n = pm, p > 3 and p is prime

(3.11)

When we divide Zn as prime powers. When combining back we obtain the general form.The resultant anomaly conditions are as follows, given by Dirac genus as:

η([X], R) = η(L(n; 1, 1, 1, 1), τn([X])R) mod Z (3.12)

Where τn([X]) generates the In/In ∩ RU0(Zn)4 elements. In our case the maximumgroup of generators are ρ1 − ρ−1 and ρ2 − ρ−2, each resulting in:

η(L(n; 1, 1, 1, 1), (ρ1 − ρ−1)ρs) = − 1

n

(A4(s+

n

2+ 1;n, 1, 1, 1, 1)− A4(s+

n

2− 1;n, 1, 1, 1, 1)

)mod Z

η(L(n; 1, 1, 1, 1), (ρ2 − ρ−2)ρs) = − 1

n

(A4(s+

n

2+ 2;n, 1, 1, 1, 1)− A4(s+

n

2− 2;n, 1, 1, 1, 1)

)mod Z

(3.13)Which results in the following condition for ρs, understood as Zn charges:∑

s

η([X1n], ρs) =∑s

− 1

6n(n2 + 3n+ 2)s3 = 0 mod Z

∑s

η([X2n], ρs) =− 1

3n

(2s3 + (n2 + 6)s

)mod Z

(3.14)

Or similarly:

(n2 + 3n+ 2)∑

s3 = 0 mod 6n, 2∑

s1 = 0 mod n (3.15)

As a special version of gauge-gravitational anomalies in U(1).

3.2.2 SpinZ2n and SpinZ4

As noted before it is no different from BZn when n is odd. However, if n is evenwe will expect the lens space to not allow spin structures. Conditions will be both kodd and n even for Lk(n) = S2k−1/Zn. In five dimensions the concerned lens space isL(m; a1, a2, a3), S5 under discrete symmetry dividing U(3) with three eiπa/n generators.

We give the results directly as ΓSpinZ4

5 groups and constraints as follows, dividing intopowers of prime numbers, for each the bordism groups are:

ΓSpinZ2n

5 =

Z8n ⊕ Zn/2 n = 2m,m ≥ 1

Z3n ⊕ Zn/3 n = 3m,

Zn ⊕ Zn n = pm, p > 3 and p is prime

(3.16)

51

And the cancellation condition put on charges:

(2n2 +n+1)∑

s3− (n+3)∑

s1 = 0 mod 48n, n∑

s3 +∑

s1 = 0 mod 2n (3.17)

The conditions will result into Z16 and Ns = 16a for massless fermions of charge 1 in ageneration if we take n = 2.

In specific, for Z4 we can refer to 3.2 with different Steenrod squares, given as Sq2w[34],

which has the relation:

Sq2w(wn) = Sq2

(wn) + w2 ^ wn =

(n(n− 1)

2+ 1

)wn+2 (3.18)

So for lower entries in E2 page we have all arrows moving two dimensions to the left. Inother words all p+ q = 5 terms are surviving into the third page. Turning another pagewill then confirm them all surviving to E∞, which gives us Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2

∼= Z16.We now reach the Z16 anomaly with an easier access.

3.2.3 Some comments on Z4 and 16

The 16 constraint will certainly require the existence of a new type of fermion sincewe have 15 fermions per generation in Standard Model so far. Such predictions meetwith various of models including Pati-Salam, SU(5)× U(1)X and in general GUT withSO(10) or Spin(10). Moreover, in Standard Model we notice one combination of chargevalue X = 2Y − 5(B −L) being present and has value 1 mod 4 for every candidates. Inshort the chart is3

Fermions BL Y X

q 13

13 −1

u −13 −4

3 −1d −1

323 3

l −1 −1 3e+ 1 2 −1νR 1 0 −5

The X charges fit in Z4 as U(1)X a possible extension. Such discussion will fitour model into breaking pattern4 of SO(10) into SU(5) as (00001)16 = (0000)1−5 +(0001)53 + (0100)10−1, where we have an anomaly free parent theory SO(10). Thebreaking patterns are similarly illustrated with R⊗ C⊗H⊗O [2], where from SO(10)we expand into Cl(10) and exterior ΛC(5) representations. In lack of time and effortswe will recommend readers to find further developments in papers mentioned for de-tails. Moreover, in the gauge part we will have a Spin(10) ∼= Cl(10) with Z2 identifiedin order to break into 16. An coincidence worth mentioning is into Pati-Salam model

3Related numbers might disagree across the presentations, we take the same numbers as [36], whilenoting in texts including [7] they may vary depending on the chirality, yet the Z4 symmetry is preserved.

4See table 58 in [39].

52

[40] we have for B − L part, into SU(4) = SU(3)× U(1) breaking it reduces into stan-dard model and loses its Z4. However, the Pati-Salam model can be elegantly writtenin (Spin(4) × Spin(6))/Z2 where Spin(4) = SU(2)L × SU(2)R and Spin(6) ∼= SU(4).Therefore in the space of gauge symmetries we obtain a Spin(4)× Z4/Z2. However, wecan only understand it as a coincidence so far since the anomalies discussed above arenot for such groups.

Another guess will be fermions in Standard Model+right-handed neutrinos, or betterknown as MSSM, can be coupled to a topological superconductor with Pin+(4) structure,as our “easy approach” gives. We however don’t attempt to expand into this theory,although recommending [37] as a further reference. With all discussions above, we closethe thesis with wide openings.

53

4 Summary

4.1 Results

We have shown with Dai-Freed theorem and APS index theorem, a (one-year-old) newtype of anomaly for fermions in d-dimensions with gauge group G is spotted as non-zerod+ 1-dimensional spin bordism group ΩSpin

d+1 (BG). The type called Dai-Freed anomaliesare when the mapping torus construction for Dai-Freed is extended to a general manifoldwith spin structure. Related spin bordism groups are well-studied by AHSS and otherspectral sequences. In this thesis as a main result, we showed ΩSpin

5 (BG) = 0 for

G = U(1) and G = SU(n) for n > 2, and ΩSpin5 (BSU(2)) = 0. A regarding important

conclusion is that in 4 dimensions Standard Model embedded in SU(5) is free from thisnew type of anomaly.

Moreover, with G a discrete group Zn and further extending the spin structure asSpinZ2n , the attempt to study Dai-Freed anomalies with related spin bordism groupsis briefly summarized. For ΩSpin

5 (BZn) groups anomalies are given maximum groupas Zn ⊕ Zn, with further reducing given an even n. Other studies with representationtheory showed us anomaly cancellation condition for Zn charges, listed in Chapter 3.Noticeably, for SpinZ4(n) structures it can be related to Pin+(n) structures by Smithhomomorphism, both giving a Z16 anomaly and Z16 constraint to number of fermions.For Pin+(4) this is the amount of 3d Majorana fermions on the boundary of a topologicalsuperconductor, while for SpinZ4(4) this might relate to Standard Model with right-handed neutrinos. The contents are meant to describe such an interest in theoreticalphysics.

In the introduction part we also made a mathematical introduction into Spin struc-tures with fiber bundle language and Clifford algebra. Towards Pin structures andSpinZ4 some introductions are attempted to be given a depth for further interest. Wealso gave introductions into exact sequence and homology groups, which are basis of ourdiscussions. Further, we should note it as a work undone and more understandings onother works other than original calculations.

4.2 Possible further developments and acknowledgement

As noted, there are branches which the author noticed but failed to cover or developbefore the completion of thesis. Here is a list of further thoughts:

1. In some recent studies covered by [29], the APS index theorem has a causalityissue. In the paper the problem was partly studied with Fujikawa’s method which

54

gave the same result. The studies also relate to domain walls which are what thethesis planned to cover, yet has to be left for further notice.

2. The studies upon Pina,b,c groups are noted here as one mislead. However it mightshow new structures for to study and provide a different understanding in thetopics. C symmetry in covered by the author’s more recent paper [41] which alsoincluded higher spin structures. The author would like to see further implementa-tions yet did not get the resource.

3. In the TQFT and type-II string sections there are recent developments with anomalystudies. Due to the ability up to writing they are not covered. However, with possi-ble chances, the author might like to reach in the fields for a more detailed and clearpicture which Dai-Freed anomaly paints in high energy and low energy theories.

4. As noted the part regarding to particle models around discrete symmetries arenot developed enough. The discussions upon Clifford algebra was planned to helpillustrate this part. While surrendering on adding it into the thesis, the authormust note that it might still be a rewarding area to look into. Moreover, beforefinishing the author noticed a new version of [7] which included discussions uponsupersymmetry fermions with the same Z4 symmetry. From Higgsinos and gaugi-nos the anomalies are not yet as nicely cancelled as SM fermions, which is worthnoticing.

4.2.1 Acknowledgement

In general, due to depth of the work, I the author finish the thesis with regrets andpities. However, despite all sorrows it is my pleasure to present this sneak peak intoimplementation of topology into particle theories. Above all, I would thank Prof. JosephMinahan for the supervision and leading me to this area when I was in a bigger confusioninto research topics. I would also thank Dr. Guido Festuccia for approving this topic,as well as Dr. March Chiodaroli for help into understanding topology. Towards themI also thank for the recommendation letters they wrote and sent. I will also thank allother Master students in classes for countless helps in life.

The work will be dedicated to my parents for their support throughout the years. Iwould also give specific thanks to Ms. Liisa Lotta Tarvainen for her emotional supportas well as shelter for me throughout the second academic year and afterwards.

55

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