alan carey the australian national university montréal 2015 · alan carey the australian national...

Spectral flow and the essential spectrum

Alan Carey

The Australian National University

Montréal2015

Alan Carey The Australian National University Spectral flow

Preamble

Motivation for this work comes from the uses of spectral flow incondensed matter theory (where operators can have very complicatedspectra and are not obviously Fredholm).

In this talk I will mention some results inspired by scattering theory.Scattering theory probes the essential spectrum.

Typically a self adjoint Fredholm operator, particularly those arising inquantum theory, will have both discrete and essential spectrum.

Index theory has tended to focus on operators with compact resolventas arises in the study of compact manifolds.

On non-compact manifolds, which is where scattering theory isstudied, the essential spectrum is both physically and mathematicallyimportant.


Also, on the mathematical side it may contain topological information(a good example is provided by the Novikov-Shubin invariants forcovering spaces).

Furthermore there are many examples of non-Fredholm operators thatarise naturally on non-compact manifolds and it is of interest to know ifspectral geometry can be developed for them.

Colleagues contributing to the work described today include NurullaAzamov, Peter Dodds, Fritz Gesztesy, Harald Grosse, Jens Kaad,Galina Levitina, Denis Potapov, Fedor Sukochev, Yuri Tomilov, DimaZanin among many others.


The story really started with some meetings with Fritz Gesztesy at theErwin Schrödinger Institute in Vienna in 2010 and 2011.

Gesztesy was using a notion from scattering theory: the spectral shiftfunction, in order to generalise a theorem of Robbin-Salamon on therelationship between the Fredholm index and spectral flow.


Part 1. Spectral flow and spectral shift

In the 1950s I. Lifshitz introduced the spectral shift function. He wasmotivated by studying the S-matrix in quantum scattering problems.This spectral shift function was made mathematically precise (in theframework of trace class perturbations) by M. Krein in his famous 1953paper.

For a pair of self-adjoint bounded operators A0 and A1 such that theirdifference (being the perturbation) is trace class, there exists a uniqueξ ∈ L1(R) satisfying the trace formula:

Tr(φ(A1)− φ(A0)) =

∫ξ(µ)φ′(µ)dµ

whenever φ belongs to a class of admissible functions.

NB: ξ is not determined pointwise in general but only almosteverywhere.


1.1 Spectral flow

At first sight this is an unrelated concept.

Recall that spectral flow was introduced in the index theory papers ofAtiyah-Patodi-Singer in the 1970’s. They give Lusztig the credit for theidea.

Consider a norm continuous path Ft; t ∈ [0, 1] of bounded self adjointFredholms joining F1 and F0.

J. Phillips in 94-95 introduced an analytic definition of spectral flow thatis more useful than the original approach of APS.

We let, for each t, Pt be the spectral projection of Ft corresponding tothe non-negative reals. Then we can write Ft = (2Pt − 1)|Ft|. Phillipsshowed that if one subdivides the path into small intervals [tj , tj+1]such that Ptj ,Ptj+1 are ‘close’ in the Calkin algebra then they form aFredholm pair and the spectral flow along {Ft} is∑

j

index(PtjPtj+1 : range Ptj+1 → range Ptj )


1.2 Recent interactions between spectral flow and spectral shift

• To my knowledge the first person to understand that spectral shiftand spectral flow are the same (when both are defined as occurs insome examples) was Werner Mueller (Bonn) in 1998.

• Note that one usually considers the spectral shift function foroperators with continuous spectrum and spectral flow for operatorswith a discrete spectrum.

• In 2007, (with Nurulla Azamov, Peter Dodds and Fedor Sukochev),we showed that under very general conditions1 that guarantee thatboth are defined, spectral shift and spectral flow are the same moduloconventions on how kernel dimensions are handled.

1and also in semi-finite von Neumann algebrasAlan Carey The Australian National University Spectral flow

• The spectral shift function is defined more generally, even fornon-Fredholm operators. Azamov posed the question: can it beregarded as giving a generalisation of spectral flow in thenon-Fredholm case?

• Pushnitski started a program of extending the Robbin-Salamonresult relating spectral flow and the Fredholm index to operators withsome essential spectrum using spectral shift function methods. Thisseems to be the most promising way to study Azamov’s question.

•We have proved some index/spectral flow formulas when theoperators in the path of self adjoint Fredholms have some essentialspectrum.

• Recently we have seen that the formulas hold without Fredholmassumptions. This raises the question of their meaning in general.


Part 2: The Witten index: a proposal for the non-Fredholm case

Witten was studying supersymmetric quantum field theory and as a toymodel introduced supersymmetric quantum mechanics in the early1980s.

Mathematically we interpret ‘supersymmetry’ as a Z2 grading.

Witten speculated that there might be a notion of index fornon-Fredholm operators and proposed a formula for this, extending anidea due to Callias.

Witten’s idea was taken up by mathematical physicists, Bollé,Gesztesy, Grosse and Simon in 1987 and Borisov, Schrader, Muellerin 1988. They were able to give it a mathematical basis.


Early results

• Bollé et al found simple examples in two dimensions of Diracoperators coupled to connections that had only continuous spectrumfor which Witten’s formula gave a finite answer.

•When these operators are Fredholm then Witten’s formula gives theFredholm index (even if the operators have some essential spectrum).In non-Fredholm situations the formula can give any real number.

• They proved a stability result that gave invariance of the Witten indexfor relatively trace class perturbations but not compact perturbations inthe non-Fredholm case.

• Callias studied a three dimensional example but unfortunately hismethods are mathematically incomplete. All other early examples ofthe Witten index were one and two dimensional.


2.1 Gesztesy-SImon

The first substantial theoretical advance was due to Gesztesy- Simon(1987).

We are given a Hilbert space equipped with a Z2 grading γ and a selfadjoint unbounded operator D such that Dγ + γD = 0.

For simplicity take the space to be H(2) := H⊕H for a fixed separableinfinite dimensional Hilbert space H. Then we may write:

γ =

(1 00 −1

)D =

(0 D−D+ 0

).

Gesztesy-Simon consider, for z in the intersection of the resolventsets, the situation where the difference

(z +D−D+)−1 − (z +D+D−)−1

is in the trace class but the individual operators are not trace class.


Then they define nonnegative self-adjoint operators

H1 = D−D+; H2 = D+D−

and introduce the spectral shift function ξ(µ;H2;H1) associated with apair (H2;H1) such that ξ(µ;H2;H1) = 0, µ < 0 and

Tr[(z +H1)−1 − (z +H2)−1] = −∫ ∞

0ξ(µ;H2;H1)(µ+ z)−2dµ

and they introduce the ‘Witten index’ of D+ as

limz→0

zTr[(z +H1)−1 − (z +H2)−1]

whenever the limit exists. They prove that in some cases it is given bylimµ↓0 ξ(µ;H2;H1).


2.2 Relation with spectral flow

Consider spectral flow along a path of self adjoint Fredholm operators{A(s)|s ∈ R} joining A0 and A1 on K.

We convert this into an index problem on a Z2 graded space byintroducing on L2(R,K) the operators:

D+ =∂

∂s+A(s), D− = D∗+

and on L2(R,K(2)) the operator

D =

(0 D−D+ 0

).

Recently it was shown by GLMST that for Fredholm operators A± thatare invertible but may have some essential spectrum there is asituation where the index of D+ is the spectral flow along the path A(s).


2.3 The GLMST trace formula.

In a long paper in Advances in Mathematics in 2011 GLMSTestablished a trace formula that relates the Fredholm index of D+ withspectral flow along a ‘sufficiently regular’ path A(s), between invertibleendpoints A− and A+, even when there is some essential spectrum.

But a very restrictive hypothesis was needed.

HYPOTHESIS: the relatively trace class perturbation assumption holdsnamely: (A+ −A−)(1 +A2

−)−1/2 is trace class.

This means their result does not apply directly to geometric situationsand differential operators in general (it is the zero dimensional case!).

The GLMST trace formula is:

trL2(R,K) z((H2 + z)−1 − (H1 + z)−1

)=

1

2trK

(gz(A+)− gz(A−)

), (1)

where gz(x) = x(x2 + z)−1/2. The operator differences on both sidesare trace class under their hypotheses.


When all operators have discrete spectrum the LHS is the Fredholmindex of D+ and the RHS is spectral flow.

More generally when there is some essential spectrum GLMST showthat in the Fredholm case the limit as z → 0 on both sides exists, andusing spectral shift function theory, one obtains:

index D+ = ξ(0, H1, H2) = ξ(0, A+, A−) = sf(A−, A+)

The first time this kind of formula appeared was in an older paper ofPushnitski.

Note that the RHS of the trace formula is not completely new. fIt isrelated to a unital spectral flow formula that I proved with Phillips many(nearly 20) years ago.

What is new is the situation where the operators in question haveessential spectrum.


This old formula, in the case where A− and A+ are unitarily equivalent,A(s) is a ‘nice path’ joining them, and we assume that (1 +A(s)2)−r istrace class for r large enough, is

sf(A−, A+) = C̃r

∫tr[dA(s)

ds(1 +A(s)2)−r]dt

where C̃r = Γ(r)Γ(r−1/2)Γ(1/2) .

If we can set r = 3/2 then C̃3/2 = 1/2 and it is true that

tr[dA(s)

ds(1 +A(s)2)−3/2] = tr[

d

ds

(g1(A(s))

)].


So we may integrate with respect to s and the old trace formula gives

sf(A−, A+) =1

2tr(g1(A+)− g1(A−)

)in agreement with GLMST.

BUT the RHS of the GLMST formula is not spectral flow in thepresence of essential spectrum.

It is only after taking the limit z → 0 that one can prove we have theFredholm index on the LHS and spectral flow on the RHS.


This GLMST formula raises lots of questions.

For example what happens when the operators on one or the otherside of the trace formula are not Fredholm?

In fact D+ is not Fredholm if one of A± has a kernel.

Can we remove the relatively trace class perturbation assumption?

This is equivalent to asking whether there is a version of the formulathat applies to DIrac type operators in higher dimensions.

Then we need a relatively Schatten class perturbation assumption.


Part 3: The trace formula

Motivated by a question of Mathai, in BCPRSW, we established aformula relating the index to spectral flow for Dirac-type operators oncovering spaces. The set-up there is that we have a simply connectedspace M̃ admitting an action by a discrete group Γ such that thequotient M = M̃/Γ is a compact manifold.

Assume A± are unitarily equivalent (by a bundle automorphism)Γ-invariant Dirac type operators on M̃ and τΓ is the Γ trace of Atiyah.

We were interested in the ‘type II spectral flow’ along a pathA(s), s ∈ [0, 1] of Γ invariant self adjoint operators joining A±.

This is defined by taking Phillips analytic approach to spectral flow andgeneralising it to the type II setting.

Note that in this type II setting operators can have continuousspectrum including zero but still be ‘Breuer-Fredholm’.


We form D± as before (that is D+ = ∂∂s +A(s)) and regard these as

differential operators on a Hilbert space H of sections of a bundle onS1 × M̃ .

Then the following trace formula holds:

τΓ,H(e−εD−D+ − e−εD+D−) =

√ε

π

∫τΓ(

dA(s)

dse−εA(s)2)ds

The LHS is the (Breuer)-Fredholm index and the RHS is semi-finitespectral flow.


Using a Laplace transform argument we obtain for r > 0: on the LHS:∫ ∞0

εre−ε(e−εD−D+−e−εD+D−)dε = Γ(r)((1+D−D+)−r−(1+D+D−)−r),

while, inserting the same function of ε and integrating on the RHS weobtain: ∫ ∞

0εr+1/2e−επ−1/2

∫dA(s)

dse−εA(s)2dsdε.

Because it is OK to interchange the order of integration we obtain

Γ(r + 1/2)π−1/2

∫dA(s)

ds(1 +A(s)2)−(r+1/2)ds.


So we have on each side of the formula respectively:

Γ(r)((1 +D−D+)−r − (1 +D+D−)−r)

andΓ(r + 1/2)π−1/2

∫dA(s)

ds(1 +A(s)2)−(r+1/2)ds.

Under the assumption that all the resolvents in both expressions aretrace class and under appropriate continuity assumptions for the pathA(s) we may take traces and then interchange the trace and integral toobtain the following trace formula.

τH,Γ((1 +D−D+)−r − (1 +D+D−)−r)

= Cr+1/2

∫τΓ(

dA(s)

ds(1 +A(s)2)−(r+1/2))ds

where Cr+1/2 = Γ(r + 1/2)π−1/2Γ(r)−1.


With Harald Grosse and Jens Kaad I have been able to prove a purelyoperator theoretic version of this trace formula that makes no Fredholmassumptions:

arXiv:1501.05453 “On a spectral flow formula for the homologicalindex".

The method of proof is analogous to what was used in BCPRSW butthe hypotheses are not inspired by NCG but by our previous work inCommun. Math. Phys. (arXiv:1310.1796) “Anomalies of Dirac typeoperators on Euclidean space".

More relevant to NCG is a version of the trace formula that is part ofwork in progress with G. Levitina, and F. Sukochev.


We have a path of self adjoint operators {A(s), s ∈ R} (on a separableHilbert space H) joining A± of the form A(s) = A− +B(s).

Assume B(s), B′(s) bounded, norm continuous in s and smooth (in thesense that iterated commutators of B(s), B′(s) with |A−| are bounded)and satisfying (A+ −A−)(1 +A2

−)−(r+1/2)) is trace class for somer > 0;

Then, under some side conditions (that guarantee the integral belowexists in trace norm), the following trace formula holds:

trL2(R,H)((1 +D−D+)−r − (1 +D+D−)−r)

= Cr+1/2

∫trH(

dA(s)

ds(1 +A(s)2)−(r+1/2))ds

where Cr+1/2 = Γ(r + 1/2)π−1/2Γ(r)−1 and D± = ±∂/∂s+A(s).


Corollaries

On a compact manifold where A± are invertible and have discretespectrum then this formula equates the index of D+ on the LHS withspectral flow along the path {A(s)} on the RHS. (That is a differentversion of the Robbin-Salamon theorem.)

When the operators on both sides have some essential spectrum thenone has to take a limit (analogous to the z → 0 limit we saw previously)and exploit the spectral shift function in order to have an index/spectralflow interpretation of both sides (proof not yet complete).

In general the LHS of the trace formula gives an expression, in thelimit, for the Witten index when it exists. The RHS limit then givessome kind of ‘generalised spectral flow’.

We are investigating the properties of this flow now.


2.4 The basic even dimensional example

Let σj ; j = 1, 2, 3 be the Pauli matrices. Consider the following operatordensely defined on L2(R2, C2):

D = σ1(∂

i∂x− ax) + σ2(

∂

i∂y− ay)

where a = (ax, ay) is a connection. Physically we are looking at afermion in an electromagnetic field.

This gives an example of a supersymmetric quantum system withgrading γ = σ3.

For one particular choice of connection the ‘Witten index’ can beshown to be given by the total magnetic flux calculated from thecurvature da. It need need not be integral. By perturbing this case onegets other examples with the same index.


These allowed perturbations are constrained by somewhatcomplicated relatively trace class conditions that arise from the factthat whereas the Fredholm index is topological and invariant undercompact perturbations the more refined invariants being considered fornon-Fredholm operators are not.

In higher dimensions they are associated to Schatten ideals and not tothe ideal of compact operators.

These refined invariants like the Witten index are associated withcyclic homology and not topological K theory. Cyclic homology isconnected to algebraic K theory in previous work of Connes-Karoubiand much later, Kaad.

However it is not yet known what kind of geometric data can bedetected by these invariants that are associated to homology and notK-theory.


2.5 Cyclic homology: the work of Jens Kaad, R.W. Carey and J.Pincus

Consider bounded operators T satisfying [T, T ∗] is trace class. Such‘almost normal operators’ formed the subject of an extensive series ofpapers by Carey-Pincus in both the Fredholm case and thenon-Fredholm case.

Carey-Pincus (1986) wrote T = X + iY where X is the real part and Yis the imaginary part of T . Then we have [T, T ∗] = 2i[X,Y ] and thuswe are dealing with an ‘almost commuting’ pair of self adjointoperators.

They then introduced their ‘principal function’, observed that it can berelated to the spectral shift function and created a functional calculusfor almost commuting pairs.

They introduced an index for such almost commuting pairs using thelimit at zero from above of the spectral shift function (when it exists)just as for the Gesztesy-Simon ‘Witten index’.


The punchline

Witten goes to Carey-Pincus via the map

D+ → T = D+(1 +D−D+)−1/2

because(1 +D−D+)−1 − (1 +D+D−)−1 = [T, T ∗]

Form the algebra A generated by T, T ∗ and consider the quotientB = A/(A ∩ Trace class) with q : A → B the quotient map..

The Witten index is defined via q(T )⊗ q(T ∗)→ Tr([T, T ∗]) which turnsout to be a function on the degree one cyclic homology group of B.

By analogy the Fredholm index would be a function on the K-theory ofA/compacts.

Higher dimensions: there is an ‘index’ constructed from

Tr((1 +D−D+)−n − (1 +D+D−)−n) = Tr((1− T ∗T )n − (1− TT ∗)n)

which is a functional on a more complicated cyclic homology group forthe algebra A.


2.6 Kaad’s viewpoint

Suppose we are given Banach algebras A and J where J is an ideal inA (not necessarily closed).

Let b : J ⊗A+A⊗ J → J be an extension of the Hochschild boundarygiven by

b : s⊗ a+ a′ ⊗ t→ sa− as+ a′t− ta′

where s, t ∈ J and a, a′ ∈ A.

Choose for example A to be the C∗-algebra generated by T, T ∗ andJ = L1(H) ∩A.

Next: there is an exact sequence X : 0→ J → A→ B → 0 wherei : J → A is the inclusion and q : A→ B is the quotient.


By the zeroth relative continuous cyclic homology group of the pair(J,A) we will understand the quotient space HC0(J,A) = J/Im(b).We also have the first cyclic homology group of B denoted HC1(B) (cfLoday). Then Kaad in Chapter 1 of his thesis proves the followingresult:

Theorem(i) The operator trace determines a well defined map on the zerothcontinuous relative cyclic homology group Tr∗ : HC0(J ;A)→ C(ii) This can be extended to a map on HC1(B) using the connectingmap

∂X : HC1(B)→ HC0(J ;A)

coming from the above exact sequence.

Notice that B is a commutative algebra. The pair q(T ); q(T ∗) defines aclass q(T )⊗ q(T ∗) in HC1(B). This class maps to the commutator[T, T ∗] under ∂X .


Part 3: Connecting all this together

The Witten index is formulated in terms of unbounded operators. It canbe reformulated as a bounded operator problem using the map on selfadjoint unbounded operators D on the space H(2) = H⊕H that takesus to the operator FD = D(1 +D2)−1/2.

Recall that we have the grading γ on H(2) and the relationγD +Dγ = 0 but we only consider the case where

z[(z +D−D+)−1 − (z +D+D−)−1]

is trace class and do not ask for the individual resolvents to be traceclass as we would in a Fredholm problem.


Set z = λ2 and rewrite the previous formula as

Tr((1 + λ−2D−D+)−1 − (1 + λ−2D+D−)−1)

so we are scaling D by λ−1 and taking λ→ 0. This is the large timelimit in the McKean-Singer situation although now for operators thatare not necessarily Fredholm.

So let us define F λD = λ−1D(1 + λ−2D2)−1/2 and then we have

1− (F λD)2 = (1 + λ−2D2)−1

Introduce the notation:

F λD =

(0 T ∗λTλ 0

).

then putting these definitions together:Tλ = λ−1D+(1 + λ−2D−D+)−1/2 and

1− (F λD)2 =

(1− T ∗λTλ 0

0 1− TλT ∗λ

).


Consequently our assumption that

(1 + λ−2D−D+)−1 − (1 + λ−2D+D−)−1

is trace class translates in the bounded picture to the assumption thatthe commutator [Tλ, T

∗λ ] is trace class.

The Witten index is thus calculating limλ→0 Tr([Tλ, T ∗λ ] when this exists.

Conclusion. The theorem quoted above from Kaad’s thesis tells usthat the Witten index is calculated by ‘scaling’ from a function definedon HC1(B).

This extends to higher dimensions.


3.1 The homological or ‘topological’ side

Joachim Cuntz asked the question of whether one can use theS-operator in cyclic homology to understand the homotopy invarianceof this number Tr([Tλ, T ∗λ ]).

As in any formulation of index theory we want to relate the analytic side(ie Gesztesy-Simon-Carey-Pincus) to a ‘topological’ formula.

With Jens’ insights we now have a homological formulation that is acounterpart to the older analytic theory from the eighties.

In addition the role of general Schatten ideals, not just the trace class,is now clear.


3.2 New examples

• Everything up until now applies to differential operators only in lowdimensions.

• So, with the help of Harald Grosse, Jens and I generalised theexample of Bollé et al to Dirac type operators on R2n.

• These operators are of the general form

DA =

(0 D +A

D∗ +A∗ 0

).

where A is a U(N) connection which has non-trivial asymptotics atinfinity.

• For the odd dimensional case with {A(s)|s ∈ R} joining Dirac typeoperators A0 and A1 on L2(R2n−1, S) (S a vector bundle over R2n−1)we introduce, on L2(R, L2(R2n−1, S), the operators:

D+ =∂

∂s+A(s), D− = D∗+


Then, as before, we have the operator

D := DA =

(0 D−D+ 0

)which is of Dirac type as in the general even dimensional case.

• But in these geometric examples we do not have(A+ −A−)(1 +A2

−)−1/2 trace class.

In fact we have (A+ −A−)(1 +A2−)−s/2 trace class only for for

s > 2n− 1.

In these examples we also have

tr((1 +D−D+)−n − (1 +D−D+)−n) <∞

although the individual terms are not trace class.


We can combine the preceding observation with our previousdiscussion of the connection between index and spectral flow and seethat we need to find the precise extension of the GLMST formula forn > 1.

Recall that we conjectured the following relation as a generalisedGLMST formula when the operators in question have essentialspectrum and are possibly even non-Fredholm:

τ((1 +D−D+)−r − (1 +D−D+)−r)

= Cr+1/2

∫tr(dA(s)

ds(1 +A(s)2)−(r+1/2)

)ds

provided both sides are finite for r sufficiently large.

While we understand the LHS in examples we are only just starting onthe RHS in the non-Fredholm case.


Support for this conjecture is provided by

• the formula is directly related to our extension of the GLMST formulafor 1 + 1 dimensions, that is with r = 1 (work in progress),

• in 2n dimensions the left hand side is precisely the homological indexof D+ that Jens, Harald and I introduced for Dirac type operators whenr = n,

• the integrand on the RHS is indeed trace class for the examples ofDirac type operators we have studied in 2n− 1 dimensions forr > n− 1.


GLMST and the spectral flow formula

We include a small calculation to support the preceding conjecture.

I claim that under the relatively trace class perturbation assumptionand with dA(s)

ds (1 +A(s)2)−1/2 trace class we have

tr(dA(s)

ds(1 +A(s)2)−3/2

)= tr

( dds

(A(s)(1 +A(s)2)−1/2))

I will prove this only for A(s) = (1− s)A+ + sA−, s ∈ [0, 1] though itholds more generally. The argument is simply to take the formula thatholds in operator norm:

d

ds(A(s)(1 +A(s)2)−1/2)

= (A+ −A−)(1 +A(s)2)−1/2 −A(s)2(1 +A(s)2)−3/2(A+ −A−)

= (A+ −A−)(1 +A(s)2)−1/2 − (1 +A(s)2)(1 +A(s)2)−3/2(A+ −A−)

+(1 +A(s)2)−3/2(A+ −A−)


and apply the trace to both sides of the last equality.

Then using cyclicity of the trace the first two terms cancel leaving onlythe third which, using cyclicity again, gives the required result.

Now we see that∫ 1

0tr(dA(s)

ds(1 +A(s)2)−3/2

)ds = tr(g1(A+)− g1(A−))

We conclude then that for r = 3/2 and under the assumptions of theoriginal GLMST paper the conjectured trace formula implies theGLMST trace formula and vice versa.


3.3 The homological index

• The map DA → FDAemployed previously gives an operator

T = D+(1 +D−D+)−1/2 with

(1− T ∗T )n − (1− TT ∗)n = (1 +D−D+)−n − (1 +D−D+)−n

in the trace class.

•We calltr((1 +D−D+)−n − (1 +D−D+)−n)

the homological index of D+

•We have that [T, T ∗] is in the nth Schatten class so we see that wehave a generalisation of the situation in Carey-Pincus.

• But [T, T ∗] being in th nth- Schatten class is not equivalent to

Tr[(1− T ∗T )n − (1− TT ∗)n] <∞ (∗)


Our main new result is that

• There is a bicomplex that defines an homology theory whichreplaces the cyclic homology discussion of Kaad’s thesis in the case ofthese higher Schatten ideals.

• The homological index is the pairing of a cycle in our bicomplex witha cocycle in a dual cohomology theory. It is homotopy invariant incyclic theory as suggested by Joachim.

•We now have an homology theory into which the old work of R.Carey and Pincus fits as the lowest degree example.

• As before we may scale DA so that there is a corresponding mapλ−1D+ → Tλ for these scaled operators and we can show that thehomological index is non-trivial by calculating the λ→∞ limit.

• The λ→∞ limit is known as the anomaly in the physics literatureand we have found a local formula for it in terms of the curvature of theconnection.


3.4 The limit λ→ 0

• In work in progress we are aiming for general results on the λ→ 0limit using homological methods.

• Otherwise one needs the spectral shift function (this relies onGesztesy-Simon and work in progress with Gesztesy-Sukochev etal...) and so far we have made progress only for differential operatorsin 1 + 1 dimensions.

• This approach uses the GLMST formula:

Witten index = limz→0

trL2(R,K2) z((H2 + z)−1 − (H1 + z)−1

)= lim

z→0

1

2trK

(gz(A+)− gz(A−)

)= generalised spectral flow.

and both sides of this relation are given by the respective spectral shiftfunctions at zero .


In higher dimensions we know that the appropriate Witten indexformula is:

limλ→0

tr((1 + λ−2D−D+)−r − (1 + λ−2D−D+)−r) (2)

for an appropriate choice of r and when the limit exists.

Based on the conjectural higher dimensional GLMST formula it followsthat there is a ‘generalised spectral flow’ given by:

limλ→0

Cr+1/2 λ−1

∫tr(dA(s)

ds(1 + λ−2A(s)2)−(r+1/2)

)ds (3)

when this limit exists. For λ > 0 we expect (2)=(3) so that the problemis existence of the limit.

We know that the RHS is spectral flow when one is in the situation ofBCPRSW.

One method of proof would be via the spectral shift function.


3.5 The bicomplex

Take the algebra A generated by T and T ∗.

We are assuming that

Tr[(1− T ∗T )n − (1− TT ∗)n] (∗)

is finite. Let I be the smallest ideal containing (1− T ∗T )n and(1− TT ∗)n and let J be the ideal generated by(1− T ∗T )n − (1− TT ∗)n so that J ⊂ I.


3.6 Relative cyclic homology.

Within the (b, B) bicomplex for A there is a subcomplex determined byI. For each k ∈ N ∪ {0}, define the subspace

Ck(A, I) := I ⊗A⊗ · · · ⊗A+ · · ·+A⊗ · · · ⊗A⊗ I,

where the algebra A appears precisely k times. The Hochschildboundary b and the Connes boundary B restrict to boundary maps

b : Ck(A, I)→ Ck−1(A, I) and B : Ck(A, I)→ Ck+1(A, I).

This subcomplex equipped with b and B gives the relative bicomplexB∗∗(A, I).Similarly we also have B∗∗(A, J)


3.7 Concatenation

For the purposes of showing that

Tr[(1− T ∗T )n − (1− TT ∗)n] (∗)

arises as the result of pairing a cocycle and a cycle we use a notion wecall concatenation applied to the two complexes B∗∗(A, I), B∗∗(A, J).

Let X and Y be first quadrant bicomplexes. The vertical and horizontalboundaries are denoted by dvX , d

hX and dvY , d

hY respectively and

dvXdhX + dhXd

vX = 0, dvY d

hY + dhY d

vY = 0.

Let X∗0 and Y∗0 denote the 0th column of X and Y respectively. Theseare chain complexes when equipped with their vertical boundaries.

Suppose that we have a chain map α : X∗0 → Y∗0 of degree 1, that is,α : Xk,0 → Yk+1,0 satisfies the relation dY α+ αdX = 0.


The concatenation of X and Y along α is the bicomplex Y∐αX which

in degree (k,m) is given by

(Y∐α

X)k,m := Yk,0 for m = 0, and X(k−1),(m−1) for m ∈ {1, 2, . . .}.

The horizontal boundary is given bydh(α) : (Y

∐αX)k,m → (Y

∐αX)k,(m−1),

dh(α) := α for m = 1, and dhX for m ≥ 2.

The vertical boundary is given bydv(α) : (Y

∐αX)k,m → (Y

∐αX)(k−1),m,

dv(α) := dvY for m = 0, and dvX for m ≥ 1.

Then the homological index is obtained from a pairing of a cycle inY∐αX with a cocycle from a dual bicomplex.


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