CERN-TH.5240/88

MORSE THEORY INTERPRETATION OFTOPOLOGICAL QUANTUM FIELD THEORIES

J. M. F. Labastida

Theory Division, CERNCH-1211 Geneva 23, Switzerland

ABSTRACT

Topological quantum field theories are interpreted as a generalized form of Morsetheory. This interpretation is applied to formulate the simplest topological quantumfield theory: topological quantum mechanics. The only non-trivial topological invariantcorresponding to this theory is computed and identified with the Euler characteristic.Using field theoretical methods this topological invariant is calculated in different waysand in the process a proof of the Gauss-Bonnet-Chern-Avez formula as well as someresults of degenerate Morse theory are obtained.

Permanent address: Instituto de Estructura de la Materia, CSIC, Serrano 119, 28006 Madrid,Spain

CERN-TH.5240/88November, 1988

– 1 –

1. Introduction

Topological quantum field theories [1,2,3] may provide a very useful tool in analyz-ing some mathematical problems and in describing a possible unbroken phase of stringtheory. The essential feature of topological quantum field theories is that they possessa symmetry which leads to the formulation of possibly non-trivial topological invari-ants. Due to the presence of this symmetry, which will be called Q-symmetry, thereare no physical excitations. Commuting bosonic degrees of freedom cancel againstanticommuting bosonic degrees of freedom, leaving only zero modes which give riseto topological invariants. This Q-symmetry, as well as the content of the theory re-sembles the BRST quantization of some theory. Typically, topological quantum fieldtheories consist of commuting bosonic fields and anticommuting bosonic ones makingan appearance very similar to the one in the BRST quantization of a classical bosonictheory. Furthermore, the operator corresponding to the Q-symmetry is anticommutingand squares to zero at least in some subspace of field configurations. Ever since topo-logical quantum field theories were created it was believed that these theories mightcorrespond to a BRST quantization of an underline gauge invariant theory.

Canonical BRST-quantization methods have been used [4,5] to make contact withtopological Yang-Mills theory [1]. In these cases, starting with a trivial action (either agaussian, zero, a topological invariant, or 137), and BRST-gauge-fixing deformations ofthe gauge potential modulo gauge transformations, a formulation very close to the onein ref. [1] was obtained. However, although close, the resulting theory and topologicalYang-Mills are not equivalent. There are two reasons for this. First, the chosen ‘gaugefixing’ leads to several Gribov copies (arguments leading to this conclusion can befound in [1,6]) and so it is not really a ‘gauge fixing’. Actually, it is because there arethese Gribov copies that topological field theories are interesting and, in fact, this isthe basis of the interpretation of these theories in terms of generalized Morse theorypresented in this paper. Second, the observables of topological Yang-Mills, once theYang-Mills gauge symmetry is fixed [7] (if one ignores for the moment possible Gribovambiguities [8]) are not trivial and, in fact, are in correspondence with the Donaldsoninvariants [9] as shown in [1]. However, the observables of the theories presented in[4,5] are trivial. For a recent analysis of this question see [10]. The Q-symmetry inthis theory does not correspond to a BRST symmetry and it has a nature similar tosupersymmetry in a supersymmetric Yang-Mills theory (as one could have concludedfrom the fact that topological Yang-Mills can be constructed from a truncated N = 2supersymmetric Yang-Mills theory [1]). In this sense, the quantization of topologicalYang-Mills proposed in [7] seems to be the right one.

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Similar arguments lead to the same conclusion in the case of topological sigmamodels [3]. The resemblance of topological quantum field theory to trivial theories ina BRST-quantized form may, however, be used to formulate new topological quantumfield theories. This has been treated without much success for the moment for the caseof topological gravity in four [11] and in two [12,13] dimensions.

In this paper it is conjectured that topological quantum field theories can be in-terpreted as a generalized form of Morse theory. This correspondence is shown in fulldetail for the case of topological quantum mechanics which seems to be the simplestof such theories. A similar construction can be carried out to formulate topologicalYang-Mills and topological sigma models. The connection with generalized Morse the-ory allows to find out the general pattern of a topological quantum field theory andpermits, in principle, its construction.

The only non-trivial topological invariant in topological quantum mechanics cor-responds to the Euler characteristic. Using methods of quantum field theory one canshow the invariance of the observables of the theory under continuous deformations ofthe potential. This allows to obtain a proof of the Gauss-Bonnet-Chern-Avez formula.In addition, making a deformation of the potential to other adequate choices, we obtainresults corresponding to ordinary degenerate Morse theory. This shows how powerfulthis type of theories can be from a mathematical point of view.

The paper is organized as follows. In sect. 2 the generalized form of Morse theory isconjectured. In sect. 3, it is applied to the case of topological quantum mechanics andits topological invariants are computed. Finally, in sect. 4, we state our conclusionsanalyzing the general characteristics of topological quantum field theories and listingsome of the possible theories.

2. Morse Theory

In this section we will briefly review the elements of Morse theory [14] which willbe utilized in the paper and we will conjecture their generalization. In the next sectionwe will discuss the validity of this generalization using methods of topological quantumfield theories.

Let us consider a compact C∞ manifold V and a smooth function f on it. A pointp ∈ V is a critical point of f if df vanishes at such a point. The Hessian at p, Hpf ,is a quadratic form on TpV , the tangent space to V at p. In local coordinates xi

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centered at p, the matrix of (Hpf)ij relative to the base ∂/∂xi at p is

(Hpf)ij =∂2f

∂xi∂xj. (2.1)

The point p is called a non-degenerate critical point of f if detHpf = 0. The indexof p is the number of negative eigenvalues of Hpf and it will be denoted by λp(f).One of the results of Morse theory is that if f is a function with a finite number ofnon-degenerate critical points the Euler characteristic of the manifold, χ(V ), can bewritten in terms of the indices associated to the critical points of f , namely,

χ(V ) =∑

p

(−1)λp(f), (2.2)

where the sum extends over all the critical points. This equation shows that such a sumis a topological invariant. Morse theory proves stronger results than this, however, it is(2.2) that is of interest to us. Our main goal in this section is to conjecture an equationlike (2.2) for a more general case. In more concrete terms we will conjecture thatan expression similar in spirit to the right-hand side of (2.2) is topological invariant.In the subsequent section we will argue that the conjecture is true using methods oftopological quantum field theories.

Let us introduce more notation to be able to deal with the case of having degeneratecritical points [15] which in fact are the central part of our discussion. LetMα ⊂ V bea connected submanifold of V . We will callMα a non-degenerate critical submanifoldfor f if df = 0 along Mα and HMα

f is non-degenerate on the normal bundle ν(Mα)ofMα. If we denote by det′HMα

f the determinant of HMαf restricted to the normal

bundle, from our definition, det′HMαf = 0.

In order to introduce our conjecture let us rewrite (2.2) as a sum of a ratio ofdeterminants:

χ(M) =∑

p

det Hpf

|det Hpf |. (2.3)

The possible generalizations of this sum for the case of having degenerate points shouldinvolve integrals over critical submanifolds. These integrals in general would not leadto topological invariants unless, possibly, if one introduces some dα-forms being dα thedimension of the critical submanifold Mα. For the case at hand, i.e., degeneratedMorse theory with non-degenerate critical submanifolds, it is known that there exist

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dα-forms Aα such that the quantity

Ξ =∑α

∫

Mα

Aα det′HMαf

|det′HMαf | (2.4)

is a topological invariant. In fact, Aα is zero if dα is odd, and it is the dα

2 exteriorproduct of the curvature 2-form Ωab onMα if dα is even.

Actually, we are interested in a further generalization of (2.2). So far, our discussionis based on the analysis of the critical submanifolds originated from the equation df =0, being f a smooth function on a manifold V . A more general situation consist of thefollowing. Let us consider two manifolds Σ and M , being Σ compact, and a topologicalclass of smooth mappings φ : Σ → M . Let V be the space of mappings containedon such a topological class. Notice that now V does not need to be a manifold. Letus consider an operator F acting on V such that it maps φ ∈ V into sections in thepullback of the tangent bundle of M , φ∗(T (M)), and such that V and ImF have thesame dimensionality. The kernel of F defines critical subspaces Mα ⊂ V . Let JMα

F

be the Jacobian of the operator F at the critical subspaceMα. This Jacobian definesan homomorphism on sections of the pullback bundle φ∗(T (M)). JMα

F will play therole of the Hessian in the previous discussion. Contrary to the case of the Hessian,JMα

F is not self-adjoint in general. We will consider situations where the Jacobian isdegenerate in the normal subspace toMα. It is typically the case that the kernel andthe cokernel of this Jacobian restricted to the normal subspace are not trivial. Thus,we should not restrict ourselves to non-degenerate critical subspaces. We will try tomake sense of (2.4) even in these situations.

We conjecture that given two manifolds Σ and M (being Σ compact), a topologicalclass of mappings V = φ|φ : Σ → M, and an operator F acting on V which mapsφ ∈ V into sections of φ∗(T (M)) such that V and ImF have the same dimensionality,there exist dα-forms Aα living in the critical dα-dimensional subspace Mα ⊂ V suchthat

Ξ =∑α

∫

Mα

Aα det′JMαF

|det′JMαF | (2.5)

is a topological invariant, i.e., Ξ depends only on the topology of Σ and M . If thekernel and cokernel of F in the normal bundle to Mα are non-trivial this expressioncontains ratios of zeros. This may seem rather undefined at this point. The formalismof topological quantum field theory will provide a way to make (2.5) well defined.Notice that in the case that Σ is a point, V is the space of mappings of that point intoM = V , and F =df , (2.5) reduces to (2.4).

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Using methods of topological quantum field theories one can show that Ξ is topo-logical invariant and, in the process, find the forms Aα. Of course (2.5) may be avacuous statement unless a good choice of F is made. The topological sigma modelsand the topological Yang-Mills theory formulated by Witten correspond to two caseswhere there is a good choice of F and where the forms Aα are constructed. In thispaper we will present in detail a simpler case to show that (2.5) holds. The treatmentof the topological sigma model is analogous to the one of this simpler model. The caseof topological Yang-Mills has special features and deserves a separated publication.However, it corresponds also to a particular case of (2.5).

3. Topological Quantum Mechanics

In this section we will evaluate the right-hand side of (2.5) for a concrete choiceof Σ, M and F . This will lead to the formulation of topological quantum mechanics.Applying methods of topological quantum field theory to this formulation we will showthat indeed Ξ is a topological invariant and we will construct the forms Aα enteringin (2.5). Topological quantum mechanics is a rather simple model and so it doesnot contain a rich quantity of topological invariants. However, the discussion of thismodel in the framework of Morse theory proposed in the previous section illustratesthe general characteristics of this type of formulations very simply.

3.1 The Model and its Field Theory Representation

Let us consider a smooth Riemannian manifold M with Euclidean signature anda smooth vector field ξi (or more precisely, a section on the tangent bundle T (M))on it. The space V introduced in the previous section consists of all the mappingsφ : S1 →M of a given topological class. Taking ui as local coordinates in M the mapφ can be locally described by functions ui(τ) where τ denotes a point on the circle S1.The operator F is chosen to be

F i(uj) = ui + ξi(uj), (3.1)

where the dot denotes a derivative with respect to τ . Notice that V and ImF have thesame dimensionality. The corresponding Jacobian at a critical subspaceMα of (3.1) is(JMα

F )ij which, acting on an arbitrary section of the pullback bundle φ∗(T (M)), λi,has the form:

(JMαF )ijλj = Dλj + (Djξ

i)λj , (3.2)

where Dk is the Riemannian covariant derivative, DkWi = ∂kW

i + ΓikjW

j , Γikj is

the standard affine connection on M , and D is the covariant derivative on sections of

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φ∗(T (M)). If λi is one of these sections, D is defined to act as

Dλi = λi + ukΓikjλ

j . (3.3)

Notice that (3.2) is correct at a critical subspace Mα. In general, JMαF contains

non-covariant looking terms which just vanish at the critical subspace. Let us restrictourselves for the moment to the case of only one critical subspace and let us call theJacobian at this subspace Di

j ,

Dij = δi

jd

dτ+ uiΓi

kj + Djξi. (3.4)

The conjectured topological invariant (2.5) can then be written as

Ξ =∫

M

A det′D|det′D| , (3.5)

where A are certain forms to be constructed.

We would like to express (3.5) as an integral over the whole space V and not justover the critical subspaceM. To do this one could think of a delta function restrictingV to the critical subspace. This could be easily accomplished since such a delta functionwould be accompanied by the absolute value of the determinant of D restricted to thenormal subspace of M and evaluated at M in the denominator. This is in whatrespect to the integration region. Another problem in extending (3.5) to an integralover the full space V is that the d-form (d is the dimension of the critical subspace) Acontains d indices which should be properly contracted. There is a satisfactory formof handling this problem which involves the expression of det′D in the numerator ina field theoretical language. One would be tempted to write such a determinant as afunctional integration over anticommuting fields i and χi (which are sections on thepullback of the tangent bundle φ∗(T (M))),

det′D ∼∫

Di Dχi exp(−∫

S1

dτ iDijχ

j). (3.6)

However, the right-hand side of (3.6) vanishes whenever the operatorDij (or its adjoint)

has zero modes. One way to remove these zeros is to soak them up. This procedurejust requires the introduction of a product of χi and i in (3.6) in such a way that,properly normalized, (3.6) becomes an equality. However, we may have also zeros inthe denominator of the integrand of (3.5) that we want to overcome so we should notsoak up all of them. Let us analyze the situation in detail.

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Given a solution to the equation

ui + ξi = 0, (3.7)

we may ask when this solution can be deformed ui → ui + δui so that it remains stilla solution. From (3.7) one finds that to lowest order in δui,

Dijδu

j = 0, (3.8)

where D is the operator defined in (3.4). The solutions of (3.8) would give us thedimension of the critical subspaceM. Actually, this is not true because there may beobstructions to make the infinitesimal deformation finite. Typically (although not inthe case at hand), if the adjoint operator of D, D∗, has zero modes, not all the solutionsto (3.8) can be made finite. The solutions of (3.8) constitute the kernel of the operatorD while the zero modes of its adjoint constitute the cokernel of D. The dimension ofthe critical subspace, d, is the difference between the dimension of the kernel and thecokernel, i.e., the index of the operator D. If we denote by p the dimension of kerD andby q the dimension of cokerD, we have d = p− q. In the particular case of topologicalquantum mechanics the index of D is zero and so p = q. However, we will carry outthe analysis for arbitrary p and q so that we can develop the general formalism.

Going back to (3.6) and using this reasoning we observe that the functional integralcontains p χi-zero modes and q i-zero modes. This means that one must introduce ap-product of χi and a q-product of i to soak up these zero modes. However, this is notwhat we want to make contact with (3.5). That ratio of determinants contains ratios ofzeros whenever the normal subspace toM is degenerate. Such is the case if q = 0. Thequantity (3.5) contains q zeros in the numerator and q zeros in the denominator. Thismeans that we do not have to soak up all the χi or i zero modes but only d = p − q

of them to make contact with (3.5). This implies that (3.5) could be written as,

Ξ =∫

Dui Di Dχi δ(ui + ξi)Ai1,...,id(ui)

χi1 ...χid exp(−∫

S1

dτ iDijχ

j),(3.9)

where Ai1,...,id is a certain tensor. Notice that if a factor i or one more χi wereintroduced in front of the exponential one would break the balance of zero modes inthe normal subspace and the expression either becomes infinity or zero. To compute

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A in (3.5) from this expression one must expand χi and i in zero modes and non-zeromodes,

χi =d∑

µ=1

xiµχµ +

p∑µ=d+1

xiµχµ + non-zero modes,

i =q∑

µ=1

rµi µ + non-zero modes,

(3.10)

where xiµ are the d zero modes which give rise to a finite deformation of a solution and

so form a basis of the tangent space of M. One of the contributions in (3.9) consistsin the integration over the non-zero modes, giving the non-vanishing part of det′D,and the integration over the χµ giving the product of zeros necessary to balance theproduct of zeros of the denominator. There are other contributions where, instead,some of the χi in front of the exponential become χµ. After adding all of them up andintegrating µ and χµ one is left left with the integration of χµ which gives the form Ain (3.5) from Ai1,...,id . Notice that since the xi

µ, µ = 1, ..., d make a basis in the tangentspace of M this way to proceed creates a canonical measure in M, i.e., if we denoteby aµ, µ = 1, ..., d the local coordinates in M, the measure (

∏dµ=1 daµ)(

∏dµ=1 dχµ) is

invariant under reparametrizations of M since aµ and χµ transforms in opposite wayas follows from (3.10). Expression (3.9), however, is still not well defined due to thepresence of q ratios of zeros. We will show below, using field theoretical techniqueshow this expression can be made well defined.

The delta function entering (3.9) can be expressed as a functional integral justintroducing a multiplier field di. As i and χi, di is a section in the pullback bundleφ∗(T (M)). We have,

Ξ =∫

Dui Di Dχi Ddi

(2π)nAi1,...,id(u

i)χi1...χid

exp(−

∫

S1

dτ (idi(ui + ξi) + iDijχ

j)).

(3.11)

Notice that we have used the normalization of the delta function as it would corre-spond to constant configurations. In this form the expression for Ξ resembles thevacuum expectation value of an operator in quantum field theory. The argument ofthe exponential should be identified as the action of such a theory. Once written inthis form we can use the machinery of quantum field theories to study the properties

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of Ξ. The first question to ask is if the action

S =∫

S1

dτ (idi(ui + ξi) + iDijχ

j), (3.12)

possesses some symmetries. We will find out that in fact this action has a BRST-like symmetry characteristic of topological quantum field theories. To look for thissymmetry it is very useful to utilize the fact that typically these theories have a similarstructure to the one of a trivial theory in which one starts with zero as a classicalaction and one gauge fixes deformation invariance in the gauge ui + ξi = 0. Of course,this is only a similarity which helps in finding the symmetries involved in topologicalquantum field theories. Topological quantum field theories are non-trivial while theother ones are. This trick, however, does not always works as is the case of topologicalgravity [11].

Using the insight discussed in the paragraph above, we start with the BRST versionof the algebra of deformations of ui:

δui =iεχi,

δχi =0,(3.13)

where ε is a constant anticommuting parameter. For the ‘gauge fixing’ of the defor-mations we should introduce an ‘antighost’ field i and a ‘Nakanishi-Lautrup’ one, di.The transformations of these fields are typically of the form,

δi =εdi + ... ,

δdi =... ,(3.14)

where the content of the dots is determined as follows. On the one hand, one wouldlike to have

δ( ∫

S1

dτ ii(ui + ξi)), (3.15)

leading to (3.12). That requirement fixes uniquely the transformation of i,

δi = ε(di − ijΓjikχ

k). (3.16)

On the other hand, one wants δ to be nilpotent so that (3.15) (and so (3.12)) is invariant.Using (3.16) that requirement determines the transformation of di,

δdi = ε(idjΓjikχ

k +12Rlk

jijχ

lχk), (3.17)

where Rijkl is the Riemann tensor Rij

kl = ∂iΓk

jl−∂jΓkil+Γk

imΓmjl−Γk

jmΓmil . By construc-

tion, (3.12) is invariant under (3.13), (3.16) and (3.17). The transformations (3.13),

– 10 –

(3.16) and (3.17) preserve U -invariance if the fields (ui, χi, i, di) have the U -number

(‘ghost’ number) assignment (0, 1,−1, 0). The action (3.12) has U -number 0.

The fundamental property of the symmetry found for the action (3.12) is that theaction itself can be written as a symmetry transformation. Defining the operator Q

such that

−iεQ, X = δX, (3.18)

where X represents any field ui, χi, i or di, one finds,

S = −iQ,V, (3.19)

where

V = i

∫

S1

dτ i(ui + ξi). (3.20)

Now that we have discovered this symmetry in the action (3.12), we are in theposition to analyze the properties of Ξ in (3.11) using methods of quantum field theory.Our aim has been all along to show that there exist some forms A such that Ξ istopological invariant. Let us investigate which ones are the properties that Ai1,...,id

have to satisfy for this to be true.

Given an operator O, we define its vacuum expectation value, < O > as,

< O >=∫

(DX)O e−S(X). (3.21)

With this definition Ξ in (3.11) is just the vacuum expectation value of an operator.To study when < O > is topological invariant let us perform a variation of the metricinM and of the vector field ξi, which we will denote it by δ. From (3.19) this variationtakes the form,

δ < O >=∫

(DX) e−S(X)(−iQ, δVO + δO

), (3.22)

and therefore, if we assume that O does not depend on the metric ofM neither on the

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vector field ξi, we conclude that < O > is topological invariant if

< −iQ, δVO >= 0. (3.23)

A sufficient condition for (3.23) to hold is,

Q,O = 0. (3.24)

The reason for this is that if (3.24) holds then (3.23) can be written as

< −iQ, δVO >=< −iQ, (δV)O >, (3.25)

and, assuming that the measure in (3.21) is invariant under a Q-symmetry transfor-mation, the right-hand side of (3.25) vanishes.

According to (3.24) and (3.25), if an operator O is such that Q,O = 0, then< O > is topological invariant. Actually, one would like to know which ones amongthe operators leading to (3.24) do not lead to a trivial vacuum expectation value. Aswe have discussed in the previous paragraph, if an operator is Q-exact, i.e., there existsanother operator λ such that O = Q, λ, and its vacuum expectation value vanishes.Therefore, to find out the non-trivial operators leading to topological invariants weare led to study the cohomology of Q. This analysis is identical to the one done in[3]. Applied to this case one finds that the cohomology classes are built out of the deRham cohomology of M . Given a de Rham cohomology class in M with representativeA = Ai1,...,indui1 ...duin, one can build two operators. The first one,

O(0)A = Ai1,...,inχi1...χin, (3.26)

satisfies

Q,O(0)A = 0, (3.27)

and constitutes the first set of Q-cohomology classes,

W(0)A = O(0)

A . (3.28)

The second one is built out of the exterior derivative (in S1) of the first one,

dO(0)A = iQ,O(1)

A , (3.29)

– 12 –

where,

O(1)A = inAi1,...,indui1χi2...χin. (3.30)

The integral over S1 of this 1-form leads to the second set of Q-cohomology classes,

W(1)A =

∫

S1

O(1)A . (3.31)

In general, the quantities which may lead to non-trivial topological invariants from aquantum field theory point of view are vacuum expectation values of the form,

<∏

i

W(ki)Ai

>, (3.32)

where ki = 0, 1 depending on which set of operators W(ki)Ai

belongs.

We are now in the position to make contact with (3.11). So far we have been veryvague about the operator Ai1,...,idχ

i1...χid in (3.11). We have not specified if it dependson points on S1 or if it involves integrations over some regions of S1. However, thisvagueness is in fact natural since our aim was to construct them. Comparing (3.11) and(3.32) we observe that this construction is about to finish. Let us define the χ-numberof an operator as the number of times that χi appears in such an operator. Noticethat since χi is anticommuting this is well defined. For example, W

(0)A and W

(1)A in

(3.31) have χ-numbers m and m − 1 respectively. If we denote by niχ the χ-number

of W(ki)Ai

, among all the vacuum expectation values of products (3.32), only the oneswhich satisfy

d =∑

i

niχ (3.33)

will lead to expressions of the form (3.11) and will be topological invariants. Noticethat if (3.33) does not hold (3.32) would vanish due to the presence of zero modes.To be more precise, the operator in (3.11) which makes Ξ a non-trivial topologicalinvariant has the general form:

Ai1,...,idχi1 ...χid =

∏i

W(ki)Ai

, (3.34)

where W(ki)Ai

are such that (3.33) holds.

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So far we have analyzed the behavior of (3.11) and we have constructed the oper-ators which could lead to topological invariants. However, when q = 0 the expression(3.11) is not well defined due to the fact that it contains a product of q ratios of zeros.We will try to make it well defined by introducing a regulator and requiring that theanswer be independent of such a regulator. Perhaps there are several ways to do this.Here we will present what we think corresponds to the simplest choice. Let us add tothe action (3.12) a term of the form 1

βdidjgij with β > 0. In the limit β → ∞ this

regulated formulation defines the previous one. Actually we would like to do betterthan that. We would like that all our arguments leading to (3.34) be valid in thisregulated form of the functional integral (3.11). In other words, we would like to havea regulator which is compatible with the Q-symmetry. This certainly can be done andthat will be the way we will proceed.

To have a term of the form 1βdidjg

ij in the regulated form of the action and to keepQ-invariance we may exploit the fact that Q2 = 0 and add the following term to theaction S in (3.12):

Sβ = S + ∆S, (3.35)

where,

∆S = −iQ,1β

dijgij. (3.36)

Using (3.13), (3.16) and (3.17) one finds,

∆S =1β

(didjgij +

12Rkl

ijχkχlij), (3.37)

and performing the change of variables

di = di −i

2β(uj + ξj)gij , (3.38)

the new action becomes,

Sβ =∫

S1

dτ( 1

βdidjg

ij +β

4gij(ui + ξi)(uj + ξj)

+ iDijχ

j +12β

Rklijχkχlij

).

(3.39)

Notice that due to the presence of the new terms the product of ratios of zeros in (3.11)is not present as long as β > 0. If one rescales the fields χi, i and di, χi →

√βχi,

– 14 –

i →√

βi and di → βdi, one finds that β factors out of the action Sβ,

Sβ → Sβ = β

∫

S1

dτ(didjg

ij +14gij(ui + ξi)(uj + ξj)

+ iDijχ

j +12Rkl

ijχkχlij

),

(3.40)

while the measure in the functional integral times Ai1,...,idχi1 ...χid as in (3.11) remains

invariant. Using (3.13), (3.16), (3.17) and (3.38), the Q-transformations of the fieldsafter rescaling ε→ ε√

βbecome,

δui =iεχi,

δχi =0,

δi =ε(di −i

2(uj + ξj)gij − ijΓ

jikχ

k),

δdi =ε[idjΓ

jikχ

k +12Rlk

jijχ

lχk

+i

2gij

((Dχj + (Dkξ

j)χk) +12(um + ξm)Γj

mkχk)]

.

(3.41)

Since the action (3.40) is Q-exact, the vacuum expectation value of any operator invari-ant under a Q-transformation is independent of β. To prove this one has just to followthe same reasoning that led to (3.25). Notice that this procedure could be done start-ing with any other expression with U -number -1 and introducing its Q-transformation.The one chosen here is rather simple because allows to integrate out one of the fields(di).

So far we have been discussing the case of having only one critical subspace. Itis straightforward to generalize the formalism to the case of having several criticalsubspaces. Clearly, the field theoretical formulation is the same for such a case. Infact, the field theoretical formulation forces to add the contributions of different criticalsubspaces. Otherwise topological invariance, as derived in the field theory language, isnot guaranteed.

In this section we have treated topological quantum mechanics in full generality inorder to develop the general formalism. However, since the index of D for this case iszero, the only non-trivial topological invariant corresponds to the operator 1. In thenext section we will carry out the computation of its vacuum expectation value.

– 15 –

3.2 Explicit Calculation of Topological Invariants

In this subsection we are going to evaluate the vacuum expectation values (3.32)leading to topological invariants for specific choices of the vector field ξi. Using thefact that the result must be independent of these choices we will be able to prove theGauss-Bonnet-Chern-Avez formula, and some results of degenerate Morse theory.

First, let us start with a potential W , i.e., a smooth scalar function on M , W :M →R, and let us assume that ξi = ∂iW . The topological classes of mappings fromS1 into M correspond to the homotopy group of M , π1(M). Let us consider the classof trivial mappings, i.e., mappings which can be continuously shrunk to a point. Inevaluating (3.32) we need to analyze first the equation

ui + ∂iW = 0. (3.42)

In particular, we have to find out the critical subspaces corresponding to this equation,and in those subspaces we have to study the eigenvectors of the Jacobian of (3.42).This, in general, may be a rather complicated problem. However, from the analysis ofthe previous subsection we know that the result of the evaluation of (3.32) is invariantunder continuous deformations of the potential. Thus, given a potential W requiringa complicated analysis one can always deform it to a more convenient one. Using thisreasoning we can assume that the potential W has a finite number of non-degeneratecritical points. The reason for this is that any smooth scalar function on M can beuniformly approximated by another one which has a finite number of non-degeneratecritical points.

Let us study the solutions to (3.42). If the critical points of W are labelled by uiα,

a set of obvious solutions to (3.42) are the constant mappings ui(τ) = uiα. Notice that

these mappings are contained in the topological class of mappings under consideration.Let us prove that in fact those are the only solutions. Let us assume that there exist anon-constant solution and let us denote it by vi(τ). This solution generates a loop inM with total length ∮

dτ vivjgij ≥ 0, (3.43)

since the manifold M has Euclidean signature. However, since vi is a solution of (3.42),

∮dτ vivjgij = −

∮dvi∂iW = 0, (3.44)

after using Stokes theorem. Therefore, vi can be only a constant.

– 16 –

Let us evaluate (3.32) for the case of the operator 1, which is the only one leadingto non-trivial topological invariants. Looking at the form of the action in (3.12), afterintegrating di one obtains a delta function that selects the solutions to (3.42). At suchsolutions the Berezin integral of i and χj is straightforward and one obtains,

Ξ =∑α

det ∂i∂jW

|det ∂i∂jW |

∣∣∣∣∣α

= χ(M), (3.45)

where in the last step we have used the standard result of Morse theory discussed in(2.2) and (2.3).

We may also compute Ξ using the regulated action (3.40). In fact, as we shownow, in doing this computation we obtain a proof of the Gauss-Bonnet-Chern-Avezformula. The computation of the partition function corresponding to the action (3.40)is complicated for an arbitrary W even in the limit β →∞. However, as discussed inthe previous subsection, the result is invariant under continuous deformations of thepotential W . In particular, one can make a continuous deformation of W such that itvanishes everywhere in M . In this case the computation of Ξ can be easily done in theβ →∞ limit. Taking (3.40), the quantity to be evaluated is

Ξ =∫

Dui Di Dχi Ddi

(2π)ne−Sβ , (3.46)

where

Sβ = β

∫

S1

dτ(didjg

ij +14giju

iuj + iDχi +12Rkl

ijχkχlij

). (3.47)

The di-integration is gaussian and can be easily performed,

∫Ddi

(2π)nexp(−βdidjg

ij) =√

g

(4πβ)n2

. (3.48)

To carry out the rest of the functional integrations just notice that in the β → ∞limit there is a contribution from the constant configurations and another from theratio of determinants for non constant modes which is known to be 1 due to the Q-symmetry. Therefore, using the result (3.48), the functional integral (3.46) reduces tothe computation of

Ξ =1

(4πβ)n2

∫

M

√gdnui

∫(

n∏p=1

dp)(n∏

q=1

dχq)exp(− β

12Rkl

ijχkχlij

), (3.49)

where the only integration left is the one corresponding to constant Grassmann vari-ables. Using the standard Berezin integration one gets that only the term with n times

– 17 –

the variable i, and n times the variable χi contributes. This implies that for compactmanifolds of odd dimension Ξ = 0, in agreement with the standard result for the Eulercharacteristic. If n is even, n = 2m, one obtains,

Ξ =(−1)m

(4π)m(m)!2m

∫

M

√g dnui εk1l1...kmlmεi1j1...imjmRk1l1i1j1 ...Rkmlmimjm

, (3.50)

which together with the result obtained in (3.42) constitutes a proof of the Gauss-Bonnet-Chern-Avez formula [16]. εi1...in is normalized such that ε1...n = 1. Noticethe cancellation of the β-dependence in the calculation of (3.50), a crucial test for thecorrectness of our arguments. The last step of this calculation resembles the com-putation of the index of a classical elliptic complex using supersymmetric quantummechanics [17]. Topological quantum mechanics is, however, different than supersym-metric quantum mechanics as can be concluded by simply analyzing their symmetryalgebras. While topological quantum mechanics is related to Morse theory, supersym-metric quantum mechanics is not.

We have been using either the regulated or the non-regulated formulations to carryout the computations, according to convenience. We may ask what happen, for exam-ple, if one tries to compute Ξ for the case ξi = 0 using (3.11). That functional integralreduces in that case to a product of n (notice that q = n) ratios of zeros which seemsrather undefined. The regulated theory seems to give a definition to such a product.Furthermore, such a definition is in agreement with expectations.

On the other hand, using (3.40) one may also compute Ξ very simply for the case inwhich W has only a finite number of non-degenerate critical points ui

α. Let us describethis computation. The functional integration of di is carried out using (3.48) as before.To perform the other integrations notice that in the limit β →∞ the main contributioncomes from bosonic configurations which are constant ui = ui

α. Let us expand ui innormal coordinates around one of these constant configurations [18]. If we denote byωi the tangent vector at ui

α which defines the geodesic from uiα to ui, we find,

Ξ =∑α

∫DωiDχiDi

(4πβ)n2

√ge−S(α)(ω

i,χi,i), (3.51)

where,

S(α) = β

∫

S1

dτ(1

4gij(ui

α)(Dikω

k)(Djlω

l) + iDijχ

j

+12Rkl

ijχkχlij + ...),

(3.52)

and the dots denote terms containing at least one ωi and two any other fields. Notice

– 18 –

that since the critical points are not degenerate there are no χi or i zero modes. Toproceed, let us rescale the fields ωi, χi and i with the vielbein and some appropriatepowers of β and other factors,

ωa =√

β

2ea

iωi,

χa =√

βeaiχ

i,

a =√

βeaii.

(3.53)

We are using a notation in which tangent space indices are denoted by Latin lettersat the beginning of the alphabet. The functional integral (3.51) becomes in the newvariables,

Ξ =∑α

∫DωaDχaDa

πn2

e−S(α)(ωa,χa,a), (3.54)

where,

S(α) =∫

S1

dτ((Da

bωb)(Dacω

c) + aDabχb

+12β

Rabcdχeχbcd + ...

).

(3.55)

In the large β limit the leading contribution comes from the ratio of determinants leftafter the integration of ωa, χa and a. All other terms contain powers of β in thedenominator and so they vanish in the limit β → ∞. Taking into account (3.4) andthe fact that at ui = ui

α the first derivative of the potential vanishes, from (3.54) weobtain,

Ξ =∑α

det ∂i∂jW

|det ∂i∂jW |

∣∣∣∣∣α

= χ(M), (3.56)

in agreement with (3.46).

Now that we have developed some machinery we are in the position to derive someresults of degenerate Morse theory. Let us deform the potential W to a new one whichhas critical submanifoldsMα such that HMα

W is non-degenerate on the normal bundleν(Mα). Using (3.40) we are going to derive the expression for the Euler characteristicin this case. This corresponds to a mixed situation between the two already discussed.The reason for this is that on Mα the corresponding vector field ∂iW vanishes. Letus choose coordinates such that the metric of M is box-diagonal i.e., if there are C

critical submanifolds each of dimension dα, the metric does not mix the coordinates of acritical submanifold with the coordinates of the others. The determinant of the metricthen becomes a C + 1 product: g = g⊥(

∏Cα=1 gα), where gα is the determinant of the

– 19 –

metric restricted to the critical submanifoldMα and g⊥ the determinant of the metricrestricted to the directions which are not tangent to any of the critical submanifolds.Let us pick one of these critical submanifolds Mα and let us suppose that its localcoordinates are ui

α, i = rα, rα +1, ..., rα +dα. Now let us expand ui around one of theseui

α in normal coordinates ωi: ui = uiα + ωi − 1

2Γijk(u

i)ωjωk + .... Using this expansionthe expression for Ξ becomes,

Ξ =∑α

∫

Mα

duiα

∫DωiDχiDi

(4πβ)n2

√ge−S(α)(ω

i,χi,i), (3.57)

where S(α) is given in (3.52). Now, depending on whether ωi lies in the directiontangent to the critical submanifold Mα or not we are in one or the other situationabove. If ωi is such that i = rα, ..., rα + dα the contribution to Ξ is dominated by theconstant configurations of ωi, χi and i. If such is the case, only the term involvingRkl

ij in (3.52) survives in the leading contribution. The integration of the non-zeromodes over these dα directions gives at leading order 1 because of the Q-symmetry.Once this is done, one has still to integrate ωi over the remaining directions. In thiscase one must proceed as in (3.53), i.e., one must rescale fields etc., to pick the leadingcontribution in β. One finds,

Ξ =∑α

∫

Mα

duiα√

gαE(mα)

(4π)mαmα!2mα

det′∂i∂jW

|det′∂i∂jW |

∣∣∣∣∣α

εk1l1...kmα lmα εi1j1...imαjmαRk1l1i1j1 ...Rkmα lmα imαjmα,

(3.58)

where mα = dα

2 and E(mα) is such that it is 1 if mα is an integer and zero otherwise,and the prime denotes that the determinant is computed in the normal bundle ν(Mα).The ratio det′∂i∂jW/|det′∂i∂jW | has the same value all along a critical submanifoldMα. In fact, it is related to the index of a non-degenerate critical submanifold. Givena non-degenerate critical submanifoldMα, its index, λα, is defined as the dimension ofthe subspace of the normal bundle ν(Mα) in which HMα

W has negative eigenvalues.In terms of this index,

det′∂i∂jW

|det′∂i∂jW |

∣∣∣∣∣α

= (−1)λα. (3.59)

Using this expression together with (3.46) and (3.50) we finally obtain,

χ(M) =∑α

(−1)λαχ(Mα), (3.60)

which is one of the standard results of degenerate Morse theory [15]. Notice that if wewhere in the situation in which W has non-degenerate critical submanifolds as well as

– 20 –

non-degenerate critical points, (3.60) would hold defining χ(Mα) as 1 when Mα is apoint.

In this paper we have concentrated in obtaining results of Morse theory usingtopological quantum mechanics. Similarly, one could study the case of vector fieldsinstead of potentials. In this case one may also obtain standard results. For example,if one takes a vector field which contains isolated zeros, repeating the steps that led toto (3.46) one obtains a proof of the Poincare-Hopf theorem [19].

4. Conclusions

In this paper we have given an interpretation of topological quantum field theoriesin terms of a generalized form of Morse theory. This connection may be very helpful inderiving results either using techniques of Morse theory or the much less well definedtechniques of quantum field theory. As a first example of this interplay we have con-sidered the case of topological quantum mechanics. Although simple, this model hasshown to be able to reveal some of the virtues of topological quantum field theories.We have been able to obtain a new proof of the Gauss-Bonnet-Chern-Avez formulaand some results of standard degenerate Morse theory. There are two lessons that welearn from this type of analysis. First, topological quantum field theories formulateinvariants in a much larger range of situations than other theories. Notice for examplethat Morse theory is able to prove the result (2.2) for a special case of functions W .However, from the point of view of topological field theories that result is valid for awider range of potentials. A problem of a different kind is which W is more suitableto carry out the functional integral. As we have seen, and this leads us to the secondlesson, even in the case that the potential W is deformed to zero we still get the rightresult. But more important, in computing the topological invariant in two differentways we are able in this case to prove a classical result: the Gauss-Bonnet-Chern-Avezformula. There may be other situations in which this kind of analysis may lead to newtheorems relating topological invariants. The case of topological quantum mechanicsconsidered in this paper is rather simple and one does not expect big surprises. Inmore complicated situations topological quantum field theories may lead to either newinvariants or to a better defined formulation of known ones, along with new theoremsrelating them.

The main ingredients of a topological field theory once the manifolds Σ and M

have been chosen, are the topological class of mappings V and the operator F . Thisoperator is such that V and ImV have the same dimensionality. For the case oftopological sigma models [3], Σ is a Riemann surface and M is an almost Hermitian

– 21 –

manifold. The operator F transforms a mapping φ ∈ V , defined by local coordinatesui into ∂αui + εα

βJ ij∂βuj , where J i

j is the almost complex structure of M and εαβ

is the complex structure of Σ. This mapping is n to n if n is the dimension of M

because the operator F has the properties of a projector. This case is much richerin topological invariants than topological quantum mechanics since the index of theoperator obtained from F (φ) after a variation of φ is in general different from zero.

Topological Yang-Mills with gauge group G corresponds to a situation in which Σis a compact 4-dimensional Riemannian manifold and M is a vector bundle over thefour-dimensional manifold with structure group G. The space of mappings V is madeout of gauge connections modulo gauge transformations on that bundle. The operatorF is such that to each gauge connection it associates the self-dual part of its fieldstrength. The operator F is 3×N to 3×N , where N is the dimension of the Lie groupG. On the one hand, the dimensionality of the space of gauge connections modulogauge transformations is (4− 1)×N , on the other hand the independent componentsof the self-dual part of the field strength are 1

2(124× 3)×N .

A similar analysis shows that topological gravity [11] might exist in four dimensions.In this case Σ is again a four-dimensional compact Riemannian manifold and V is madeout of metrics modulo reparametrizations and Weyl transformations. The operator F issuch that it transforms a metric into the self-dual part of the corresponding Weyl tensor.The dimensionality of V is 10-4-1=5, while the number of independent components ofthe self-dual part of the Weyl tensor is 1

210 = 5. However, for the moment, no successfultopological quantum field theory for topological gravity has been obtained [11]. It seemsthat it is not possible to obtain an action with a Q-symmetry (where Q squares to zeromodulo reparametrizations and Weyl transformations) which is also reparametrizationand Weyl invariant.

In two dimensions there are situations where the counting of dimensionalities in-dicates that there may be interesting topological quantum field theories. For the caseof gravity this analysis indicates why the attempts in [12,13] have been unsuccessful.In two dimensions, the space of metrics modulo reparametrizations and Weyl trans-formations has zero dimensionality, 3-2-1=0. One is then left with ordinary Morsetheory on modular space. However, Yang-Mills in two dimensions seems to be muchricher. In this case the space V is made out of gauge connections of a given vectorbundle with structure group G modulo gauge transformations, which has dimensional-ity (2 − 1) × N = N . The operator F is the one that transforms a gauge connectioninto its field strength, which has N independent components. The construction of thetopological quantum field theory for this case is entirely analogous to topological Yang-Mills in four dimensions. One does not expect to obtain new topological invariants but

– 22 –

it would be a very interesting exercise to work out this simple case in detail and toidentify the resulting topological invariants. It may provide some insight about howto perform explicit calculations in topological quantum field theory. Furthermore, theextension of this case to consider non-compact gauge groups, in a similar spirit to theone in the formulation of 2+1 dimensional gravity in [20], may lead to obtain topo-logical invariants related to the Mumford classes (which was one of the motivations in[12]) within the framework of topological quantum field theories.

ACKNOWLEDGEMENTS

I am grateful to L. Alvarez-Gaume for encouragement and many valuable dis-cussions. I would also like to thank E. Gozzi, M. Pernici and R. Stora for helpfuldiscussions.

– 23 –

REFERENCES

[1] E. Witten, Comm. Math. Phys. 117 (1988) 353

[2] M.F. Atiyah, ‘New Invariants of Three and Four Dimensional Manifolds’, to ap-pear in the proceedings of the Symposium on the Mathematical Heritage of Her-man Weyl (Chapel Hill, May, 1987), ed. R. Wells et al.

[3] E. Witten, Comm. Math. Phys. 118 (1988) 601

[4] J.M.F. Labastida and M. Pernici, Phys. Lett. 212B (1988) 56

[5] L. Baulieu and I.M. Singer, Talk at the “LAPP Meeting on Conformal FieldTheories and Related Topics”, Annecy-le-Vieux, 1988

[6] D. Birmingham, M. Rakowski and G. Thompson, ‘Topological Field Theories,Nicoli Maps and BRST Quantization’, ICTP preprints IC/88/100, IC/88/149,1988; P. Mansfield, ‘The First Donaldson Invariant as the Winding Number of aNicoli Map’, Oxford preprint, 1988

[7] J.H. Horne, ‘Superspace Versions of Topological Theories’, Princeton preprint,PUPT-1096, June 1988

[8] V.N. Gribov, Nucl. Phys. B139 (1978) 1; I.M. Singer, Comm. Math. Phys. 60

(1978) 7

[9] S. Donaldson, J. Diff. Geom. 18 (1983) 269, ibid. 26 (1987) 397 and ‘PolynomialInvariants for Smooth Manifolds’, Oxford preprint

[10] S. Ouvry, R. Stora and P. van Baal, ‘On the Algebraic Characterization ofWitten’s Topological Yang-Mills Theory’, CERN preprint, CERN-TH.5224/88,November, 1988

[11] E. Witten, Phys. Lett. 206B (1988) 601; J.M.F. Labastida and M. Pernici, ‘ALagrangian for Topological Gravity and its BRST Quantization’, IAS preprint,IASSNS-HEP-88/21, May ,1988

[12] J.M.F. Labastida, M. Pernici and E. Witten, ‘Topological Gravity in two Dimen-sions’, IAS preprint, IASSNS-HEP-88/29, June 1988

[13] D. Montano and J. Sonnenschein, ‘Topological Strings’, SLAC preprint, SLAC-PUB-4664, June 1988

[14] J. Milnor, Morse Theory, Annals of Mathematics, Studies No. 51, (PrincetonUniversity Press, 1963)

[15] See, for example, M.F. Atiyah and R. Bott, Phil. Trans. R. Soc. London A308

(1982) 523, and references therein

– 24 –

[16] S.S. Chern, Ann. Math. 46 (1942) 674

[17] L. Alvarez-Gaume, Comm. Math. Phys. 90 (1983) 161; D. Friedan and P.Widney, Nucl. Phys. B235 (1984) 395

[18] L. Alvarez-Gaume, D.Z. Freedman and S. Mukhi, Ann. Phys. 134 (1981) 85

[19] See, for example, J. Milnor, Topology from the Differentiable Viewpoint, (Univer-sity Press of Virginia, Charlottesville, 1965)

[20] E. Witten, ‘2+1 Dimensional Gravity as an Exactly Soluble System’, IASpreprint, IASSNS-HEP-88/32, October 1988