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Equivariant Morse theory and quantum integrability

We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincar\'e-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology and localization techniques, and is closely related to the formalism developed by Matthai and Quillen in their approach to Gaussian shaped Thom forms.


Introduction
Equivariant cohomology is presently attracting much interest both in Physics and Mathematics. This is largely due to its relevance to various localization formulas originally introduced by Duistermaat and Heckman, and its connections to cohomological topological field theories. More recently, equivariant cohomology has also found applications in the investigation of W -gravity and the formalism is also relevant in a geometric loop space approach to Poincaré supersymmetric theories.
The original localization formula by Duistermaat and Heckman [1], [2] (for a review, see [3]) concerns exponential integrals over a 2n dimensional symplectic manifold M i.e. classical partition functions of the form Z = 1 (2π) n (iφ) n 1 n! ω n e iφH = 1 (2π) n exp{iφ(H + ω)} (1) where ω is the symplectic two-form. If the Hamiltonian H determines a global symplectic action of a circle S 1 ∼ U(1) -or more generally the global action of a torus -on the manifold with isolated and nondegenerate fixed points p, the integral (1) localizes to these points, Here η H is the η-invariant of the matrix ∂ µν H when viewed as a linear operator on T M p , i.e. if we denote the dimensions of the eigenspaces of the matrix ∂ µν H at a (non-degenerate) critical point p by dimT + p and dimT − p , As explained by Berline and Vergne [4], the integration formula (2) can be An infinite dimensional generalization of the Duistermaat-Heckman formula was presented by Atiyah and Witten [5]. They considered loop space equivariant cohomology described by the loop space differential operator Qż = d + iż and they were interested in evaluating a supersymmetric path integral that describes the Atiyah-Singer index of a Dirac operator on a Riemannian manifold.
The crucial idea in their work is that the fermionic bilinear in the supersymmetric action can be interpreted as a loop space symplectic two-form, and integration over the fermions yields the loop space Liouville measure. Their approach was generalized by Bismut [6] to twisted operators, and to the computation of the Lefschetz number of a Killing vector field acting on the manifold.
In [7], [8] the action of four dimensional topological Yang-Mills theory was interpreted in terms of equivariant cohomology and Weil algebra. In [9] Atiyah and Jeffrey gave an interpretation of its partition function as a regularized Euler class on a vector bundle associated to the bundle A → A/G of connections on a principal bundle. This interpretation of cohomological topological field theories is an infinite dimensional generalization [9], [10] of the formalism developed earlier by Matthai and Quillen [11], who explained how representatives of the Thom class of vector bundles can be constructed using equivariant cohomology. They also applied this formalism to establish a direct connection between the Poincaré-Hopf and Gauss-Bonnet-Chern theorems in classical Morse theory using localization methods.
In a series of papers [12]- [19] the quantum mechanics of circle actions of isometries on symplectic manifolds was considered using a different method of loop space localization. The derivation of the twisted Atiyah-Singer index theorem and its generalizations was also considered in this formalism [14], [15] and the ensuing equivariant cohomology and loop space symplectic geometry was applied to formulate general Poincaré-supersymmetric quantum field theories in a geometrical framework [14], [16].
In the present paper we shall be interested in applying the equivariant cohomology and localization techniques developed in [12]- [19]  In order to describe in more detail the results that we shall derive here, we first recall some aspects of classical Morse theory (from the present point of view see e.g. [3], [10], [20]). For this we consider critical points p of a smooth function F -the Morse function -on a compact oriented 2n dimensional manifold M, If these critical points p ∈ M of F are isolated and non-degenerate, the Poincaré-Hopf theorem states that the Euler characteristic X (M) of M is related to these critical points by with x µ local coordinates around p. In particular, (3) means that the sum over the critical points of F is a topological invariant of the manifold, independently of the function F .
Here we shall be interested in generalizations of (3) to sums that are of the where φ is a parameter. We shall find that such generalizations of (3) are also related to an invariant of the manifold M which is an equivariant version of the Euler class. Furthermore, we shall explain how sums like (4) arise in the loop space, and we also present the appropriate degenerate versions.
In section 2. we shall introduce some background material on equivariant cohomology. In section 3. we present a derivation of the Duistermaat-Heckman integration formula (2)

Equivariant cohomology on a symplectic manifold
In this section we introduce relevant concepts in equivariant cohomology. We shall be interested in a 2n dimensional compact symplectic manifold M, with local coordinates x µ and Poisson bracket Here ω µν is the inverse matrix to the closed symplectic two-form on M, so that locally we can introduce a one-form ϑ called the symplectic potential such In the case of non-free G-actions, more elaborated methods are needed to compute the equivariant cohomology. Three different approaches have been introduced to model H * G (M), using differential forms on M together with polynomial functions and forms on the Lie algebra g of G. The two classical approaches are the Cartan and Weil models, and they are interpolated by the BRST model as described e.g.
in [19], [21]. In the following we shall mainly need the Cartan model.
The G-action on M is generated by vector fields X α , α = 1, ..., m that realize the commutation relations of the Lie-algebra g, with f αβγ the structure constants of g. With X a generic vector field on M, we denote contraction along X by i X . In particular, the basis of contractions corresponding to the Lie algebra generators {X α } is denoted by i Xα ≡ i α . The pertinent Lie-derivatives are We shall assume that the action of G is symplectic so that for all α Provided the one-forms i α ω are exact (for this the triviality of H 1 (M, R) is sufficient), we can then introduce the corresponding momentum map where g * is the dual Lie algebra. This yields a one-to-one correspondence between the vector fields X α (and corresponding Lie-derivatives L α ) and certain functions H α on M, the components of the momentum map where {φ α } is a (symmetric) basis of the dual Lie algebra g * , and i α ω = − dH α or in local coordinates In the sequel we shall only consider the simplest example of a group action on the symplectic manifold M, the action of a circle G = U(1) = S 1 . It is determined by a momentum map H corresponding to a Hamiltonian vector field X H as the only generator of the Lie algebra u(1) of U(1), We introduce the following equivariant exterior derivative operator on M, Here the factor φ is a real parameter that we interpret as a generator of the algebra In terms of these variables the exterior derivative, contraction and Lie derivative can be realized by the Poisson bracket actions of Finally, since (φ = 1) we conclude that H + ω is an element of H * U (1) (M), that is it determines an equivariant cohomology class. This is an equivalence class consisting of elements in Λ(M) which are linear combinations of zero-and two-forms and can be represented

Duistermaat-Heckman Integration Formula
The integration formula by Duistermaat and Heckman concerns the exact evaluation of the classical partition function where H is a hamiltonian function that determines the global symplectic action of S 1 ∼ U(1) on the phase space M, and in physical applications φ is identified as the inverse temperature. The integration measure is the phase space Liouville measure which is a canonically invariant measure on M.
If the critical points of H are isolated and nondegenerate, the integration formula by Duistermaat and Heckman states that (9) localizes to the critical points of H, where for simplicity we have included the phase factor that appears in (2) to the definition of the determinant of H, and we have also absorbed the additional factors that appear in (2) to the normalization of Z.
In this section we shall present a derivation of the integration formula (10) and explain how it generalizes to the case where the critical points of H are degenerate.
In the subsequent sections we then use the techniques that we introduce here to derive new integration formulas for integrals that are more general than (9).
Using the fact that integration picks up the 2n-form, modulo an overall normalization we can write (9) as or in local coordinates In order to prove the integration formula (10), we introduce the following generalization of (11) Here ψ is a one-form and λ is a parameter. We shall first argue that if ψ is in the the integral (12) does not depend on λ. Since the integrals (11) and (12) coincide for λ → 0, this implies that these integrals coincide for all values of λ. By evaluating (12) in the λ → ∞ limit we then obtain the integration formula (10).
Notice in particular, that the λ-independence of (12) means that (11) only depends on the equivalence class that H +ω determines in the equivariant cohomology In order to establish the λ-independence of (12), we consider an infinitesimal variation λ → λ + δλ and show that For this we introduce the following infinitesimal change of variables in (12): where δψ = δλ · ψ As a consequence of (8) and (13) the exponential in (12) is invariant under the change of variables (14). However, the Jacobian is nontrivial: and the classical partition function (11) depends only on the equivalence class that . The λ-independence of (12) implies (10): For this we first observe that since the group U(1) is compact we may construct a metric tensor g µν on M for which the canonical flow of H is an isometry, L H g = 0 (15) or in components, Such a metric is obtained by selecting an arbitrary Riemannian metric g µν on M, and averaging it over the group U(1). A converse is also true: Since M is compact the isometry group of g µν must be compact.
We select As a consequence of (15), and d H ψ = K + Ω and the integral is independent of λ. Here we have defined which is called the Riemannian momentum map [3]. Here ∇ µ is the covariant and the last relation in (18) follows from antisymmetry.
We note that since Ω determines a closed two-form on M it can be viewed as a (degenerate) symplectic two-form. Furthermore, we find that (H, ω) and (K, Ω) defines a bi-hamiltonian pair in the sense that their classical trajectories coincide, which is consistent with the classical integrability of H.
Explicitly, (17) is and the integration formula (10) follows immediately when we recall that which localizes (19) onto (10), Here we have used on the critical points dH = 0 and included the phase factor that appears in (2) to the definition of the determinant of ∂ µν H.
We now generalize (10) for Hamiltonians H with degenerate critical points. We denote the critical submanifold of H in M by M 0 and by N ⊥ its normal bundle in M. In a neighborhood near M 0 we write the local coordinates x µ as and δx µ are local coordinates in N ⊥ . Similarly, we introduce η µ =η µ + δη µ whereη µ are one-forms in T * M 0 and δη µ are one-forms in T * N ⊥ . In particular, for all µ.
For large λ the integral (19) localizes exponentially to the vicinity of M 0 , and consequently we can extend integration over all values of δx µ . We introduce the following change of variables, Since the Jacobians for the δx µ and for the δη µ cancel, the corresponding Jacobian is trivial.
Consider the last term in (19), If we perform the change of variables (22) and expand (23) in powers of √ λ we find that the term which is proportional to λ η µ Ω µνη ν vanishes by (21). Similarly, the terms which are proportional to √ λ vanish as a consequence of (20), (21) and the Lie derivative condition (15) for the metric tensor.
We shall now proceed to investigate the O(1) contribution in this expansion, A lengthy but straightforward calculation reveals, that these terms can be com- is the Riemann tensor. Hence in the λ → ∞ limit (23) reduces to (24).
in powers of √ λ, after we first perform the change of variables (22). We find, that in the λ → ∞ limit the only term that survives is From (23) and (24) we conclude that in the λ → ∞ limit the integrals over δx µ and δη µ in (19) become Gaussian, and evaluating these integrals we get where the det and Pf are evaluated over the normal bundle N ⊥ . This result generalizes (10) to the degenerate case. Indeed, if we take the limit where M 0 becomes a set of isolated and nondegenerate critical points and carefully account for the sign of the Pfaffian in (26), we find that in this limit (26) reproduces (10).
If we set H = 0 in (26) we recognize in the numerator the Chern class of the symplectic two-form ω and in the denominator the Euler class of the curvature two-form R µν . With nonzero H, we can then identify [3] Ch(H + ω) = exp{iφ(H + ω)} as the equivariant Chern class of M 0 and as the equivariant Euler class of M 0 , evaluated over the normal bundle N ⊥ . Equivariant characteristic classes [3] are generalizations of ordinary characteristic classes to the equivariant context, and provide representatives of equivariant cohomology.
In particular, we note that equivariant characteristic classes are independent of the connection which implies that (26) is independent of the metric tensor consistently with our general arguments.

S * M and generalization of the Matthai-Quillen formalism
We shall now proceed to explain how the previous construction can be extended to derive more general integration formulas. For this, instead of integrals over the cotangent bundle M ⊗ T * M with local coordinates x µ and η µ we shall in the following consider integrals that are defined over all four variables x µ , η µ , p µ ,η µ that we have introduced in (6).
Originally, we introduced p µ as a canonical realization of the local basis for the tangent bundle T M andη µ as a canonical realization of the local basis of the contraction dual bundle of T * M. In the following we shall instead interpret these variables as follows: We view x µ andη µ as local coordinates on a supermanifold that we denote by S * M. In analogy with (7), we interpret η µ as (part of) a local basis for the cotangent bundle of S * M with the identification η µ ∼ dx µ . However, instead of viewing p µ as a local basis for the tangent bundle T S * M we shall in the following interpret p µ as (part of) a local basis for the cotangent bundle of S * M with the identification p µ ∼ dη µ . The pertinent exterior derivative is and it is a nilpotent operator on the exterior algebra Λ(S * M).
We also introduce a basis i µ , π µ of contractions on the exterior algebra Λ(S * M), dual to the basis η µ , p µ of one-forms, In order to construct the Matthai-Quillen formalism we consider a conjugation of d by a functional Φ, Since this conjugation is an invertible transformation, the cohomologies of (27) and (28) coincide. Using the Levi-Civita connection Γ ρ µν we define which yields for the conjugated exterior derivative In particular, we have where we identify the transformation laws of the standard (N = 1) deRham supersymmetric quantum mechanics, with p µ the auxiliary field (see e.g. [20]). As explained in [10], this means that (30) can be related to the Matthai-Quillen formalism [11]. In particular, the corresponding quantum mechanical path integral determines an infinite dimensional version of the Matthai-Quillen formalism [10] and reduces to the original, finite dimensional Matthai-Quillen formalism by a regularization procedure. As explained in [10] we can use such a path integral and localization methods to derive the Gauss-Bonnet-Chern and Poincaré-Hopf theorems of classical Morse theory. We shall not reproduce these derivations here, but refer to [10] for details. Instead we shall now proceed to generalize the previous construction of the Matthai-Quillen formalism to include the action of a nontrivial vector field on S * M, which corresponds to the global circle action on the original symplectic manifold M generated by our Hamiltonian H.
We first recall the canonical realization (7) of the Lie derivative along the Hamiltonian vector field X H on the exterior algebra Λ(M), Notice that since it acts by the Poisson brackets (6), it is in fact defined on the exterior algebra Λ(S * M). In particular, on our canonical variables the Poisson bracket action of L H is In order to generalize the Matthai-Quillen formalism, as a first step we realize this action by a Lie derivative that acts as a differential operator on the exterior algebra Λ(S * M). For this we define the following vector field on T S * M and introduce the corresponding nilpotent contraction operator on Λ(S * M), and the corresponding equivariant exterior derivative We find that the ensuing Lie-derivative then reproduces the transformation laws (31) on the variables (x µ ,η µ , η µ , p µ ).
Next, we introduce the conjugation (28) for Q X using the functional Φ defined in (29). This yields for the equivariant exterior derivative (32) and for the Lie-derivative (33) we find Here we recognize in the last term Consequently we find that (34) simplifies into and instead of (31) we have the following transformation laws for the local variables on S * M, In particular, comparing (31) and (37)  In particular, the last one has the appropriate homogeneous form of the generally covariant transformation law for a covariant derivative on M.
Obviously, the formalism that we have developed here can be viewed as a generalization -or equivariantization -of the Matthai-Quillen formalism [11], [10], and we now proceed to apply it to derive new integration formulas.

Equivariant Morse theory
We shall first apply our generalization of the Matthai-Quillen formalism to derive a new finite dimensional integration formula for integrals on Λ(S * M) that are of the form Here the integration measure is the Liouville measure in the extended phase space (x µ , p µ , η µ ,η µ ) with Poisson brackets (6). This is an invariant integration measure on Λ(S * M). The Hamiltonian H and the symplectic two-form ω are as before, i.e. defined in the original phase space M so that H depends only on x µ while ω is a function of x µ and a bilinear in η µ . The equivariant exterior derivative Q X is defined in (32) and ψ is an element in the subspace of Λ(S * M) that satisfies where L X is the Lie derivative defined in (36). In particular, since (36) assumes In order to evaluate (38), we need to construct an appropriate functional ψ. For this, we observe that since the basic variables transform in the homogeneous and generally covariant manner (37) under the Lie derivative, any generally covariant quantity which is built from p µ ,, η µ ,η µ and invariant tensors on M such as the metric tensor g µν , the Hamiltonian vector field X µ H , the symplectic two-form ω µν , the covariant derivative ∇ µ etc. automatically satisfies the Lie-derivative condition (40) on S * M. Notice that this is in a marked contrast with the Duistermaat-Heckman case, where we have no general rule for the construction of functionals ψ that satisfy the condition (13) on M.
We shall first assume that the critical points of H are isolated and nondegenerate. We select As a generally covariant quantity, this automatically satisfies the condition (40). Explicitly, where ∇ µ denotes the covariant derivative with respect to the H-invariant Levi-Civita connection Γ µ on M. Substituting in the integral (38) we get We evaluate the p µ integrals and the integrals over the anticommuting variables where on the r.h.s. we recognize an equivariant version of the quantity that appears in the Poincaré-Hopf theorem.
Next we consider ψ = g µν p µην As a generally covariant quantity, this again satisfies the condition (40) and we have When we substitute this in (38), evaluate the Gaussian integral over p µ and the integral over the anticommutingη µ we get We identify this as an equivariant version of the quantity that appears in the Gauss-Bonnet-Chern theorem.
Combining (41) and (42) we obtain the following equivariant generalization of the familiar relation between the Poincaré-Hopf and Gauss-Bonnet-Chern theo- In particular, in the φ → 0 limit (43) reduces to the standard relation Indeed, as we have explained in section 3. these quantities are the equivariant generalizations of the Chern class and Euler class on M, respectively. Using our standard arguments we then conclude that the following generalization of the integral in (43) over M, which can be viewed as an equivariant version of the Gauss-Bonnet-Chern formula in the degenerate case.

Duistermaat-Heckman formula in the loop space
We shall now proceed to generalize the previous results to loop space i.e. path integrals. We again start by first considering the loop space generalization of the Duistermaat-Heckman integration formula, and in the subsequent sections we continue to more general loop space integration formulas.
In the Duistermaat-Heckman case we are interested in evaluating the standard path integral Here we have represented the path integral Liouville measure using anticommuting variables η µ . We again assume that the Hamiltonian H generates the action of a circle S 1 on the classical phase space M.
In order to evaluate (45), we interpret [12] it as an integral in the loop space LM over the phase space M. This loop space is parametrized by the time evolution . As before, we identify η µ (t) as a basis of loop space one-forms, η µ ∼ dx µ and define exterior derivative on LM by lifting the exterior derivative from M, In the following all time integrals will be implicit, and e.g. instead of (46) we write Similarly various other quantities on M can be lifted to LM. For example, by defining the loop space symplectic two-form we have a symplectic structure on LM.
We interpret the bosonic part of the action in (45) as a Hamiltonian functional on the loop space. The corresponding loop space hamiltonian vector field has components In particular, its critical points coincide with the classical trajectories.
We introduce a basis i µ of loop space contractions dual to the basis of one-forms and define the loop space equivariant exterior derivative along the loop space vector field (47), The corresponding Lie-derivative is Again, we conclude [12] that if we add to the exponential in (45) a d S exact term, where ψ is in the nilpotent subspace the corresponding path integral is invariant under local variations of ψ and coincides with (45). By selecting ψ properly the path integral can then be evaluated by the localization method.
In order to enumerate the different possibilities, we consider the following example, This path integral is defined on the sphere S 2 and yields the character of SU (2) in the spin-j + 1 2 representation [22]. The loop space Hamiltonian vector field (47) of the action in (52) has two components, In order to localize (51), we need to construct an appropriate ψ. For this we lift the H-invariant metric g µν on M to the loop space and define As a consequence of (15) the Lie-derivative condition (50) is satisfied. The evaluation of the λ → ∞ limit follows closely (that in section 3.: We write the loop space variable x µ (t) as wherex µ (t) solves the classical equations of motion and δx µ (t) denotes the fluctuations. Similarly, we define where theη µ (t) are zeroes of the loop space Riemannian momentum map In particular this implies that theη µ are Jacobi fields, i.e. satisfy the fluctuation We define the loop space measure in (51) by and introduce the path integral change of variables The corresponding Jacobian in the path integral measure is one. We are interested in the λ → ∞ limit. By generalizing the computation in section 3. to the loop space, we find that in this limit the path integral (51) reduces to an integral over the moduli space M 0 of classical solutions, Here the Pfaffian is evaluated over the fluctuation modes δx µ , and R µ ν is the Riemannian curvature of the metric g µν evaluated on the classical solutionx µ (t).
Notice in particular, that the measure in ( ) is an invariant measure over the moduli space of classical solutions which is itself a symplectic manifold.
In the limit, where we assume that the solutions to (54) are isolated and nondegenerate (for example if the critical points of H itself are isolated and nondegerate and if the period T is such that the boundary condition x µ (0) = x µ (T ) only admits constant loops as solutions to the classical equations of motion) we can further reduce (55) to In conclusion, we have here reduced the path integral to an integral over the moduli space of classical solutions. In general this moduli space has a complicated, T dependent structure. We shall now proceed to derive an alternative integration formula for (45) which is applicable independently of the structure of the moduli space of classical solutions. It also has the advantage, that it can be directly generalized to loop space equivariant Morse theory, as we shall see in the following sections.
We select As a consequence of (15) the condition (50) is satisfied and the corresponding action (51) is where Γ σ µρ is again the (metric) Levi-Civita connection. By the ψ independence, the path integal ( 51) remains invariant under the scaling g µν → λg µν of the metric.
If we evaluate it in the λ → ∞ limit we find [17] that the result is an ordinary integral over the classical phase space M, We evaluate the determinant using e.g. ζ-function regularization. The result is is the equivariantÂ-genus. This is our final integration formula for the path Since the equivariantÂ-genus and the equivariant Chern class are both equivariantly closed with respect to d H , we can further reduce (57) to the critical point set M 0 of H by repeating the steps that led to (26). For this, instead of (57) we consider the more general integral If ψ is in the invariant subspace of L H , according to our standard arguments (57) and (58) coincide. If we select (16) and repeat the steps in section 3. we find that (57) localizes to the following integral over the critical point set M 0 of the classical Hamiltonian H, which generalizes (26) to path integrals. In particular, if the critical point set M 0 of H is isolated and nondegenerate this reduces further to the following generalization of (10),

Classical Morse theory and path integrals
We shall now proceed to generalize the loop space localization methods of the previous section to path integrals that are defined over the supermanifold S * M.
In the present section we shall consider the infinite dimensional Matthai-Quillen formalism described in [10] to derive some standard results in nondegenerate Morse theory. In the following section we then generalize these results to the equivariant and degenerate cases.
The various quantities constructed in section 4. can be directly lifted to the loop space L(S * M) over S * M. In particular, we define the loop space equivariant exterior derivative (in the following time integrals are implicit) which is equivariant with respect to the natural action of the circle S 1 : x µ (t) → x µ (t + τ ) etc. on L(S * M). The corresponding Lie derivative which acts on the exterior algebra over L(S * M) is We again introduce the conjugation (28), (29) which yields for the equivariant exterior derivative (59) while the Lie derivative (60) remains intact.
We are interested in deriving localization formulas for the following path integral over L(S * M) which is of the standard form of (cohomological) topological path integral [20].
According to our general arguments, (61) is invariant under local variations of ψ provided these variations are in the subspace But as a consequence of (60) any functional ψ which is single-valued over the loops satisfies (62) since where g µν is a metric tensor on the original phase space M. This gives for the action in (61) According to our standard arguments the corresponding path integral is independent of λ, and we evaluate it in the λ → ∞ limit. For this we introduce the following local coordinates on the loop space, with x µ 0 , . . .η µ0 the constant modes of x µ (t), . . .η µ (t) and x µ t , . . .η µ t the tdependent fluctuation modes. We define the path integral measure by [dp][dη] = dx µ 0 dη µ 0 dp µ0 dη µ0 t dx µ t dη µ t dp µt dη µt and introduce the change of variables At least formally, the corresponding Jacobian is trivial. In the λ → ∞ limit we can evaluate the path integrals over all fluctuation modes and the ordinary integrals over the constant modes p µ0 andη µ0 . In this way we find that (61) reduces to the integral of the Pfaffian of the curvature two-form over the constant modes x µ 0 and i.e. the path integral (61) yields the Euler number of the phase space M.
In order to relate (61) to the non-degenerate version of the Poincaré-Hopf theorem, we introduce an arbitrary smooth vector field on M with components V µ such that its zeroes are isolated and nondegenerate. (Notice that e.g. in [10] only gradient vector fields were considered.) We then select This yields for the action in (61) We again introduce the change of variables (64) for the fluctuation modes. In the λ → ∞ limit the integral over p µ (t) yields a δ-function that localizes the path integral over x µ (t) to the zeroes of V µ . The remaining path integrals are Gaussians, and evaluating these we obtain Combining (65) and (66)  sign(det ||∇ µ V ν ||) = dxdη Pf(R µνρσ η ρ η σ ) We note that a generalization of (67) to the degenerate case can be derived by directly generalizing the computations in the previous sections.

Equivariant Morse theory in loop space
In order to derive path integral versions of the equivariant Poincaré-Hopf and Gauss-Bonnet-Chern theorems we need a loop space version of the equivariant exterior derivative Q X in (32). For this, we combine (32) and (59) to the following equivariant exterior derivative on the loop space L(S * M), Notice that the zeroes of the i µ -components of the loop space vector field in (68) yield the equations of motionẋ µ − X µ H = 0 for the classical action S = ϑ µẋ µ − H while the zeroes of the π µ -components of the vector field in (68) determines the Jacobi equation, We again assume that L H Γ µ = 0 and introduce the conjugation (28) which yields for (68) The pertinent conjugated Lie derivative is a linear combination of (33) and (60), and it determines the action of the vector field in 68) on the exterior algebra over the loop space L(S * M).
We are interested in deriving localization formulas for the path integral We first introduce the functional where g µν is again a metric tensor on the original phase space M which is Liederived by the Hamiltonian vector field of H, As a local and generally covariant quantity (72) then satisfies the Lie-derivative condition (71) . A direct computation yields for the last term in the action in (70) Q S ψ 1 = g µν p µ p ν − 1 2 R ρ µσν η ν η ση ρ g µκη κ −η µ (g µν ∂ t + ∂ ρ X µ H g ρν )η ν +η µ g µσ (ẋ ρ − X ρ H Γ ν ρσ )Γ ν ρση ν Next, we introduce where λ is a parameter. This functional is also generally covariant and single valued on the loop space, and consequently it satisfies the condition (71). Explicitly, we find for the last term in (70) We substitute both (72) and (73) in (70) and take the λ → ∞ limit. By assuming that the solutions to (54) are isolated and nondegenerate and repeating the localization procedure that led to (56), we find that in this limit (70) reduces to We shall now consider the following functional As a generally covariant and single valued loop space functional, this satisfies the Lie-derivative condition (71). Explicitly, we get for the last term in (70) Q S ψ 3 = λ 2 (ẋ µ − X µ H )g µνẋ ν + λ 2 η µ (g µν ∂ t +ẋ ρ g µσ Γ σ ρν )η ν in terms of equivariant cohomology on the classical moduli space of the theory.
In particular, our work indicates that there should be an intimate relationship between cohomological topological field theories and quantum integrable models.