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Topological Gauge Theories

arXiv:hep-th/9112064v1 20 Dec 1991MZ-TH/91-42December 1991Topological Gauge Theoriesfrom Supersymmetric Quantum Mechanicson Spaces of ConnectionsMatthias Blau1NIKHEF-HP.O. Box 41882, 1009 DB AmsterdamThe NetherlandsGeorge Thompson2Institut f¨ur PhysikJohannes-Gutenberg-Universit¨at MainzStaudinger Weg 7, D-6500 Mainz, FRGAbstractWe rederive the recently introduced N = 2 topological gauge the-ories, representing the Euler characteristic of moduli spaces M ofconnections, from supersymmetric quantum mechanics on the infi-nite dimensional spaces A/G of gauge orbits.To that end we dis-cuss variants of ordinary supersymmetric quantum mechanics whichhave meaningful extensions to infinite-dimensional target spaces andintroduce supersymmetric quantum mechanics actions modelling theRiemannian geometry of submersions and embeddings, relevant to theprojections A →A/G and inclusions M ⊂A/G respectively.

We ex-plain the relation between Donaldson theory and the gauge theory offlat connections in 3d and illustrate the general construction by other2d and 4d examples.1e-mail: t75@nikhefh.nikhef.nl, 22747::t752e-mail: thompson@vipmzt.physik.uni-mainz.de

1IntroductionFrom its historical development it is evident that topological field theory isclosely related to supersymmetric quantum mechanics with infinite dimen-sional target spaces. Nevertheless this aspect of topological field theory hasattracted little attention in subsequent developments and the purpose of thispaper is to fill this gap and to illustrate the usefulness of this perspective.In particular, we will use supersymmetric quantum mechanics on thespace A/G of gauge orbits of connections to give a rather pedestrian deriva-tion of the topological gauge theory models introduced recently in [1] anddiscussed there from several other points of view (e.g.

via the Mathai-Quillenformalism [2, 3] and in terms of topological N = 2 superfields). The presentderivation is aimed at clarifying the origin of the fundamental property ofthese theories (obviously reminiscent of supersymmetric quantum mechan-ics): that the partition function of the action SM associated to a given modulispace M of connections equals the Euler characteristic χ(M) of M,Z(SM) = χ(M) .

(1)Moreover, as a side-result, we will also see in this paper that the action ofDonaldson theory [4], the prototype of a cohomological topological field the-ory (see [5] for a review), can be identified term by term with the standardaction SX,V of supersymmetric quantum mechanics on a manifold X cou-pled to a potential V for X = A3/G3 (the space of gauge orbits in threedimensions) and V = CS (the Chern-Simons functional). In view of theconsiderations in [6] this is almost tautologically true, but we have includedthis result here because we have never seen it spelled out explicitly (i.e.

interms of the Riemannian metric, connection, and curvature of A3/G3) and anumber of things can be learned from this construction.The main reason why this way of constructing and looking at topologicalgauge theories has enjoyed only limited popularity at best, is that Donaldsontheory has some features which are not available in general. In particular,the existence of a potential on A3/G3, the Chern-Simons functional, whichleads to a description of moduli spaces MI ⊂A4/G4 of instantons in 4d (viaits gradient flow) is a fortuitous coincidence which reflects the richness ofDonaldson theory.

The fact that, used in this way, supersymmetric quantum1

mechanics on An/Gn leads to a theory in n + 1 dimensions, in general pre-vents one from applying this method to the construction of topological gaugetheories based on moduli spaces M ⊂An+1/Gn+1 as no suitable potentialfunction on An/Gn, nor any other obvious means of exerting control over Mfrom an n-dimensional point of view, will exist.In this letter we show that the n-dimensional topological gauge theoryactions SM associated with a given moduli space M ⊂An/Gn can be con-structed directly from supersymmetric quantum mechanics on An/Gn, theintermediate theory on An+1/Gn+1 playing only an auxiliary role. Not onlyis this method manifestly covariant (in contrast with the above constructionwhich leads to a (3 + 1)-dimensional description of Donaldson theory), butalso (and more importantly) it frees us from the necessity of having to findan n-dimensional description of a moduli space of connections in n + 1 di-mensions.

This also completes the circle of ideas relating Floer cohomology,the Chern-Simons functional, instanton moduli spaces, the Casson invariant,and moduli spaces of flat connections from the point of view of [1].Our construction will be based on the standard localization and indextheory arguments of supersymmetric quantum mechanics. In particular, wewill make use of the fact that the theory SX,V localizes to the critical pointset XV of V to derive a topological gauge theory of flat connections in threedimensions (the critical points of CS) from Donaldson theory.

In order todescribe moduli spaces of connections which are not of the form XV for someV we introduce a new variant of supersymmetric quantum mechanics in finitedimensions which localizes onto an arbitrary given submanifold Y ⊂X anddescribes the Riemannian geometry of embeddings via the Gauss-Codazziequations. The corresponding action SY ⊂X also makes sense when applied toX = An/Gn and any finite-dimensional moduli subspace Y = M of An/Gn.In section 2 we review the pertinent features of supersymmetric quantummechanics (path integral representation of the Euler characteristic, evalua-tion of the partition function of SX,V , localization) and introduce the Gauss-Codazzi quantum mechanics action SY ⊂X.

In section 3 we describe the Rie-mannian geometry of (moduli) spaces of connections and its field-theoreticrealization. We also establish the relation between Donaldson theory andsupersymmetric quantum mechanics on A3/G3, and construct the topolog-ical gauge theory of flat connections in three dimensions.

In section 4 we2

illustrate the use of SY ⊂X by treating moduli spaces of instantons and of flatconnections in two dimensions.2Review of supersymmetric quantum me-chanicsThe fundamental action of supersymmetric quantum mechanics modellingthe de Rham complex of a Riemannian manifold (X, g), is (we are using theconventions of [5])SX =Z β0 dt[i ˙xµBµ + 12gµνBµBν + 14Rµνρσ ¯ψµψρ ¯ψνψσ −i ¯ψµ∇tψµ] . (2)Here xµ are the coordinates of the Riemannian manifold (X, g) with curvaturetensor RX = (Rµνρσ), ψµ and ¯ψµ are Grassmann odd coordinates, and thecovariant derivative ∇t is the pull-back of the covariant derivative on X to theone dimensional space with Euclidean time coordinate t. Upon integratingout the auxiliary field B, one recovers the action of [7, 8, 9, 10] with thespinors appearing there decomposed into their components.

We choose ¯ψand ψ to be independent real fields (instead of complex conjugates). Thesupersymmetry of the action (2) isδxµ=ψµ ,δ ¯ψµ = Bµ −Γνµρ ¯ψνψρδψµ=0 ,δBµ = ΓνµρBνψρ −12Rνµρσ ¯ψνψρψσ .

(3)SX can be written as the supersymmetry variation ofR β0 dt[ ¯ψµ(i ˙xµ+ 12gµνBν)]and this has far-reaching consequences. In particular, reinterpreting δ as aBRST operator, this demonstrates that the ground state reduction of super-symmetric quantum mechanics is topological and that the theory is indepen-dent of the coefficient of B2 (cf.

[5, pp. 140-176] for a detailed discussion ofsupersymmetric quantum mechanics in the context of topological field theo-ries).As is well known, the partition function Z(SX) of (2) with periodic bound-ary conditions on all the fields is the Euler number χ(X) of X.

The way to seethis is to start with the definition of χ(X) as the Euler characteristic of the de3

Rham complex of X, χ(X) = Pk(−1)kbk(X) (where bk(X) = dim Hk(X, R)is the k’th Betti number of X) and to rewrite this as the Witten indexχ(X) = tr(−1)F exp(−βH) of the Laplace operator H ≡∆= dd∗+ d∗d ondifferential forms. One then uses the Feynman-Kac formula to represent thisas a supersymmetric path integral with the action (2) and periodic bound-ary conditions on the anticommuting variables ψµ (due to the insertion of(−1)F).The partition function Z(SX) can be evaluated explicitly to give a pathintegral proof of the Gauss-Bonnet theorem which expresses χ(X) as anintegral over X of the Pfaffian of the curvature RX,Z(SX) = χ(X) =ZX Pf(RX) .

(4)The crucial fact responsible for the reduction of the integral over the loopspace LX of X (the path integral) to an integral over X (the Gauss-Bonnetintegral) is the β-independence of the Witten index. This permits one toevaluate the partition function in the limit where the radius β of the circletends to zero.

In this limit it can be seen that only the Fourier zero modes(e.g. ˙x = 0) of the fields are relevant, the contributions from the other modescancelling identically between the bosonic and fermionic fields.

All this is,of course, also an immediate consequence of the BRST symmetry and topo-logical nature of supersymmetric quantum mechanics mentioned above. Itis the analogue in infinite dimensions of this observation that allows us toconstruct topological gauge theories in n (instead of n + 1) dimensions fromsupersymmetric quantum mechanics on An/Gn.As they stand, the partition function of (2) and the right hand side of (4)do not make sense for infinite dimensional target spaces.

There are, however,two generalizations of (2) which turn out to have meaningful counterpartson A/G. The first of these involves a choice of potential V (x) on X. Thecorresponding actionSX,V=Zdt[i( ˙xµ + sgµν∂νV (x))Bµ + 12gµνBµBν + 14Rµνρσ ¯ψµψρ ¯ψνψσ−i ¯ψµ(δµν ∇t + sgµρ∇ρ∂νV )ψν] .

(5)(s is a parameter) arises by replacing the exterior derivative d by dsV ≡exp(−sV )d exp(sV ). As there is a one-to-one correspondence between d- and4

dsV -harmonic forms this also represents χ(X) (independently of s). In thiscase the additional freedom in the choice of s allows one to reduce Z(SX,V )to an integral over the set of critical points of V in the limit s →∞(al-ternatively one uses the fact that the partition function is also indepen-dent of the coefficient of B2 and observes that ˙xµ + sgµν∂νV (x) = 0 implies˙xµ = ∂νV (x) = 0 for any s ̸= 0 by squaring and integrating).

In the casethat the critical points of V are isolated and non-degenerate one arrives atthe classical Poincar´e-Hopf-Morse theoremχ(X) =Xxk:dV (xk)=0(±1)(6)which calculates χ(X) as the signed sum of critical points of V . More gener-ally (6) holds, and can be derived from supersymmetric quantum mechanicsfor the sum over the zeros of a generic vectorfield on X.

If the critical pointsare not isolated then, by a combination of the arguments leading to (4) and(6), one findsχ(X) =X(k)χ(X(k)V ) ,(7)where the X(k)Vare the connected components of the critical point set of V .The relevance of this for our purposes is that the right hand side of (7) maybe well defined, even if X is infinite dimensional, provided that XV is finitedimensional. In that case χ(XV ) is well defined and can be regarded as aregularized Euler number of X (this is the point of view adopted in [3]).

Theadvantage of our construction is that it permits an a priori identification ofthis V -dependent regularized Euler number of X with the Euler number ofXV .Although this looks like a satisfactory state of affairs, we may not alwaysbe so fortunate to have a potential at our disposal whose critical points defineprecisely the (moduli) subspace Y ⊂X we are interested in. In fact, it followsfrom (7) that in finite dimensions χ(Y ) = χ(X) is a necessary condition forthis to be possible.

Moreover, it is by now well known that even on a compactfour-manifold there are critical points of the vacuum Yang-Mills functionalother than instantons. Thus a suitable potential is unlikely to exist e.g.

forthe instanton moduli spaces MI ⊂A4/G4. We thus require a generalizationof (2) which calculates the Euler number χ(Y ) for any submanifold Y ⊂Xregardless of whether Y = XV for some V or not.5

In that setting we have the classical Gauss-Codazzi equations which relatethe intrinsic curvature RY of Y (with the induced metric) to RX restricted toY and the extrinsic curvature (second fundamental form) of the embedding i :Y ֒→X. The second fundamental form KY of (Y, i) is defined by KY (v, w) =(∇i∗vi∗w)⊥, where v, w ∈TY , ∇is the Levi-Civit`a connection on X, and(.

)⊥denotes projection onto the normal bundle to TY in TX|Y . If Y is ahypersurface in X, this reduces to the more mundane statement that theextrinsic curvature is essentially the normal derivative of the induced metric.The Gauss equation now states that⟨RY (u, v)z, w⟩=⟨RX(u, v)z, w⟩+(⟨KY (v, z), KY (u, w)⟩−(u ↔v)) .

(8)Our construction of an action SY ⊂X calculating χ(Y ) via the Gauss-Bonnettheorem applied to (8) will be modelled on (8) itself.Essentially, it willconsist of the action SX (2) plus a Lagrange multiplier term enforcing therestriction to Y ⊂X. Provided that this restriction is performed in a wayconsistent with the supersymmetries of de Rham supersymmetric quantummechanics this will automatically give rise to the second term of (8).More concretely, assume that Y ⊂X is (locally) given byY = {x ∈X : F a(x) = 0, a = 1, .

. .

, dim(X) −dim(Y )}(the relation between the formulae arising from this implicit description andthat in terms of an explicit embedding yk(xµ) is explained e.g. in [11]).

Wethen group the fields appearing in (2) into a topological N = 2 superfieldXµ(t, θ, ¯θ) = xµ(t) + θψµ(t) + ¯θgµν ¯ψν(t) −θ¯θ(gµνBν(t) + gβνΓµβλ ¯ψνψλ) (9)(θ and ¯θ are Grassmann odd scalars). This choice of superfields is designedto reproduce the supersymmetry transformations (3).

As we will see below,the second term of the θ¯θ-component moreover leads to a Taylor expansionof superfields in terms of covariant derivatives so that superspace actions aremanifestly covariant. We also introduce N = 2 Lagrange multiplier fieldsΛa(t, θ, ¯θ) = λa(t) + θσa(t) + ¯θ¯σa(t) + θ¯θba(t) ,(10)and choose the action to beSY ⊂X = SX + αZdtZdθ d¯θΛa(t, θ, ¯θ)F a(X(t, θ, ¯θ)) ,(11)6

so that the integration over the Λa imposes the superconstraints F a(X) = 0.The argument given above leading to the elimination of the non-constantmodes is not affected by the addition of this term and thus, upon Taylorexpanding F a(X), (11) becomes (all ‘fields’ are now time independent)SY ⊂X=β[12gµνBµBν + 14Rµνρσ ¯ψµψρ ¯ψνψσ](12)+α[baF a −σa ¯ψµ∂µF a + ¯σaψµ∂µF a + λa(Bµ∂µF a −gβν∇β∂λF a ¯ψνψλ)] .We see that the integral over b restricts the bosonic coordinates to Y whilethe integrals over σ and ¯σ constrain the fields ψµ and ¯ψµ to be tangent toY . It is now a simple matter to perform the Gaussian integrals over theremaining auxiliary fields Bµ and λa with the resultSY ⊂X = (14Rµνρσ + 12gµαgνβ∇α∂ρF a(F −1)ab∇β∂σF b) ¯ψµψρ ¯ψνψσ .

(13)Here F ab is the matrix F ab = ∂µF a∂νF bgµν and the description of Y ⊂Xin terms of the F a is valid at points where det(F ab) ̸= 0 so that F ab isindeed invertible there. We see that α has dropped out (as it should) andwe have rescaled the ¯ψ’s by β1/2 to eliminate all β-dependence from boththe measure and the action.

Equation (13) is precisely the Gauss equation(8) which we have thus derived from supersymmetric quantum mechanics.Therefore, upon expanding the path integral to soak up the dim(Y ) fermionicψ and ¯ψ zero modes, we will indeed findZ(SY ⊂X) = χ(Y ) ,(14)now valid for arbitrary submanifolds Y ⊂X (not necessarily of the formXV ). This is the generalization we need to be able to apply supersymmetricquantum mechanics to spaces of connections.

We also see that, in a certainsense, the action SX,V (5) is a special case of the action SY ⊂X (11), the zeromode of B playing the role of the multiplier b.Finally we mention that one can also construct supersymmetric quant-um mechanics actions SZ→X for Riemannian submersions Z →X instead ofembeddings, deriving the O’Neill equations [12] in this case instead of theGauss-Codazzi equations. This is most effortlessly done when the submersionis actually a fibration.

Instead of developing the full mashinery here, we willillustrate this in passing in the following section.7

3Donaldson theory and flat connections in3dWe will now introduce the data entering into the construction of the super-symmetric quantum mechanics actions SX,V and SY ⊂X on spaces of connec-tions (see e.g. [13, 14, 15]).

Let (N, g) be a compact, oriented, Riemanniann-manifold, P →N a principal G bundle over N, G a compact semisimpleLie group and g its Lie algebra. We denote by A the space of (irreducible)connections on P, by G the infinite dimensional gauge group of vertical auto-morphisms of P (modulo the center of G), by Ωk(N, g) the space of k-formson N with values in the adjoint bundle ad P := P ×ad g and by dA the co-variant exterior derivative.

The spaces Ωk(N, g) have natural scalar productsdefined by the metric g on N (and the corresponding Hodge operator ∗) andan invariant scalar product tr on g, namely⟨X, Y ⟩=ZM tr(X ∗Y ) ,X, Y ∈Ωk(N, g) . (15)The tangent space TAA to A at a connection A can be identified withΩ1(N, g) and (15) thus defines a metric gA on A.

At each point A ∈A,TAA can be split into a vertical part VA = Im(dA) (tanget to the orbit of Gthrough A) and a horizontal part HA = Ker(d∗A) (the orthogonal complementof VA with respect to the scalar product (15)). Explicitly this decompositionof X ∈Ω1(N, g) into its vertical and horizontal parts isX=dAG0Ad∗AX + (X −dAG0Ad∗AX) ,≡vAX + hAX ,(16)where G0A = (d∗AdA)−1 is the Greens function of the scalar Laplacian (whichexists if A is irreducible).

We will identify the tangent space T[A]A/G withHA for some representative A of the gauge equivalence class [A]. Then gAinduces a metric gA/G on A/G viagA/G([X], [Y ]) = gA(hAX, hAY ) ,(17)where X, Y ∈Ω1(N, g) project to [X], [Y ] ∈T[A]A/G.

With the same nota-tion the Riemannian curvature of A/G is⟨RA/G([X], [Y ])[Z], [W]⟩=⟨∗[hAX, ∗hAW], G0A ∗[hAY, ∗hAZ]⟩−(X ↔Y )+2⟨∗[hAW, ∗hAZ], G0A ∗[hAX, ∗hAY ]⟩. (18)8

The last ingredient we would need to be able to write down the action (2)or (5) is the Christoffel symbols of gA/G or, rather, particular componentsthereof. We will sketch the required calculation below.

Equipped with allthis we can now exhibit the relation between Donaldson theory and super-symmetric quantum mechanics on A3/G3.The action of Donaldson theory on a four-manifold N in equivariant form(i.e. prior to the introduction of gauge ghosts) is [4]S=ZNB+FA + χ+dAψ −B2+/2 + ηdA ∗ψ+¯φdA ∗dAφ + ¯φ[ψ, ∗ψ] −φ[χ+, χ+]/2.

(19)Here FA = dA + 12[A, A] is the curvature of the connection A, ψ ∈Ω1(N, g),the superpartner of A, is a Grassmann odd Lie algebra valued one-form withghost number 1, (B+, χ+) are self-dual two-forms with ghost numbers (0, −1)(Grassmann parity (even,odd)), and (φ, ¯φ, η) are elements of Ω0(N, g) withghost numbers (2, −2, −1) and parity (even,even,odd).The equivariantlynilpotent BRST-symmetry of (19) isδA=ψδψ = −dAφδχ+=B+δB+ = [φ, χ+]δ ¯φ=ηδη = [φ, ¯φ]δφ=0δ2 = δφ(20)where δφ denotes a gauge variation with respect to φ. The action (19) is farfrom being unique.

In particular, by standard arguments of topological fieldtheory many δ-exact terms can be added to the action without changing thepartition function or correlation functions (the Donaldson invariants in thiscase). We will make use of this freedom below.

For many of the other thingsthat can and should be said about (19,20) we refer to [4] and [5, pp. 199-235].If N is of the form N = M × S1 (where we think of S1 as the ‘time’direction) we can perform a (3 + 1)-decomposition of the action.

Identifyingthe self-dual two-forms B+ and χ+ with (time-dependent) elements B and ¯ψof Ω1(M, g), reserving henceforth the notation A for the spatial part of theconnection, and renaming A0 →u and ψ0 →¯η we find that (19) takes theformS=ZMZdtB ∗( ˙A −dAu −∗FA) −B ∗B/2 + ¯ψdAψ + ˙ψ ∗¯ψ + ¯φdA ∗dAφ9

+u[ψ, ∗¯ψ] + ηdA ∗ψ + ¯ηdA ∗¯ψ + ¯φ[ψ, ∗ψ] −φ[ ¯ψ, ∗¯ψ]/2. (21)In going from (19) to (21) we have, for later convenience, also subtracted theBRST exact term (D0 ≡∂0 + [u, ])δ(¯φD0¯η) = ηD0¯η + ¯φ[¯η, ¯η] −¯φD0D0φ .We now perform the following elementary manipulations (Gaussian inte-grals):• Integration over η and ¯η forces ψ and ¯ψ to be horizontal, hAψ = ψ,hA ¯ψ = ¯ψ, i.e.

to represent tangent vectors to A/G• Integraton over ¯φ yields φ = −G0A ∗[ψ, ∗ψ], giving rise to a term12⟨∗[ ¯ψ, ∗¯ψ], G0A ∗[ψ, ∗ψ]⟩in the action• The equation of motion for u readsu = G0A(d∗A ˙A + ∗[ψ, ∗¯ψ])and plugging this back into (21) one obtains12⟨hA( ˙A−∗FA), hA( ˙A−∗FA)⟩+12⟨∗[ψ, ∗¯ψ], G0A∗[ψ, ∗¯ψ]⟩+⟨∗[ψ, ∗¯ψ], G0Ad∗A ˙A⟩Putting all this together we see that the combination of Greens functionsappearing is precisely that entering the equation (18) for the curvature ten-sor RA/G while the kinetic term for the gauge fields is exactly gA/G([ ˙A −∗FA], [ ˙A −∗FA])/2 (eq. 17).

Recalling that ∗FA is the (automatically hori-zontal) gradient vector field of the Chern-Simons functional CS(A) and ∗dAits second derivative, we find perfect term by term agreement with the actionSX,V = SA/G,CS (5) provided that ⟨∗[ψ, ∗¯ψ], G0Ad∗A ˙A⟩correctly reproduces theaffine terms appearing in the second line of (5). That this is indeed the casecan be seen by noting that the variation of the metric gA/G with respect to Aarises solely from the variation of the projectors hA.

As ψ and ¯ψ are horizon-tal the only variation that will therefore contribute is that of the first A in10

the vertical projector dAG0Ad∗A giving rise to the above term when contractedwith ˙A as in ⟨¯ψ, ∇tψ⟩. By the same argument the affine term in ∇∂V doesnot contribute as ∗FA, ψ and ¯ψ are all horizontal and one of them will beannihilated by a d∗A appearing in the variation of the metric.In summary, we have seen thus far that the standard action (19) of Don-aldson theory on a four-manifold of the form M ×S1 is precisely the quantummechanics action (5) on A3/G3 rewritten, as in [3], in local form with thehelp of auxiliary fields.

Conversely, the action (19), for which several otherconstructions are also available [5], could have been used to derive the met-ric and curvature on A/G from those on A. It is in this sense that (21),without the (model dependent) terms coming from the potential, provides arealization SA→A/G of the Riemannian submersion action SZ→X mentionedat the end of section 2.

This part of the action is universal, i.e. common toall supersymmetric quantum mechanics actions on A/G, and is the counter-part of the universal action of N = 2 topological gauge theory describing theRiemannian geometry of A/G and discussed in [1].The general strategy for the construction of such actions, at least in thecase of fibrations, should now also be clear: one modifies the nilpotent BRSTsymmetry (3) to an equivariantly nilpotent symmetry squaring (as in (20))to translations along the fibers parametrized by a new (ghost number 2) fieldφ.

Rest as before.Our next goal is to show how to obtain a topological gauge theory offlat connections in 3d from the supersymmetric quantum mechanics actionSA/G,CS in 3d. This is straightforward since, by the arguments of section 2,only the time-independent modes contribute to the partition function of (21)which is thus the same as the partition function of the action1SM=ZM12FA ∗FA + 12dAu ∗dAu −dA ¯φ ∗dAφ + ¯ψdAψ+u[ψ, ∗¯ψ] + ηdA ∗ψ + ¯ηdA ∗¯ψ + ¯φ[ψ, ∗ψ] −φ[ ¯ψ, ∗¯ψ]/2.

(22)This is precisely the action obtained in [1] as a field theoretic realization of theEuler number χ(M) of the moduli space M = M(M, G) of flat connections.1To arrive at this action, multiply (21) by 1/β and scale t by β so that the circle hasunit-radius. Then β will appear only in terms with time-derivatives.

To eliminate these,scale the non-constant modes of A and ψ by β. In the limit β →0 these modes decoupleand one is left with (22).11

From the present derivation of this action it is obvious that Z(SM) = χ(M)while in [1] we verified this by calculating the partition function. To that endwe integrate over φ, ¯φ, u, η, and ¯η as above to obtain RA/G.

To evaluate theintegral over the remaining fields A, ψ, and ¯ψ we expand them about theirclassical configurations. By standard arguments we may restrict ourselves toa one-loop approximation and to this order the remaining terms in the actionbecomeZM12FA ∗FA + ¯ψdAψ→ZM(12dAcAq ∗dAcAq + [ ¯ψc, ψc]Aq) .

(23)where we can choose FAc = 0 as the coefficient of B2 in (21) is arbitrary.Integration over Aq yields12⟨∗[ ¯ψc, ψc], G1Ac ∗[ ¯ψc, ψc]⟩,(24)where G1Ac is the Greens function of the Laplacian dAcd∗Ac + d∗AcdAc on one-forms, composed with a projector onto the orthogonal complement of thespace of dAc-harmonic one-forms. Now the extrinsic curvature of M ⊂A3/G3isKM([X], [Y ]) = −d∗AG1A ∗[ ¯X, ¯Y ](see [16, 1]) with ¯X and ¯Y satisfying the linearized flatness and horizontalityequations dA ¯X = d∗A ¯X = dA ¯Y = d∗A ¯Y = 0.

Therefore the above term (24)is precisely the K 2M contribution to the Gauss equation (8) for RM and thepartition function of SM is indeed the Euler characteristic χ(M).We want to draw attention to the double-role played by the multiplierfield B: its exact part couples to u and gives rise to the submersion actionSA→A/G and the O’Neill equations, while its coexact part couples to thegradient ∗FA of the potential and is responsible for the K-part of the Gauss-Codazzi equations.The reader may be puzzled at this point by the fact that Donaldsontheory on M ×S1 apparently calculates the Euler number of the moduli spaceM(M, G) of flat connections in three dimensions although it is known (andwas invented) to describe moduli spaces of instantons in four dimensions.How this happens is explained in detail in [1]. It is essentially due to the factthat the index of the instanton deformation complex (the formal dimension12

of the instanton moduli space) on a four-manifold of the form M × S1 isnon-zero in the topologically non-trivial sector so that the partition functionvanishes there, while irreducible ‘instantons’ in the trivial sector correspondto flat connections in 3d.We end this section with the remark that, by a result of Taubes [17], thepartition function of (22) formally equals the Casson invariant of M if M isa homology three-sphere [18]. This, combined with the above considerations,has led us to propose χ(M) as a candidate for the definition of the Cas-son invariant of more general three-manifolds (see [1] for some preliminaryconsiderations).4Other examplesWe will now briefly discuss the corresponding constructions for those modulispaces M which are not of the form XV for some potential V on X = A/G.

Inthat case we will, as discussed in section 2, use the Gauss-Codazzi supersym-metric quantum mechanics action SM⊂A/G (5) which will be the sum of SA/G(or its local counterpart SA→A/G), and the N = 2 supersymmetric delta func-tion constraints onto M ⊂A/G. Alternatively, to construct e.g.

the actionassociated to the moduli space M2 of flat connections in two dimensions wecan simply dimensionally reduce the action (21) assuming that M = Σ × S1.This makes the double role played by B particularly transparent: the actionSA3/G3,CS reduces to the Gauss-Codazzi action SM⊂A2/G2 viaSA3/G3,CS=ZB ∗( ˙A −dAu −∗FA) + . .

.→SM2⊂A2/G2=ZB ∗( ˙A −dAu) + bFA + . .

. (25)(b is the scalar (time) component of B).

Now B evidently represents thesubmersion (O’Neill) part of the action, while b represents the embedding(Gauss-Codazzi) part. Proceding as above to extract the two-dimensional(zero mode) action from (25) one findsSM2=ZΣbFA + 12dAu ∗dAu + u[ψ, ∗¯ψ] −dA ¯φ ∗dAφ + ¯χdAψ −χdA ¯ψ + 12B ∗B+ηdA ∗ψ + ¯ηdA ∗¯ψ + ¯φ[ψ, ∗ψ] −φ[ ¯ψ, ∗¯ψ]/2 + u(dAB + [ψ, ¯ψ]).

(26)13

Once again, this is precisely the action derived in [1] satisfying Z(SM2) =χ(M2). This can, of course, also be established by direct calculation.

Thiscalculation is somewhat simpler here than in three dimensions as we havea delta function constraint instead of a Gaussian around M2 so that thepartition function can be calculated directly without performing a classical-quantum split. In particular, it is now the integration over u and B that willproduce the extrinsic curvature contribution which is (cf.

(24)) 12⟨∗[ ¯ψ, ψ], G0A∗[ ¯ψ, ψ]⟩, in agreement with the calculations in [16, 1].We also want to mention that the non-degeneracy condition det(F ab) ̸= 0we encountered in our discussion of Gauss-Codazzi supersymmetric quantummechanics in section 2 is just the condition that the Laplacian on zero-formsbe invertible, i.e. that the connection be irreducible, as we have assumedall along.

It is, of course, only at those points that the condition FA = 0gives a non-singular description of M2. It appears likely that a non-singulardescription of the reducible points (automatically gauge fixing the residualsymmetry there) can be obtained by adding a term γΛ2 (10) to the actionSM⊂A/G (11) and carefully taking the limit γ →0.This, as well as theother suggestions for dealing with reducible connections put forward in [1],is currently under investigation.A related issue is the question, what the path integral calculates if thetarget spaces X, Y ⊂X or XV ⊂X (and, in particular, the moduli spacesM ⊂A/G) are not smooth manifolds but perhaps orbifolds or orbifold strat-ifications.

In the case of orbifolds one expects to obtain the vitual Eulercharacteristic via Satake’s Gauss-Bonnet theorem for V -manifolds [19]. Onthe other hand, the equivariant orbifold Euler characteristic familiar fromstring theory appears to arise upon reduction of supersymmetric quantummechanics on X = Y × S1 to Y with twisted boundary conditions on boththe ‘temporal’ and ‘spatial’ circles.Finally, in order to obtain a theory modelled on the moduli spaces MIof instantons in four dimensions, we can construct the corresponding Gauss-Codazzi quantum mechanics action on A4/G4.

The construction is almostidentical to that for M2 (essentially because the deformation complex is‘short’ in both examples so that no additional gauge fixing is required) ,and the resulting 4d action is obtained from (26) by replacing the scalarb-multiplet by a multiplet of self-dual two-forms.14

AcknowledgementsWe thank the Bundesministerium f¨ur Forschung und Technologie (Bonn,FRG)and the Stichting voor Fundamenteel Onderzoek der Materie (Utrecht, NL)for financial support.References[1] M. Blau and G. Thompson, N = 2 topological gauge theory, the Eu-ler characteristic of moduli spaces, and the Casson invariant, preprintNIKHEF-H/91-28, MZ-TH/91-40 (November 1991), submitted to Com-mun. Math.

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