Construction of the Supersymmetric Path Integral: A Survey
aa r X i v : . [ m a t h . DG ] J a n Construction of the Supersymmetric Path Integral:A Survey
Matthias Ludewig ∗ February 3, 2020
This is a survey based on the joint work [17, 19] with Florian Hanisch andBatu Güneysu reporting on a rigorous construction of the supersymmetric pathintegral associated to compact spin manifolds.
A way to understand the geometry of the loop space L X of a manifold X is by studyingits differential forms. On finite-dimensional manifolds, a key feature of differential formsis that they can be integrated , which gives a linear functional on the space of differentialforms . Of course, one of the fundamental properties of this integration functional is thatit is only non-zero on forms of highest degree; at first glance, this seems to make the taskof defining such an integration functional in the infinite-dimensional context of the loopspace impossible, as there is no top degree.However, if we fix a Riemannian metric on X and define the canonical two-form on L X by setting ω [ v, w ] := ˆ T (cid:10) v ( t ) , ∇ ˙ γ w ( t ) (cid:11) d t (1.1)for γ ∈ L X , v, w ∈ T γ L X = C ∞ ( T , γ ∗ T X ) , it turns out that there is a natural way to makesense of the top-degree component of the wedge product e ω ∧ θ for suitable forms θ , byusing simple analogies to the finite-dimensional situation (here e ω denotes the exponentialof ω in the algebra Ω( L X ) of differential forms). This top degree component [ e ω ∧ θ ] top should be seen as the pairing of e ω ∧ θ with the volume form corresponding to the L -metric on L X (which must remain heuristic as there is no such volume form). Using thisnotion, an integration functional can then be defined using the Wiener measure. ∗ The University of Adelaide. [email protected] Throughout, we denote T = S = R / Z ; the smooth loop space is then defined by L X = C ∞ ( T , X ) . We remark that even in the finite-dimensional case, if the manifold is non-compact, then the inte-gration functional will only be defined on a suitable subset of integrable forms . The same will be true inthe infinite-dimensional case. elation to the Atiyah-Singer index theorem. There is another side to this storyof integrating differential forms on the loop space, which is our main motivation. Morethan 30 years ago, it was observed by Atiyah and Witten [3] that there is a very short andconceptual, but formal, i.e. non-rigorous, proof of the Atiyah-Singer index theorem usinga supersymmetric version of the Feynman path integral. In physics terms, this is the pathintegral of the N = 1 / supersymmetric σ -model [1]. Reformulating the supergeometryappearing in the work of Alvarez-Gaumé in the language of differential forms, Atiyah wasled to consider the differential form integral I [ θ ] formally = ˆ L X e − S + ω ∧ θ (1.2)over the loop space of a Riemannian (spin) manifold X , for suitable differential forms θ ∈ Ω( L X ) , where S ( γ ) = 12 ˆ T | ˙ γ ( t ) | d t (1.3)is the usual energy functional, and ω is the canonical two-form defined in (1.1). Atiyahproceeds with a series of formal manipulations allowing him to rewrite (1.2) as a Wienerintegral. Then, using the Feynman-Kac formula, he identifies this Wiener integral withthe supertrace of the heat semigroup associated to the Dirac operator and thus (via theMcKean-Singer formula) with the index of the Dirac operator.On the other hand, the loop space has a natural T -action by rotation of loops, and thedifferential form S − ω is closed with respect to the equivariant differential d K := d + ι K , (1.4)where ι K denotes insertion of the generating vector field K ( γ ) = ˙ γ of the rotation action.Hence, if the given θ also satisfies d K θ = 0 , then the composite differential form e − S + ω ∧ θ considered above is equivariantly closed as well. Motivated by this observation, Atiyahformally applies a Duistermaat-Heckmann type formula [7, 11], in order to localize theintegral to the fixed point set with respect to the rotation action, which is precisely theset of constant loops. Now there is an obvious inclusion map i : X → L X identifying X with this fixed point set, and one has the localization formula I [ θ ] formally = ˆ X b A ( X ) ∧ i ∗ θ. (1.5)It was later observed by Bismut [8] that this can be used to (formally) prove the twistedAtiyah-Singer index theorem, by considering special differential forms on L X defined fromthe data of a vector bundle with connection on X , which today are called Bismut-Cherncharacters . Our work.
In this survey, we give an account of a recent project [19, 17] that carries outa rigorous construction of the supersymmetric path integral map I described above. Themap should have the following properties. Meaning that one pretends that L X is finite-dimensional. I is defined on some large subset of Ω( L X ) of integrable forms , which atleast includes the Bismut-Chern characters defined by Bismut [8].(ii) For any integrable differential form θ with d K θ = 0 , the map I satisfies the local-ization formula (1.5).We remark that in particular, (ii) implies that I is coclosed with respect to d K ; in physicslanguage, this means that the path integral is supersymmetric , where the idea is that thefunctional is invariant under the odd symmetry generated by d K . Of course, the properties(i)-(ii) do not fix I uniquely, since e.g. the functional I ( θ ) , just defined as the right handside of (1.5) satisfies both requirements tautologically. To obtain a reasonable problem,we therefore add the following rather heuristic requirement.(iii) The map I is given by formula (1.2) in a suitable sense.In our work, we construct such a map I . Notice that property (ii) follows if I is homologousto the map I defined by the right hand side of (1.5); however, we emphasise that ourconstruction is geometric . In other words, we construct I as a cochain rather than anequivalence class in cohomology.In fact, we provide two different constructions of the map I : In [19], a stochastic approachis taken to construct I starting from property (iii); it is not necessarily apparent from thisapproach, however, that the map constructed that way has property (ii). This is fixedin [17], where we use methods from non-commutative geometry to define a map which − using a fancy version of Getzler rescaling − can be shown to satisfy (ii). The equivalenceof these constructions is then established in [19].In this survey, we proceed by highlighting the first construction, as described in [19];afterwards, we discuss the second construction, as given in [17]. In the final section, weconnect the two approaches and discuss the localization formula (1.5) and its applicationto the Bismut-Chern characters. In this section, we give a quick overview of the construction of the path integral map I portrayed in the introduction, following [19]. The Wiener measure.
The construction is essentially based on the Wiener measure W , a certain measure on the continuous loop space L c X = C ( T , X ) of a Riemannianmanifold X . The Wiener integral of so-called cylinder functions is easy to describe.These are functions F : L c X → C of the form F ( γ ) = f (cid:0) γ ( τ ) , . . . , γ ( τ N ) (cid:1) (2.1) In fact, the Wiener measure is defined on any space of paths, but here we restrict to the loop space. f ∈ C ( M N ) and ≤ τ < · · · < τ N < ; the formula for their Wiener integral W [ F ] is W [ F ] def = ˆ X · · · ˆ X f ( x , . . . , x N ) N Y j =1 p τ j +1 − τ j ( x j , x j +1 ) ! d x · · · d x N , (2.2)where p t ( x, y ) is the heat kernel of X , i.e. the fundamental solution to the heat equation.By the extension theorem and the continuity theorem of Kolmogorov, this determines W [ F ] uniquely for all bounded functions F on L c X .For X = R n , one has the explicit formula p t ( x, y ) = (2 πt ) − n/ exp (cid:18) − | x − y | t (cid:19) for the heat kernel. After inserting this into (2.2) for F a cylinder function, some elemen-tary manipulations give the result W [ F ] = N Y j =1 (cid:0) π ( τ j − τ j − ) (cid:1) − n/ ! ˆ R n · · · ˆ R n F ( γ ) e − S ( γ ) d x, (2.3)where γ = γ x is the piecewise linear loop with γ ( τ j ) = x j and as usual, S is the energyfunctional (1.3). This formula has an extension to manifolds [2, 4, 22]. In fact, formulaslike (2.3) go all the way back to Feynman [13], constituting the starting point for his pathintegral approach to quantum mechanics. Taking the limit over N , (2.3) leads to theheuristic formula W [ F ] formally = 1 C ˆ L c X F ( γ ) e − S ( γ ) d γ (2.4)for a suitable constant C ; in other words, the slogan is that the Wiener measure hasthe density function e − S with respect to the “Riemannian volume measure” d γ on theloop space L c X . Of course, there are several well-known problems with this formula thatmake it remain heuristic, first and foremost the non-existence of the measure d γ and theinfinitude of the constant C . Formal definition of the path integral map.
Ignoring the difference between thesmooth and the continuous loop space for the moment, we record that we do not know yethow to integrate differential forms, but at least the Wiener measure enables us to integrate functions over the loop space. However, if M is an oriented (for now finite-dimensional)Riemannian manifold, integrating differential forms and functions is essentially the samething: The two are related by the formula ˆ M θ = ˆ M [ θ y ] top d y (2.5) In the formula, we adopt the notation τ N +1 := 1 + τ and x N +1 := x . θ , where the left hand side is to be understood as a differential formintegral (determined by the orientation) and the right hand side is the integration mapfor functions determined by the Riemannian structure. While the latter integration mapdoes not depend on the choice of orientation, the integrand [ θ y ] top def = h θ y , vol y i , a function on M called the top degree component of θ , does, as the sign of the volumeform vol depends on the orientation. In supergeometry, this functional is often called Grassmann or Berezin integral , after [5].The idea is now to apply the observation above to the heuristic formula (1.2) for the pathintegral map. Starting from this formula, we obtain the chain of identifications ˆ L X e − S + ω ∧ θ formally = ˆ L X [ e ω ∧ θ γ ] top e − S ( γ ) d γ formally = W (cid:2) [ e ω ∧ θ ] top (cid:3) ; (2.6)here in the first step, we formally applied (2.5) to this infinite-dimensional example, whilein the second step, we recognized the right hand side of the heuristic formula (2.4), forthe integrand F ( γ ) = [ e ω ∧ θ γ ] top .With a view on the right hand side of (2.6), the non-trivial task that remains is to providemeaning for the top degree component [ e ω ∧ θ ] top of the differential form e ω ∧ θ as a W -integrable function on L c X ; this is the main achievement of the paper [19]; we outline theconstruction below. Remark 2.1.
The formal manipulations conducted in (2.6) are more or less well-known.However, in the literature, the differential form θ is either constant equal to one (see [3]) ortaken to be a Bismut-Chern character (see [8]). In both cases, the top degree component [ e ω ∧ θ ] top can be defined (and computed) using ad hoc methods. The novelty of theapproach taken in our paper [19] is that we allow a very general class of differential forms θ to be plugged into our top degree functional, in order to obtain a general definition ofthe path integral. Definition of the top degree functional.
To explain the definition of our top degreefunctional, notice that the canonical two-form ω has the form ω [ v, w ] = h v, Aw i L in termsof the L scalar product, where A = ∇ ˙ γ , a skew-adjoint operator on C ∞ ( T , γ ∗ T X ) . Nowif V is an arbitrary finite-dimensional, oriented Euclidean vector space and ω ∈ Λ V ′ hasthe form ω [ v, w ] = h v, Aw i V for an invertible skew-adjoint operator A on V , one has theresult [ e ω ∧ ϑ ∧ · · · ∧ ϑ N ] top = pf( A ) pf (cid:16) h ϑ a , A − ϑ b i V (cid:17) ≤ a,b ≤ N , (2.7)for ϑ , . . . , ϑ N ∈ V ′ , where pf stands for the Pfaffian of a skew-symmetric matrix, c.f. [21,Prop. 1]. In the case that A is not invertible, there is a similar, slightly more complicatedformula, for details, c.f. [19]. This allows to define the top degree functional on the5nfinite-dimensional Euclidean vector space V = C ∞ ( T , γ ∗ T X ) by analogy: In case that A = ∇ ˙ γ is invertible, we can set q ( θ ∧ · · · ∧ θ N ) def = pf ζ ( ∇ ˙ γ )pf (cid:16) h θ a , ∇ − γ θ b i V (cid:17) ≤ a,b ≤ N (2.8)for θ , . . . , θ N ∈ C ∞ ( T , γ ∗ T ′ X ) and if ∇ ˙ γ is not invertible, it is invertible on the orthog-onal complement of its (always finite-dimensional) kernel, which allows to employ thegeneralization of the formula (2.7) mentioned above. Hence heuristically, q ( θ ) is the “topdegree component” of the differential form e ω ∧ θ .In (2.8), pf ζ ( ∇ ˙ γ ) denotes the zeta-regularized Pfaffian of ∇ ˙ γ , a square root of its zeta-regularized determinant. This quantity is not a number but rather an element of the Pfaffian line Pf γ , a certain one-dimensional real vector space canonically associated to γ ; this reflects the fact that there is no naïve concept of orientation on the infinite-dimensional vector space V . These Pfaffian lines glue together to the so-called Pfaffianline bundle Pf on L X , which is related to the spin condition: By the work of Stolz-Teichnerand Waldorf [23, 24], a spin structure on X gives an orientation of the loop space L X , inthe sense that it provides a canonical trivialization of the Pfaffian line bundle and turnsthe top degree component (2.8) into an honest number. This is the reason why the spincondition is important to define our path integral. Remark 2.2.
This is analogous to the fact that on a finite-dimensional non-oriented manifold, the top degree component is also a section of a real line bundle, the orientationbundle , which is trivialized by an orientation.The main result of our paper [19] is then the following formula, which provides a way toactually compute its value and the value of the integral map.
Theorem 2.3.
Suppose that X is a spin manifold with spinor bundle Σ . Then the top-degree component defined above is canonically a number, and it is given by the formula q ( θ N ∧ · · · ∧ θ ) = 2 − N/ X σ ∈ S N sgn( σ ) ˆ ∆ N str [ γ k τ N ] Σ N Y a =1 c (cid:0) θ σ a ( τ a ) (cid:1) [ γ k τ a τ a − ] Σ ! d τ. (2.9)Here ∆ N = { ≤ τ ≤ · · · ≤ τ N ≤ } is the standard simplex, [ γ k τ a τ a − ] Σ denotes paralleltranslation in the spinor bundle along the loop γ and c denotes Clifford multiplication.Moreover, str is the supertrace of the spinor bundle . Rigorous definition of the path integral map.
At this point, the top degree mapassigns to a certain class of differential forms θ on the loop space L X of a spin manifold X the smooth function q ( θ ) on L X , to be interpreted as the “top degree component” of e ω ∧ θ . The problem now is that we need q ( θ ) to be a function on the continuous loop Throughout, we take the real spinor bundle , a bundle of irreducible
Cl(
T X ) - Cl n -bimodules. In anydimension, the space End Cl n (Σ) of endomorphisms of Σ commuting with the right Cl n -action carries acanonical supertrace. L c X , in order to be able to integrate with respect to the Wiener measure, as in(2.6).One problem here when looking at formula (2.9) is that a loop has to be absolutelycontinuous in order to define the parallel transport along it. A solution to this problemis provided by the notion of stochastic parallel transport : As ultimately, I is defined byWiener integration, it suffices to have q ( θ ) defined as a measurable function only (withrespect to the Wiener measure W ). This is achieved by interpreting the occurrences of theparallel transport in (2.9) in the stochastic sense, which provides a stochastic extension e q of the function q ; for details on the stochastic parallel transport, see e.g. [12, 16, 18].To discuss the possible integrands θ , notice that since L c X ⊂ L X is dense, we can consider Ω( L c X ) as a subspace of Ω( L X ) . Notation 2.4.
Denote my D ⊆ Ω( L c X ) ⊂ Ω( L X ) the space of differential forms θ thatare wedge products of one forms that are uniformly bounded.For elements θ ∈ D , the function e q ( θ ) is a well-defined measurable function on L c X . Sinceit is also bounded by boundedness of θ , it is moreover integrable, and we define I : D → R by the formula I [ θ ] def = W (cid:20)e q ( θ ) exp (cid:18) − ˆ T scal (cid:0) γ ( τ ) (cid:1) d τ (cid:19)(cid:21) . (2.10)The main difference of this definition to the formal version (2.6) is the appearance of theexponential including the scalar curvature. While this may seem strange at glance, thisis an important “quantum correction” to the definition, c.f. Remark 2.5 below. Examples.
We now give some examples of differential forms that are contained in D ,together with their I -integrals. We assume X to be a compact spin manifold in thissection. Given a differential form ϑ ∈ Ω ℓ ( X ) and τ ∈ T , we can produce a differential ℓ -form ϑ ( τ ) ∈ Ω ℓ ( L X ) by setting ϑ ( τ ) γ [ v , . . . , v ℓ ] def = ϑ γ ( τ ) (cid:2) v ( τ ) , . . . , v ℓ ( τ ) (cid:3) (2.11)for v , . . . , v ℓ ∈ T γ L X . Moreover, for any function ϕ ∈ C ∞ ( T ) , we can construct anotherdifferential form ϑ ∈ Ω ℓ ( L X ) by setting ϑ = ˆ T ϑ ( τ )d τ or ϑ γ [ v , . . . , v ℓ ] = ˆ T ϑ γ ( τ ) (cid:2) v ( τ ) , . . . , v ℓ ( τ ) (cid:3) d τ. (2.12)If ϑ ∈ Ω ( X ) , then ϑ ∈ Ω ( L X ) as defined in (2.12) satisfies the assumptions (i) and (ii)of Notation 2.4, hence sums of wedge products of forms of this type are contained in thedomain D .On these forms, the integral map is given as follows: Given ϑ a ∈ Ω ( X ) , a = 1 , . . . , N ,and correspondingly ϑ a ∈ Ω ( L X ) defined by (2.12), then ϑ ∧ · · · ∧ ϑ N ∈ D , and thecorresponding integral is given by the combinatoric formula I (cid:2) ϑ ∧ · · · ∧ ϑ N (cid:3) = 2 − N/ X σ ∈ S N sgn( σ ) ˆ ∆ N Str (cid:16) e − τ H N Y a =1 c ( ϑ σ a ) e − ( τ a − τ a − ) H (cid:17) d τ (2.13)7here H = D / , with D the Dirac operator. This follows from the explicit formula(2.9); the Wiener integral in (2.10) is then evaluated using a vector-valued Feynman-Kacformula, see e.g. [16]. Remark 2.5.
The scalar curvature factor of (2.10) is needed because of the Lichnerowiczformula; without it, formula (2.13) would feature the operator H − scal / instead of H ,which has no good cohomological properties: It turns out that the scalar curvature termis necessary in order to make the functional coclosed (or, in physics lingo: to make thepath integral supersymmetric). The above construction of the integral map was achieved by a naïve reformulation (2.6) ofthe heuristic path integral formula (1.2). Its disadvantage is that the domain D ⊂ Ω( L X ) where it is defined is quite small; for example it does not contain the Bismut-Cherncharacters considered below, which are the most interesting integrands due to their rôleplayed in relation to the index theorem. We therefore now give a different constructionof a path integral map, which has a much larger domain of definition and turns out toextend the previous one. A complete account can be found in the paper [17]. The bar construction.
To set things up, we have to introduce the following algebraicmachinery: The bar complex associated to a differential graded algebra Ω is the gradedvector space B (cid:0) Ω (cid:1) = ∞ M N =0 Ω[1] ⊗ N . (3.1)The elements of B (Ω) are called bar chains and denoted by ( ϑ , . . . , ϑ N ) for ϑ a ∈ Ω ,suppressing the tensor product sign in notation for convenience. B (Ω) has a distinguishedsubspace B ♮ (Ω) , which consists of those elements of B (Ω) that are invariant under gradedcyclic permutation of the tensor factors. B (Ω) has two differentials, a differential d comingfrom the differential of Ω and the bar differential b ′ ; they are given by d ( ϑ , . . . , ϑ N ) = N X k =1 ( − n k − ( ϑ , . . . , ϑ k − , dϑ k , . . . , ϑ N ) b ′ ( ϑ , . . . , ϑ N ) = − N − X k =1 ( − n k ( ϑ , . . . , ϑ k − , ϑ k ϑ k +1 , ϑ k +2 , . . . , ϑ N ) where n k = | ϑ | + · · · + | ϑ k | − k . The above differentials satisfy db ′ + b ′ d = 0 , henceturn B (Ω) and B ♮ (Ω) into bicomplexes with total differential d + b ′ . Dually, we have the codifferential ( δℓ )[ ϑ , . . . , ϑ N ] def = − ℓ (cid:2) ( d + b ′ )( ϑ , . . . , ϑ N ) (cid:3) . (3.2) Here Ω( X )[1] equals Ω( X ) as a vector space, but with degrees shifted by one: We have ϑ ∈ Ω k +1 ifand only if ϑ ∈ Ω[1] k .
8n the space of linear maps ℓ : B (Ω) → C . The iterated integral map.
Remember the definition (2.11) of cylinder forms above.
Chen’s iterated integral map [10] also constructs differential forms on the loop space fromdifferential forms on X , this time taking as input elements of the bar complex B (Ω( X )) .For our purposes, we need an extension of this, introduced by Getzler, Jones and Petrack[15]. We consider the differential graded algebra Ω T ( X ) def = Ω( X × T ) T , the space of differential forms on X × T which are constant in the T -direction. Elements ϑ ∈ Ω T ( X ) will always be decomposed into ϑ = ϑ ′ + dt ∧ ϑ ′′ , where ϑ ′ , ϑ ′′ ∈ Ω( X ) . Thedifferential of Ω T ( X ) is d T := d − ι ∂ t , where ι ∂ t denotes insertion of the canonical vectorfield ∂ t on the T factor and d denotes the de-Rham differential on X × T . In other words,we have d T ϑ = d T ( ϑ ′ + dt ∧ ϑ ′′ ) = dϑ ′ − dt ∧ dϑ ′′ − ϑ ′′ , where now on the right hand side, d denotes the de-Rham differential on X . The versionof the extended iterated integral map used in this survey is a map taking B (Ω T ( X )) to Ω( L X ) ; it is defined by the formula ρ ( ϑ , . . . , ϑ N ) = ˆ ∆ N (cid:0) ι K ϑ ′ ( τ ) + ϑ ′′ ( τ ) (cid:1) ∧ · · · ∧ (cid:0) ι K ϑ ′ N ( τ N ) + ϑ ′′ N ( τ N ) (cid:1) d τ (3.3)for ϑ , . . . , ϑ N ∈ Ω T ( L X ) , where we recall that K ( γ ) = ˙ γ is the canonical velocity vectorfield. This allows to produce many examples of differential forms on the loop space.The crucial fact about ρ is that its restriction ρ ♮ to cyclic chains ρ ♮ : B (cid:0) Ω T ( X ) (cid:1) ⊃ B ♮ (cid:0) Ω T ( X ) (cid:1) −→ Ω( L X ) T ⊂ Ω( L X ) is a chain map in the sense that ρ ♮ sends the total differential d T + b ′ to the equivariantdifferential d K (defined in (1.4)). Moreover, notice that the degree shift in the definitionof B (Ω T ( X )) ensures that ρ is in fact degree-preserving. The Chern character.
Let now X be a compact spin manifold. Our second constructionof the path integral map I is based on the construction of a closed cochain Ch D : B ♮ (cid:0) Ω T ( X ) (cid:1) −→ R , called the Chern character in [17]. It has the property that it vanishes on the kernel ker( ρ ) of the iterated integral map (3.3), hence Ch D can be pushed forward to a functional onthe image of the iterated integral map inside Ω( L X ) .To define Ch D , we define a cochain F on B (Ω T ( X )) with values in the algebra of linearoperators on L ( X, Σ) . Explicitly, F is given on homogeneous elements by the formula F [ ϑ ] def = c ( ϑ ′′ ) + [ D , c ( ϑ ′ )] − c ( dϑ ′ ) ,F [ ϑ , ϑ ] def = ( − | ϑ ′ | (cid:0) c ( ϑ ′ ) c ( ϑ ′ ) − c ( ϑ ′ ∧ ϑ ′ ) (cid:1) , D is the Dirac operator; moreover, we set F [ ϑ , . . . , ϑ k ] = 0 whenever k ≥ . Theformula for the Chern character is now Ch D [ ϑ , . . . , ϑ N ] = 2 − n N / X s ∈ P N ˆ ∆ M Str (cid:16) e − τ H M Y a =1 F [ ϑ s a − +1 , . . . ϑ s a ] e − ( τ a − τ a − ) H (cid:17) d τ. (3.4)Here P N denotes the set of all partitions of { , . . . , N } , given by a sequence of numbers s = { s < s < · · · < s M = N } . In particular, as F vanishes when one inputs morethan two elements, a summand corresponding to a partition s is zero as soon as thereexists an index a with s a − s a − ≥ . Remark 3.1.
The name Chern character stems from the fact that Ch D can be interpretedas the version of a Chern character in non-commutative geometry, namely that of aFredholm module given by the Dirac operator on X . For details, see [17]. Properties of the Chern character.
One of the advantages of the second approachto the supersymmetric path integral map is that due to the algebraic character of theconstruction, it is easier to investigate its properties. As mentioned above, one of theresults is that Ch D is Chen normalized [17, Thm. 5.5], meaning that it vanishes on thekernel ker( ρ ♮ ) of the iterated integral map, restricted to cyclic chains. This means thatwe can define its push-forward I ′ : Ω( L X ) ⊃ im( ρ ♮ ) −→ R , I ′ [ θ ] = Ch D (cid:2) ρ ♮ ( ϑ , . . . , ϑ N ) (cid:3) if θ = ρ ♮ ( ϑ , . . . , ϑ N ) ; notice that this is well-defined as Ch D is Chen normalized. Thisgives a second functional on the space of differential forms on the loop space, with domain im( ρ ♮ ) . One of the main features of the construction is the fact that Ch D is coclosed,meaning that δ Ch D = 0 , (3.5)where δ is the codifferential (3.2); c.f. [17, Thms 4.2, 5.3]. By the compatibility of ρ ♮ withrespect to the differentials, this implies that I ′ is coclosed with respect to the equivariantdifferential d K . In other words, for any differential form θ ∈ im( ρ ♮ ) , we have the followingversion Stokes’ theorem I ′ [ d K θ ] = 0 , stating that exact forms have vanishing integral. In physics slang, this is the supersym-metry of the path integral.However, much more is true. The operator-theoretic formula (3.4) for Ch D makes itaccessible to Getzler’s rescaling technique; a souped up version of this machinery thenenables to show the following [17, Thm. 9.1]. Theorem 3.2. Ch D is cohomologous, as a Chen normalized cochain on B ♮ (Ω T ( X )) , tothe Chen normalized cochain µ , defined by µ ( X )[ ϑ , . . . , ϑ N ] def = 1(2 π ) n/ N ! ˆ X ˆ A ( X ) ∧ ϑ ′′ ∧ · · · ∧ ϑ ′′ N , (3.6)10here ˆ A ( X ) is the Chern-Weil representative of the ˆ A -genus of X .Remember here that we say that a cochain is Chen normalized if it vanishes on the kernelof ρ . Since ˆ A ( X ) is a closed differential form, this implies (3.5), by the usual Stokestheorem. As we discuss below, (3.6) essentially implies the localization formula (1.5) forsuitable differential forms. Comparison to the previous definition.
Inspecting formula (3.3), we see that ρ ( ϑ , . . . , ϑ N ) = ˆ ∆ N ϑ ′′ ( τ ) ∧ · · · ∧ ϑ ′′ N ( τ N )d τ, whenever ϑ ′ a = 0 for each a . If each ϑ ′′ a has degree one, we have X σ ∈ S N sgn( σ ) ρ ( ϑ σ , . . . , ϑ σ N ) = ϑ ′′ ∧ · · · ∧ ϑ ′′ N , which is contained in D , hence has a well-defined path integral, as defined in (2.10). Onthe other hand, an inspection of the formula (3.4) yields Ch D [ ϑ , . . . , ϑ N ] = 2 − N/ ˆ ∆ N Str (cid:16) e − τ H N Y a =1 c ( ϑ ′′ a ) e − ( τ a − τ a − ) H (cid:17) d τ. After anti-symmetrization, this coincides with the formula for I [ ϑ ′′ ∧· · ·∧ ϑ ′′ N ] , as calculatedin (2.13). In this sense, the two versions of the integral map agree, and we from now on,we will use the notation I instead of I ′ . As usual, throughout this section X denotes a Riemannian manifold, which is assumedto be compact and spin for all statements related to the path integral map. Periodic cyclic cohomology.
Throughout, a general differential form on the loop spaceis the direct sum of its homogeneous components, in other words, we denote Ω( L X ) def = ∞ M ℓ =0 Ω ℓ ( L X ) . It is well-known however [20], that in the equivariant cohomology of the loop space,it is important to allow differential forms that are an infinite sum of its homogeneouscomponents, in other words, elements of the direct product of the Ω ℓ ( L X ) . This gives the11 eriodic equivariant cohomology h T ( L X ) of the loop space, which is the cohomology of the Z -graded complex b Ω( L X ) T = b Ω + ( L X ) T ⊕ b Ω − ( L X ) T , where b Ω + ( L X ) def = ∞ Y ℓ =0 Ω ℓ ( L X ) , b Ω − ( L X ) def = ∞ Y ℓ =0 Ω ℓ +1 ( L X ) . The corresponding differential is the equivariant differential d K , c.f. (1.4), which exchangesthe even and the odd part. Bismut-Chern-characters.
Maybe the most prominent example of such a differentialform are the Bismut-Chern-characters, defined as follows.
Definition 4.1.
Let E be a Hermitean vector bundle with connection ∇ over the man-ifold X . The Bismut-Chern-character associated to this data is the equivariantly closeddifferential form
Ch( E, ∇ ) ∈ b Ω + ( L X ) given by the formula Ch( E, ∇ ) γ = ∞ X N =0 ( − N ˆ ∆ N tr E [ γ k τ N ] E N Y a =1 R ( τ a ) γ [ γ k τ a τ a − ] E ! d τ at γ ∈ L X , where R is the curvature of the connection ∇ .Explicitly, the degree N -component Ch N of Ch( E, ∇ ) is given by Ch N [ v N , . . . , v ] = 2 − N X σ ∈ S N ˆ ∆ N tr E [ γ k τ N ] E N Y a =1 R (cid:0) v σ a ( τ a ) , v σ a − ( τ a ) (cid:1) [ γ k τ a τ a − ] E ! d τ. The main properties of the Bismut-Chern-character is that it is equivariantly closed, d K Ch( E, ∇ ) = 0 , in other words, d Ch N = ι K Ch N +1 , and that its pullback along theinclusion i : X → L X is the ordinary Chern character of ( E, ∇ ) , defined using ChernWeyl-theory: i ∗ Ch( E, ∇ ) = ch( E, ∇ ) . (4.1)Formally, the following theorem has been observed by Bismut [8] and was his originalmotivation for the definition of these differential forms. Of course, by the usual argumentof McKean-Singer [6, Thm. 3.50], the right hand side of (4.2) below equals ind( D E ) , thegraded index of the twisted Dirac operator D E . Theorem 4.2.
We have the formula I (cid:2) Ch( E, ∇ ) (cid:3) = Str( e − D E / ) . (4.2) It customary in this context to introduce a formal variable of degree and its inverse in order todefine the periodic cyclic cohomology. The effect is that the complex and its cohomology are Z -graded,but -periodic. Here we reduce modulo right away.
12n (4.2), we take I to be the path integral map constructed in Section 3. This makes senseas by the results of [15, §6], Ch( E, ∇ ) can be written as an iterated integral, i.e. thereexists elements c N ∈ B N (Ω T ( X )) , N = 0 , , , . . . such that ρ ( c N ) = Ch N . In particular,each Ch N is contained in the domain of the integral map ρ ♮ ∗ Ch D constructed in Section 3.The identity (4.2) is then proven using Prop. 8.2 of [17].We remark that Ch does not directly lie in the domain of the integral map I defined inSection 2; in fact, Ch is not even a smooth differential form on L c X , due to the presenceof the parallel transport in its definition. However, interpreting the parallel transport inthe stochastic sense, one obtains a differential form on L c X with measurable coefficients.The top degree map can be applied to this measurable differential form, which yields ameasurable function on L c X . One can then compute I [Ch( E, ∇ )] by employing a suitableversion of the Feynman-Kac-formula, which gives the same result. Entire cohomology.
In the discussion of Thm. 4.2, we have so far omitted the factthat the Bismut-Chern-characters are not contained in Ω( L X ) , but only in the extension b Ω( L X ) that allows infinite sums of homogeneous forms. In particular, it is not at all clear a priori that I [Ch( E, ∇ )] , defined as the sum of the individual integrals I [Ch N ] makesany sense. This issue is best discussed in our second approach to the integral map, whereit is related to the entire cohomology of Connes.For a differential graded algebra Ω , we denote by b B (Ω) the complex defined by the sameformula (3.1) as B (Ω) , but with a direct product replacing the direct sum. In otherwords, its elements are arbitrary sums P ∞ N =0 θ ( N ) , with θ ( N ) ∈ Ω[1] ⊗ N , without anyconvergence requirement. The entire bar complex B ǫ (Ω) is then a certain subcomplexof b B (Ω) , containing chains that satisfy a certain growth condition; for details, we referto [17]. One can then show that for any Bismut-Chern-character Ch( E, ∇ ) , the chain c = P ∞ N =0 c N ∈ b B (Ω T ( X )) such that Ch( E, ∇ ) = ρ ( c ) , constructed by Getzler-Jones-Petrack, is entire. Dually, the following result is shown in [17, Thms 4.1, 5.2]: Theorem 4.3.
The Chern character Ch D has a continuous extension to B ǫ (Ω) .Together with the discussion before, this gives an a priori reason why the left hand sideof (4.2) is well-defined. The localization formula and the index theorem.
We now explain how to rigorouslyconduct the proof of the Atiyah-Singer index theorem envisioned by Atiyah [3] and Bismut[8] using our results. The first result is the following localization formula. We say that θ ∈ b Ω( L X ) is an entire iterated integral, if there exists c ∈ B ♮ǫ (Ω T ( X )) such that θ = ρ ♮ ( c ) . Theorem 4.4.
Let θ ∈ b Ω( L X ) be equivariantly closed, i.e. d K θ = 0 , and assume that itis an entire iterated integral. Then I [ θ ] = (2 π ) − n/ ˆ X ˆ A ( X ) ∧ i ∗ θ. (4.3)13 roof. By the assumption on θ , there exists c ∈ B ♮ǫ (Ω T ( X )) with ρ ♮ ( c ) = θ . Define I : b Ω( L X ) → R by setting I [ θ ] to be the right hand side of (4.3) and notice that by thedefinition (3.3) of the iterated integral map, we have I [ θ ] = µ ( X )[ c ] , where µ ( X ) is defined in (3.6). Now by Thm. 3.2, there exists a Chen normalized cochain µ ′ such that Ch D − µ ( X ) = δµ ′ . Therefore, I [ θ ] − I [ θ ] = ( I − I )[ ρ ♮ ( c )] = (cid:0) Ch D − µ ( X ) (cid:1) [ c ]= δµ ′ [ c ] = − µ ′ (cid:2) ( d T + b ′ ) c (cid:3) . Since ρ ♮ is a chain map and θ is equivariantly closed, the calculation ρ ♮ (cid:0) ( d T + b ′ ) c (cid:1) = d K ρ ♮ ( c ) = d K θ = 0 shows that ( d T + b ′ ) c ∈ ker( ρ ♮ ) , hence µ [( d T + b ′ ) c ] = 0 , as µ is Chen normalized. (cid:3) The localization formula (4.3) is an infinite-dimensional version of the localization formulaof equivariant cohomology in finite dimensions, c.f. [7, 11]. Applying it to a Bismut-Cherncharacter (which is both equivariantly closed and can be represented as an entire iteratedintegral, as discussed above), we get I (cid:2) Ch( E, ∇ ) (cid:3) = (2 π ) − n/ ˆ X ˆ A ( X ) ∧ i ∗ Ch( E, ∇ ) = (2 π ) − n/ ˆ X ˆ A ( X ) ∧ ch( E, ∇ ) , (4.4)where in the last step, we used (4.1). Together with our previous formula (4.2) and theMcKean-Singer formula, this proves the Atiyah-Singer index theorem. Odd dimensions.
We remark that nowhere in the above, it was necessary to restrictto even-dimensional manifolds. Of course, in odd dimensions, both (4.4) and (4.2) arezero; for I [Ch( E, ∇ )] , this is true because the path integral is an odd functional in thiscase, in other words, it evaluates as zero on even-dimensional forms. To obtain a non-trivial result in this case, one uses the odd Bismut-Chern character Ch( g ) of Wilson [25],an equivariantly closed, odd element of b Ω( L X ) associated to a map g : X → U( k ) , theunitary group of order k , for some k . This can be represented by an entire iterated integralfollowing the work of Cacciatori-Güneysu [9]. A result similar to Thm. 4.2 connects thisto the spectral flow to the family D s = D + s c ( g − dg ) of Dirac operators on Σ ⊗ C k . Thisrecovers the odd index theorem of Getzler [14]. References [1] L. Alvarez-Gaumé. Supersymmetry and the Atiyah-Singer index theorem.
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