aa r X i v : . [ m a t h . F A ] N ov Heat Kernels and Critical Limits
Doug Pickrell
Mathematics Department, University of ArizonaTucson, AZ, USA 85721 [email protected]
Summary.
This article is an exposition of several questions linking heat kernelmeasures on infinite dimensional Lie groups, limits associated with critical Sobolevexponents, and Feynmann-Kac measures for sigma models. The first part of thearticle concerns existence and invariance issues for heat kernels. The main examplesare heat kernels on groups of the form C ( X, F ), where X is a Riemannian manifoldand F is a finite dimensional Lie group. These measures depend on a smoothnessparameter s > dim ( X ) /
2. The second part of the article concerns the limit s ↓ dim ( X ) /
2, especially dim ( X ) ≤
2, and how this limit is related to issues arising inquantum field theory. In the case of X = S , we conjecture that heat kernels convergeto measures which arise naturally from the Kac-Moody-Segal point of view on loopgroups, as s ↓ / Given a finite dimensional real Hilbert space f , there is an associated Lebesguemeasure λ f , a (positive) Laplace operator ∆ f , and a convolution semigroupof heat kernel measures ν f t = e − t ∆ f δ = (2 πt ) − dim ( f ) / e − t | x | f dλ f ( x ) . (1)The map f → { ν f t } is functorial, in the sense that if P : f → f is anorthogonal projection, then P ∗ ν f t = ν f t . (2)More generally, given a finite dimensional Lie group F , with a fixed innerproduct on its Lie algebra f , there is an induced left invariant Riemannianmetric, a normalized Haar measure λ F , a Laplace operator ∆ F , and a convo-lution semigroup of heat kernel measures, ν F t = e − t∆ F / δ = lim N →∞ ( exp ∗ ( ν f t/N )) N (3) Doug Pickrell (the power refers to convolution of measures on F ). The map F → { ν F t } isfunctorial, in the sense that if P : F → F and the induced map of Liealgebras f → f is an orthogonal projection, then P ∗ ν F t = ν F t . (4)In Section 2 we will recall how this construction can be generalized to in-finite dimensions. The form of the linear generalization is well-known: given aseparable real Hilbert space h , there is an associated convolution semigroup ofGaussian measures { ν h t } . An essential complication arises when dim ( h ) = ∞ ,namely the ν h t are only finitely additive, viewed as cylinder measures on h (see [St] for orientation on this point). To overcome this, following Gross, weconsider an abstract Wiener space h → b ; in this framework, the Gaussiansare realized as countably additive measures ν h ⊂ b t on b , a Banach space com-pletion of h . Geometrically speaking, one imagines that the support of theheat kernel diffuses into the enveloping space b .In the larger group category, the objects are abstract Wiener groups: suchan object consists of an inclusion of a separable Hilbert Lie group into aBanach Lie group, H → G , (5)such that the induced map of Lie algebras h → g is an abstract Wiener space.With some possible restrictions, the corresponding heat kernels ν H ⊂ G t can beconstructed as in the finite dimensional case, and they form a convolutionsemigroup of inversion-invariant probability measures on G (this use of Ito’sideas in an infinite dimensional context apparently originated in [DS]). Thisconstruction is functorial, in the same sense as in finite dimensions.In Section 3 we consider invariance properties of heat kernels. The Cameron-Martin-Segal theorem asserts that for t >
0, the measure class [ ν h ⊂ b t ] is in-variant with respect to translation by h ∈ b if and only if h ∈ h . In thelarger group category, we conjecture that [ ν H ⊂ G t ] is invariant with respect totranslation by h ∈ G if and only if h ∈ H and Ad ( h ) ∗ Ad ( h ) − h . The evidence in favor of this conjecture is based ondeep results of Driver.In Section 4, and in the remainder of the paper, we focus on groups ofmaps. Suppose that X is a compact manifold and W is a real Hilbert Sobolevspace corresponding to degree of smoothness s > dim ( X ) /
2. In this contextthe Sobolev embedding W → C ( X ; R ) (6)is a canonical example of an abstract Wiener space. If F is a finite dimensionalLie group with a fixed inner product on its Lie algebra f , then H = W ( X, F ) → G = C ( X, F ) (7)is an abstract Wiener group. In this example the measure ν H ⊂ G t is determinedby its distributions corresponding to evaluation at a finite number of points of eat Kernels 3 X ; these distributions involve a nonlocal Green’s function interaction betweenpairs of points.When X = S , F = K , a simply connected compact group with Ad ( K )-invariant inner product, and s = 1, Driver and collaborators have proven thatthe measure class of ν H ⊂ G t equals the measure class of a Wiener measure.This latter measure has a heuristic expression in terms of a local kineticenergy functional exp ( − t Z X h g − dg ∧ ∗ g − dg i ) Y X dλ ( g ( θ )) (8)(see [Dr2]). When dim ( X ) = 2, s = 1 is the critical exponent, the heat kernelsare defined for s >
1, and the heuristic expression (8) is the Feynmann-Kacmeasure for the quantum field theory with fields g : X → K (the sigma modelwith target K ).This motivates the following questions, considered in Sections 5 and 6: isthere an analogue of (7) when s = dim ( X ) /
2, and is there a sense in whichheat kernels have limits as s ↓ dim ( X ) /
2? When s = dim ( X ) /
2, the Sobolevembedding fails (because of scale invariance), and fields cannot be evaluatedat individual points. Moreover for fields with values in a curved space such as K , there is not an obvious notion of generalized function.In the case of X = S , there is at least one possible resolution. In additionto accepting that the support diffuses outside of ordinary functions, one alsohas to accept that it diffuses in noncompact directions in G , the complexifica-tion of K . For various reasons (e.g. the Kac-Moody-Segal point of view on loopgroups) one is led to consider, in place of (7), the hyperfunction completion W / ( S , K ) → Hyp ( S , G ) . (9)A generic point in the hyperfunction completion is represented by a formalRiemann-Hilbert factorization g = g − · g · g + , (10)where g − ∈ H ( ∆ ∗ , G, g ∈ G , g + ∈ H ( ∆, G, ∆ ( ∆ ∗ ) is theopen unit disk centered at 0 ( ∞ ). There is a family of measures on Hyp ( S , G ),indexed by a level l , having heuristic expressions for their densities involvingToeplitz determinants. Whereas heat kernels (and Wiener measures) are de-termined by distributions corresponding to points on S , these latter measuresare in theory determined by distributions corresponding to points off of S ;in reality, an enlightening way to display these measures has not been found.Based on symmetry considerations, we conjecture that the heat kernels pa-rameterized by s and t converge to a measure with level l = 1 / t as s ↓ / X = ˆ Σ , a closed Riemannian surface, becauseof (8) and the central role of sigma models in Physics. This is intertwinedwith the one dimensional case, by the trace map W ( ˆ Σ ) → W / ( S ), when S Doug Pickrell is a 1-manifold (space) embedded in ˆ Σ (a Euclidean space-time). In section 6I have tried to point out a few mathematical facts about sigma models whichmight be relevant to understanding how to formulate a critical limit as s ↓ Notation 1 (1) Throughout this paper, all spaces are assumed to be separableand real, unless explicitly noted otherwise.(2) We will frequently encounter continuous maps of Hilbert spaces h → f such that the quotient Hilbert space equals the Hilbert space structure onthe target. If f is a subspace of h itself, then such a map is an orthogonalprojection. In this paper we will refer to such a map h → f as a projection.(3) Suppose that h is a Hilbert space. Let L denote the symmetricallynormed ideal of Hilbert-Schmidt operators on h . For an invertible operator A on h , the two conditions AA − t ∈ L , and A − A − t ∈ L (11) are equivalent. We let GL ( h ) ( L ) denote the group of invertible operators sat-isfying these two conditions (in the same way, we can define GL ( h ) ( I ) , forany symmetrically normed ideal I ).(4) Given a measure ν on a space X , and a Borel space Y , a map f : X → Y will be said to be ν -measureable if f − ( E ) is ν -measureable for each Borelset E ⊂ Y . The measure class of ν will be denoted by [ ν ] .(5) Lebesgue measure will be denoted by dλ , and Haar measure will bedenoted by dλ G , for a group G . Suppose that h is a Hilbert space. The corresponding convolution semi-groupof Gaussian measures, { ν h t } t> , has the heuristic expression dν h t = (2 πt ) − dim ( h ) / e − t | x | h dλ h ( x ) , (12)where λ h denotes the ‘Lebesgue measure for the Hilbert space h ’. The mea-sure λ h and the expression (12) have literal meaning only when h is finitedimensional.To go beyond this, following Gross, suppose that h → b is a continuousdense inclusion of h into a Banach space. We can consistently define a finitelyadditive probability measure ν h ⊂ b t on b in the following way.A cylinder set is a Borel subset of b of the form p − ( E ), where p : b → f , p | h : h → f is a finite rank projection, and E is a Borel subset of f . The set eat Kernels 5 of cylinder sets is an algebra; it is a σ -algebra if and only if h = b is finitedimensional. The functorial property of finite dimensional Gaussian measures,(2), implies that there exists a well-defined finitely additive measure ν h ⊂ b t onthe algebra of cylinder sets satisfying p ∗ ν h ⊂ b t = ν f t , (13)for all p as above. Definition 2.1. (a) The inclusion h → b is called an abstract Wiener spaceif the finitely additive measure ν h ⊂ b t has a (necessarily unique) countably ad-ditive extension to a Borel probability measure on b , for some t > (andhence all t > , because ν h ⊂ b t ( E ) = ν h ⊂ b ( E/ √ t ) ).(b) A map of abstract Wiener spaces, ( h ⊂ b ) → ( h ⊂ b ) (14) is a map b → b such that the restriction to h is a projection h → h .(c) The cylindrical Schwarz space of h ⊂ b is S ( h ⊂ b ) = lim p p ∗ S ( f ) , (15) where the limit is a directed limit over all finite rank maps of Wiener spaces p : b → f .(d) The Laplace operator acting on a cylindrical Schwarz function is givenby ∆ h ⊂ b ( p ∗ φ ) = p ∗ ( ∆ f φ ) (16)For a map of abstract Wiener spaces, the corresponding Gaussian semi-groups push forward.In many ways, the restriction to Banach space completions of h is artificial.However the Banach framework seems elegant and natural, in part because ofthe following characterization and examples. Theorem 2.2.
The inclusion h → b is an abstract Wiener space if and onlyif for each ǫ > there exists a finite dimensional subspace f ǫ ⊂ h such that ν h ⊂ b t { x ∈ b : | px | f > ǫ } < ǫ (17) for all p : b → f vanishing on f ǫ , where p : h → f is a finite rank projection. If b is itself a Hilbert space, this is equivalent to the condition that the inclusion h → b is a Hilbert-Schmidt operator. For a discussion of this theorem, see section 3.9 of [B].
Examples 1 (a) If f is a finite dimensional Hilbert space, then f = h = b isan abstract Wiener space, and ν f t is given literally by (12). Doug Pickrell (b) Suppose that X is a compact manifold, possibly with boundary, and W is a Hilbert Sobolev space of degree of smoothness s > dim ( X ) / . Then theSobolev embedding W → C ( X ; R ) (18) is an abstract Wiener space. This is closely related to the fact that W → W is a Hilbert-Schmidt operator precisely when s is above the critical exponent.(c) For W as in (b), if h → b is an abstract Wiener space, then W ( X, h ) = W ⊗ h → C ( X, b ) (19) is also an abstract Wiener space (see . . of [B]). I do not know of a good reference for (b). However the basic ideas are in[CL]. A special case of (c) is the Wiener space analogue of the path functor, W ⊗ h = W ([0 , T ] , h , → C ([0 , T ] , b ,
0) (20)with inner product h x, y i W = h ˙ x, ˙ y i L = Z T h ˙ x ( τ ) , ˙ y ( τ ) i h dτ. (21)To stay within the Banach category, we have choosen T < ∞ . By letting T → ∞ , we obtain a Gaussian semigroup on the semi-infinite path space C ([0 , ∞ ) , b , . (22)For notational simplicity, we will write ν h ⊂ b = ν W ⊗ h ⊂ C ([0 , ∞ ) , b , , (23)and we will refer to this as the Brownian motion associated to h ⊂ b . Givena partition V : 0 < t < .. < t n , there is an evaluation map Eval V : C ([0 , ∞ ) , b , → n Y b : x → ( x i ) , x i = x ( t i ) . (24)In terms of the coordinates x i ,( Eval V ) ∗ ν h ⊂ b = dν h ⊂ b t ( x ) × dν h ⊂ b t − t ( x − x ) × .. × dν h ⊂ b t n − t n − ( x n − x n − ) . (25)In particular the distribution of ν h ⊂ b at time t is ν h ⊂ b t . The semigroup prop-erty of ν h ⊂ b t is equivalent to the consistency of these distributions for ν h ⊂ b when the partition is refined.The path space construction can be iterated. Thus given h ⊂ b , there is aBrownian motion, a Brownian sheet, etc., associated to h ⊂ b . eat Kernels 7 (a) An abstract Wiener group is an inclusion of a separableHilbert Lie group H into a Banach Lie group G such that the associated Liealgebra map h → g is an abstract Wiener space.(b) A map of abstract Wiener groups is a map of Lie groups G → G such H → H and the induced Lie algebra map is a projection.(c) An abstract Wiener group H ⊂ G with the property that finite rankmaps of Wiener groups p : G → F separate the topology of G is called local.(d) For a local abstract Wiener group, we define the cylindrical Schwarzspace by S ( H ⊂ G ) = lim p p ∗ S ( F ) (26) and the (left-invariant) Laplacian acting on a cylindrical Schwarz function by ∆ H ⊂ G ( p ∗ φ ) = p ∗ ∆ F φ. (27)Given an abstract Wiener group H ⊂ G , there is an induced left-invariantRiemannian metric on H , and a left-invariant Finsler metric on G . Theseinduce separable complete metrics compatible with the topologies of H and G . Examples 2 (a) If F is a finite dimensional Lie group, with an arbitraryinner product on its Lie algebra f , then F = H = G is an abstract Wienergroup.(b) If W is as in (b) of Examples 1, given an abstract Wiener group H ⊂ G , W ( X, H ) ⊂ C ( X, G ) (28) is a local abstract Wiener group.(c) The construction in (b) can be adapted to local gauge transformationsof a nontrivial principal bundle P → X , provided that the structure group K is a compact Lie group with a fixed Ad ( K ) -invariant inner product.(d) There are also nonlocal examples involving infinite classical matrixgroups; see [Go]. The following would be the ideal existence result.
Conjecture 2.4.
For fixed t ≥
0, the sequence of probability measures exp ∗ ( ν h ⊂ g t/N ) N (29)(the N -fold convolution on G ) has a weak limit with respect to BC ( G ),bounded continuous functions. The limits, denoted ν H ⊂ G t , form a convolutionsemigroup of inversion-invariant probability measures on G . Given a map P : ( H ⊂ G ) → ( H ⊂ G ) (30)of abstract Wiener groups, P ∗ ν H ⊂ G t = ν H ⊂ G t , for each t ≥ Doug Pickrell
This conjecture is known to be true if one additionally assumes that theinclusion h ⊂ g is 2-summable (see [BD]).I have not stated this result in a standard form, and this obscures animportant idea. To explain this idea, fix t >
0. Consider the map C ([0 , ∞ ) , g , → G : x → g N ( t ) = e x e x − x ..e x N − x N − , (31)where x i = x ( it/N ). The g N ( t )-distribution of the Brownian motion mea-sure ν h ⊂ g is the N -fold convolution (29), because the x i − x i − are indepen-dent ( g -valued) Gaussian variables with variance t/N . Thus the conjecture isequivalent to the assertion that g N ( t ) has a limit as N → ∞ , say g ( t ), with ν h ⊂ g -probability one. The latter statement implies that the heat kernel is animage of a Gaussian measure, an insight due to Ito (the standard approachgoes a step further, and constructs g ( t ), as a continuous function of t , byshowing that it solves a stochastic differential equation). Examples 3 (a) If F is a finite dimensional real Lie group, with an arbi-trary inner product on the Lie algebra f , then for ∆ the Laplacian for the leftinvariant Riemannian metric, ν F t = e − t∆/ δ , (32) This measure is absolutely continuous with respect to Haar measure, and hasa real analytic density.(b) Consider the path construction H = W ( I, F , → G = C ( I, F ,
1) (33) where I = [0 , T ] , and h has the norm (21) (with f in place of h in (20)). Inthis case ν H ⊂ G t is Brownian motion for F , with variance t . The path spacefor G is the double path space C ( I × I, { } × I ∪ I × { } ; F , , (34) and ν H ⊂ G is the F -valued Brownian sheet. The Brownian sheet is relevant to Physics in the following way. Suppposethat F = K , a compact Lie group, and the inner product is Ad ( K )-invariant.Suppose that Σ is a closed surface with an area element. The Yang-Millsmeasure is the measure on the space of K -connections which has the heuristicexpression e − R h F A ∧∗ F A i dλ ( A ) (35)for A ∈ Ω ( Σ ; k ), where F A is the curvature for the connection d + A , and dλ denotes a heuristic Lebesgue measure on the linear space of k -valued oneforms. This measure can be (roughly) defined as a finitely additive measurein the following way. Let p t ( g ) dλ ( g ) denote the heat kernel for K . Given atriangulation of Σ , one considers the projection eat Kernels 9 Ω ( Σ, k ) → K E : A → ( g e ) (36)where E denotes the set of edges for the triangulation, and g e ∈ K representsparallel translation for the connection d + A . The image of the Yang-Millsmeasure is Y F p tArea ( f ) ( g ∂f ) Y E dg e , (37)where g ∂f denotes the holonomy around the face f ∈ F . The fact that thesemeasures are coherent with respect to refinement of the triangulation followsdirectly from the semigroup property of the heat kernel. When projected togauge equivalence classes, this measure can be expressed as a finite dimen-sional conditioning of the Brownian sheet (see [Sen]). The linear invariance properties of Gaussian measures are summarized by thefollowing
Theorem 3.1.
Suppose that h → b is an abstract Wiener space. Let ν = ν h ⊂ b t for some t > .(a) The subset of translations in b which fix [ ν ] is h .(b) The group of linear transformations which fix [ ν ] is GL ( h ) ( L ) , in thefollowing sense: given a ν -measureable linear map ˆ T : b → b which fixes [ ν ] , T = ˆ T | h ∈ GL ( h ) ( L ) ; conversely, given T ∈ GL ( h ) ( L ) , there exists a ν -measureable linear map ˆ T : b → b which fixes [ ν ] and satisfies ˆ T | h = T . These results, and their history, are discussed in a very illuminating way in[B] (where one can also find expressions for the Radon-Nikodym derivatives).Now suppose that H ⊂ G is an abstract Wiener group, and g ∈ G . Whenis [( L g ) ∗ ν H ⊂ G t ] = [ ν H ⊂ G t ] , (38)where L g is left translation? Necessarily g ∈ G , the identity component(because ν H ⊂ G t is supported in the identity component). The abelian casesuggests that g ∈ H . Because ν H ⊂ G t is inversion invariant, translation invari-ance implies that [ ν H ⊂ G t ] is g -conjugation invariant. Thus if (38) holds forall t , this suggests that [ ν h ⊂ g t ] is Ad ( g )-invariant, and this implies that, as anoperator on h , Ad ( g ) ∈ GL ( h ) ( L ) . These considerations suggest the following Conjecture 3.2.
For
T > h ∈ G , [ ν H ⊂ G T ] is left and right h -invariant ifand only if h ∈ H and Ad ( h ) ∈ GL ( h ) ( L ) .Hilbert-Schmidt criteria are universal in unitary representation-theoreticquestions of this type. In the next section we will consider evidence in favorof this conjecture. Before leaving this abstract setting, I will mention two other related ques-tions.When t → ∞ , a Gaussian measure is asymptotically invariant in the fol-lowing sense introduced by the Malliavins: given ν t = ν h ⊂ b t and h ∈ h , foreach p < ∞ , Z | − dν t ( b + h ) dν t ( b ) | p dν t ( b ) → t → ∞ (in this linear context, this integral reduces to a one-dimensionalintegral, which is easily estimated). Now suppose that Conjecture 3.2 is true,and let ν t = ν H ⊂ G t and h satisfy the conditions in Conjecture 3.2. Is it truethat for each p < ∞ , Z | − dν t ( gh ) dν t ( g ) | p dν t ( g ) → t → ∞ ? The Malliavins proved that Wiener measure on C ( S , K ) isasymptotically invariant in this sense (see chapter 4, Part III, of [Pi1] for aquantitative version of this result, and an example of how this result is usedto prove existence of an invariant measure for the loop group).The second question is the following. Suppose that we change the innerproduct on h in a way which (by (b) of Theorem 3.1) does not change themeasure class of ν h ⊂ g t : h x, y i = h Ax, y i , x, y ∈ h (41)where A − ∈ L ( h ). Write h for h with this inner product depending on A . Is [ ν H ⊂ G t ] = [ ν H ⊂ G t ], for each t ? M ap ( X, F) Let X denote a compact Riemannian manifold, and fix a Sobolev space (witha fixed inner product) of continuous real-valued functions W ֒ → C ( X ) , (42)necessarily corresponding to some degree of smoothness s > dim ( X ) /
2. Forthe sake of clarity, we will suppose that ∂X is empty, and the inner productfor W is of the form h φ , φ i W = Z X ( P s/ ( φ ))( P s/ ( φ )) dV (43)where P denotes a positive elliptic psuedodifferential operator of order 2, e.g. P = ( m + ∆ ), where ∆ is a (positive) Laplace type operator. We will write G ( x, y ) for the Green’s function of P s ; thus eat Kernels 11 P sy ( G ( x, y ) dV ( y )) = δ x ( y ) (44)in the sense of distributions. Evaluation at x ∈ X defines a continuous linearfunctional δ x : W → R : φ → φ ( x ); (45)this functional is represented by G ( x, · ) ∈ W : φ ( x ) = h φ, G ( x, · ) i W . (46)Let F denote a finite dimensional abstract Wiener group. Then H = W ( X, F ) → G = C ( X, F ) (47)is an abstract Wiener group, where h = W ⊗ f , as a Hilbert space. It is localbecause we can evaluate at points of X .Given a finite set of points V ⊂ X , let Eval V : C ( X, F ) → F V : g → ( g ( v )) (48)denote the evaluation map. This is a map of abstract Wiener groups, where f V has the Hilbert space structure induced by the Lie algebra map Eval V : W ( X, f ) → f V . (49)It follows from (46) that this inner product on f V is given by h ( x v ) , ( y w ) i = X v,w ∈ V G v,w h x v , y w i f , (50)where ( G v,w ) is the matrix inverse to the covariance matrix ( G ( v, w )) v,w ∈ V .More generally, given an embedded submanifold S ⊂ X , restriction inducesa map of abstract Wiener groups C ( X, F ) → C ( S, F ) , (51)where the inner product on W ( S, f ) corresponds to a degree of smoothness s − codim ( S ) /
2. This is very reminiscent of functorial properties which areexploited at a heuristic level for Feynmann-Kac measures (see Section 6).If f , .., f N denotes an orthonormal basis for f , and if f ( v ) i denotes the leftinvariant vector field on F V corresponding to the v -coordinate, then the leftinvariant Laplacian on F V determined by the inner product (50) is given by ∆ V = X v,w ∈ V N X j =1 G ( v, w ) f ( v ) i f ( w ) j , (52)and ( Eval V ) ∗ ν H ⊂ G t = e − t∆ V / δ ( F V )1 . (53) These finite dimensional distributions determine the measure ν H ⊂ G t . We willcompare this with Wiener measure below, when X = S .We now want to understand the content of Conjecture 3.2. Suppose that g ∈ W ( X, F ). We need to compute the adjoint for Ad ( g ), denoted Ad ( g ) ∗ W ,acting on the Hilbert Lie algebra W ( X, f ). Let Ad ( g ) t denote the adjoint for Ad ( g ) acting on L ( X, dV ) ⊗ f . Then h Ad ( g ) ∗ W ( f ) , f i h = Z P s f Ad ( g )( f ) dV (54)= Z P s ( P − s Ad ( g ) t P s )( f ) f dV. (55)Thus Ad ( g ) ∗ W = P − s ◦ Ad ( g ) t ◦ P s , (56)and Ad ( g ) + Ad ( g ) ∗ W = ( Ad ( g ) P − s + P − s Ad ( g ) t ) P s . (57)In general this is simply a zeroth order operator. However suppose that Ad ( g )has values in O ( f ), the orthogonal group. Then Ad ( g ) + Ad ( g ) ∗ W = [ Ad ( g ) , P − s ] P s , (58)and because of the commutator, the order drops to − L + d denote the ideal of compactoperators T on h satisfyingsup N log ( N ) N X s n ( T ) , (59)where s ≥ s ≥ .. are the eigenvalues of | T | . The fact that (58) has order − L + d . This implies the following statement. Theorem 4.1.
Let d = dim ( X ) .(a) Suppose that g ∈ W ( X, f ) and Ad ( g ( x )) ∈ O ( f ) , for each x ∈ X . Then Ad ( g ) ∈ GL ( h ) ( L + d ) . (60) (b) Suppose that F = K , or F = K C , where K is a compact Lie group, and theinner product on f is Ad ( K ) -invariant. Assuming the truth of Conjecture 3.2, [ ν H ⊂ G t ] is W ( X, K ) -invariant when X = S , and noninvariant for dim ( X ) > . The important point is that dim ( X ) = 2 is the critical case, where thecriterion marginally fails.We now want to consider evidence in support of Conjecture 3.2.Suppose that X = S , and P = m − ( ∂∂θ ) . For s > / eat Kernels 13 G ( θ , θ ) = G ( θ − θ ) = 1 π X n ≥ ( m + n ) − s cos ( n ( θ − θ )) . (61)We will write ν s,m t for the corresponding heat kernels. When s = 1, we cancompare ν s,m t with Wiener measure ω t . This latter measure has distributions( Eval V ) ∗ ω t = Y E p t ( g v g − w ) Y V dλ K ( g v ) , (62)where E denotes the set of edges (between vertices in V ), and ν Kt = p t dλ K .The important point is that this is a nearest neighbor interaction, which isfar more elementary than the interaction involving all pairs of points for heatkernels, as in (53).The following summarizes some of the deep results of Driver and collabo-rators on heat kernels for loop groups. Theorem 4.2.
Suppose that X = S and F = K , a simply connected compactLie group with Ad ( K ) -invariant inner product on f .(a) For s = 1 , [ ν s,m t ] = [ ω t ] .(b) For all s > / , [ ν s,m t ] is W ( S , K ) -biinvariant. A relatively short and illuminating discussion of (a) can be found in [Dr2].Part (b) is a long story. For s = 1, part (b) follows from (a) and the relativelywell-understood fact that [ ω t ] is W ( S , K )-biinvariant. Driver gave a directproof of (b), for s = 1 ([Dr1]), and the ideas were shown to extend to all s > / X is a closed Riemannian surface, we expect that the heat kernelsfor W s,m ( X, K ) → C ( X, K ) to just miss being translation quasiinvariant,for all s >
1. It is interesting to ask whether there is another natural con-struction which yields a quasiinvariant measure. This should be comparedwith Shavgulidze’s construction of quasiinvariant measures for
Dif f ( X ), for X of any dimension (see page 312 of [B]). X = S For s = dim ( X ) /
2, the Sobolev embedding fails, and in general a W dim ( X ) / -function on X is not bounded (for references to the finer properties of genericfunctions with critical Sobolev smoothness, see [H]). This leads to a subtle sit-uation. On the one hand, while W dim ( X ) / ( X, k ) is a Hilbert space, unless k isabelian, it is not a Lie algebra. On the other hand, because K is assumed com-pact, a map g ∈ W dim ( X ) / ( X, K ) is automatically bounded. Consequently W dim ( X ) / ( X, K ), with the Sobolev topology (we do not impose uniform con-vergence), is a topological group, but it is not a Lie group (the study of thealgebraic topology of spaces with topologies defined by critical Sobolev normsis nascent, see [Br]).For nonabelian k , in general it is simply not clear how to form an analogueof (47), when s = dim ( X ) / X = S . We will freelyuse facts about loop groups, as presented in [PS]. For simplicity of exposition,we will assume that K is simply connected, k is simple, and the inner product(on the dual) satisfies h θ, θ i = 2, where θ is a long root (the abelian case isessentially trivial, but requires qualifications). The complexification of K willbe denoted by G .For each s > /
2, there is a (Kac-Moody) universal central extension ofLie groups 0 → T → ˆ W s ( S , K ) → W s ( S , K ) → . (63)This extension exists for s = 1 / K ⊂ U ( N ), a loop g : S → K can be viewed as a unitary multiplicationoperator M g on H = L ( S , C N ). Relative to the Hardy polarization H = H + ⊕ H − , where H + consists of functions with holomorphic extension to thedisk, M g = (cid:18) A BC D (cid:19) , (64)where B (or C ) is the Hankel operator, and A (or D ) is the Toeplitz operator,associated to the loop g . The condition g ∈ W / is equivalent to B ∈ L .The Toeplitz operator defines a map A : W / ( S , K ) → F red ( H + ) . (65)The lift of the left action of W / ( S , K ) on itself to A ∗ Det, the pullback of thedeterminant line bundle
Det → F red ( H + ), induces (a power of) the extension(63). The upshot is that, from an analytic point of view, Kac-Moody theoryis intimately related to the critical exponent and Toeplitz determinants.The group H ( S , G ) is a complex Lie group. An open, dense neighborhoodof the identity consists of those loops which have a unique (triangular orBirkhoff or Riemann-Hilbert) factorization g = g − · g · g + , (66)where g − ∈ H ( D ∗ , ∞ ; G, g ∈ G , g + ∈ H ( D, G, D and D ∗ denote the closed unit disks centered at 0 and ∞ , respectively. A model forthis neighborhood is H ( D ∗ , g ) × G × H ( D, g ) , (67) eat Kernels 15 where the linear coordinates are determined by θ + = g − ( ∂g + ), θ − =( ∂g − ) g − − . The (left or right) translates of this neighborhood cover H ( S , G ).The hyperfunction completion, Hyp ( S , G ), is modelled on the space H ( ∆ ∗ , g ) × G × H ( ∆, g ) , (68)where ∆ and ∆ ∗ denote the open disks centered at 0 and ∞ , respectively, andthe transition functions are obtained by continuously extending the transitionfunctions for the analytic loop space. The group H ( S , G ) acts naturally fromboth the left and right of Hyp ( S , G ) (see ch. 2, Part III of [Pi1]).There is a holomorphic line bundle L →
Hyp ( S , G ) with a mapˆ W / ( S , K ) → L↓ ↓ W / ( S , K ) → Hyp ( S , G ) (69)which is equivariant with respect to natural left and right actions by ˆ H ( S , K ),where H ( S , K ) denotes real analytic maps into K . The line bundle L ∗ has adistinguished holomorphic section σ ; restricted to W / loops, L ∗ m = A ∗ Det and σ m = detA , the pullback of the canonical holomorphic section of Det → F red ( H + ), where m is the ratio of the C N -trace form and the normal-ized inner product on k . We will write |L| for the line bundle L ⊗ ¯ L ; we canform real powers of this bundle, because the transition functions are positive. Theorem 5.1.
For each l ≥ , there exists a H ( S , K ) -biinvariant measure dµ |L| l with values in the line bundle |L| l → Hyp ( S , G ) such that dµ l = ( σ ⊗ ¯ σ ) l dµ |L| l (70) is a probability measure. In particular there exists a H ( S , K ) -biinvariantprobability measure dµ = dµ |L| on Hyp ( S , G ) . This is a refinement of the main result in [Pi1] (see also [Pi2]). Thesebundle-valued measures are conjecturally unique. Uniqueness would implyinvariance with respect to the natural action of analytic reparameterizationsof S . This independence of scale is the hallmark of the critical exponent.One motivation for constructing the measure µ |L| l is to prove a Peter-Weyl theorem of the schematic form H ∩ L ( L ∗⊗ l ) = X level ( Λ )= l H ( Λ ) ⊗ H ( Λ ) ∗ (71)where the H ( Λ ) are the positive energy representations of level l ∈ Z + (Thisstatement requires more explanation. Here we will simply say, Kac and Peter-son proved an algebraic generalization of the Peter-Weyl theorem (see § Because | σ | = det | A | /m , restricted to W / ( S , K ), µ l has a heuristicexpression dµ l = 1 Z det | A | l/m ) dµ. (72)This begs two questions: (1) why do Toeplitz determinants have anything todo with (limits of) heat kernels and (2) how do we write the background µ ina way which suggests how to think about it analytically?It is instructive to first consider the abelian case. When K = T , the theo-rem is valid provided l > µ does not exist, reflecting a lackof ‘compactness’ in this flat case), and we consider identity components. Anelement of Hyp ( S , C ∗ ) can be written uniquely as exp ( X n< x n z n ) · exp ( x ) · exp ( X n> x n z n ) (73)where the sums represent holomorphic functions in the open disks. If (73)represents a loop g ∈ W / ( S , T ) , then det | A ( g ) | l = exp ( − l X n> n | x n | ) (74)(the Helton-Howe formula). Consequently dµ l = dλ T ( exp ( x )) Y k> Z k exp ( − lk | x k | ) dλ ( x k ) , (75)and there is no diffusion in noncompact directions: x − k = − x ∗ k a.e. . Since dν s,m t = dν T m − s t ( exp ( x )) Y k> Z k exp ( − t ( k + m ) s | x k | ) dλ ( x k ) , (76)(75) is the limit of heat kernels (with l = 1 / t ) as s ↓ /
2, provided that wealso send the mass m to zero.Now suppose that K is simply connected. Since K and its loop group fitinto the Kac-Moody framework, it is natural to compare µ to Haar measure dλ K for K .Fix a triangular decomposition g = n − ⊕ h ⊕ n + . (77)A generic g ∈ K (or G ), can be written uniquely as g = lmau, (78)where l ∈ N − = exp ( n − ), m ∈ T = exp ( h ∩ k ), a ∈ A = exp ( h R ), and u ∈ N + = exp ( n + ). This implies that there is a K -equivariant isomorphismof homogeneous spaces, K/T → G/B + and l ∈ N − is a coordinate for the eat Kernels 17 top stratum. Harish-Chandra discovered that in terms of this coordinate, theunique K -invariant probability measure on K/T can be written as a δ dλ ( l ) = Y | σ i ( g ) | dλ ( l ) = 1 | l · v δ | dλ ( l ) , (79)where ( ?? ) implicitly determines a = a ( gT ) as a function of l , the σ i are thefundamental matrix coefficients, dλ ( l ) denotes a properly normalized Haarmeasure for N − , 2 δ denotes the sum of the positive complex roots for thetriangular factorization ( ?? ), and v δ is a highest weight vector in the highestweight representation corresponding to the dominant integral functional δ .The point of the third expression is that it shows the denominator for thedensity is a polynomial in l , hence the integrability of (79) is quite sensitive.It also follows from work of Harish-Chandra that for λ ∈ h ∗ R Z a − iλ dλ K g = Y α> h δ, α ih δ − iλ, α i (80)(the right hand side is Harish-Chandra’s c -function for G/K ). In particularthis determines the integrability of powers of a : for ǫ > Z a (1+ ǫ )2 δ dλ ( l ) = ǫ − ( d − r ) / < ∞ , (81) d and r denote the dimension and rank of g , respectively. The formula (79) canbe derived by calculating a Jacobian in a straightforward way. The formula(80) can be deduced from the Duistermaat-Heckman localization principle (inparticular log ( a ) is a momentum map), using a (Drinfeld-Evens-Lu) Poissonstructure which generalizes to the Kac-Moody framework (see [Pi3]).Now consider the loop situation. Recall that θ − is a coordinate for H ( ∆ ∗ , G,
1) (which is similar to N − ). The loop analogue of the for-mula (79) leads to the following heuristic expression for the θ − distribu-tion of µ l , where initially we think of θ − as corresponding to a unitary loop g ∈ W / ( S , K ): ( θ − ) ∗ µ l = 1 Z | σ ( g ) | g + l ) dλ ( θ − ) (82)= 1 Z det | A ( g ) | g + l ) /m dλ ( θ − ) (83)where ˙ g is the dual Coxeter number, and dλ ( θ − ) is a heuristic Lebesgue back-ground. The important point is the shift by the dual Coxeter number, whichreflects a regularization of the ‘sum over all positive roots’. In a similar man-ner the Duistermaat-Heckman approach to (80) applies in a heuristic way tocompute the g distribution of µ l : Z a ( g ) − iλ dµ l = Y α> sin (
12( ˙ g + l ) h δ, α i ) sin (
12( ˙ g + l ) h δ − iλ, α i ) . (84) The function on the right hand side is an affine analogue of Harish-Chandra’s c -function.The heuristic expression (83) explains why µ l is expected to be invari-ant with respect to P SU (1 ,
1) (the Toeplitz determinant and the backgroundLebesgue measure are conformally invariant). It also points to a way of con-structing the θ − (or g − ) distribution of µ l , by imposing a cutoff and taking alimit: d (( g − ) ∗ µ l )( θ − ) = lim N ↑∞ Z P N det | A ( g ( P N θ − )) | g + l ) /m dλ ( P N θ − ) , (85)where P N θ − is the projection onto the first N coefficients, g − is related to P θ − by P θ − = ( ∂g − ) g − − , and g is a unitary loop having Riemann-Hilbertfactorization (66). Many, but not all, of the details of this construction havebeen worked out. The main idea is that log ( det | A ( g ) | ) is part of a momentummap. Consequently localization can be used to compute integrals involvingthe determinant against a symplectic volume element, on finite dimensionalapproximations.The formula (84) for the g distribution remains a conjecture, but it seemsto point in a fruitful direction. The zero-mode g roughly arises from theprojection H ( ˆ C , , ∞ ; G, → Hyp ( S , G ) generic → G (86)More generally, given a closed Riemann surface ˆ Σ and a real analytic embed-ding S → ˆ Σ , a G -hyperfunction induces a holomorphic G -bundle on ˆ Σ , andconsequently there is a bundle H ( ˆ Σ \ S , G ) → Hyp ( S , G ) → H ( ˆ Σ, O G ) , (87)where H ( ˆ Σ, O G ) denotes the set of isomorphism classes of holomorphic G -bundles. The stable points of H ( ˆ Σ, O G ) identify with the irreducible pointsof H ( ˆ Σ, K ), which has a canonical symplectic structure (see [AB]). Thissuggests the following
Question 5.2.
Does dµ project to the (normalized) symplectic volume on thestable points of H ( ˆ Σ, O G )?There is a natural generalization of this to include dµ |L| l . But it is notclear how to incorporate reduction at points, as in (87).Continuous loops have Riemann-Hilbert factorizations, and as a conse-quence there is a diagram W s ( S , K ) → C ( S , K ) ↓ ↓ W / ( S , K ) → Hyp ( S , G ) (88)for each s > /
2. We consider the W s norm eat Kernels 19 X n ( m + n ) s/ | x n | , (89)and ν s,m t , the corresponding heat kernels, which we view as measures on Hyp ( S , G ). Conjecture 5.3.
The measures ν s,m t have a limit as s ↓ /
2, and the measureclass of this limit equals the measure class of dµ l , where l = 1 / t . When m ↓
0, this limit converges to µ l .The intuition is simply that for m = 0, the heat kernel should relax toa configuration which is conformally invariant, as s ↓ /
2. The existing hardevidence is slight. It is true in the abelian case. The heat kernel measure class[ ν s,m t ] is biinvariant with respect to W s ( S , K ), s > /
2. It is expected that[ µ l ] will be biinvariant with respect to W / ( S , K ), in the sense that thenatural induced unitary representation H ( S , K × K ) → U ( L ( dµ l )) (90)will extend continuously to W / ( S , K ) (this is true for the discrete part ofthe spectrum in (71)). In two dimensions the question of how to formulate a notion of critical limitfor heat kernels is possibly related to the question of how to mathematicallyformulate sigma models. To give the discussion some structure, I will focuson one aspect of quantum field theory (qft), following Segal.Fix a dimension d , thought of as the dimension of (Euclidean) space-time.As in section 4 of [Se], let C metric denote the category for which the objects areoriented closed Riemannian ( d − d -manifolds with totally geodesic boundaries. Definition 6.1.
A primitive d -dimensional unitary qft is a representation of C metric by separable Hilbert spaces and Hilbert-Schmidt operators such thatdisjoint union corresponds to tensor product, orientation reversal correspondsto adjoint, and C metric -isomorphisms correspond to natural Hilbert space iso-morphisms. It is an interesting question to what extent this definition captures themeaning of locality in qft. Segal has recently advocated additional axioms,and a priori there could be primitive theories which are not truly local.To my knowledge there does not exist an example of a nonfree qft which hasbeen shown to satisfy Segal’s primitive axioms in dimension d >
2. However,in two dimensions, the space of all theories is definitely large. From one point of view, this space is the configuration space of string theory, and it is expectedto have a remarkable geometric structure; see [Ma].Before discussing sigma models, we will recall, in outline, how a ‘generic’theory is constructed, using constructive qft techniques (see [Pi4]).Fix (a bare mass) m >
0, and a polynomial P : R → R which is boundedbelow. Given a closed Riemannian surface ˆ Σ , the P ( φ ) -action is the localfunctional A : F ( ˆ Σ ) → R : φ → Z ˆ Σ ( 12 ( | dφ | + m φ ) + P ( φ )) dA, (91)where F ( ˆ Σ ) is the appropriate domain of R -valued fields on ˆ Σ for A . Aheuristic expression for the P ( φ ) -Feynmann-Kac measure is exp ( −A ( φ )) Y x ∈ ˆ Σ dλ ( φ ( x )) . (92)In this two dimensional setting, there is a rigorous interpretation of (92) as afinite measure on generalized functions,1 det ζ ( m + ∆ ) / e − R ˆ Σ : P ( φ ): C dφ C , (93)where C = ( m + ∆ ) − , dφ C is the Gaussian probability measure with co-variance C (corresponding to the critical Sobolev space W ( ˆ Σ )), R : P ( φ ) : C denotes a regularization of the nonlinear interaction, and det ζ denotes thezeta function determinant.Suppose that S is a closed Riemannian 1-manifold. Given an inner productfor W / ( S ), there is an associated Gaussian measure on generalized functions.We only need the measure class, and hence we only require that the principalsymbol of the operator defining the inner product is compatible with length on S . Associated to this measure class, say C ( S ), there is a Hilbert space H ( S ),the space of half-densities associated to C ( S ) (see the appendix to [Pi4]).Suppose that Σ is a Riemannian surface with totally geodesic boundary S . We consider the closed Riemannian surface ˆ Σ = Σ ∗ ◦ Σ obtained by gluing Σ to its mirror image Σ ∗ along S . Because dφ C corresponds to the criticalexponent s = 1, typical configurations for the Feynmann measure (93) arenot ordinary functions. However typical configurations are sufficiently regularso that it makes sense to restrict them to S . Consequently the Feynmannmeasure can be pushed forward to a finite measure on generalized functionsalong S . Essentially because the underlying map of Hilbert spaces is the tracemap W ( ˆ Σ ) → W / ( S ), this finite measure lands in the measure class C ( S ).By taking a square root (which is required because Σ is half of ˆ Σ ), we obtaina half density Z ( Σ ) ∈ H ( S ). Theorem 6.2.
The maps S → H ( S ) and Σ → Z ( Σ ) define a representationof Segal’s category. eat Kernels 21 This construction extends to vector space-valued fields in a routine way.But severe problems arise for qfts with classical fields having values in non-linear targets.Consider the sigma model with target K . A naive idea is to use heatkernels, in place of Gaussians, as backgrounds for an analogous construction.This introduces a new element: we must nudge s >
1, then take a criticallimit. From this point of view, the renormalization group (see section 3 of[Ga]) should be a prescription for how to take this limit.We will augment the sigma model action to include a topological term,the Wess-Zumino-Witten ‘B-field’. Ignoring s -regularization temporarily, theabove outline reads as follows: we represent the Feynmann-Kac measure exp ( − t Z ˆ Σ h g − dg ∧ ∗ g − dg i + 2 πilW ZW ( g )) Y x dλ K ( g ( x )) (94)as a density against a critical heat kernel background; we use the naturalityof heat kernels to push the Feynmann-Kac measure forward along a map ofgeneralized abstract Wiener spaces, induced by the trace map W ( ˆ Σ, K ) → W / ( S, K ), ( W ( ˆ Σ, K ) ⊂ ?) → ( W / ( S, K ) ⊂ Hyp ( S, G )); (95)and we obtain a Hilbert space of half-densities, and a vector, by taking thepositive square root of the pushforward measure.The point of introducing the
W ZW term is that when the level l is apositive integer, and t = 1 / l , this ( W ZW l ) model is solvable in many senses(see chapter 4 of [Ga], and [We]). It is believed that the model satisfies Segal’saxioms (see the Foreword to [Se]). There also is the belief, far more speculative,that there should be a one parameter family of theories which interpolatesbetween W ZW l and the original sigma model. This deformation intuitivelyarises by letting t ↓ s >
1, and then taking a limit as s ↓ W ZW l model: The Hilbert space of the W ZW l model is given by theright hand side of (71). The vacuum of the theory (the vector correspondingto a disk) is det ( A ) l , which is a ”holomorphic square root” of dµ l (this isfar more complicated than the positive square root for P ( φ ) ). The proof ofsewing (in the holomorphic sectors, involving the W ZW l modular functor,see page 468 of [Se]), ultimately hinges on a generalized Peter-Weyl theorem(our conjectural analytical version (71) would fit perfectly with Segal’s globalapproach).The bare sigma model is believed also to be solvable, but in a completelydifferent sense: in terms of scattering (see [ORW]).This raises the following questions: Is there a reasonable way to regularize(94) with respect to heat kernels (for s > prescription for understanding some aspect of the critical limit s ↓
1? Thereis possibly an important hint arising from work of Uhlenbeck (see [U]). Theclassical theory corresponding to (94) is ‘completely integrable’, for all valuesof t and l . This involves reformulating the classical equations in terms ofthe connection one-form A = g − dg and a spectral parameter (which evolvesinto the deformation parameter at the quantum level). In considering thehyperfunction completion in one dimension, we abandoned unitarity. In twodimensions it apparently is necessary to relax unitarity and the zero-curvaturecondition. References
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