VIEWS: 17 PAGES: 42 CATEGORY: Technology POSTED ON: 5/8/2010 Public Domain
Free loop spaces in topology and physics Kathryn Hess What is the space of free loops? Free loop spaces in topology and physics Enumeration of geodesics Hochschild and cyclic homology Kathryn Hess Homological conformal ﬁeld theories Institute of Geometry, Algebra and Topology Ecole Polytechnique Fédérale de Lausanne Meeting of the Edinburgh Mathematical Society Glasgow, 14 November 2008 Free loop spaces The goal of this lecture in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and cyclic homology An overview of a few of the many important roles played Homological conformal ﬁeld by free loop spaces in topology and mathematical theories physics. Free loop spaces Outline in topology and physics Kathryn Hess What is the space of free loops? 1 What is the space of free loops? Enumeration of geodesics Hochschild and 2 Enumeration of geodesics cyclic homology Homological conformal ﬁeld theories 3 Hochschild and cyclic homology 4 Homological conformal ﬁeld theories Cobordism and CFT’s String topology Loop groups Free loop spaces The functional deﬁnition in topology and physics Kathryn Hess What is the space of free loops? Let X be a topological space. Enumeration of geodesics The space of free loops on X is Hochschild and cyclic homology LX = Map(S , X ). 1 Homological conformal ﬁeld theories If M is a smooth manifold, then we take into account the smooth structure and set LM = C ∞ (S 1 , M). Free loop spaces The pull-back deﬁnition in topology and physics Let X be a topological space. Let PX = Map [0, 1], X . Kathryn Hess Let q : PX → X × X denote the ﬁbration given by What is the space of free loops? Enumeration of q(λ) = λ(0), λ(1) . geodesics Hochschild and cyclic homology Homological conformal ﬁeld Then LX ﬁts into a pull-back square theories LX / PX e q X ∆ / X × X, where e(λ) = λ(1) for all free loops λ : S 1 → X . Note that the ﬁber of both e and q over a point x0 is ΩX , the space of loops on X that are based in x0 . Free loop spaces Structure: the circle action in topology and physics Kathryn Hess What is the space of free loops? Enumeration of The free loop space LX admits an action of the circle geodesics group S 1 , given by rotating the loops. Hochschild and cyclic homology More precisely, there is an action map Homological conformal ﬁeld theories κ : S 1 × LX → LX , where κ(z, λ) : S 1 → X : z → λ(z · z ). Free loop spaces Structure: the power maps in topology and physics Kathryn Hess What is the space of free loops? For any natural number r , the free loop space LX admits Enumeration of geodesics an r th -power map Hochschild and cyclic homology r : LX → LX Homological conformal ﬁeld theories given by r (λ) : S 1 → X : z → λ(z r ), i.e., the loop r (λ) goes r times around the same path as λ, moving r times as fast. Free loop spaces A related construction in topology and physics Let U and V be subspaces of X . Kathryn Hess The space of open strings in X starting in U and ending What is the space of free loops? in V is Enumeration of geodesics PU,V X = λ : [0, 1] → X | λ(0) ∈ U, λ(1) ∈ V , Hochschild and cyclic homology Homological which ﬁts into a pull-back diagram conformal ﬁeld theories PU,V X / PX . ¯ q q (prU ,prV ) U ×V /X ×X Both the free loop space and the space of open strings are special cases of the homotopy coincidence space of a pair of maps f : Y → X and g : Y → X . Free loop spaces The enumeration problem in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and cyclic homology Question Homological conformal ﬁeld Let M be a closed, compact Riemannian manifold. theories How many distinct closed geodesics lie on M? Free loop spaces Betti numbers and geodesics in topology and physics Kathryn Hess What is the space of free loops? For any space X and any ﬁeld k, let Enumeration of geodesics bn (X ; k) = dimk H n (X ; k). Hochschild and cyclic homology Homological conformal ﬁeld theories Theorem (Gromoll & Meyer, 1969) If there is ﬁeld k such that bn (LM; k) n≥0 is unbounded, then M admits inﬁnitely many distinct prime geodesics. Proof by inﬁnite-dimensional Morse-theoretic methods. Free loop spaces The rational case in topology and physics Kathryn Hess What is the space of free loops? Theorem (Sullivan & Vigué, 1975) Enumeration of geodesics If Hochschild and cyclic homology M is simply connected, and Homological the graded commutative algebra H ∗ (M; Q) is not conformal ﬁeld theories monogenic, then bn (LM; Q) n≥0 is unbounded, and therefore M admits inﬁnitely many distinct prime geodesics. Proof using the Sullivan models of rational homotopy theory. Free loop spaces The case of homogeneous spaces I in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Theorem (McCleary & Ziller, 1987) Hochschild and If M is a simply connected homogeneous space that is cyclic homology Homological not diffeomorphic to a symmetric space of rank 1, then conformal ﬁeld theories bn (LM; F2 ) n≥0 is unbounded and therefore M admits inﬁnitely many distinct prime geodesics. Proof by explicit spectral sequence calculation, given the classiﬁcation of such M. Free loop spaces The case of homogeneous spaces II in topology and physics Kathryn Hess What is the space of free loops? Remark Enumeration of geodesics It is easy to show that if M is diffeomorphic to a Hochschild and symmetric space of rank 1, then bn (LM; k) n≥0 is cyclic homology bounded for all k, but Homological conformal ﬁeld theories Hingston proved that a simply connected manifold with the rational homotopy type of a symmetric space of rank 1 generically admits inﬁnitely many closed geodesics, and Franks and Bangert showed that S 2 admits inﬁnitely many geodesics, independently of the metric. Free loop spaces A suggestive result for based loop spaces in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Theorem (McCleary, 1987) Hochschild and cyclic homology If X is a simply connected, ﬁnite CW-complex such that Homological H ∗ (X ; Fp ) is not monogenic, then bn (ΩX ; Fp ) n≥0 is conformal ﬁeld theories unbounded. Proof via an algebraic argument based on the Bockstein spectral sequence. Free loop spaces A conjecture and its consequences in topology and physics Kathryn Hess What is the space of free loops? Conjecture Enumeration of geodesics If X is a simply connected, ﬁnite CW-complex such that Hochschild and H ∗ (X ; Fp ) is not monogenic, then bn (LX ; Fp ) n≥0 is cyclic homology Homological unbounded. conformal ﬁeld theories Corollary If there is a prime p such that H ∗ (M; Fp ) is not monogenic, then M admits inﬁnitely many distinct closed geodesics. Free loop spaces Proof strategy in topology and physics Kathryn Hess (Joint work with J. Scott.) What is the space of free loops? Construct “small” algebraic model Enumeration of geodesics B /A Hochschild and cyclic homology Homological conformal ﬁeld theories C ∗ LX / C ∗ ΩX of the inclusion of the based loops into the free loops. By careful analysis of McCleary’s argument, show that representatives in A of the classes in H ∗ (ΩX , Fp ) giving rise to its unbounded Betti numbers lift to B, giving rise to unbounded Betti numbers for LX . Free loop spaces Hochschild (co)homology of algebras in topology and physics Kathryn Hess Let A be a (perhaps differential graded) associative What is the space of free loops? algebra over a ﬁeld k. Enumeration of geodesics The Hochschild homology of A is Hochschild and cyclic homology A⊗Aop HH∗ A = Tor∗ (A, A) Homological conformal ﬁeld theories and the Hochschild cohomology of A is HH ∗ A = Ext∗ op (A, A ), A⊗A where A = homk (A, k). If A is a (differential graded) Hopf algebra, then HH ∗ A is naturally a graded algebra. Free loop spaces HH and free loop spaces in topology and physics Kathryn Hess Theorem (Burghelea & Fiedorowicz, Cohen, What is the space of free loops? Goodwillie) Enumeration of If X is a path-connected space, then there are k-linear geodesics Hochschild and isomorphisms cyclic homology Homological HH∗ C∗ (ΩX ; k) ∼ H∗ (LX ; k) = conformal ﬁeld theories and HH ∗ C∗ (ΩX ; k) ∼ H ∗ (LX ; k). = Theorem (Menichi) The isomorphism HH ∗ C∗ (ΩX ; k) ∼ H ∗ (LX ; k) respects = multiplicative structure. Free loop spaces Power maps: the commutative algebra case in topology and physics Kathryn Hess What is the space Theorem (Loday, Vigué) of free loops? Enumeration of If A is a commutative (dg) algebra, then HH∗ A admits a geodesics natural “r th -power map” that is topologically meaningful in Hochschild and cyclic homology the following sense. Homological conformal ﬁeld theories If A is the commutative dg algebra of rational piecewise-linear forms on a simplicial complex X , then there is an isomorphism HH−∗ A ∼ H ∗ (LX ; Q) = that commutes with r th -power maps. Free loop spaces Power maps: the cocommutative Hopf in topology and physics algebra case Kathryn Hess What is the space of free loops? Theorem (H.-Rognes) Enumeration of geodesics If A is a cocommutative (dg) Hopf algebra, then HH∗ A Hochschild and cyclic homology admits a natural “r th -power map” that is topologically Homological meaningful in the following sense. conformal ﬁeld theories Let K be a simplicial set that is a double suspension. If A is the cocommutative dg Hopf algebra of normalized chains on GK (the Kan loop group on K ), then there is an isomorphism HH∗ A ∼ H∗ (L|K |) = that commutes with r th -power maps. Free loop spaces Cyclic homology of algebras in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics The cyclic homology of a (differential graded) algebra A, Hochschild and cyclic homology denoted HC∗ A, is a graded vector space that ﬁts into a Homological conformal ﬁeld long exact sequence (originally due to Connes) theories I S B − → → ... → HHn A → HCn A − HCn−2 A − HHn−1 A → .... Free loop spaces HC and free loop spaces in topology and physics Kathryn Hess What is the space of free loops? Enumeration of For any G-space Y , where G is a topological group, let geodesics YhG denotes the homotopy orbit space of the G-action. Hochschild and cyclic homology Homological conformal ﬁeld Theorem (Burghelea & Fiedorowicz, Jones) theories For any path-connected space X , there is a k-linear isomorphism HC∗ C∗ (ΩX ; k) ∼ H∗ (LX )hS 1 ; k . = Free loop spaces Generalizations: ring spectra I in topology and physics Kathryn Hess [Bökstedt, Bökstedt-Hsiang-Madsen] What is the space of free loops? Let R be an S-algebra (ring spectrum), e.g., the Enumeration of geodesics Eilenberg-MacLane spectrum HZ or S[ΩX ], the Hochschild and suspension spectrum of ΩX , for any topological space X . cyclic homology Homological Topological Hochschild homology conformal ﬁeld theories THH(R) and topological cyclic homology (mod p) TC(R; p) are important approximations to the algebraic K-theory of R. Free loop spaces Generalizations: ring spectra II in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Let X be a topological space, and let R = S[ΩX ]. Hochschild and cyclic homology Homological Then TC(R; p) can be constructed from conformal ﬁeld theories S[LX ] and S (LX )hS 1 , using the pth -power map p : LX → LX . Free loop spaces Generalizations: (derived) schemes in topology and physics Kathryn Hess What is the space of free loops? [Weibel, Weibel-Geller] Enumeration of geodesics Hochschild and Hochschild and cyclic homology can be generalized in a cyclic homology natural way to schemes, so that there is still a Homological conformal ﬁeld Connes-type long exact sequence relating them. theories [Toën-Vezzosi] Hochschild and cyclic homology can then be further generalized to derived schemes and turns out to be expressible in terms of a “free loop space” construction. Free loop spaces The closed cobordism categories C and HC in topology and physics Kathryn Hess What is the space of free loops? The objects of C and of HC are all closed Enumeration of 1-manifolds (disjoint unions of circles), which are in geodesics bijective correspondance with N. Hochschild and cyclic homology Homological conformal ﬁeld C(m, n) = C∗ (Mm,n ) and HC(m, n) = H∗ (Mm,n ) , theories where Mm,n is the moduli space of Riemannian Cobordism and CFT’s String topology cobordisms from m to n circles. Loop groups Both C and HC are monoidal categories, i.e., endowed with a “tensor product,” which is given by disjoint union of circles (equivalently, by addition of natural numbers) and disjoint union of cobordisms. Free loop spaces Cobordisms as morphisms in topology and physics Kathryn Hess A 3-to-2 cobordism What is the space of free loops? Enumeration of geodesics A 1-to-1 cobordism Hochschild and cyclic homology Homological conformal ﬁeld theories Cobordism and CFT’s String topology Loop groups Free loop spaces Composition of cobordisms in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and cyclic homology = Homological o conformal ﬁeld theories Cobordism and CFT’s String topology Loop groups Free loop spaces Topological CFT’s in topology and physics Kathryn Hess Let k be a ﬁeld, and let Chk denote the category of chain What is the space of free loops? complexes of k-vector spaces. Enumeration of geodesics A closed TCFT is a linear functor Φ : C → Chk that is Hochschild and cyclic homology monoidal up to chain homotopy. Homological conformal ﬁeld In particular, for all n, m ∈ N, theories Cobordism and CFT’s String topology Loop groups Φ(n) is a chain complex; there is a natural chain equivalence → Φ(n) ⊗ Φ(m) − Φ(n + m); there are chain maps C(m, n) ⊗ Φ(m) → Φ(n). Free loop spaces Homological CFT’s in topology and physics Let grVectk denote the category of graded k-vector Kathryn Hess spaces. What is the space of free loops? A closed HCFT is a linear functor Ψ : HC → grVectk that Enumeration of geodesics is strongly monoidal. Hochschild and cyclic homology In particular, for all n, m ∈ N, Homological conformal ﬁeld theories Ψ(n) is a graded vector space; Cobordism and CFT’s String topology Loop groups there is a natural isomorphism ∼ = → Ψ(n) ⊗ Ψ(m) − Ψ(n + m); there are graded linear maps HC(m, n) ⊗ Ψ(m) → Ψ(n). If Φ : C → Chk is a closed TCFT, then H∗ Φ is a closed HCFT Free loop spaces Folklore Theorem in topology and physics Kathryn Hess If Ψ : HC → grVectk is a closed HCFT, then Ψ(1) is a bicommutative Frobenius algebra, i.e., there exists What is the space of free loops? Enumeration of a commutative, unital multiplication map geodesics Hochschild and cyclic homology µ : Ψ(1) ⊗ Ψ(1) → Ψ(1) Homological conformal ﬁeld theories and Cobordism and CFT’s String topology a cocommutative, counital comultiplication map Loop groups δ : Ψ(1) → Ψ(1) ⊗ Ψ(1) such that (µ⊗1)(1⊗δ) = δµ = (1⊗µ)(δ ⊗1) : Ψ(1)⊗Ψ(1) → Ψ(1)⊗Ψ(1). Free loop spaces The geometry of µ and δ in topology and physics Let Ψ : HC → grVectk be a closed HCFT. Using the Kathryn Hess isomorphism Ψ(1) ⊗ Ψ(1) ∼ Ψ(1 + 1), we get: = What is the space of free loops? Enumeration of geodesics µ=Ψ( ):Ψ(1) Ψ(1) Ψ(1) Hochschild and cyclic homology Homological conformal ﬁeld theories Cobordism and CFT’s String topology δ=Ψ( ):Ψ(1) Ψ(1) Ψ(1) Loop groups Free loop spaces Generalizations in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and There are open-closed cobordism categories, in which cyclic homology the objects are all compact, 1-dimensional oriented Homological conformal ﬁeld manifolds (disjoint unions of circles and intervals). The theories Cobordism and CFT’s notion of open-closed conformal ﬁeld theories then String topology Loop groups generalizes in an obvious way that of closed CFT’s. Free loop spaces Philosophy in topology and physics Kathryn Hess What is the space of free loops? String topology is the study of the (differential and Enumeration of algebraic) topological properties of the spaces of smooth geodesics paths and of smooth loops on a manifold, which are Hochschild and cyclic homology themselves inﬁnite-dimensional manifolds. Homological conformal ﬁeld The development of string topology is strongly driven by theories Cobordism and CFT’s analogies with string theory in physics, which is a theory String topology Loop groups of quantum gravitation, where vibrating “strings” play the role of particles. As we will see, string topology provides us with a family of HCFT’s, one for for each manifold M. Free loop spaces Compact manifolds and intersection products in topology and physics Kathryn Hess Let M be a smooth, orientable manifold of dimension n. What is the space ∼ = of free loops? Let δM :H n−p M − → Hp M denote the Poincaré duality Enumeration of isomorphism (the cap product with the fundamental class geodesics of M). Hochschild and cyclic homology The intersection product on H∗ M is given by the Homological conformal ﬁeld composite theories Cobordism and CFT’s String topology −1 −1 δM ⊗δM / H n−p M ⊗ H n−q M / H 2n−p−q M Loop groups ∪ Hp M ⊗ Hq M YY YYYYYY YYYYYY YYYYYY YYYYYY δM • YYYYYY YY, Hp+q−n M and endows H∗ M := H∗+n M with the structure of a Frobenius algebra. Free loop spaces The Chas-Sullivan product in topology and physics Kathryn Hess Theorem (Chas & Sullivan, 1999) Let M be a smooth, orientable manifold of dimension n. What is the space of free loops? There is a commutative and associative “intersection” Enumeration of product geodesics Hp LM ⊗ Hq LM → Hp+q−n LM Hochschild and cyclic homology Homological that conformal ﬁeld theories endows H∗ LM := H∗+n LM with the structure of a Cobordism and CFT’s String topology Frobenius algebra and Loop groups is compatible with the intersection product on H∗ M, i.e., the following diagram commutes. Hp LM ⊗ Hq LM / Hp+q−n LM e∗ ⊗e∗ e∗ Hp M ⊗ Hq M / Hp+q−n M Free loop spaces From string topology to HCFT’s in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Theorem (Godin, Cohen-Jones, Harrelson, Ramirez, Hochschild and cyclic homology Lurie) Homological For any closed, oriented manifold M, there is an HCFT conformal ﬁeld theories Cobordism and CFT’s String topology ΨM : HC → grVectk Loop groups such that ΨM (1) = H∗ LM. Free loop spaces “Algebraic” string topology and HCFT’s in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Theorem (Costello, Kontsevich-Soibelman) Hochschild and cyclic homology If A is an A∞ -symmetric Frobenius algebra (e.g., if A is a Homological bicommutative Frobenius algebra), then there is an HCFT conformal ﬁeld theories Cobordism and CFT’s String topology ΨA : HC → grVectk Loop groups such that ΨA (1) = HH∗ A. Free loop spaces Positive-energy representations in topology and physics If G is a connected, compact Lie group, then LG is the Kathryn Hess loop group of G. What is the space A projective representation of free loops? Enumeration of ϕ : LG → PU(H), geodesics Hochschild and where H is an inﬁnite-dimensional Hilbert space, is of cyclic homology Homological positive energy if there is a smooth homomorphism conformal ﬁeld u : S 1 → PU(H) such that theories Cobordism and CFT’s String topology u×ϕ / U(H) × PU(H) Loop groups S × LG 1 κ conj. LG ϕ / PU(H) commutes, and H= Hn , n≥0 where u(eiθ )(x) = einθ · x for every x ∈ Hn . Free loop spaces The Verlinde ring in topology and physics Kathryn Hess There is a “topological” equivalence relation on the set of What is the space projective, positive-energy representations of LG. of free loops? Enumeration of Let R ϕ (G) denote the group completion of the monoid of geodesics Hochschild and projective, positive-energy representations that are cyclic homology equivalent to a given representation ϕ : LG → PU(H). Homological conformal ﬁeld theories Verlinde deﬁned a commutative multiplication–the fusion Cobordism and CFT’s product–on R ϕ (G), giving it the structure of a String topology Loop groups commutative ring. In fact, R ϕ (G) ⊗ C is a Frobenius algebra, and there is an HCFT Ψϕ : HC → grVectk such that Ψϕ (1) = R ϕ (G) ⊗ C. Free loop spaces The topology behind the algebra in topology and physics Kathryn Hess What is the space of free loops? Enumeration of geodesics Hochschild and Theorem (Freed-Hopkins-Teleman) cyclic homology Let G and ϕ : LG → PU(H) be as above. Homological conformal ﬁeld theories There is a ring isomorphism from R ϕ (G) to a twisted Cobordism and CFT’s String topology version of the equivariant K -theory of G acting on itself by Loop groups conjugation. Free loop spaces Free loop spaces and twisted K -theory in topology and physics Kathryn Hess Let What is the space of free loops? PG = {f ∈ C ∞ (R, G) | ∃x ∈ G s.t. f (θ + 2π) = x · f (θ) ∀θ ∈ R}. Enumeration of geodesics Consider the principal LG-ﬁbre bundle Hochschild and cyclic homology Homological p : PG → G : f → f (2π)f (0)−1 , conformal ﬁeld theories Cobordism and CFT’s where LG acts freely on PG by right composition. String topology Loop groups Together, ϕ and p give rise to a twisted Hilbert bundle PG × P(H) → G, LG where P(H) = H/S 1 . The twisted equivariant K -theory of G is given in terms of sections of this bundle.