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 1 What is the space of free loops? 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 Let X be a topological space. of free loops? Enumeration of The space of free loops on X is geodesics Hochschild and cyclic homology 1 LX = Map(S , X). Homological conformal field theories
If M is a smooth manifold, then we take into account the smooth structure and set
LM = C∞(S1, 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
What is the space Let q : PX → X × X denote the fibration given by 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 : S1 → 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 S1, given by rotating the loops. Hochschild and cyclic homology More precisely, there is an action map Homological conformal field theories κ : S1 × LX → LX, where κ(z, λ): S1 → X : z0 7→ λ(z · z0). Free loop spaces Structure: the power maps in topology and physics
Kathryn Hess
What is the space of free loops?
Enumeration of For any natural number r, the free loop space LX admits geodesics th an r -power map Hochschild and cyclic homology
Homological `r : LX → LX conformal field theories given by 1 r `r (λ): S → X : z 7→ λ(z ),
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 n o 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 n bn(X; ) = dim H (X; ). Hochschild and k k k 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 M is simply connected, and cyclic homology ∗ Homological the graded commutative algebra H (M; ) is not conformal field Q 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 Theorem (McCleary & Ziller, 1987) geodesics 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 It is easy to show that if M is diffeomorphic to a geodesics Hochschild and symmetric space of rank 1, then bn(LM; k) n≥0 is cyclic homology bounded for all , but Homological k conformal field Hingston proved that a simply connected manifold theories with the rational homotopy type of a symmetric space of rank 1 generically admits infinitely many closed geodesics, and Franks and Bangert showed that S2 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; ) is not monogenic, then b (ΩX; ) is conformal field Fp n Fp n≥0 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 ∗ cyclic homology H (X; Fp) is not monogenic, then bn(LX; Fp) is n≥0 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
Hochschild and B / A 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 = ExtA⊗Aop (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 Goodwillie) of free loops? Enumeration of If X is a path-connected space, then there are k-linear geodesics Hochschild and isomorphisms cyclic homology Homological HH C (ΩX; ) =∼ H (LX; ) conformal field ∗ ∗ k ∗ k 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 admits a natural “r th-power map” that is topologically cyclic homology 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 ... → HHnA −→ HCnA −→ HCn−2A −→ HHn−1A → .... 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)hS1 ; 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 Eilenberg-MacLane spectrum HZ or S[ΩX], the geodesics 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)hS1 ,
th using the p -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 . Hochschild and N cyclic homology
Homological conformal field C(m, n) = C∗(Mm,n) and HC(m, n) = H∗(Mm,n) , theories Cobordism and CFT’s where Mm,n is the moduli space of Riemannian 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 be a field, and let Ch denote the category of chain What is the space k k 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 monoidal up to chain homotopy. cyclic homology Homological conformal field In particular, for all n, m ∈ , theories N 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 grVect denote the category of graded -vector Kathryn Hess k k spaces. What is the space of free loops? A closed HCFT is a linear functor Ψ: HC → grVect that k Enumeration of is strongly monoidal. geodesics 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 → grVect is a closed HCFT, then Ψ(1) is a k What is the space bicommutative Frobenius algebra, i.e., there exists 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 → grVect be a closed HCFT. Using the k Kathryn Hess isomorphism Ψ(1) ⊗ Ψ(1) =∼ Ψ(1 + 1), we get: What is the space of free loops?
Enumeration of geodesics
Hochschild and µ=Ψ( ):Ψ(1) Ψ(1) Ψ(1) 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 Hochschild and paths and of smooth loops on a manifold, which are 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 ∼ n−p = of free loops? Let δM : H M −→ HpM 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 δ ⊗δ Loop groups M M n−p n−q ∪ HpM ⊗ HqM / H M ⊗ H M / 2n−p−q Y H M YYYYYY YYYYYY YYYY δM • YYYYYY YYYYYY YYYYY, Hp+q−nM
and endows H∗M := H∗+nM with the structure of a Frobenius algebra. Free loop spaces The Chas-Sullivan product in topology and physics Theorem (Chas & Sullivan, 1999) Kathryn Hess 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 Hochschild and HpLM ⊗ HqLM → Hp+q−nLM cyclic homology
Homological that conformal field theories endows H∗LM := H∗+nLM 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.
HpLM ⊗ HqLM / Hp+q−nLM
e∗⊗e∗ e∗ HpM ⊗ HqM / Hp+q−nM 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 Lurie) cyclic homology Homological For any closed, oriented manifold M, there is an HCFT conformal field theories Cobordism and CFT’s String topology Ψ : HC → grVect Loop groups M k
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 anA -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 Ψ : HC → grVect Loop groups A k
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 : S1 → PU(H) such that theories Cobordism and CFT’s String topology u×ϕ Loop groups S1 × LG / U(H) × PU(H)
κ conj. ϕ LG / PU(H) commutes, and M H = Hn, n≥0 iθ inθ where u(e )(x) = e · 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 Rϕ(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 ϕ String topology product–on R (G), giving it the structure of a Loop groups commutative ring. ϕ In fact, R (G) ⊗ C is a Frobenius algebra, and there is an HCFT Ψ : HC → grVect ϕ k ϕ such that Ψϕ(1) = R (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 ϕ theories There is a ring isomorphism from R (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
Hochschild and Consider the principal LG-fibre bundle cyclic homology
−1 Homological p : PG → G : f 7→ f (2π)f (0) , 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/S1. The twisted equivariant K -theory of G is given in terms of sections of this bundle.