Free loop spaces in topology and physics by ttg74934

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									                                                       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.

								
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