# 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
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);

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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