An application of
quantization
ideas to pure mathematics: TQFT, Hitchin's connection and
Toeplitz operators,
Joergen
Andersen
In the talk we
will review the geometric gauge theory
construction of the vector spaces that the ReshetikhinTuraev TQFT
associates to a closed oriented surface. Hence we will build the
Hitchin connection in certain vector bundles over Teichmüller
space. The Hitchin connection is obtained by applying the procedure
called geometric quantization to the moduli
space of flat connections on the surface. This will be followed by a
discussion of the relation between the Hitchin connection, special
operators introduced by Toeplitz and the states in quantum
theory
called coherent states. The talk will end with a discussion of
our proof that for the mapping class groups that the trivial
irreducible representation is approximately contained in nontrivial
irreducible unitary representations, a longstanding open
question (Ivanov, Sullivan,...early 80's) This uses the geometric
construction of the ReshetikhiTuraev TQFT's together
with some asymptotic analysis.
Developments in a
noncommutative BatalinVilkovisky formalism,
Serguei Barannikov
The BatalinVilkovisky formalism is an integration theory for
polyvector (multivector) fields as opposed to the integration of
differential forms. This talk is an introduction into my noncommutative
version of the BatalinVilkovisky
formalism, which leads, in particular, to a higher dimensional
generalization of the celebrated matrix Airy integrals, which were
shown 20 years ago to describe intersection of psiclasses on moduli
spaces.
The asymptotic expansions of my matrix integrals provide a
combinatorial construction of various natural cohomology classes of
compactified moduli spaces of curves of arbitrary genus. In the
"treelevel" limit the noncommutative BV formalism incorporates the
BVstructure on polyvectors on CalabiYau manifolds, or, equivalently,
the BVstructure on Hochschild cochains, whose relation with g=0
GromovWitten invariants I've studied in the 90s. If the time
permits I'll explain how an idea from the ncBV formalism, applied back
to the commutative BVsetup, implies equivariant
localization of the ChernSimons theory in arbitrary dimension.
The talk is based on my 2006 papers hal00102085 and
doi:10.1093/imrn/rnm075.
Unexpected statistical
structure in the selfintersections of curves on surfaces,
Moira Chas,
Consider the set of free homotopy classes of oriented closed curves on
a surface, namely the set of equivalence classes of maps from the
circle into the surface, where two such maps are equivalent if the
corresponding directed curves can be deformed one into the other. There
is a canonical bijection between this set and the set of conjugacy
classes in the fundamental group of the surface.
Given
a free homotopy class one can ask what is the
minimum number of
times, counted with multiplicity, a curve in that class intersects
itself. Call this the self intersection number of the class of curves.
In
this talk, several problems related to the self
intersection number of a class will be discussed. We will
address
such
questions as: the maximal and minimal selfintersection number for a
given combinatorial
length, the number of conjugacy classes with given selfintersection
and given length, and finally the unanticipated distribution
of the selfintersection number among the conjugacy classes of
a
given combinatorial length. One part of this work is joint
with
Anthony Phillips and another
part
is joint with Steve Lalley.
Orbifold Singularities,
Lie
Algebras of the Third Kind, LATkes , and Pure YangMills with Matter,
Tamar
Friedman.
There is a much
studied correspondence between
codimension 4 orbifold singularities in Calabi Yau or special holonomy
spaces that serve as the extra dimensions in physical theories with
spacetime dimension higher than 4, and certain gauge theories whose
gauge group is a Lie group of the ADE series. Alas, there is no
analogous correspondence for codimension 2n orbifold singularities,
where n>2. In this lecture, I will show how my search for
such
analogs led me from the singularities to the definition
and constructions of Lie Algebras of the Third Kind
(LATKes). I will
also introduce the example of the algebra that arises from a certain
singularity C^3/( three fold symmetry) and prove it to be
simple and unique. I will then discuss the
application of these results in physics, particularly to
string theory and particle physics.
Abelian ChernSimons
theory and
categorical algebra,
Anton
Kapustin,
The Feynman
functional integral is a central notion in
Quantum
Field Theory but so far it has not been rigorously defined in a
sufficiently general situation. There was some success in axiomatizing
properties of the functional integral in the special case of
Topological Field Theories (TFT), i.e. field theories which are
independent of the metric on spacetime. Recently it became clear that
for spacetime dimension greater than two axioms of TFT are best
formulated in terms of higher categorical structures (ncategories with
n>1). From the physical viewpoint, ncategories encode
properties of
observables localized on submanifolds of dimension n. My goal will be
to explain this fact in the simplest possible setting: ChernSimons
theory in three dimensions with an abelian gauge group.
Monodromy of knot contact
homology,
Jason
McGibbon
Knot contact homology
(KCH) is a topologically or rather
combinatorially defined invariant of smooth knots introduced by Ng.
Work of Ekholm, Etnyre, Ng and M.Sullivan shows that KCH is
the contact homology of the unit conormal lift of the knot.
In this talk we describe a monodromy result for knot contact
homology,namely that associated to a path of knots there is a
connecting homomorphism which is invariant under homotopy.
The proof of this result suggests a conjectural interpretation for KCH
via open strings, which we will describe.
Twisted tensor
products
and String Topology,
Micah
Miller
Given an
infinity cocommutative coalgebra C, a strict
Hopf algebra H, and
a twisting cochain t (in the sense of E.H. Brown, Annals 1957) mapping
C
into H whose image lies in the subspace of primitive elements of
H, we describe a procedure for obtaining an infinity
coassociative coalgebra structure on C tensor H.
This is an extension of Brown's work on twisted tensor products where
only the additive structure of the models was considered.
We apply this procedure to
obtain an infinity
coassociative coalgebra model for the
chains on the free loop space of a simply connected finite
complex M. We take for C the infinity cocommutative coalgebra
structure on the homology induced by the diagonal map of M into MxM and
for H the universal enveloping algebra of the Whitehead Lie
algebra of rational homotopy groups calculated say from a minimal
model. The twisting cochain t will be described in
the lecture.
When C has a cyclic infinity
cocommutative
coalgebra structure, for example defined by the intersection numbers
when M is a closed oriented manifold, we describe an infinity
associative algebra
structure on C tensor H . This is used to give an explicit
infinity associative algebra model of the chain level string topology
loop product.
Furthermore, this model is realized as the universal enveloping algebra
of a Lie infinity algebra.
21dimensional Euclidean
field
theories and noncommutative
L^p  spaces.
Dmitri Pavlov
A conjecture by
Stolz and Teichner states that
concordance classes of
21dimensional Euclidean field theories are in bijective
correspondence with cohomology classes of the cohomology theory TMF
(topological modular forms). Here a field theory is a functor from the
bicategory of 21dimensional Euclidean bordisms to the bicategory of
von Neumann algebras, L^pbimodules, and their morphisms.
A
significant amount of labor is required to make the definitions of the
two bicategories mentioned above precise.Most of the talk will be
devoted to a rigorous definition of the algebraic bicategory of von
Neumann algebras,L^pbimodules, and their morphisms,which involves
proving several theorems about noncommutative L^pspaces.
If time
permits, I will also explain how the study of
21dimensional Euclidean field theories naturally leads to consider
such interesting structures as oneparameter semigroups of bimodules
and twoparameter semigroups of bimodule endomorphisms further
parametrized by the moduli space of elliptic curves.

String Topology and
Compactified Moduli Spaces,
Katherine
Poirier.
The
goal of this work is to solve the master
equation dX
+ X*X = 0 where X is a direct sum over g nonnegative and over k and j
positive of k to j operations on the chain complexes of closed
multistrings in a dmanifold M. The symbol *
refers to all
ways up to homeomorphism of splitting a connected
surface of
genus g with k labeled input circle boundaries and j labeled output
circle boundaries
into two other connected surfaces with input and output circle
boundaries.
The operation corresponding to
a triple (g,k,j) has degree (d3)(euler) 1 where euler is
the
quantity
22gkj, the euler characteristic of the corresponding surface, and
only triples (g,k,j) with (euler) negative yield non zero string
topology operations.
The homology of these chain complexes can be
expressed in
terms of the equivariant homology of the free loop space mod constant
loops.
The construction of the solution of the master
equation,
the subject of this talk,
proceeds by building pseudomanifolds of string diagrams with
levels which have prescribed input boundary. The string topology
construction for manifolds M
describes the action of cellular chains
of these pseudomanifolds
on the
above chain complexes of closed strings. Furthermore, each
pseudomanifold is homeomorphic
to a compactification of the corresponding moduli space of Riemann
surfaces. One application, which will not be
discussed in
the lecture, of the existence of a solution of the master
equation here is the following:
Corollary:The solution X can be deformed
to give k to
l operations
Y(g,k,j) on the reduced equivariant free loop space homology whose sum
Y
satisfies Y*Y = 0. This yields one quadratic
relation for each (g,k,j) among the operations
corresponding
to splittings of (g,k,j). The two operations for
(g,k,j) equal to (0,2,1) and (0,1,2) satisfy four quadratic
relations corresponding to the various splittings of (g,k,j) in the set
{(0,3,1), (0,1,3),
(0,2,2), (1,1,1)}. These precisely define an involutive lie bialgebra
structure on the reduced equivariant homology of the free loop space of
the manifold. For d=2 this stucture is all there is and it was
discovered by Goldman and Turaev in the 80's. The further
operations for d > 2 yield a higher algebraic
structure
extending this involutive Lie bialgebra structure.
Construction: A second smaller
compactification will be mentioned over which (conjecturally)
string topology
operations also extend. "Poincare duality at the chain level,"
Nathaniel Rounds
Poincare duality at the
chain level,
Nathaniel Rounds
Closed oriented manifolds satisfy
Poincare duality. This duality is reflected in the chains and cochains
of the manifold. Considered naively on homology, this duality is
not enough to help us distinguish two nonhomeomorphic manifolds in the
same homotopy type. However, there are invariants described by surgery
theory which allow us to distinguish manifolds in a homotopy
type.These invariants may be interpreted in terms of duality at
the chain or cochain level.This is the subject of this thesis.
We can describe homotopy types and the
manifolds structures within them in the following way. We
consider chain complexes with a fixed basis satisfying certain axioms.
We show that a homotopy type of based chain complexes determines a
homotopy type of spaces. If such a homotopy type satisfies Poincare
duality, we show, using Ranicki's algebraic reformulation of surgery
theory, that topological manifold structures in the homotopy type are
in one to one correspondence with local inverses to the Poincare'
duality map. Defining the word local in our setting in a homotopy
invariant way is the key point of our theory and Ranicki's.
Mandell has shown that homotopy types of spaces are
determined by good cochain functors. We are hopeful that
Mandell's notion of good cochain functor can be synthesized with our
theory of based chain complexes to give an enriched Mandell cochain
functor
which determines the ingredients of our classification above and
therefore homeomorphism types of the manifolds. This would establish an
old conjecture of Sullivan.
Determining the
obstructions to
forming a quantum theory,
Yuan
Shen
Costello studied
a quantum field theory based on
holomorphic maps of a two torus (i.e. an elliptic curve) into
the
cotangent bundle of a complex manifold endowed with its
holomorphic symplectic structure.
As a corollary Costello obtains a QFT interpretation of an
invariant refining the A^ characteristic class in a more subtle theory
called elliptic cohomology theory. This construction of the Witten
genus is mathematically rigorous and is based on Costello's treatment
of renormalization, the method of dealing with the divergences in
perturbative QFT, but will not be discussed here.
This talk will consider the QFT associated to
holomorphic
maps between the unit two disk and a target holomorphic
cotangent
bundle up to holomorphic isomorphism of the source. The goal of the
talk will be to explain the construction of the obstructiondeformation
complex of local action functionals in this setting. The upshot will be
that obstructions to quantization of the classical theory will vanish
if the first two chern classes with rational coefficients
vanish.
Axioms for uniqueness of differential cohomology
Andrew Stimpson
Simons and Sullivan [0] studied a notion of equivalence
between two complex vector bundles with unitary connections such that
both the Ktheory class and the Chern character differential forms are
constant on an equivalence class. The Chern character form of these
equivalence classes (called "structured bundles") defines a natural
transformation into a class of special closed total even forms,
tautologically those total even forms whose total cohomology
class is the chern character of a vector bundle.By considering this
natural transformation into forms, as well as the natural
transformation given by the taking the Ktheory class of a structured
bundle, we get half of a commuting square of natural
transformations. The other half is the map of special forms into real
total even cohomology and the chern character map from K theory into
same.
Hopkins and Singer showed in [1] that
functors that, like structured bundles, fit into analogous diagrams of
natural transformations, with Ktheory replaced with any
generalized cohomology theory (namely functors which satisfy the
Eilenberg Steenrod axioms save the concentration in degree zero for a
point space).
This talk will discuss to what extent a
certain class of functors similar to these can be classified by axioms
pertaining to this commutative square supplemented by an axiom
consistent with both the suspension isomorphism for exotic cohomology
theories and the integration along circle fibres map for differential
forms..
[0] arXiv:0810.4935
[1] arXiv:math/0211216
Traces in monoidal
categories ,
Stephan Stolz
This talk is about joint
results with Peter Teichner which are
motivated by the following question: Let W' be a ddimensional bordism
from a closed manifold Y to itself, and let W be the closed dmanifold
obtained by
identifying the two copies of Y in the boundary of W'. Suppose that E
is a ddimensional field theory in the sense of AtiyahSegal; i.e., E
associates to Y a topological vector space E(Y), to the bordism
W' a continuous operator E(W') mapping E(Y) to E(Y) and
to
W a complex number E(W).
Question: How can the number E(W) be calculated
from the
operator E(W')?
The expected answer is that E(W) is the trace of
E(W'),
but the difficulty is to verify that the operator satisfies the
properties required to have a meaningful trace. We'll describe an
approach to traces in monoidal categories which is a common
generalization of what has been done in the category of locally convex
topological vector spaces and for monoidal categories in which all
objects are dualizable. This construction is an important step in the
proof of our result that the partition function of a super symmetric
2dimensional Euclidean field theory is a modular function. If time
permits, I'll give an indication of how traces are used in that proof.
A Stable
(infinity,1)category
of Lagrangian Cobordisms,
Hiro
Tanaka,
Given a
symplectic manifold M with some additional data,
we
define a category whose objects are Lagrangian submanifolds of M, and
whose morphisms are cobordisms between them. We will review
what
it means for a category to be stable, in the sense of Jacob Lurie, and
sketch a proof showing that this category is stable. (One
implication of this is that the homotopy category is
triangulated.) We will then discuss some conjecturesone is
about possible connections to the Fukaya Category of M, and another is
about a localtoglobal method of computing the
category.
This
is
joint work with David Nadler.
From a TQFT to a link quantum
field theory
Oleg Viro
A construction which turns a (2+1)dimensional TQFT into link
invariants will be presented. For a TQFT that is defined via state sums
using the representation category specialized to roots of unity of the
quantum deformation of SL(2,R) the construction gives a vector space
with a linear operator whose trace is a value of the colored Jones
polynomial at q which is a root of unity.
Functoriality of this construction will be discussed.
Constructible sheaves in
Mirror
Symmetry,
Eric
Zaslow,
Mirror symmetry is a conjectural equivalence
between two
categories carrying different kinds of geometric information, either
symplectic or algebraic (complex). We will approach this conjecture
through an intermediate category which is sensitive to topological
information, the category of constructible (roughly,piecewise locally
constant) sheaves. I will describe two theorems: one relates
the
symplectic (Fukaya) category to constructible sheaves; the other
relates the complex category of coherent sheaves on toric varieties to
constructible sheaves. Together these theorems prove a
(slightly
nonstandard) version of homological mirror symmetry.
This is based on joint work with David Nadler, and
with
Bohan Fang,
ChiuChu Melissa Liu and David Treumann.
