FRG Workshop: "Mathematical 2D-field theory and the algebraic topology of closed manifolds"

August 2nd-6th, Stony Brook Math Department

Satellite to the Simons Workshop - July 26th to August 20th. 







9:00 -10:00 in Math Common Room

Informal Discussion

and Participant's Breakfast in Math Common Room with Simons Workshop

Informal Discussion 

and Participant's Breakfast in Math Common Room with Simons Workshop

Informal Discussion and Participant's Breakfast in Math Common Room  

with Simons Workshop

Assigning Beach Rides

Informal Discussion

and Participant's

Breakfast in Math

Common  Room with Simons Workshop

Informal Discussion 

and Participant's 

Breakfast in Math

Common  Room with 

Simons Workshop

10:00 - 11:00 in S-240

Discussion in S-240

Hiro Tanaka
( On the structure of the  category of Lagrangian

Beach with Simons Workshop

Micah Miller

 (Extending Brown's 

twisting cochain models 

for total spaces of fibrations

to include the coalgebra structure and also

String Topology)

Dmitri Pavlov

 ( L^p spaces which are bimodules over operator algebras that are closed 

for the topology of 

pointwise weak 


11:15-12:15 in S-240

Katherine Poirier

11:15 to 12:15

Thesis defense

(Extending String Topology to operations  built using the top chain of compactified moduli spaces.)

Jason McGibbon

invariants in the 

space of knots.)


Simons Workshop 

Lecture by

Nathan Seiberg

(...constraints on supergravity)

Yuan Shen

(Calculation of 

 obstructions to
 quantization for

 specific theories.)

Andrew Stimpson
( Eilenberg Steenrod type axioms for  differential cohomology.)

12:30 - 2:00 in S-240

Discussion and Participant's Lunch in S-240

Discussion and  Participant's Lunch

in S-240

Veranda Lunch

at the Beach with

Simons Workshop

Discussion and 

Participant's Lunch 

in S-240

Participant's Lunch and

Discussion in S-240

Nathaniel Rounds

12:30 to 1:30
Thesis defense

 ( Complete cochain 

invariants of a manifold.)

2:00 - 3:00 in S-240


Anton Kapustin

(Explaining the  Feynman path integral using n-categories.)

Eric Zaslow
( Connecting the two sides of mirror symmetry  topologically using constructible sheaves.)

Beach with 

Simons Workshop

Serguei Barannikov

 (Integration formalism related to quantum theories both commutative and non commutative - the BV formalism)

Stephan Stolz
(Defining traces for 
operators arising from
monoidal categories.)

3:00 - 4:00 in S-240


and coffee in S-240

Discussion  and coffee  in S-240

Beach with
Simons Workshop


and coffee in S-240

Discussion  and coffee  in S-240

4:00 - 5:00 in S-240

Moira Chas

 (Experimental results about self intersection numbers of curves on surfaces leading to provable theorems.)

Oleg Viro

(A simple presentation of certain 3D quantum topology invariants)

 Return from beach

Tamar Friedman

(New Lie Algebras 

arising from 


Discussion:  Stephan Stolz 

next FRG meetings.

Berkeley 1st week of 

January 2011.(small)

 Notre Dame Spring or 

Summer 2011 (open).  

5:00 - 7:30    

Restaurant suggestions here  

Restaurant suggestions here  

Return from beach

Restaurant suggestions here  

Restaurant suggestions


7:30-8:30 in S-240

Joergen Andersen

(A physics related property of the automorphism 

group of a surface)

Discussion lead by Kevin Costello and Dennis Sullivan

 FRG Workshop

Banquet 7:00pm 

at  the Curry Club

Simons Workshop
 at Martin Rocek's








Titles and abstracts of the talks 

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 Reshetikhin-Turaev 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 Reshetikhi-Turaev TQFT's together with some asymptotic analysis.   

Developments in a noncommutative Batalin-Vilkovisky formalism,

Serguei Barannikov

The Batalin-Vilkovisky 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 Batalin-Vilkovisky
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 psi-classes 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 "tree-level" limit the noncommutative BV formalism incorporates the BV-structure on polyvectors on Calabi--Yau manifolds, or, equivalently, the BV-structure on Hochschild cochains, whose relation with g=0 Gromov-Witten invariants I've studied in the 90s.  If the time permits I'll explain how an idea from the nc-BV formalism, applied back to the commutative BV-setup, implies equivariant
localization of the Chern-Simons theory in arbitrary dimension.
The talk is based on my 2006 papers hal-00102085 and doi:10.1093/imrn/rnm075.

Unexpected statistical structure in the self-intersections 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 self-intersection number for a given combinatorial length, the number of conjugacy classes with given self-intersection and given length,  and finally the unanticipated distribution of the self-intersection 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 Yang-Mills with Matter

Tamar Friedman.

 There is a much studied correspondence between co-dimension 4 orbifold singularities in Calabi Yau or special holonomy spaces that serve as the extra dimensions in physical theories with space-time 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 co-dimension 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 Chern-Simons 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 space-time. Recently it became clear that for space-time dimension greater than two axioms of TFT are best formulated in terms of higher categorical structures (n-categories with n>1). From the physical viewpoint, n-categories encode properties of observables localized on submanifolds of dimension n. My goal will be to explain this fact in the simplest possible setting: Chern-Simons 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.  

2|1-dimensional Euclidean field theories and noncommutative L^p - spaces.

  Dmitri Pavlov

  A conjecture by Stolz and Teichner states that concordance classes of 2|1-dimensional 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 2|1-dimensional Euclidean bordisms to the bicategory of von Neumann algebras, L^p-bimodules, 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^p-bimodules, and their morphisms,which involves proving several theorems about noncommutative L^p-spaces. 
  If time permits, I will also explain how the study of 2|1-dimensional Euclidean field theories naturally leads to consider such interesting structures as one-parameter semigroups of bimodules and two-parameter 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  multi-strings in a  d-manifold 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  (d-3)(euler) -1 where euler is the quantity 2-2g-k-j, 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 non-homeomorphic 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 obstruction-deformation 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 K-theory 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 K-theory 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 K-theory 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 d-dimensional bordism from a closed manifold Y to itself, and let W be the closed d-manifold obtained by identifying the two copies of Y in the boundary of W'. Suppose that E is a d-dimensional field theory in the sense of Atiyah-Segal; 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 2-dimensional 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 conjectures--one is about possible connections to the Fukaya Category of M, and another is about a local-to-global 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, Chiu-Chu Melissa Liu and David Treumann.