# Monday

10:00-10:05 | Opening | |

10:05-10:55 | Garoufalidis | Chern-Simons theory and arithmetic |

10:55-11:25 | Coffee | |

11:25-12:15 | Thompson | Chern-Simons Theory On Seifert 3-Manifolds |

12:15-14:10 | Lunch | |

14:10-15:00 | Kirk | Non-abelian representations, homology 3-spheres, and knot concordance |

15:00-16:00 | Marino | Chern-Simons theory, the 1/N expansion, and string theory |

16:00-16:30 | Coffee | |

16:30-17:20 | Meusburger | Getting physics from 3d gravity: What does an observer in 3d gravity see? |

# Tuesday

9:00-10:00 | Khovanov | Categorification of quantum groups |

10:00-10:30 | Coffee | |

10:30-11:20 | Mikovic | Invariants of spin networks embedded in 3-manifolds |

11:20-12:10 | Herald | An SU(3) Casson invariant for rational homology spheres |

12:10-14:10 | Lunch | |

14:10-15:00 | Kauffman | Khovanov Homology and the Potts Model |

15:00-16:00 | Kashaev | On rings associated with ideal triangulations of knot complements |

16:00-16:30 | Coffee | |

16:30-17:20 | Sengupta | Functional Integrals in Low-Dimensional Gauge Theories |

# Wednesday

9:00-9:50 | Lescop | On the cube of the equivariant linking pairing for closed 3-manifolds of rank one |

9:50-10:40 | Guadagnini | Functional integration and abelian link invariants |

10:40-11:10 | Coffee | |

11:10-12:00 | Weitsman | Fermionization and Convergent Perturbation Expansions in Chern-Simons Gauge Theory |

12:00-12:50 | Murakami | SL(2;C)-representations and asymptotic behaviors of the colored Jones polynomial of a knot |

12:50-14:00 | Lunch | |

14:00-19:00 | Excursion | |

# Thurdsay

9:00-10:00 | Penner | Fatgraphs and finite type invariants |

10:00-10:30 | Coffee | |

10:30-11:20 | Klemm | Chern-Simons Theory and Topological String theory on non-compact Calabi-Yau manifolds |

11:20-12:10 | Auckly | Gauge-string duality and the structure of large rank Chern-Simons invariants |

12:10-14:10 | Lunch | |

14:10-15:00 | Beasley | Localization for Wilson Loops in Chern-Simons Theory |

15:00-16:00 | Witten | Branes and Quantization |

16:00-16:30 | Coffee | |

16:30-17:20 | Bar-Natan | Convolutions on Lie Groups and Lie Algebras and Ribbon 2-Knots |

17:20-18:10 | Schwarz | Generalizations of Chern-Simons theory |

# Friday

9:00-9:50 | Masbaum | Integral structures in TQFT and the mapping class group |

9:50-10:40 | Hikami | WRT invariants and modular forms |

10:40-11:10 | Coffee | |

11:10-12:00 | Fine | A geometric alternative to gauge fixing in Chern-Simons theory on ^1 imes \Sigma$.es, including recent results of Hahn and Haro |

12:00-12:50 | Gukov | Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group |

# Abstracts

**Dave Auckly (Kansas State University): Gauge-string duality and the structure of large rank Chern-Simons invariants**

Gauge-string duality predicts that many gauge theories are equivalent to ’dual’ string theories. For Chern-Simons theory, the best understood manifistation of this duality relates the Chern-Simons invariants of the three sphere to the Gromov-Witten of the resolved conifold, a the total space of a rank two complex vector over the projective line. This talk will review this duality and discuss what it might imply about the structure and unification of the theory.

**Dror Bar-Natan (University of Toronto): Convolutions on Lie Groups and Lie Algebras and Ribbon 2-Knots**

I’ll start my talk by describing the Kashiwara-Vergne Conjecture (1978, Proven Alekseev-Meinrenken, 2006), which states that for any finite dimensional Lie group, certain convolutions on the group are equal to certain convolutions on its Lie algebra, and I’ll end my talk in knot theory. Going backward from the hard to the easy, here’s the whole thing in one paragraph: Convolutions are integrals so we need to prove an equality of integrals. This we do (roughly) by finding a Measure Preserving Transformation (MPT) which carries one integrand to the other. This has to work for any Lie group, so the MPT better be given as a universal formula. At this generality all we have to work with is the Lie bracket, and bracket-made formulas can be pictured as certain trivalent diagrams (which in themselves are subject to the Jacobi, or "IHX" relation). Thus we are really seeking a certain big sum V of trivalent diagrams, which, in order to describe our desired MPT, must satisfy certain equations mod IHX. Our space Aw of trivalent diagrams turns out to be the "associated graded" space of the space Kw of ribbon 2-knots in 4-space, and the equations V needs to solve are the equations one needs to solve to get a well-behaved "expansion" Z:Kw-->Aw. Thus the Kashiwara-Vergne Conjecture, itself a witness to the famed "orbit method", is more or less the same as a natural problem in knot theory, and since knot theory is related to associators and to quantum groups, so is the Kashiwara-Vergne Conjecture. Cool, eh?

**Chris Beasley (SUNY): Localization for Wilson Loops in Chern-Simons Theory**

As noted long ago by Atiyah and Bott, the classical Yang-Mills action on a Riemann surface admits a beautiful symplectic interpretation as the norm-square of the moment map associated to the Hamiltonian action by gauge transformations on the affine space of connections. In this talk, I will explain how certain Wilson loop observables in Chern-Simons gauge theory on a Seifert three-manifold can be given an analogous symplectic description. Among other results, this fact implies that the stationary-phase approximation to the Wilson loop path integral is exact for torus knots, an observation made empirically by Lawrence and Rozansky prior to this work.

**Dana Fine (University of Massachusett): A geometric alternative to gauge fixing in Chern-Simons theory on ^1 imes \Sigma$.es, including recent results of Hahn and Haro**

The space of connections modulo gauge transformations on S1 x ? may be viewed as an infinite-dimensional bundle over \map(?, G) with affine-linear fiber. In this view the (heuristic) path integral simplifies significantly: the Wilson loop expectations of certain links can be calculated as elementary expectations for WZW models. This talk will review this construction and its relation to gauge-fixing approach.

**Stavros Garoufalidis (Georgia Institute of Technology): Chern-Simons theory and arithmetic**

The Chern-Simons invariants of a closed 3-manifold are secondary characteristic numbers that are given in terms of a finite set of phases in the unit circle. Hyperbolic 3-dimensional geometry links these phases with arithmetic, and identifies them with values of the Rogers dilogarithm at algebraic numbers. The quantization of these invariants are the famous Witten-Reshetikhin-Turaev invariants of 3-manifolds, constructed by the Jones polynomial. In the talk we will review conjectures regarding the asymptotics of the quantum 3-manifold invariants, and their relation to Chern-Simons theory and arithmetic. We will review progress on those conjectures, theoretical, and experimental.

**Enore Guadagnini (University of Pisa): Functional integration and abelian link invariants**

For the abelian Chern-Simons quantum field theory, it will be shown how to compute the link invariants by means of an explicit functional integration in the case of nontrivial 3-manifolds. The connections between the abelian link invariants and the homology invariants will be discussed.

**Sergei Gukov (California Institute of Technology): Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group**

Chern-Simons gauge theory with complex gauge group has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. Yet, it remains quite a mystery compared to a much better understood theory with compact gauge group. In this talk, I will present several methods that allow to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We shall explore various aspects of these invariants and present an example where we compute them explicitly to very high loop order.

**Chris Herald (University of Nevada): An SU(3) Casson invariant for rational homology spheres**

In this talk I will describe a gauge theoretic formulation of an SU(3) Casson invariant for rational homology spheres, defined in joint work with Hans Boden. This invariant extends the integer valued invariant defined in earlier work of Boden-H-Kirk for integral homology spheres. After perturbing to obtain a flat moduli space consisting of finitely many points, we begin with an algebraic count of the irreducible points, and then add suitable correction terms for the different reducible strata so that the result is perturbation independent.

**Kazuhiro Hikami (University of Tokyo): WRT invariants and modular forms**

Generalizing a result of Lawrence & Zagier, who showed a relationship between the SU(2) WRT invariant for Poincare homology sphere and modular form, we will point out a relationship between the Seifert manifolds and the Ramanujan mock theta functions. We will also study Arnold’s strange duality from this view point.

**Rinat Kashaev (University of Geneva): On rings associated with ideal triangulations of knot complements**

The combinatorics of a truncated tetrahedron can be encoded in a delta-groupoid (a groupoid with some extra structure). In this way, one can associate delta-groupoids to ideal triangulations of knot complements. On the other hand, there exist at least two functors from the category of rings to the category of delta-group(oid)s, admitting left-adjoints. As a result, to any ideal triangulation one can associate rings in at least two different ways. Certain localizations of these rings can be constructed from consideration of a restricted class of representations of the knot group into two-by-two matrices over arbitrary rings.

**Louis H. Kauffman (Math Dept, University of Illinois at Chicago): Khovanov Homology and the Potts Model**

The Potts model for plane graphs can be reformulated as a state summation that puts it in a two-parameter family with the Jones polynonmial. This makes it possible to compare the expression of the Potts partition function with the graded Euler characteristics that are natural for the Jones polynomial. This talk discusses such formulations for the Potts model and speculates about the possible uses of categorification homology in statistical mechanics.

**Mikhail Khovanov (Columbia University): Categorification of quantum groups**

I’ll go over the joint work with Aaron Lauda on categorification of positive halves of quantum groups and Beilinson-Lusztig-MacPherson idempotented version of sl(n).

**Paul Kirk (Indiana University): Non-abelian representations, homology 3-spheres, and knot concordance**

I’ll survey different approaches to knot concordance and homology cobordism that start with non abelian fundamental group representations, including the Fintushel-Stern/Furuta theorem and Reidemeister Torsion interpretations of Casson-gordon invariants. If time permits, I’ll discuss a few new results.

**Albrecht Klemm (University of Bonn): Chern-Simons Theory and Topological String theory on non-compact Calabi-Yau manifolds**

SU(N) Chern-Simons theory on S3 is equivalent to open topological string theory on T^*S3. Large duality relates this open topological string theory to closed topolopical string theory on O(-1)? O(-1)? P1. This setting is a topological realization of the large gauge theory string theory duality of ’t Hooft and Maldacena. Generalization it lead to a solution of topological string theory on non-compact Calabi-Yau toric manifolds. A key building block in this construction is the topological vertex. Mirror symmetry relates the above topological A-model string duality to a duality relating a matrix model to the topological B-model.

**Christine Lescop (University of Grenoble I): On the cube of the equivariant linking pairing for closed 3-manifolds of rank one**

As predicted by Witten, the asymptotics of the Chern-Simons theory lead to a series of invariants of framed 3--manifolds studied by many authors including Axelrod, Bott, Cattaneo, Kontsevich, Singer and Taubes.

The simplest non-trivial of these invariants is an invariant of homology 3-spheres N (equipped with an appropriate trivialisation) that may be written as the integral of the cube of a closed 2-form ? over a configuration space C2(N), where the cohomology class of ? represents the linking form of N. This invariant is associated to the ?--graph, G. Kuperberg and D. Thurston proved that it is the Casson invariant.

We shall present a similar invariant for closed oriented 3-manifolds with first Betti number one, in an equivariant setting.

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube ?(\KK) of this Blanchfield pairing with respect to a framed knot \KK that generates H1(M;\ZZ)/Torsion. The invariant ?(\KK) takes its values in a polynomial ring R over \QQ, it is equivalent to the two--loop part of the rational Kontsevich integral of the core of the surgery on \KK.

We determine the rational subvector space of R generated by the variations (?(\KK’)-?(\KK)) for pairs of framed knots as above, and we discuss the properties of the invariant ?(M) of M that is the class of ?(\KK) in the quotient of R by this subspace.

The invariant ?(M) is equivalent to a special case of invariants recently defined by Ohtsuki using the LMO invariant. It detects the connected sums with rational homology spheres with non trivial Casson-Walker invariant.

**Marcos Marino (University of Geneva): Chern-Simons theory, the 1/N expansion, and string theory**

In this talk I will review recent results on the 1/N expansion of Chern-Simons theory with classical gauge groups. This approach was first proposed by ’t Hooft in order to understand QCD, and has played a central role in many recent developments in theoretical physics. Many ideas related to the 1/N expansion which are difficult to implement in other gauge theories can be tested in detail in Chern-Simons theory, like for example analyticity of the amplitudes. Most importantly, it is by now clear that Chern-Simons theory on various three-manifolds is equivalent to a string theory. This leads to a string theoretic interpretation of the colored HOMFLY and Kauffman polynomials which "explains" many of their properties and leads to surprising predictions.

**Gregor Masbaum (Université Paris 7): Integral structures in TQFT and the mapping class group**

The mapping class group representations associated to the Witten-Reshetikhin-Turaev SO(3)-TQFTs at odd primes preserve a natural lattice defined over a ring of cyclotomic integers. This integral structure can be used to approximate the representations at a fixed prime p by representations into finite groups. I will also show how to take an Ohtsuki-style limit of the representations as p goes to infinity. This is expected to be related to Witten’s asymptotic expansion conjecture.

**Catherine Meusburger (University of Hamburg): Getting physics from 3d gravity: What does an observer in 3d gravity see?**

3d gravity has a direct but subtle relation to Chern-Simons theory and provides an interesting physical application by relating its quantisation to quantum gravity. However, it has been difficult to give a clear physical interpretation of its observables in terms of geometrical quantities measured by an observer.

We adress this issue for 3d gravity with vanishing cosmological constant. By considering observers, who probe the geometry of the spacetime by emitting lightrays that return to them at a later time, we obtain realistic quantities that could be measured by observers - the eigentime elapsed between the emission and the return of the lightray, the directions associated with returning lightrays and the redshift of the lightray at its return. We give explicit expressions for these quantities as functions on the phase space and relate them to the fundamental observables of the theory (Wilson loops). We discuss their physical interpretation and show how an observer can reconstruct the full geometry of the spacetime from these measurements. Ref: Class.Quant.Grav.26: 055006, 2009, arXiv:0811.4155v2 [gr-qc]

**Aleksandar Mikovic (Lisbon University): Invariants of spin networks embedded in 3-manifolds**

We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev's definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.

**Hitoshi Murakami (Tokyo Institute of Technology): SL(2;C)-representations and asymptotic behaviors of the colored Jones polynomial of a knot**

As a generalization of the volume conjecture, I will talk about a possible relation between asymptotic behaviors of the colored Jones polynomial of a knot and SL(2;C)-representations of its fundamental group.

**Robert Penner (USC and Aarhus University): Fatgraphs and finite type invariants**

Recent joint work with Andersen, Bene, and Meilhan provides a new link between 2d geometry and 3d quantum topology. Specifically, we introduce a finite type invariant of suitable 3-manifolds which is universal for homology cylinders over a surface with boundary. This new invariant depends upon a choice of fatgraph spine for the surface, and representations of its Ptolemy groupoid in the automorphism group of certain Jacobi diagrams evolve from this dependence. Applications include a TQFT-type expression for the Le-Murakami-Ohtsuki invariant and a canonical cocyle, which originated in joint work with Morita, representing the first Johnson homomorphism.

**Ambar Sengupta (Louisiana State University): Functional Integrals in Low-Dimensional Gauge Theories**

An overview of some rigorous formulations of Chern-Simons and 2-dimensional Yang-Mills functional integrals will be described. The Chern-Simons functional integral appears as an infinite-dimensional distribution while the Yang-Mills measure in 2 dimensions is a genuine measure. Methods of stochastic analysis underlie the rigorous formulation of such infinite-dimensional integrals. Open questions and difficulties with Chern-Simons integrals for Wilson loop variables will be discussed.

**George Thompson (International Centre for Theoretical Physics ICTP): Chern-Simons Theory On Seifert 3-Manifolds**

We show that the Chern-Simons path integral on a Seifert manifold can be expressed as a path integral of Yang-Mills type on the underlying Riemann surface. The resulting path integral has a very suggestive form which appears to be related to the non-Abelian localization of Beasley and Witten.

**Jonathan Weitsman (Northeastern University): Fermionization and Convergent Perturbation Expansions in Chern-Simons Gauge Theory**

We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to conjecture that Chern-Simons gauge theory also gives rise to convergent perturbation expansions, which would give a mathematically well-defined construction of the theory. A similar method when applied to Yang-Mills theory in four dimensions gives a polynomial decay conjecture for correlations of fields

**Edward Witten (Institute for Advanced Study): Branes and Quantization**