Schedule for: 18w5007 - Modular Forms and Quantum Knot Invariants

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A gaussian group is a Pontryagin self-dual locally compact abelian group together with a fixed gaussian exponential that is a symmetric second order character associated with a non-degenerate self-pairing. I will explain how a quantum dilogarithm can be used for construction of projective unitary representations of the mapping class groups of punctured surfaces of negative Euler characteristic.

Quantum link invariants lie at the intersection of hyperbolic geometry, 3- dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important quantum link invariant. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the colored Jones polynomial in view of the Slope Conjectures by Garoufalidis and Kalfagianni-Tran and the Coarse Volume Conjecture by Futer-Kalfagianni-Purcell, and we will explore the potential connection to the number-theoretic properties of the polynomial. In particular, I will indicate how my recent joint work with Garoufalidis and van der Veen on the Slope Conjectures for Montesinos knots is related to the existence of multiple tails of the polynomial for Montesinos knots.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Refined Chern-Simons invariants of torus knots can be defined by using modular matrices associated to Macdonald polynomials or DAHA, generalizing colored quantum invariants. The theory was originally defined in string theory, and conifold transition in string theory leads to a positivity conjecture of refined Chern-Simons invariants of torus knots. This conjecture connects refined Chern-Simons theory to enumerative geometry.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The slope conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjectureproposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. We prove the strong Slope Conjecture for a twisted, generalized Whitehead double of a knot $K$ whenever $K$ satisfy the strong Slope Conjecture and certain extra condition. This is joint work with Kenneth L. Baker and Kimihiko Motegi.

The Mock Theta Conjectures were identities stated by Ramanujan for his so called fifth order mock theta functions. Andrews and Garvan showed how two of these fifth order functions are related to rank differences mod 5. Hickerson was first to prove these identities and was also able to relate the three Ramanujan seventh order mock theta functions to rank differences mod 7. Based on work of Zwegers, Zagier observed that the two fifth order functions and the three seventh order functions are holomorphic parts of real analytic vector modular forms on $SL_2(Z)$.  Zagier gave an indication how these functions could be generalized.  We give details of these generalizations and show how Zagier's 11th order functions are related to rank differences mod 11.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

For a hyperbolic link in the 3-sphere, the hyperbolic volume of its complement is an interesting and well-studied geometric link invariant. Similarly, the determinant of a link is one of the oldest diagrammatic link invariant. In previous work we studied the asymptotic behavior of volume and determinant densities for alternating links, which led us to conjecture a surprisingly simple relationship between the volume and determinant of an alternating link, called the Vol-Det Conjecture. In this talk we outline an interesting method to prove the Vol-Det Conjecture for infinite families of alternating links using a variety of techniques from the theory of dimer models, Mahler measures of 2-variable polynomials and the hyperbolic geometry of link complements in the thickened torus.

We will discuss various ways to define and compute new q-series invariants that have integer powers and integer coefficients. After a quick review of the physical framework, we show how it explains and generalizes the observations of Lawrence-Zagier and Hikami et.al. to arbitrary 3-manifolds. If time permits, we will talk about a modular tensor category MTC[M3] responsible for the modularity properties of \hat Z_a (q). There are many unexpected and intriguing connections with various counting problems as well as with the works of Beliakova-Blanchet-Le and Garoufalidis-Le.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The fundamental group of a knot complement is called a knot group.  A way to present a knot groups is the Wirtinger presentation, which is given by a conjugation action at each crossing of the knot.  This presentation is also given by a conjugate quandle, which matches well to the Hopf algebra structure of the group ring of the knot group.  Here we introduce the braided conjugate quandle corresponding to the braided Hopf algebra, which is a deformation of a Hopf algebra.  A typical example of the braided Hopf algebra is the braided  SL(2)  introduced by S. Majid, and so it may give a q-deformation of a SL(2) representation of the knot group.  This is joint with Roland van der Veen.

This is a report on work-in-progress on higher depth quantum modular forms arising from sl_3 WRT invariants of the Poincare homology sphere, following the work of Bringmann.

In this talk, based on joint work with Bringmann, we introduce the notion of a quantum Jacobi form, marrying Zagier's notion of a quantum modular form with that of a Jacobi form. We also offer a number of two-variable combinatorial generating functions as first examples of quantum Jacobi forms, including certain rank generating functions studied by Bryson-Ono-Pitman-Rhoades, Hikami-Lovejoy, and Kim-Lim-Lovejoy. These combinatorial functions are also duals to partial theta functions studied by Ramanujan. Additionally, we show that all of these examples satisfy the stronger property that they exhibit mock Jacobi transformations in $\mathbb C \times \mathbb H$ as well as quantum Jacobi transformations in $\mathbb Q \times \mathbb Q$. Finally, we discuss applications of these quantum Jacobi properties which yield new, simple expressions for the aforementioned combinatorial generating functions as two-variable polynomials when evaluated at pairs of rational numbers, and yield similarly simple evaluations of certain Eichler integrals.

Physicists have long been arguing that gauge theories at large rank are related to topological string theories. As a concrete example, I will describe a correspondence between the colored HOMFLY-PT polynomials of knots and the motivic DT invariants of certain symmetric quivers, which was recently proposed by Kucharski-Reineke-Stosic-Sulkowski. I will outline a proof of this correspondence for 2-bridge knots and then speculate about how much of the HOMFLY-PT skein theory might carry over to the realm of DT quiver invariants. This is joint work with Marko Stosic.

Supported by numerical evidence, Chen and I conjectured that at the root of unity$\exp(2 \pi \sqrt{-1}/r)$, instead of the usually considered root $exp(\pi \sqrt{-1}/r)$, the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Witten's invariants (that grow polynomially by the Asymptotic Expansion Conjecture), which may indicate a different geometric interpretation of those invariants than the SU(2) Chern-Simons gauge theory. In this talk, I will introduce the conjecture and show further supporting evidences, including recent joint works with Detcherry-Kalfagianni.

The 3D-index of Dimofte-Gaiotto-Gukov is a collection of q-series with integer coefficients which is defined for 1-efficient ideal triangulations, and gives topological invariants of hyperbolic manifolds, in particular counts the number of genus 2 incompressible and Heegaard surfaces. We give an extension of the 3Dindex to a meromorphic function defined for all ideal triangulations, and invariant under all Pachner moves. Joint work with Rinat Kashaev.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

According to a conjecture of Andersen, Masbaum and Ueno, the Witten-Reshetikhin-Turaev quantum representations of mapping class groups send pseudo-Anosov mapping classes to infinite order elements, when the level is big enough. We relate this conjecture to a properties about the growth rate of Turaev-Viro invariants, and derive infinite families of pseudo-Anosov mapping classes that satisfy the conjecture, in all surfaces with n boundary components and genus g>n>=2. These families are obtained as monodromies of fibered links containing some specific sublinks.

The Turaev-Viro invariant is a 3-manifold invariant (TQFT) defined at roots of unity q using a handle decomposition. Following Frohman and Kania Bartoszynska we consider extending the variable q into the unit disk. We will discuss under what conditions one still gets an invariant and give some explicit examples of the ensuing q-series.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

An important problem in conformal field theory is to describe modular transformation properties of characters of representations of a vertex algebra. By now this is well-understand for "nice" vertex algebras (e.g. if the category of modules is semisimple). But most vertex algebras have non-semisimple category of representations so modular properties of their characters is difficult to formulate.  In my talk I'll focus on a family of W-algebras coming from certain extensions of affine W-algebras. Their irreducible characters have recently been proposed and studied. I'll explain two approaches to modular invariance and Verlinde formula for their modules, both based on iterated integrals of modular forms. In the first approach, modular invariance is formulated with the help of regularized variables. In the second approach, we replace the characters with better behaved non-holomorphic integrals. This approach is intimately linked to mock and quantum modular forms. We shall see how the two approaches can be understood from a unified viewpoint.

This talk will discuss the patterns and stability in the coefficients of colored Jones polynomial. While much work has been done looking at the leading sequence of coefficients, we will discuss work moving towards understanding the middle coefficients. This will include small steps like looking at the second N coefficients of the Nth colored Jones polynomial and larger steps like looking at the growth rate of the coefficients and looking at the patterns present in the coefficients under various re-normalizations.

A well-known result is that modules of a rational vertex algebra form a modular tensor category and that the modular group action on graded traces coincides with the categorical one. Prime examples are affine vertex algebras at positive integer level.  I would like to explain the state of the art for affine vertex algebras at admissible level and our knowledge is mainly restricted to the case of sl(2). From the character point of view three types of traces arise: vector-valued modular forms, meromorphic Jacobi forms and formal distributions. There are also three types of categories one can associate to the affine vertex algebra and categorical action of the modular group seems to coincide with the one on characters. 

Representation theory of affine Lie algebras and more generally, vertex operator algebras, leads to interesting q-series identities. The q-series related to VOAs often exhibit deep modularity properties. Some of these q-series may also be of interest to the knot theorists. In this talk, I'll present some new q-series conjectures obtained jointly with Matthew Russell, and with Debajyoti Nandi and Matthew Russell

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.