Refined invariants in geometry, topology and string theory (13w5134)

Arriving in Banff, Alberta Sunday, June 2 and departing Friday June 7, 2013

Organizers

(University of British Columbia)

Duiliu Emanuel Diaconescu (University of Alberta)

(EPF Lausanne)

(Oxford University)

Objectives

Our proposed workshop would aim to survey approaches to, and results on,
the following motivating questions.

{it Refined topological strings}.
The topological vertex formalism for topological string amplitudes on
toric Calabi--Yau threefolds is an example
of the Donaldson--Thomas/Gromov--Witten
correspondence. Three partition Hodge integrals on the moduli space of
curves are related to a certain weighted count of three dimensional
partitions; Iqbal--Kozcaz--Vafa's refined topological vertex builds on this
combinatorial model. As pointed out by Dijkgraaf and Vafa, this
formalism is related to certain deformations of matrix models. Study these
correspondences in more depth and understand consequences for mirror symmetry
and the topological B-model.

{it Quiver theories and wall crossing}.
Quiver theories provide a useful testing ground where refined DT theory can
be rigurously defined and in many cases computed. Study wall crossing of
refined invariants, and relations to Hall algebras and geometric examples.

{it Connections between refined Chern-Simons theory, refined GV and
DT invariants of local curves and Nekrasov instanton sums}.
Study the connections between the aformentioned refined generating series
$F_{GV}$, $Z_{DT}$, $Z_{CS}$ and $Z_{gauge}$ associated to local curves,
coming respectively from Gopakumar--Vafa theory, DT theory, Chern--Simons
theory and gauge theory. Find mathematical connections between the
constructions.

{it Refined knot invariants and Hilbert schemes}.
Study the conjectural connection discovered by Oblomkov and Shende between
enumerative imvariants of Hilbert schemes of points of plane curve
singularities and the HOMFLY polynomials of their links.
Find a physical interpretation of this relation, and explore its
mathematical consequences.

{it Cohomology of Hitchin fibrations}. Understand and explicitly describe
the cohomology of Hitchin fibrations, together with the natural perverse
filtration. Compute generating series of perverse Hodge numbers in terms of
Macdonald polynomials and orbital integrals.

{it Refinement and categorification}. Study the relation between numerical
and motivic refinements of generating series and categorification,
lifting properties of generating series of invariants to actions of
symmetry algebras on cohomology spaces (and beyond). In particular, find
a DAHA action the cohomology of the Hitchin fibration. Explain the ubiquitous
appearance of Macdonald polynomials in the local curve story using this
action.

As it should be clear from our Overview and list of Objectives, the proposed
topics touch on a diverse variety of fields. Our Workshop would
bring together specialists with a common interest but working in different
areas including topological strings, enumerative invariants, moduli of
bundles over curves and geometric representation theory. The particular
subject chosen is extremely topical; several of the above-mentioned works
(notably those of Diaconescu et al, Aganagic et al, Migliorini et al, etc)
have appeared during the last few months, and in the time between writing
the proposal and the workshop itself, we expect further major developments
(which we would certainly aim to incorporate).