8 research outputs found
Wall-crossing, open BPS counting and matrix models
We consider wall-crossing phenomena associated to the counting of D2-branes
attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both
from M-theory and matrix model perspective. Firstly, from M-theory viewpoint,
we review that open BPS generating functions in various chambers are given by a
restriction of the modulus square of the open topological string partition
functions. Secondly, we show that these BPS generating functions can be
identified with integrands of matrix models, which naturally arise in the free
fermion formulation of corresponding crystal models. A parameter specifying a
choice of an open BPS chamber has a natural, geometric interpretation in the
crystal model. These results extend previously known relations between open
topological string amplitudes and matrix models to include chamber dependence.Comment: 25 pages, 8 figures, published versio
A & B model approaches to surface operators and Toda theories
It has recently been argued by Alday et al that the inclusion of surface
operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions
of certain degenerate operators in the dual Liouville theory. So far only the
insertion of a single surface operator has been treated (in a semi-classical
limit). In this paper we study and generalise this proposal. Our approach
relies on the use of topological string theory techniques. On the B-model side
we show that the effects of multiple surface operator insertions in 4d N=2
gauge theories can be calculated using the B-model topological recursion
method, valid beyond the semi-classical limit. On the mirror A-model side we
find by explicit computations that the 5d lift of the SU(N) gauge theory
partition function in the presence of (one or many) surface operators is equal
to an A-model topological string partition function with the insertion of (one
or many) toric branes. This is in agreement with an earlier proposal by Gukov.
Our A-model results were motivated by and agree with what one obtains by
combining the AGT conjecture with the dual interpretation in terms of
degenerate operators. The topological string theory approach also opens up new
possibilities in the study of 2d Toda field theories.Comment: 43 pages. v2: Added references, including a reference to unpublished
work by S.Gukov; minor changes and clarifications
A-polynomial, B-model, and Quantization
Exact solution to many problems in mathematical physics and quantum field
theory often can be expressed in terms of an algebraic curve equipped with a
meromorphic differential. Typically, the geometry of the curve can be seen most
clearly in a suitable semi-classical limit, as , and becomes
non-commutative or "quantum" away from this limit. For a classical curve
defined by the zero locus of a polynomial , we provide a construction
of its non-commutative counterpart using the
technique of the topological recursion. This leads to a powerful and systematic
algorithm for computing that, surprisingly, turns out to be much
simpler than any of the existent methods. In particular, as a bonus feature of
our approach comes a curious observation that, for all curves that come from
knots or topological strings, their non-commutative counterparts can be
determined just from the first few steps of the topological recursion. We also
propose a K-theory criterion for a curve to be "quantizable," and then apply
our construction to many examples that come from applications to knots,
strings, instantons, and random matrices.Comment: 58 pages, 5 figures, minor modifications, references adde
Challenges of beta-deformation
A brief review of problems, arising in the study of the beta-deformation,
also known as "refinement", which appears as a central difficult element in a
number of related modern subjects: beta \neq 1 is responsible for deviation
from free fermions in 2d conformal theories, from symmetric omega-backgrounds
with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from
eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in
Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras
etc. The main attention is paid to the context of AGT relation and its possible
generalizations.Comment: 20 page
A Matrix model for plane partitions
We construct a matrix model equivalent (exactly, not asymptotically), to the
random plane partition model, with almost arbitrary boundary conditions.
Equivalently, it is also a random matrix model for a TASEP-like process with
arbitrary boundary conditions. Using the known solution of matrix models, this
method allows to find the large size asymptotic expansion of plane partitions,
to ALL orders. It also allows to describe several universal regimes.Comment: Latex, 41 figures. Misprints and corrections. Changing the term TASEP
to self avoiding particle porces