15 research outputs found
Unified genus-1 potential and parametric P/NP relation
We study a parametric deformation of the unified genus-1 anharmonic potential
and derive a parametric form of perturbative/non-perturbative (P/NP) relation,
applicable across all parameter values. We explicitly demonstrate that the
perturbative expansion around the perturbative saddle is sufficient to generate
all the nonperturbative information in these systems. Our results confirm the
known results in the literature, where the cubic and quartic anharmonic
potentials are reproduced under extreme parameter values, and go beyond these
known results by developing the nonperturbative function of real and complex
instantons solely from perturbative data
Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
We study the partition function of the compactified 5D U(1) gauge theory (in
the Omega-background) with a single adjoint hypermultiplet, calculated using
the refined topological vertex. We show that this partition function is an
example a periodic Schur process and is a refinement of the generating function
of cylindric plane partitions. The size of the cylinder is given by the mass of
adjoint hypermultiplet and the parameters of the Omega-background. We also show
that this partition function can be written as a trace of operators which are
generalizations of vertex operators studied by Carlsson and Okounkov. In the
last part of the paper we describe a way to obtain (q,t) identities using the
refined topological vertex.Comment: 40 Page
Link Homologies and the Refined Topological Vertex
We establish a direct map between refined topological vertex and sl(N)
homological invariants of the of Hopf link, which include Khovanov-Rozansky
homology as a special case. This relation provides an exact answer for
homological invariants of the of Hopf link, whose components are colored by
arbitrary representations of sl(N). At present, the mathematical formulation of
such homological invariants is available only for the fundamental
representation (the Khovanov-Rozansky theory) and the relation with the refined
topological vertex should be useful for categorifying quantum group invariants
associated with other representations (R_1, R_2). Our result is a first direct
verification of a series of conjectures which identifies link homologies with
the Hilbert space of BPS states in the presence of branes, where the physical
interpretation of gradings is in terms of charges of the branes ending on
Lagrangian branes.Comment: 38 pages, 5 figure
The Refined Topological Vertex
We define a refined topological vertex which depends in addition on a
parameter, which physically corresponds to extending the self-dual graviphoton
field strength to a more general configuration. Using this refined topological
vertex we compute, using geometric engineering, a two-parameter (equivariant)
instanton expansion of gauge theories which reproduce the results of Nekrasov.
The refined vertex is also expected to be related to Khovanov knot invariants.Comment: 70 Pages, 23 Figure
Refined Hopf Link Revisited
We establish a relation between the refined Hopf link invariant and the
S-matrix of the refined Chern-Simons theory. We show that the refined open
string partition function corresponding to the Hopf link, calculated using the
refined topological vertex, when expressed in the basis of Macdonald
polynomials gives the S-matrix of the refined Chern-Simons theory.Comment: 17 page
Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
Recently Alday and Tachikawa proposed a relation between conformal blocks in
a two-dimensional theory with affine sl(2) symmetry and instanton partition
functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the
presence of a certain surface operator. In this paper we extend this proposal
to a relation between conformal blocks in theories with affine sl(N) symmetry
and instanton partition functions in conformal N=2 SU(N) quiver gauge theories
in the presence of a surface operator. We also discuss the extension to
non-conformal N=2 SU(N) theories.Comment: 40 pages. v2: minor changes and clarification
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