18 research outputs found
BPS Monopole Equation in Omega-background
We study deformed supersymmetries in N=2 super Yang-Mills theory in the
Omega-backgrounds characterized by two complex parameters . When one of the -parameters vanishes, the theory has
extended supersymmetries. We compute the central charge of the algebra and
obtain the deformed BPS monopole equation. We examine supersymmetries preserved
by the equation.Comment: 14 pages, typos corrected, published version in JHE
Surface Operator, Bubbling Calabi-Yau and AGT Relation
Surface operators in N=2 four-dimensional gauge theories are interesting
half-BPS objects. These operators inherit the connection of gauge theory with
the Liouville conformal field theory, which was discovered by Alday, Gaiotto
and Tachikawa. Moreover it has been proposed that toric branes in the A-model
topological strings lead to surface operators via the geometric engineering. We
analyze the surface operators by making good use of topological string theory.
Starting from this point of view, we propose that the wave-function behavior of
the topological open string amplitudes geometrically engineers the surface
operator partition functions and the Gaiotto curves of corresponding gauge
theories. We then study a peculiar feature that the surface operator
corresponds to the insertion of the degenerate fields in the conformal field
theory side. We show that this aspect can be realized as the geometric
transition in topological string theory, and the insertion of a surface
operator leads to the bubbling of the toric Calabi-Yau geometry.Comment: 36 pages, 14 figures. v2: minor changes and typos correcte
Generalized matrix models and AGT correspondence at all genera
We study generalized matrix models corresponding to n-point Virasoro
conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT
correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge
theories with generalized quiver diagrams. We obtain the generalized matrix
models from the perturbative evaluation of the Liouville correlation functions
and verify the consistency of the description with respect to degenerations of
the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2
gauge theory as the spectral curve of the generalized matrix model, thus
providing a check of AGT correspondence at all genera.Comment: 19 pages; v2: version to appear in JHE
On "Dotsenko-Fateev" representation of the toric conformal blocks
We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the
original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal
blocks in the same sense that the spherical blocks are given by the integral
representation of arXiv:1001.0563 with a peculiar choice of open integration
contours for screening insertions. In other words, we provide some evidence
that the toric conformal blocks are reproduced by appropriate beta-ensembles
not only in the large-N limit, but also at finite N. The check is explicitly
performed at the first two levels for the 1-point toric functions.
Generalizations to higher genera are briefly discussed.Comment: 10 page
Quantization of Integrable Systems and a 2d/4d Duality
We present a new duality between the F-terms of supersymmetric field theories
defined in two- and four-dimensions respectively. The duality relates N=2
supersymmetric gauge theories in four dimensions, deformed by an
Omega-background in one plane, to N=(2,2) gauged linear sigma-models in two
dimensions. On the four dimensional side, our main example is N=2 SQCD with
gauge group SU(L) and 2L fundamental flavours. Using ideas of Nekrasov and
Shatashvili, we argue that the Coulomb branch of this theory provides a
quantization of the classical Heisenberg SL(2) spin chain. Agreement with the
standard quantization via the Algebraic Bethe Ansatz implies the existence of
an isomorphism between the chiral ring of the 4d theory and that of a certain
two-dimensional theory. The latter can be understood as the worldvolume theory
on a surface operator/vortex string probing the Higgs branch of the same 4d
theory. We check the proposed duality by explicit calculation at low orders in
the instanton expansion. One striking consequence is that the Seiberg-Witten
solution of the 4d theory is captured by a one-loop computation in two
dimensions. The duality also has interesting connections with the AGT
conjecture, matrix models and topological string theory where it corresponds to
a refined version of the geometric transition.Comment: 51 pages, 7 figures. Additional comments, minor improvements and
references adde
The Liouville side of the Vortex
We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a byproduct we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks
A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
We give an explicit differential equation which is expected to determine the
instanton partition function in the presence of the full surface operator in
N=2 SU(N) gauge theory. The differential equation arises as a quantization of a
certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and
Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion,
appendix and references adde
Vertices, Vortices & Interacting Surface Operators
We show that the vortex moduli space in non-abelian supersymmetric N=(2,2)
gauge theories on the two dimensional plane with adjoint and anti-fundamental
matter can be described as an holomorphic submanifold of the instanton moduli
space in four dimensions. The vortex partition functions for these theories are
computed via equivariant localization. We show that these coincide with the
field theory limit of the topological vertex on the strip with boundary
conditions corresponding to column diagrams. Moreover, we resum the field
theory limit of the vertex partition functions in terms of generalized
hypergeometric functions formulating their AGT dual description as interacting
surface operators of simple type. Analogously we resum the topological open
string amplitudes in terms of q-deformed generalized hypergeometric functions
proving that they satisfy appropriate finite difference equations.Comment: 22 pages, 4 figures; v.2 refs. and comments added; v.3 further
comments and typo
Deformed planar topological open string amplitudes on Seiberg-Witten curve
We study refined B-model via the beta ensemble of matrix models. Especially,
for four dimensional N=2 SU(2) supersymmetric gauge theories with N_f=0,1 and 2
fundamental flavors, we discuss the correspondence between deformed disk
amplitudes on each Seiberg-Witten curve and the Nekrasov-Shatashvili limit of
the corresponding irregular one point block of a degenerate operator via the
AGT correspondence. We also discuss the relation between deformed annulus
amplitudes and the irregular two point block of the degenerate operator, and
check a desired agreement for N_f=0 and 1 cases.Comment: 27 pages. v2: minor changes and references added. v3: minor
corrections and one reference 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