139 research outputs found
Hilbert Schemes, Separated Variables, and D-Branes
We explain Sklyanin's separation of variables in geometrical terms and
construct it for Hitchin and Mukai integrable systems. We construct Hilbert
schemes of points on for \Sigma = {\IC}, {\IC}^{*} or elliptic
curve, and on and show that their complex deformations
are integrable systems of Calogero-Sutherland-Moser type. We present the
hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent
bundles to the higher genus curves, utilizing the results of Hurtubise,
Kronheimer and Nakajima. Finally we discuss the connections to physics of
-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte
Duality in Integrable Systems and Gauge Theories
We discuss various dualities, relating integrable systems and show that these
dualities are explained in the framework of Hamiltonian and Poisson reductions.
The dualities we study shed some light on the known integrable systems as well
as allow to construct new ones, double elliptic among them. We also discuss
applications to the (supersymmetric) gauge theories in various dimensions.Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHE
On Microscopic Origin of Integrability in Seiberg-Witten Theory
We discuss microscopic origin of integrability in Seiberg-Witten theory,
following mostly the results of hep-th/0612019, as well as present their
certain extension and consider several explicit examples. In particular, we
discuss in more detail the theory with the only switched on higher perturbation
in the ultraviolet, where extra explicit formulas are obtained using
bosonization and elliptic uniformization of the spectral curve.Comment: 24 pages, 1 figure, LaTeX, based on the talks at 'Geometry and
Integrability in Mathematical Physics', Moscow, May 2006; 'Quarks-2006',
Repino, May 2006; Twente conference on Lie groups, December 2006 and
'Classical and Quantum Integrable Models', Dubna, January 200
Darboux coordinates, Yang-Yang functional, and gauge theory
The moduli space of SL(2) flat connections on a punctured Riemann surface
with the fixed conjugacy classes of the monodromies around the punctures is
endowed with a system of holomorphic Darboux coordinates, in which the
generating function of the variety of SL(2)-opers is identified with the
universal part of the effective twisted superpotential of the corresponding
four dimensional N=2 supersymmetric theory subject to the two-dimensional
Omega-deformation. This allows to give a definition of the Yang-Yang
functionals for the quantum Hitchin system in terms of the classical geometry
of the moduli space of local systems for the dual gauge group, and connect it
to the instanton counting of the four dimensional gauge theories, in the rank
one case.Comment: 25 pages, 11 figures, v1. in the proceedings of Cargese conference
"String Theory: Formal Developments and Applications" (Jun 21-Jul 3, 2010);
reported also at six other conferences in 2010, v2. references correcte
Dualities in integrable systems and N=2 theories
We discuss dualities of the integrable dynamics behind the exact solution to
the N=2 SUSY YM theory. It is shown that T duality in the string theory is
related to the separation of variables procedure in dynamical system. We argue
that there are analogues of S duality as well as 3d mirror symmetry in the
many-body systems of Hitchin type governing low-energy effective actions.Comment: 16 pages, Latex, Talk given at QFTHEP-99, Moscow, May 27-June
Relating Gauge Theories via Gauge/Bethe Correspondence
In this note, we use techniques from integrable systems to study relations
between gauge theories. The Gauge/Bethe correspondence, introduced by Nekrasov
and Shatashvili, identifies the supersymmetric ground states of an N=(2,2)
supersymmetric gauge theory in two dimensions with the Bethe states of a
quantum integrable system. We make use of this correspondence to relate three
different quiver gauge theories which correspond to three different
formulations of the Bethe equations of an integrable spin chain called the tJ
model.Comment: 30 pages, published in JHEP. LaTeX problem correcte
Angular Momentum and Gravimagnetization of the SYM vacuum
In this note we discuss the gravimagnetization of the SYM vacuum
in the -background. It is argued that the Seiberg-Witten prepotential
is related to the vacuum density of the angular momentum in the Euclidean
space. The possible role of the dyonic instantons as the microscopic angular
momentum carriers which could yield the spontaneous vacuum gravimagnetization
is conjectured. We interpret the dyonic instanton as a kind of the Euclidean
bounce in similar to one responsible for the Schwinger pair creation. The
induced angular momentum in is also briefly considered in the dual
Liouville formulation of theory via AGT relation.Comment: 20 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
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