DSpace Academic/Research Unit: School of MathematicsSchool of Mathematicshttp://hdl.handle.net/2262/12014-04-18T06:22:32Z2014-04-18T06:22:32ZOn the central Haagerup tensor product and completely bounded mappings of a C*-algebra,TIMONEY, RICHARDhttp://hdl.handle.net/2262/684622014-04-10T14:50:01Z2005-01-01T00:00:00ZTitle: On the central Haagerup tensor product and completely bounded mappings of a C*-algebra,
Author: TIMONEY, RICHARD
Abstract: We consider the natural contractive map from the central Haagerup tensor product of a unital C*-algebra A with itself to the space of completely bounded maps CB ( A ) on A. We establish the necessity of the known sufficient condition for isometry of the map, namely that all Glimm ideals of A are primal. However, when the map is restricted to tensors with length bounded by a fixed quantity, a weaker necessary and sufficient condition is established.
Description: PUBLISHED2005-01-01T00:00:00ZQuantization of Integrable Systems and Four Dimensional Gauge TheoriesSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684492014-04-09T14:03:31Z2010-01-01T00:00:00ZTitle: Quantization of Integrable Systems and Four Dimensional Gauge Theories
Author: SHATASHVILI, SAMSON
Abstract: We study four dimensional
N
= 2 supersymmetric gauge theory in the Ω-
background with the two dimensional
N
= 2 super-Poincare invariance. We explain
how this gauge theory provides the quantization of the class
ical integrable system
underlying the moduli space of vacua of the ordinary four dim
ensional
N
= 2 theory.
The
ε
-parameter of the Ω-background is identified with the Planck
constant, the
twisted chiral ring maps to quantum Hamiltonians, the super
symmetric vacua are
identified with Bethe states of quantum integrable systems.
This four dimensional
gauge theory in its low energy description has two dimension
al twisted superpotential
which becomes the Yang-Yang function of the integrable syst
em. We present the
thermodynamic-Bethe-ansatz like formulae for these funct
ions and for the spectra
of commuting Hamiltonians following the direct computatio
n in gauge theory. The
general construction is illustrated at the examples of the m
any-body systems, such
as the periodic Toda chain, the elliptic Calogero-Moser sys
tem, and their relativistic
versions, for which we present a complete characterization
of the
L
2
-spectrum. We
very briefly discuss the quantization of Hitchin system
Description: PUBLISHED2010-01-01T00:00:00ZQuantum integrability and supersymmetric vacuaSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684482014-04-09T13:45:03Z2009-01-01T00:00:00ZTitle: Quantum integrability and supersymmetric vacua
Author: SHATASHVILI, SAMSON
Abstract: Supersymmetric vacua of two dimensional
N
= 4 gauge theories with matter, softly broken by the
twisted masses down to
N
= 2, are shown to be in one-to-one correspondence with the eig
enstates
of integrable spin chain Hamiltonians. Examples include: t
he Heisenberg
SU
(2)
XXX
spin chain
which is mapped to the two dimensional
U
(
N
) theory with fundamental hypermultiplets, the
XXZ
spin chain which is mapped to the analogous three dimensiona
l super-Yang-Mills theory
compactified on a circle, the
XYZ
spin chain and eight-vertex model which are related to the fo
ur
dimensional theory compactified on
T
2
. A consequence of our correspondence is the isomorphism
of the quantum cohomology ring of various quiver varieties,
such as
T
∗
Gr(
N,L
) and the ring
of quantum integrals of motion of various spin chains. The co
rrespondence extends to any spin
group, representations, boundary conditions, and inhomog
eneity, it includes Sinh-Gordon and
non-linear Schr ̈odinger models as well as the dynamical spi
n chains like Hubbard model. We give
the gauge-theoretic interpretation of Drinfeld polynomia
ls and Baxter operators. The two-sphere
compactifications of the four dimensional
N
= 2 theories lead to the instanton corrected Bethe
equations. We suggest the Yangian, quantum affine, and ellipt
ic algebras are a completely novel
kind of symmetry of the (collections of the) interacting qua
ntum field theories
Description: PUBLISHED2009-01-01T00:00:00ZSupersymmetric vacua and Bethe ansatzSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684472014-04-09T11:56:11Z2009-01-01T00:00:00ZTitle: Supersymmetric vacua and Bethe ansatz
Author: SHATASHVILI, SAMSON
Abstract: This note is a short announcement of some results of a longer p
aper where the supersymmet-
ric vacua of two dimensional
N
= 4 gauge theories with matter, softly broken by the twisted
masses down to
N
= 2, are shown to be in one-to-one correspondence with the eig
enstates of
integrable spin chain Hamiltonians. The Heisenberg
SU
(2)
XXX
spin chain is mapped to the
two dimensional
U
(
N
) theory with fundamental hypermultiplets, the
XXZ
spin chain is mapped
to the analogous three dimensional super-Yang-Mills theor
y compactified on a circle, the
XY Z
spin chain and eight-vertex model are related to the four dim
ensional theory compactified on
T
2
.
The correspondence extends to any spin group, representati
ons, boundary conditions, and inho-
mogeneity, it includes Sinh-Gordon and non-linear Schr ̈od
inger models as well as the dynamical
spin chains such as the Hubbard model. Compactifications of f
our dimensional
N
= 2 theories
on a two-sphere lead to the instanton-corrected Bethe equat
ions. We propose a completely novel
way for the Yangian, quantum affine, and elliptic algebras to a
ct as a symmetry of a union of
quantum field theories
Description: PUBLISHED2009-01-01T00:00:00ZTwo-dimensional gauge theories and quantum integrable systemsSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684462014-04-09T11:27:29Z2008-01-01T00:00:00ZTitle: Two-dimensional gauge theories and quantum integrable systems
Author: SHATASHVILI, SAMSON
Abstract: In this paper the relation between 2d topological gauge theo
ries and Bethe Ansatz
equations is reviewed.
1
In addition we present some new results and clarifications. W
e
hope the relations discussed here are particular examples o
f more general relations between
quantum topological fields theories in dimensions
d
≤
4 and quantum integrable systems
Description: PUBLISHED2008-01-01T00:00:00ZHiggs Bundles, Gauge Theories and Quantum GroupsSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684432014-04-09T02:15:02Z2008-01-01T00:00:00ZTitle: Higgs Bundles, Gauge Theories and Quantum Groups
Author: SHATASHVILI, SAMSON
Abstract: The appearance of the Bethe Ansatz equation for the Nonlinea
r Schr ̈odinger equa-
tion in the equivariant integration over the moduli space of
Higgs bundles is revisited.
We argue that the wave functions of the corresponding two-di
mensional topological
U
(
N
)
gauge theory reproduce quantum wave functions of the Nonlin
ear Schr ̈odinger equation in
the
N
-particle sector. This implies the full equivalence betwee
n the above gauge theory
and the
N
-particle sub-sector of the quantum theory of Nonlinear Sch
r ̈odinger equation.
This also implies the explicit correspondence between the g
auge theory and the represen-
tation theory of degenerate double affine Hecke algebra. We pr
opose similar construction
based on the
G/G
gauged WZW model leading to the representation theory of the
double
affine Hecke algebra. The relation with the Nahm transform and
the geometric Langlands
correspondence is briefly discussed
Description: PUBLISHED2008-01-01T00:00:00ZTowards Integrability of Topological Strings I: Three Froms on Calabi-SHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684422014-04-08T13:12:22Z2004-01-01T00:00:00ZTitle: Towards Integrability of Topological Strings I: Three Froms on Calabi-
Author: SHATASHVILI, SAMSON
Abstract: The precise relation between Kodaira-Spencer path integra
l and a particular wave
function in seven dimensional quadratic field theory is esta
blished. The special properties
of three-forms in 6d, as well as Hitchin’s action functional
, play an important role. The
latter defines a quantum field theory similar to Polyakov’s fo
rmulation of 2d gravity; the
curious analogy with world-sheet action of bosonic string i
s also pointed out.
Description: PUBLISHED2004-01-01T00:00:00ZStringy Higgs Mechanizm and Fate of Open StringsSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684412014-04-08T12:48:59Z2001-01-01T00:00:00ZTitle: Stringy Higgs Mechanizm and Fate of Open Strings
Author: SHATASHVILI, SAMSON
Abstract: We propose a refinement of the physical picture describing di
fferent vacua in bosonic
string theory. The vacua with closed strings and open string
s are connected by the string
field theory version of the Higgs mechanism, generalizing th
e Higgs mechanism of an abelian
gauge field interacting with a complex scalar. In accordance
with Sen’s conjecture, the
condensation of the tachyon is an essential part of the story
. We consider this phenomenon
from the point of view of both a world-sheet sigma-model and t
he target-space theory. In
the Appendix the relevant remarks regarding the choice of th
e coordinates in the back-
ground independent open string field theory are given
Description: PUBLISHED2001-01-01T00:00:00ZQuantum geometry and quiver gauge theoriesSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684402014-04-08T12:43:04Z2013-01-01T00:00:00ZTitle: Quantum geometry and quiver gauge theories
Author: SHATASHVILI, SAMSON
Abstract: We study macroscopically two dimensional
N
= (2
,
2)
supersymmetric
gauge theories constructed by compactifying the quiver gau
ge theories with eight su-
percharges on a product
T
d
×
R
2
ǫ
of a
d
-dimensional torus and a two dimensional
cigar with
Ω
-deformation. We compute the universal part of the effective
twisted
superpotential. In doing so we establish the correspondenc
e between the gauge the-
ories, quantization of the moduli spaces of instantons on
R
2
−
d
×
T
2+
d
and singular
monopoles on
R
2
−
d
×
T
1+
d
, for
d
= 0
,
1
,
2
, and the Yangian
Y
ǫ
(
g
Γ
)
, quantum affine
algebra
U
aff
q
(
g
Γ
)
, or the quantum elliptic algebra
U
ell
q,p
(
g
Γ
)
associated to Kac-Moody
algebra
g
Γ
for quiver
Γ
Description: PUBLISHED2013-01-01T00:00:00ZFactorization Conjecture and Open/Closed String CorrespondenceSHATASHVILI, SAMSONhttp://hdl.handle.net/2262/684392014-04-08T12:39:33Z2005-01-01T00:00:00ZTitle: Factorization Conjecture and Open/Closed String Correspondence
Author: SHATASHVILI, SAMSON
Abstract: We present evidence for the factorization of the world-shee
t path integrals for 2d con-
formal field theories on the disk into bulk and boundary contr
ibutions. This factorization
is then used to reinterpret a shift in closed string backgrou
nds in terms of boundary de-
formations in background independent open string field theo
ry. We give a proof of the
factorization conjecture in the cases where the background
is represented by WZW and
related models
Description: PUBLISHED2005-01-01T00:00:00Z