3,604 research outputs found
q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
We characterise the quantum group gauge symmetries underlying q-deformations
of two-dimensional Yang-Mills theory by studying their relationships with the
matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge
theories, together with their refinements and supersymmetric extensions. We
develop uniqueness results for quantum deformations and refinements of gauge
theories in two dimensions, and describe several potential analytic and
geometric realisations of them. We reconstruct standard q-deformed Yang-Mills
amplitudes via gluing rules in the representation category of the quantum group
associated to the gauge group, whose numerical invariants are the usual
characters in the Grothendieck group of the category. We apply this formalism
to compute refinements of q-deformed amplitudes in terms of generalised
characters, and relate them to refined Chern-Simons matrix models and
generalized unitary matrix integrals in the quantum beta-ensemble which compute
refined topological string amplitudes. We also describe applications of our
results to gauge theories in five and seven dimensions, and to the dual
superconformal field theories in four dimensions which descend from the N=(2,0)
six-dimensional superconformal theory.Comment: 71 pages; v2: references added; final version to be published in
Nuclear Physics
Branes, Rings and Matrix Models in Minimal (Super)string Theory
We study both bosonic and supersymmetric (p,q) minimal models coupled to
Liouville theory using the ground ring and the various branes of the theory.
From the FZZT brane partition function, there emerges a unified, geometric
description of all these theories in terms of an auxiliary Riemann surface
M_{p,q} and the corresponding matrix model. In terms of this geometric
description, both the FZZT and ZZ branes correspond to line integrals of a
certain one-form on M_{p,q}. Moreover, we argue that there are a finite number
of distinct (m,n) ZZ branes, and we show that these ZZ branes are located at
the singularities of M_{p,q}. Finally, we discuss the possibility that the
bosonic and supersymmetric theories with (p,q) odd and relatively prime are
identical, as is suggested by the unified treatment of these models.Comment: 72 pages, 3 figures, improved treatment of FZZT and ZZ branes, minor
change
Konishi Anomalies and Curves without Adjoints
Generalized Konishi anomaly relations in the chiral ring of N=1
supersymmetric gauge theories with unitary gauge group and chiral matter field
in two-index tensor representations are derived. Contrary to previous
investigations of related models we do not include matter multiplets in the
adjoint representation. The corresponding curves turn out to be hyperelliptic.
We also point out equivalences to models with orthogonal or symplectic gauge
groups.Comment: 21 pages, v2: References added, misprints correcte
On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks
We review explicit solutions to the stationary axisymmetric Einstein-Maxwell
equations which can be interpreted as disks of charged dust. The disks of
finite or infinite extension are infinitesimally thin and constitute a surface
layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in
the presence of one Killing vector are obtained by using a projection
formalism. The SU(2,1) invariance of the stationary Einstein-Maxwell equations
can be used to construct solutions for the electro-vacuum from solutions to the
pure vacuum case via a so-called Harrison transformation. It is shown that the
corresponding solutions will always have a non-vanishing total charge and a
gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in
the stationary axisymmetric case are completely integrable, large classes of
solutions can be constructed with techniques from the theory of solitons. The
richest class of physically interesting solutions to the pure vacuum case due
to Korotkin is given in terms of hyperelliptic theta functions. The Harrison
transformed hyperelliptic solutions are discussed.Comment: 44 pages, 11 figures, revie
Fusion rules in conformal field theory
Several aspects of fusion rings and fusion rule algebras, and of their
manifestations in twodimensional (conformal) field theory, are described:
diagonalization and the connection with modular invariance; the presentation in
terms of quotients of polynomial rings; fusion graphs; various strategies that
allow for a partial classification; and the role of the fusion rules in the
conformal bootstrap programme.Comment: 68 pages, LaTeX. changed contents of footnote no.
Super Elliptic Curves
A detailed study is made of super elliptic curves, namely super Riemann
surfaces of genus one considered as algebraic varieties, particularly their
relation with their Picard groups. This is the simplest setting in which to
study the geometric consequences of the fact that certain cohomology groups of
super Riemann surfaces are not freely generated modules. The divisor theory of
Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group,
but this map is a projection, not an isomorphism as it is for ordinary tori.
The geometric realization of the addition law on Pic via intersections of the
supertorus with superlines in projective space is described. The isomorphisms
of Pic with the Jacobian and the divisor class group are verified. All possible
isogenies, or surjective holomorphic maps between supertori, are determined and
shown to induce homomorphisms of the Picard groups. Finally, the solutions to
the new super Kadomtsev-Petviashvili (super KP) hierarchy of Mulase-Rabin which
arise from super elliptic curves via the Krichever construction are exhibited.Comment: 27 page
Modular functors, cohomological field theories and topological recursion
Given a topological modular functor in the sense of Walker
\cite{Walker}, we construct vector bundles over ,
whose Chern classes define semi-simple cohomological field theories. This
construction depends on a determination of the logarithm of the eigenvalues of
the Dehn twist and central element actions. We show that the intersection of
the Chern class with the -classes in is
computed by the topological recursion of \cite{EOFg}, for a local spectral
curve that we describe. In particular, we show how the Verlinde formula for the
dimensions is retrieved from the
topological recursion. We analyze the consequences of our result on two
examples: modular functors associated to a finite group (for which
enumerates certain -principle
bundles over a genus surface with boundary conditions specified by
), and the modular functor obtained from Wess-Zumino-Witten
conformal field theory associated to a simple, simply-connected Lie group
(for which is the Verlinde
bundle).Comment: 50 pages, 2 figures. v2: typos corrected and clarification about the
use of ordered pairs of points for glueing. v3: unitarity assumption waived +
discussion of families index interpretation of the correlation functions for
Wess-Zumino-Witten theorie
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