19 research outputs found
Integrable Lattice Models and Holography
We study four-dimensional Chern-Simons theory on (where
is a disk), which is understood to describe rational solutions of the
Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find
that the theory is dual to a boundary theory, that is a three-dimensional
analogue of the two-dimensional chiral WZW model. This boundary theory gives
rise to a current algebra that turns out to be an "analytically-continued"
toroidal Lie algebra. In addition, we show how certain bulk correlation
functions of two and three Wilson lines can be captured by boundary correlation
functions of local operators in the three-dimensional WZW model. In particular,
we reproduce the leading and subleading nontrivial contributions to the
rational R-matrix purely from the boundary theory.Comment: 22 pages, 8 figures. Additional discussions and minor improvements.
Published in JHE
Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory
We explain how, starting with a stack of D4-branes ending on an NS5-brane in
type IIA string theory, one can, via T-duality and the topological-holomorphic
nature of the relevant worldvolume theories, relate (i) the lattice models
realized by Costello's 4d Chern-Simons theory, (ii) links in 3d
analytically-continued Chern-Simons theory, (iii) the quantum geometric
Langlands correspondence realized by Kapustin-Witten using 4d N = 4 gauge
theory and its quantum group modification, and (iv) the Gaitsgory-Lurie
conjecture relating quantum groups/affine Kac-Moody algebras to Whittaker
D-modules/W-algebras. This furnishes, purely physically via branes in string
theory, a novel bridge between the mathematics of integrable systems, geometric
topology, geometric representation theory, and quantum algebras.Comment: 31 pages. Minor improvements, typos corrected, and reference adde
Little Strings, Quasi-topological Sigma Model on Loop Group, and Toroidal Lie Algebras
We study the ground states and left-excited states of the A_{k-1} N=(2,0)
little string theory. Via a theorem by Atiyah [1], these sectors can be
captured by a supersymmetric nonlinear sigma model on CP^1 with target space
the based loop group of SU(k). The ground states, described by L^2-cohomology
classes, form modules over an affine Lie algebra, while the left-excited
states, described by chiral differential operators, form modules over a
toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS
sectors of the M5-brane worldvolume theory.Comment: 32 pages. Minor imprecisions and typos corrected. To appear in
Nuclear Physics
Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory
We elucidate the relationship between 2d integrable field theories and 2d
integrable lattice models, in the framework of the 4d Chern-Simons theory. The
2d integrable field theory is realized by coupling the 4d theory to multiple 2d
surface order defects, each of which is then discretized into 1d defects. We
find that the resulting defects can be dualized into Wilson lines, so that the
lattice of discretized defects realizes integrable lattice models. Our
discretization procedure works systematically for a broad class of integrable
models (including trigonometric and elliptic models), and uncovers a rich web
of new dualities among integrable field theories. We also study the
anomaly-inflow mechanism for the integrable models, which is required for the
quantum integrability of field theories. By analyzing the anomalies of chiral
defects, we derive a new set of bosonization dualities between generalizations
of massless Thirring models and coupled Wess-Zumino-Witten (WZW) models. We
study an embedding of our setup into string theory, where the thermodynamic
limit of the lattice models is realized by polarizations of D-branes.Comment: 145 pages, 22 figure
Matrix Regularization of Classical Nambu Brackets and Super -Branes
We present an explicit matrix algebra regularization of the algebra of
volume-preserving diffeomorphisms of the -torus. That is, we approximate the
corresponding classical Nambu brackets using
-matrices equipped with
the finite bracket given by the completely anti-symmetrized matrix product,
such that the classical brackets are retrieved in the
limit. We then apply this approximation to the super -brane in
dimensions and give a regularized action in analogy with the matrix
regularization of the supermembrane. This action exhibits a reduced gauge
symmetry that we discuss from the viewpoint of -algebras in a slight
generalization to the construction of Lie -algebras from Bagger-Lambert
-algebras.Comment: 36 pages. Further clarifications. To appear in JHE
4d Chern-Simons Theory as a 3d Toda Theory, and a 3d-2d Correspondence
We show that the four-dimensional Chern-Simons theory studied by Costello,
Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a
boundary theory that is a three-dimensional analogue of Toda theory with a
novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory
in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills
theory on a manifold with corners, we argue that this three-dimensional Toda
theory is dual to a two-dimensional topological sigma model with A-branes on
the moduli space of solutions to the Bogomolny equations. This furnishes a
novel 3d-2d correspondence, which, among other mathematical implications, also
reveals that modules of the 3d W-algebra are modules for the quantized algebra
of certain holomorphic functions on the Bogomolny moduli space.Comment: 27 pages. Presented at "String Math 2020
Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde Formula
Generalizing our ideas in [arXiv:1006.3313], we explain how
topologically-twisted N=2 gauge theory on a four-manifold with boundary, will
allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture,
(ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii)
their monopole counterparts, and (iv) their higher rank generalizations. In the
case where the boundary is a Seifert manifold, one can also relate its
instanton Floer homology to modules of an affine algebra via a 2d A-model with
target the based loop group. As an offshoot, we will be able to demonstrate an
action of the affine algebra on the quantum cohomology of the moduli space of
flat connections on a Riemann surface, as well as derive the Verlinde formula.Comment: 44 pp. Typos corrected. To appear in ATM