2,312 research outputs found
Mirror symmetry in two steps: A-I-B
We suggest an interpretation of mirror symmetry for toric varieties via an
equivalence of two conformal field theories. The first theory is the twisted
sigma model of a toric variety in the infinite volume limit (the A-model). The
second theory is an intermediate model, which we call the I-model. The
equivalence between the A-model and the I-model is achieved by realizing the
former as a deformation of a linear sigma model with a complex torus as the
target and then applying to it a version of the T-duality. On the other hand,
the I-model is closely related to the twisted Landau-Ginzburg model (the
B-model) that is mirror dual to the A-model. Thus, the mirror symmetry is
realized in two steps, via the I-model. In particular, we obtain a natural
interpretation of the superpotential of the Landau-Ginzburg model as the sum of
terms corresponding to the components of a divisor in the toric variety. We
also relate the cohomology of the supercharges of the I-model to the chiral de
Rham complex and the quantum cohomology of the underlying toric variety.Comment: 50 pages; revised versio
Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums
The construction of Mellin-Barnes (MB) representations for non-planar Feynman
diagrams and the summation of multiple series derived from general MB
representations are discussed. A basic version of a new package AMBREv.3.0 is
supplemented. The ultimate goal of this project is the automatic evaluation of
MB representations for multiloop scalar and tensor Feynman integrals through
infinite sums, preferably with analytic solutions. We shortly describe a
strategy of further algebraic summation.Comment: Contribution to the proceedings of the Loops and Legs 2014 conferenc
On equivalence of Floer's and quantum cohomology
(In the revised version the relevant aspect of noncompactness of the moduli
of instantons is discussed. It is shown nonperturbatively that any BRST trivial
deformation of A-model which does not change the ranks of BRST cohomology does
not change the topological correlation functions either) We show that the Floer
cohomology and quantum cohomology rings of the almost Kahler manifold M, both
defined over the Novikov ring of the loop space LM of M, are isomorphic. We do
it using a BRST trivial deformation of the topological A-model. As an example
we compute the Floer = quantum cohomology of the 3-dimensional flag space Fl_3.Comment: 28 pages, HUTP-93/A02
Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems
A new class of non-homogeneous state-affine systems is introduced for use in
reservoir computing. Sufficient conditions are identified that guarantee first,
that the associated reservoir computers with linear readouts are causal,
time-invariant, and satisfy the fading memory property and second, that a
subset of this class is universal in the category of fading memory filters with
stochastic almost surely uniformly bounded inputs. This means that any
discrete-time filter that satisfies the fading memory property with random
inputs of that type can be uniformly approximated by elements in the
non-homogeneous state-affine family.Comment: 41 page
Classical/quantum integrability in AdS/CFT
We discuss the AdS/CFT duality from the perspective of integrable systems and
establish a direct relationship between the dimension of single trace local
operators composed of two types of scalar fields in N=4 super Yang-Mills and
the energy of their dual semiclassical string states in AdS(5) X S(5). The
anomalous dimensions can be computed using a set of Bethe equations, which for
``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified
approach to the long wavelength Bethe equations, the classical ferromagnet and
the classical string solutions in the SU(2) sector and present a general
solution, governed by complex curves endowed with meromorphic differentials
with integer periods. Using this solution we compute the anomalous dimensions
of these long operators up to two loops and demonstrate that they agree with
string-theory predictions.Comment: 49 pages, 5 figures, LaTeX; v2: complete proof of the two-loop
equivalence between the sigma model and the gauge theory is added. References
added; v4,v5,v6: misprints correcte
Three-point functions in the SU(2) sector at strong coupling
Extending the methods developed in our previous works (arXiv:1110.3949,
arXiv:1205.6060), we compute the three-point functions at strong coupling of
the non-BPS states with large quantum numbers corresponding to the composite
operators belonging to the so-called SU(2) sector in the
super-Yang-Mills theory in four dimensions. This is achieved by the
semi-classical evaluation of the three-point functions in the dual string
theory in the spacetime, using the general one-cut finite
gap solutions as the external states. In spite of the complexity of the
contributions from various parts in the intermediate stages, the final answer
for the three-point function takes a remarkably simple form, exhibiting the
structure reminiscent of the one obtained at weak coupling. In particular, in
the Frolov-Tseytlin limit the result is expressed in terms of markedly similar
integrals, however with different contours of integration. We discuss a natural
mechanism for introducing additional singularities on the worldsheet without
affecting the infinite number of conserved charges, which can modify the
contours of integration.Comment: 128 pages (A summary is given in section 1); v2 minor improvement
An Exact Elliptic Superpotential for N=1^* Deformations of Finite N=2 Gauge Theories
We study relevant deformations of the N=2 superconformal theory on the
world-volume of N D3 branes at an A_{k-1} singularity. In particular, we
determine the vacuum structure of the mass-deformed theory with N=1
supersymmetry and show how the different vacua are permuted by an extended
duality symmetry. We then obtain exact, modular covariant formulae (for all k,
N and arbitrary gauge couplings) for the holomorphic observables in the massive
vacua in two different ways: by lifting to M-theory, and by compactification to
three dimensions and subsequent use of mirror symmetry. In the latter case, we
find an exact superpotential for the model which coincides with a certain
combination of the quadratic Hamiltonians of the spin generalization of the
elliptic Calogero-Moser integrable system.Comment: 55 pages, 5 figures, latex with JHEP.cl
Derived Equivalences of K3 Surfaces and Twined Elliptic Genera
We use the unique canonically-twisted module over a certain distinguished
super vertex operator algebra---the moonshine module for Conway's group---to
attach a weak Jacobi form of weight zero and index one to any symplectic
derived equivalence of a projective complex K3 surface that fixes a stability
condition in the distinguished space identified by Bridgeland. According to
work of Huybrechts, following Gaberdiel--Hohenegger--Volpato, any such derived
equivalence determines a conjugacy class in Conway's group, the automorphism
group of the Leech lattice. Conway's group acts naturally on the module we
consider.
In physics the data of a projective complex K3 surface together with a
suitable stability condition determines a supersymmetric non-linear sigma
model, and supersymmetry preserving automorphisms of such an object may be used
to define twinings of the K3 elliptic genus. Our construction recovers the K3
sigma model twining genera precisely in all available examples. In particular,
the identity symmetry recovers the usual K3 elliptic genus, and this signals a
connection to Mathieu moonshine. A generalization of our construction recovers
a number of the Jacobi forms arising in umbral moonshine.
We demonstrate a concrete connection to supersymmetric non-linear K3 sigma
models by establishing an isomorphism between the twisted module we consider
and the vector space underlying a particular sigma model attached to a certain
distinguished K3 surface.Comment: 62 pages including 7 pages of tables; updated references and minor
editing in v.2; to appear in Research in the Mathematical Science
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