782 research outputs found
S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
A recent claim that the S-duality between 4d SUSY gauge theories, which is
AGT related to the modular transformations of 2d conformal blocks, is no more
than an ordinary Fourier transform at the perturbative level, is further traced
down to the commutation relation [P,Q]=-i\hbar between the check-operator
monodromies of the exponential resolvent operator in the underlying
Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the
conformal blocks as eigenfunctions of the monodromy check operators, what is
especially simple in the case of one-point toric block. The kernel of the
modular transformation is then defined as the intertwiner of the two
monodromies, and can be obtained straightforwardly, even when the eigenfunction
interpretation of the blocks themselves is technically tedious. In this way, we
provide an elementary derivation of the old expression for the modular kernel
for the one-point toric conformal block.Comment: 15 page
Knot invariants from Virasoro related representation and pretzel knots
We remind the method to calculate colored Jones polynomials for the plat
representations of knot diagrams from the knowledge of modular transformation
(monodromies) of Virasoro conformal blocks with insertions of degenerate
fields. As an illustration we use a rich family of pretzel knots, lying on a
surface of arbitrary genus g, which was recently analyzed by the evolution
method. Further generalizations can be to generic Virasoro modular
transformations, provided by integral kernels, which can lead to the Hikami
invariants.Comment: 29 page
Some Nonexistence Results for Systems of Nonlinear Partial Differential Inequalities
We obtain nonexistence results for systems of stationary and evolutional partial differential inequalities that involve
p-Laplacian and similar nonlinear operators as well as gradient
nonlinearities. Our proofs are based on the nonlinear capacity
method
On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
We concentrate on a treatment of a Higgs-Coulomb duality as an absence of
manifest phase transition between ordered and disordered phases of 2d
theories. We consider these examples of QFTs in the
Schr\"odinger picture and identify Hilbert spaces of BPS states with morphisms
in triangulated Abelian categories of D-brane boundary conditions. As a result
of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are
equivalent, this resembles an analog of homological mirror symmetry. Following
construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one
is able to introduce interface defects in these theories and associate them to
D-brane parallel transport functors. We concentrate on surveying simple
examples, analytic when possible calculations, numerical estimates and simple
physical picture behind curtains of geometric objects. Categorification of
hypergeometric series analytic continuation is derived as an Atiyah flop of the
conifold. Finally we arrive to an interpretation of the braid group action on
the derived category of coherent sheaves on cotangent bundles to flag varieties
as a categorification of Berry connection on the Fayet-Illiopolous parameter
space of a sigma-model with a quiver variety target space.Comment: 118 pages, 15 figures, v2: minor modifications: corrections,
references and comment
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