71,803 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Continuous and Random Vapnik-Chervonenkis Classes
We show that if is a dependent theory then so is its Keisler
randomisation . In order to do this we generalise the notion of a
Vapnik-Chervonenkis class to families of -valued functions (a
\emph{continuous} Vapnik-Chervonenkis class), and we characterise families of
functions having this property via the growth rate of the mean width of an
associated family of convex compacts
Line defect Schur indices, Verlinde algebras and fixed points
Given an superconformal field theory, we reconsider the Schur
index in the presence of a half line defect . Recently
Cordova-Gaiotto-Shao found that admits an expansion in terms
of characters of the chiral algebra introduced by Beem et al.,
with simple coefficients . We report a puzzling new feature of
this expansion: the limit of the coefficients is
linearly related to the vacuum expectation values in
-invariant vacua of the theory compactified on . This relation can
be expressed algebraically as a commutative diagram involving three algebras:
the algebra generated by line defects, the algebra of functions on
-invariant vacua, and a Verlinde-like algebra associated to
. Our evidence is experimental, by direct computation in the
Argyres-Douglas theories of type , , , and . In the latter two theories, which have flavor
symmetries, the Verlinde-like algebra which appears is a new deformation of
algebras previously considered.Comment: 64 pages, 21 figures. v2 published version, references update
Infrared Computations of Defect Schur Indices
We conjecture a formula for the Schur index of N=2 four-dimensional theories
in the presence of boundary conditions and/or line defects, in terms of the
low-energy effective Seiberg-Witten description of the system together with
massive BPS excitations. We test our proposal in a variety of examples for
SU(2) gauge theories, either conformal or asymptotically free. We use the
conjecture to compute these defect-enriched Schur indices for theories which
lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate
in various examples that line defect indices can be expressed as sums of
characters of the associated two-dimensional chiral algebra and that for
Argyres-Douglas theories the line defect OPE reduces in the index to the
Verlinde algebra.Comment: 63 pages + appendices, 15 figures. v2 published version, references
added, representations of SO(8) Kac-Moody discusse
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