208 research outputs found
Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations
We develop the theory of irregular conformal blocks of the Virasoro algebra.
In previous studies, expansions of irregular conformal blocks at regular
singular points were obtained as degeneration limits of regular conformal
blocks; however, such expansions at irregular singular points were not clearly
understood. This is because precise definitions of irregular vertex operators
had not been provided previously. In this paper, we present precise definitions
of irregular vertex operators of two types and we prove that one of our vertex
operators exists uniquely. Then, we define irregular conformal blocks with at
most two irregular singular points as expectation values of given irregular
vertex operators. Our definitions provide an understanding of expansions of
irregular conformal blocks and enable us to obtain expansions at irregular
singular points.
As an application, we propose conjectural formulas of series expansions of
the tau functions of the fifth and fourth Painlev\'e equations, using
expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page
Quantum Painlevé Equations: from Continuous to Discrete
We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
Topological Hochschild homology and the Bass trace conjecture
Abstract. We use the methods of topological Hochschild homology to shed new light on groups satisfying the Bass trace conjecture. Factorization of the Hattori-Stallings rank map through the Bökstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G wherein every subgroup isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G
Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
Classical conformal blocks naturally appear in the large central charge limit
of 2D Virasoro conformal blocks. In the correspondence, they
are related to classical bulk actions and are used to calculate entanglement
entropy and geodesic lengths. In this work, we discuss the identification of
classical conformal blocks and the Painlev\'e VI action showing how
isomonodromic deformations naturally appear in this context. We recover the
accessory parameter expansion of Heun's equation from the isomonodromic
-function. We also discuss how the expansion of the
-function leads to a novel approach to calculate the 4-point classical
conformal block.Comment: 32+10 pages, 2 figures; v3: upgraded notation, discussion on moduli
space and monodromies, numerical and analytic checks; v2: added refs, fixed
emai
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