Mahler’s Measure and the Dilogarithm (I)

Boyd, David W.; Rodríguez-Villegas, Fernando

doi:10.4153/cjm-2002-016-9

Cited by 63 publications

(134 citation statements)

References 7 publications

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“…Such triangulations are automatically "hard"; but more seriously, they fail to describe the moduli space of flat connections on M (cf. [55], Section 10.3, and [56,57]), and should never be expected to produce the correct theory T M or its index.…”

Section: -Manifolds and 3d Indices 1005mentioning

confidence: 99%

3-Manifolds and 3d indices

Dimofte

Gaiotto

Gukov

2013

Advances in Theoretical and Mathematical Physics

319

642

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We identify a large class R of three-dimensional N = 2 superconformal field theories. This class includes the effective theories T M of M5-branes wrapped on 3-manifolds M , discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories with (possibly non-perturbative) superpotential. Mathematically, class R might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class R is a quantum-field-theoretic "2 to 3 move." We proceed to study the supersymmetric index of theories in class R, uncovering its physical and mathematical properties, including relations to algebras of line operators and to 4d indices. For 3-manifold theories T M , the index is a new topological invariant, which turns out to be equivalent to non-holomorphic SL(2, C) Chern-Simons theory on M with a previously unexplored "integration cycle."

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Section: -Manifolds and 3d Indices 1005mentioning

confidence: 99%

3-Manifolds and 3d indices

Dimofte

Gaiotto

Gukov

2013

Advances in Theoretical and Mathematical Physics

319

642

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“…This choice, called the choice of polarization, means that one needs to pick a foliation of the phase space by Lagrangian submanifolds parametrized by a maximal set of mutually commuting "coordinates" (with the remaining variables understood as their conjugate momenta). In the problem at hand, the (complex) phase space is 2-dimensional, with the symplectic form (1.2), 34) so that the ambiguity associated with the choice of polarization is described by one functional degree of freedom, say, a choice of function f (u, v) that one regards as a "coordinate." Thus, in most of the present paper we make a natural 14 choice (1.7) treating u as the "coordinate" and v as the momentum.…”

Section: Jhep02(2012)070mentioning

confidence: 99%

A-polynomial, B-model, and quantization

Gukov

Sułkowski

2012

J. High Energ. Phys.

131

225

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Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as → 0, and becomes non-commutative or "quantum" away from this limit. For a classical curve defined by the zero locus of a polynomial A(x, y), we provide a construction of its non-commutative counterpart A( x, y) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing A that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a Ktheory criterion for a curve to be "quantizable," and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices.

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“…In [9], Jones showed that irreducible representations of B n are indexed by Young diagrams, and the ones with at most two columns are the T L n representations. By abuse of notation, let T L n also denote the algebra with coefficients in Q(A), and generators and relations as in (2). The Q(A)-basis is the same Z[A ±1 ]-basis described above, so the bilinear pairing extends linearly to Q(A) such that for any tangles,…”

Section: Introductionmentioning

confidence: 99%

On the Mahler measure of Jones polynomials under twisting

Champanerkar

Kofman

2005

Algebr. Geom. Topol.

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We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a 1 , . . . , a n ), we show that the Mahler measure of the Jones polynomial converges if all a i → ∞, and approaches infinity for a i = constant if n → ∞, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.

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Mahler’s Measure and the Dilogarithm (I)

Cited by 63 publications

References 7 publications

3-Manifolds and 3d indices

3-Manifolds and 3d indices

A-polynomial, B-model, and quantization

On the Mahler measure of Jones polynomials under twisting

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