Continuum Schwinger-Dyson Equations and Universal Structures in Two-Dimensional Quantum Gravity

Fukuma, Masafumi; Kawai, Hiroyuki; Nakayama, Ryuichi

doi:10.1142/9789814365802_0058

Cited by 73 publications

(143 citation statements)

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“…According to G.Belyi and A.Grothendieck [21][22][23][24][25], existence of such representation is necessary and sufficient for arithmeticity of the curve and arithmetic curves are in one-to-one correspondence with the equilateral triangulations (dessins d'enfant). Thus, enumeration of Belyi pairs is a typical matrix model problem (see more on relations between counting the Belyi maps, Hurwitz numbers and matrix models in [26][27][28]), though equivalence of matrix model [29,30] and sumover-metrics descriptions [31,32], proved in [33,34] on the lines of [35][36][37][38][39][40][41] remains a big mystery from the point of view of the complicated embedding of moduli space of arithmetic curves into the entire moduli space, see [42] and, for a related consideration, [43]. The Belyi pairs are enumerated by the triple Hurwitz numbers N∆ 0 ,∆ 1 ,∆∞ , but no adequate language is still found to describe the full generating function…”

Section: Jhep11(2014)080mentioning

confidence: 99%

On KP-integrable Hurwitz functions

Alexandrov

Миронов

Морозов

et al. 2014

J. High Energ. Phys.

101

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There is now a renewed interest [1]-[4] to a Hurwitz τ -function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz τ -functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model τ -functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W -representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz τ -functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.

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Section: Jhep11(2014)080mentioning

confidence: 99%

On KP-integrable Hurwitz functions

Alexandrov

Миронов

Морозов

et al. 2014

J. High Energ. Phys.

101

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“…Therefore we can also express (3.10) equivalently as the Virasoro constraints [37,38] L n Z = 0, n ≥ −1.…”

Section: Loop Equations and Collective Fieldsmentioning

confidence: 99%

“…There are also higher order relations [37,38]. In the case of A r there is leading spin r + 1 current that with a suitable basis of vectors ϕ 0 , .…”

Section: Quiver Theories and W -Constraintsmentioning

confidence: 99%

On geometry and matrix models

2002

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We point out two extensions of the relation between matrix models, topological strings and N = 1 supersymmetric gauge theories. First, we note that by considering double scaling limits of unitary matrix models one can obtain large N duals of the local CalabiYau geometries that engineer N = 2 gauge theories. In particular, a double scaling limit of the Gross-Witten one-plaquette lattice model gives the SU (2) Seiberg-Witten solution, including its induced gravitational corrections. Secondly, we point out that the effective superpotential terms for N = 1 ADE quiver gauge theories is similarly computed by large N multi-matrix models, that have been considered in the context of ADE minimal models on random surfaces. The associated spectral curves are multiple branched covers obtained as Virasoro and W -constraints of the partition function.

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“…As explained in [27], the suitable operation, which categorifies the specializations a = q N , involves taking homology in the triply-graded theory with respect to differentials d N , N ∈ Z. Indeed, the "extra terms" in the superpolynomial P R (a, q, t) that are not part of P sl(N ),R (q, t) and otherwise would cancel upon setting t = −1 always come in pairs, so that a more proper version of (3.8) reads where R sl(N ),R (a, q, t) and Q sl(N ),R (a, q, t) are polynomials with non-negative coefficients, such that the sl(N ) homological invariant is a specialization of the "remainder" (not the full superpolynomial as in (3.8)): 10) whereas the extra pairs of terms in (3.9) that come from Q sl(N ),R (a, q, t) are killed by the differential d N of (a, q, t)-degree (α, β, γ).…”

Section: Sl(1)r Ij (K)mentioning

confidence: 99%

Volume conjecture: refined and categorified

Fuji

Gukov

Sułkowski

et al. 2012

Advances in Theoretical and Mathematical Physics

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The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial A(x, y). Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted q or ; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation e-print archive: http://lanl.arXiv.org/abs/1203.2182v1 * Author of an appendix "ATMP-16-6-A3-FUJ" -2013/5/25 -9:47 -page 1670 -#2 ( x, y; q, t). We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

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Continuum Schwinger-Dyson Equations and Universal Structures in Two-Dimensional Quantum Gravity

Cited by 73 publications

References 0 publications

On KP-integrable Hurwitz functions

On KP-integrable Hurwitz functions

On geometry and matrix models

Volume conjecture: refined and categorified

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