Quantization of Harer-Zagier formulas

Morozov, Alexei; Popolitov, Aleksandr; Shakirov, Shamil

doi:10.48550/arxiv.2008.09577

Cited by 3 publications

(5 citation statements)

References 75 publications

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“…One of the most fundamental properties of matrix models is the genus expansion, when the diagrams of perturbation theory are interpreted as ribbon graphs, and the entire series is interpreted as a summation over all topologies. Similar expansion in 𝑞-case is more tricky and there is no final answer what to count as "expansion over genuses" in that case yet [64]. However, the dimer models might shed some light on this.…”

Section: Discussionmentioning

confidence: 99%

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Semenyakin¹

2022

Preprint

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Important illustration to the principle "partition functions in string theory are 𝜏 -functions of integrable equations" is the fact that the (dual) partition functions of 4𝑑 𝒩 = 2 gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve 𝑞-difference equations of non-autonomous dynamics of the "cluster-algebraic"integrable systems.We explain in details the "solutions" side of the proposal. In the simplest non-trivial example we show how 3𝑑 box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of "flux" 𝑞. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear 𝑞difference Kasteleyn operator. Using WKB method in the "melting" 𝑞 → 1 limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding 5𝑑 gauge theory. The "equations" side of the correspondence remains the intriguing topic for the further studies.

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Section: Discussionmentioning

confidence: 99%

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Semenyakin¹

2022

Preprint

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“…The proof of this identity is and combinatorial, very similar in spirit to that of [29]. The main point is seen already at q = 1: according to (9), in this case we deal just with the dimension of representation Schur R {N }, whose Laplace transform is factorizable only for single-hook R, while factorization is lost beyond one-hook (and thus for multi-trace averages), e.g.…”

Section: Other Single-trace Correlatorsmentioning

confidence: 96%

“…Also various generation functions w.r.t. the k-variables can be introduced, but while they lead to simplification at q = 1, they seem to blur matters in the q-deformed case [29].…”

Section: Knot Matrix Modelsmentioning

confidence: 99%

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Harer-Zagier formulas for knot matrix models

Morozov,

Popolitov,

Shakirov

2021

Preprint

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Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank -which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit β = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

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“…The manifestation that Harer-Zagier generating functions [30,31] are present for a given matrix model is that the Laplace transform of the correlators nicely factorizes [32,33]. Taking (24) as the definition of the single-trace correlators we immediately see that in a number of simplest examples there is no factorization of putative HZ-correlators…”

Section: Harer-zagier Generating Functionsmentioning

confidence: 99%

The "Null-A" superintegrability for monomial matrix models

Barseghyan¹,

Popolitov²

2022

Preprint

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We find that superintegrability (character expansion) property persists in the exotic sector of the monomial non-Gaussian matrix model, with potential Tr X r , in pure phase, where the naive partition function 1 vanishes. The role of the (anomaly-corrected) partition function is played by χρ -the Schur average of the suitably chosen square partiton ρ; such partitions are well-known to correspond to singular vectors of the Virasoro algebra. Further, non-zero are only Schur averages χµ for such µ that have ρ as their r-core, and superintegrability formula features the value of the skew Schur function χ µ/ρ at special point. The associated topological recursion and Harer-Zagier formula generalizations so far remain obscure.

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Quantization of Harer-Zagier formulas

Cited by 3 publications

References 75 publications

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Harer-Zagier formulas for knot matrix models

The "Null-A" superintegrability for monomial matrix models

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