Refined open intersection numbers and the Kontsevich-Penner matrix model

Alexandrov, A.; Buryak, Alexandr; Tessler, Ran J.

doi:10.1007/jhep03(2017)123

Cited by 22 publications

(29 citation statements)

References 18 publications

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“…If we view Σ as a two-manifold a separate parameter ws for each s. This corresponds to including in the matrix integral a factor s∈S (det(zs −Φ)) ws . Such generalizations have been treated in [45].…”

Section: The Anomalymentioning

confidence: 99%

Developments in topological gravity

Dijkgraaf

Witten

2019

Topology and Physics

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This note aims to provide an entrée to two developments in two-dimensional topological gravity -that is, intersection theory on the moduli space of Riemann surfaces -that have not yet become well-known among physicists. A little over a decade ago, Mirzakhani discovered [1,2] an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler [3] (with further developments in [4][5][6]) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint -it corresponds to adding vector degrees of freedom to the matrix model -constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.

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Section: The Anomalymentioning

confidence: 99%

Developments in topological gravity

Dijkgraaf

Witten

2019

Topology and Physics

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“…The matrix Y = diag(y 1 , ..., y n ) is a diagonal matrix satisfying Re y j > 0 so that the integrals in (1.1) converge absolutely. This matrix integral belongs to the family of the generalized Kontsevich models [KMM + 92]; the choice of the potential as in (1.1) [Kon92,Pen88] has recently attracted some interest [Ale15b, Ale15a,BH15] as it is conjectured [ABT17] that the correlators of the model described by (1.1) provide open intersection numbers.…”

mentioning

confidence: 99%

The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers

Bertola

Ruzza

2018

Ann. Henri Poincaré

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We identify the Kontsevich-Penner matrix integral, for finite size n, with the isomonodromic tau function of a 3 × 3 rational connection on the Riemann sphere with n Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann-Hilbert boundary value problem, we can pass to the limit n → ∞ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of a KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann-Hilbert approach.The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formulae for the correlators of this matrix model in terms of the solution of the Riemann Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.Lemma 2.22. The jump matrix J(λ) (2.44) admits formal meromorphic extension in D + . More precisely

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“…The parity property is already seen in (4.38) of section 4.2 where GF F o (0) (t, s) is odd in s. The general proof can be done using the dimension of the moduli space given in (2.21): g + k must always be odd [36]. Since k denotes the power of s in GF, one concludes (4.38).…”

Section: Higherḡ-expansionmentioning

confidence: 63%

“…For example,ḡ = 2 contains two different geometries: pants and kettle. For the open intersection theory, the contributions of different types of surfaces can be traced by the extension of the generating function [36,38]. However, the computations of the generating function of the minimal gravity with boundaries with topological structure different form the sphere with arbitrary number of boundaries is still not known, but can be extracted from the relation in terms of the closed GF [31,32] or the matrix model computations [14,17].…”

Section: Summary and Discussionmentioning

confidence: 99%

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From minimal gravity to open intersection theory

2019

Self Cite

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We investigated the relation between the two-dimensional minimal gravity (Lee-Yang series) with boundaries and open intersection theory. It is noted that the minimal gravity with boundaries is defined in terms of boundary cosmological constant µ B and the open intersection theory in terms of boundary marked point generating parameter s. Based on the conjecture that the two different descriptions of the generating functions are related by the Laplace transform, we derive the compact expressions for the generating function of the intersection theory from that of the minimal gravity on a disk and on a cylinder.

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Refined open intersection numbers and the Kontsevich-Penner matrix model

Cited by 22 publications

References 18 publications

Developments in topological gravity

Developments in topological gravity

The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers

From minimal gravity to open intersection theory

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