Matrix model generating function for quantum weighted Hurwitz numbers

Harnad, J.; Runov, Boris

doi:10.1088/1751-8121/ab6142

Cited by 7 publications

(11 citation statements)

References 39 publications

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“…They indeed imply the equations of motion ǫ γα D γγ f αβ = ǫ γα D γγ w α = 0. We have therefore shown that the constraints (7), (8) for A A imply the superfield equations (9) for the superfieldstrength w α and superfield vector-potential A αβ . These in turn imply the ordinary space supersymmetric self-duality equations for the component fields (which we denote by the same symbols as the superfields of which they are the leading components)…”

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confidence: 76%

“…The converse, that given a set of component fields satisfying these component equations, one can reconstruct superfields satisfying (9), and in turn the superconnection A A satisfying (7), (8) is also true. The proof closely follows the methods of [8].…”

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confidence: 87%

“…Our main purpose here, however, is to introduce yet another piece of data corresponding to the above three: a 'holomorphic' prepotential in harmonic superspace. We shall show that the constraints (7), (8) imply generalised Cauchy-Riemann (CR) conditions for a prepotential in harmonic superspace and that any superconnection satisfying (7), (8) may be expressed in terms of such holomorphic prepotentials. (We shall use 'holomorphic' in this generalised sense; to describe solutions of these generalised CR conditions) .…”

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confidence: 99%

“…4. Equations (11), (12) are therefore equivalent to the constraints (7), (8). Let us now choose a coordinate basis, which we shall call the analytic frame, in which the derivatives take the forms…”

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confidence: 99%

“…Note that the third equation is a consequence of the other two. We have shown that to any solution of the super self-duality constraints (8), there corresponds a chiral holomorphic superfield V ++ taking values in the gauge algebra and having component expansion…”

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confidence: 99%

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Super-self-duality as analyticity in harmonic superspace

Devchand¹,

Ogievetsky²

Supersymmetries and Quantum Symmetries

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confidence: 76%

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confidence: 87%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Super-self-duality as analyticity in harmonic superspace

Devchand¹,

Ogievetsky²

Supersymmetries and Quantum Symmetries

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Weighted Hurwitz Numbers, "Equation missing" -Functions, and Matrix Integrals

Harnad

2020

Quantum Theory and Symmetries

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Weighted Hurwitz Numbers and Topological Recursion

Alexandrov

Chapuy

Eynard

et al. 2020

Commun. Math. Phys.

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The KP and 2D Toda τ -functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations. * graphs such as maps, dessins d'enfants, or more generally, constellations [54]. There has also been important progress in relating Hurwitz numbers to other classes of enumerative geometric invariants [5,19,23,30,37,51,61,64,65,68] and matrix models [1,2,4,9,10,15,19,22,57,60,64,65].A key development was the identification by Pandharipande [68] and Okounkov [61] that certain special τ -functions for integrable hierarchies of the KP and 2D Toda type may serve as generating functions for simple (single and double) Hurwitz numbers (i.e., those for which all branch points, with the possible exception of one, or two, have simple ramification profiles). It was shown subsequently [9,10,38,39,41,44] that all weighted (single and double) Hurwitz numbers have KP or 2D Toda τ -functions of the special hypergeometric type [63, 66, 67] as generating functions.An alternative approach, particularly useful for studying genus dependence and recursive relations between the invariants involved [37], consists of using multicurrent correlators as generating functions for weighted Hurwitz numbers having a fixed ramification profile length n. These may be defined in a number of equivalent ways: either as the coefficients in multivariable Taylor series expansions of the τ -function about suitably defined n-parameter families of evaluation points, in terms of pair correlators, or as fermionic expectation values of products of current operators evaluated at n points [7].A very efficient way of computing Hurwitz numbers, which provides strong results about their structure, follows from the method of Topological Recursion (TR), introduced by Eynard and Orantin in [27]. This approach, originally inspired by results arising naturally in random matrix theory [24], has been shown applicable to many enumerative geometry problems, such as the counting of maps [25] or computation of Gromov-Witten invariants [18]. It has received a great deal of attention in recent years and found to have many far-reaching implications. The fact that simple Hurwitz numbers satisfy the TR relations was conjectured by Bouchard and Mariño [19], and proved in [15,26]. This provides an algorithm that allows them to...

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Matrix model generating function for quantum weighted Hurwitz numbers

Cited by 7 publications

References 39 publications

Super-self-duality as analyticity in harmonic superspace

Super-self-duality as analyticity in harmonic superspace

Weighted Hurwitz Numbers, "Equation missing" -Functions, and Matrix Integrals

Weighted Hurwitz Numbers and Topological Recursion

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