Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

Eskin, Alex; Okounkov, Andreĭ

doi:10.1007/s002220100142

Cited by 152 publications

(229 citation statements)

References 30 publications

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“…Using an appropriate version of the Lagrange-Burmann inversion formula to write the explicit expansion of x as a power series in 1/ √ µ, after some combinatorial calculations one can arrive at an explicit formula for the coefficients ξ 2h−1 of the strong-coupling saddlepoint expansion as polynomials in Bernoulli numbers, thereby determining the exact asymptotics (11) [13]. The resulting formula agrees exactly with that obtained in [14] from rather involved combinatorial techniques, but our saddle-point method provides a much simpler and efficient way of extracting the asymptotics of simple Hurwitz numbers.…”

Section: Asymptotics Of Hurwitz Numberssupporting

confidence: 59%

“…Therefore, there is a one-to-one correspondence between simple branched covers of T 2 , and hence of terms in the chiral Gross-Taylor series, and pairs (Σ h , du). The collection of isomorphism classes of pairs (Σ h , du), with du a holomorphic one-form with 2h − 2 simple zeroes on a Riemann surface Σ h of genus h, is called a (principal) moduli space M h of holomorphic differentials [14,15]. We can coordinatize this moduli space by defining a local chart φ : M h → C 4h−3 through the period map φ(Σ h , du) = ( γ 1 du, .…”

Section: Moduli Spaces Of Holomorphic Differentialsmentioning

confidence: 99%

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Double scaling string theory of QCD in two dimensions

Seminara

Szabo²

2005

Fortschritte der Physik

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We describe a novel double scaling limit of large N Yang-Mills theory on a two-dimensional torus and its relation to the geometry of the principal moduli spaces of holomorphic differentials. Is Noncommutative QCD a String Theory?One of the most sought goals of modern theoretical high-energy physics is to recast the quantum field theory of the strong interactions as a string theory. There are various indications that this might be a more tractable problem in the context of noncommutative QCD. While noncommutative field theories are in general most naturally induced in string theory, they possess many unconventional stringy properties themselves reflecting the non-locality that their interactions retain [1]. For instance, the large θ expansion of a noncommutative field theory organises itself into planar and non-planar Feynman diagrams in exactly the same way that the large N expansion of a multicolour field theory does. They can be represented and analysed exactly as matrix models, indicating a potential connection to non-critical strings. Their fundamental quanta are electric dipoles, extended rigid rods whose lengths are proportional to their momenta, and their interactions are thus governed by string-like degrees of freedom. Finally, some of these theories admit novel soliton and instanton solutions which have no counterparts in ordinary field theory and can be naturally interpreted as D-branes.These aspects become particularly interesting in two dimensions, because ordinary Yang-Mills theory on a Riemann surface has a very precise interpretation as a string † Based on invited talk given by R.J.S. at "Recent Developments in String/M-Theory and Field Theory", 37th International Symposium Ahrenshoop on the Theory

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Section: Asymptotics Of Hurwitz Numberssupporting

confidence: 59%

Section: Moduli Spaces Of Holomorphic Differentialsmentioning

confidence: 99%

Double scaling string theory of QCD in two dimensions

Seminara

Szabo²

2005

Fortschritte der Physik

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“…It defines a meromorphic function which is absolutely convergent in the domain of the complex plane defined by the conditions e −λ < |w i 1 · · · w i k | < e λ for any subset and ϑ (2m) (1, τ ) = 0 ∀m ≥ 1. These formulas demonstrate explicitly the quasi-modularity of the free energy that we described in Section 5.1 [53,61,84], and moreover give, at least in principle, the explicit expansion of the free energy into the Eisenstein series basis for the ring M of quasi-modular forms.…”

Section: B Free Fermion Representationmentioning

confidence: 86%

“…By using the identity Once (B.14) is determined, we may use the following combinatorial trick to compute the genus h free energy (5.9) [61]. Let Π + h be the set of all partitions of the set {1, .…”

Section: B Free Fermion Representationmentioning

confidence: 99%

Instantons, fluxons, and open gauge string theory

Seminara¹,

Szabo²

2005

Advances in Theoretical and Mathematical Physics

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Abstract:We use the exact instanton expansion to illustrate various string characteristics of noncommutative gauge theory in two dimensions. We analyse the spectrum of the model and present some evidence in favour of Hagedorn and fractal behaviours. The decompactification limit of noncommutative torus instantons is shown to map in a very precise way, at both the classical and quantum level, onto fluxon solutions on the noncommutative plane. The weak-coupling singularities of the usual Gross-Taylor string partition function for QCD on the torus are studied in the instanton representation and its double scaling limit, appropriate for the mapping onto noncommutative gauge theory, is shown to be a generating function for the volumes of the principal moduli spaces of holomorphic differentials. The noncommutative deformation of this moduli space geometry is described and appropriate open string interpretations are proposed in terms of the fluxon expansion.

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“…The associated elements, the completed cycles, have been considered previously in Hurwitz theory -the term completed cycle first appears in [12] following unnamed appearances of the associated elements in [1], [11]. In fact, completed cycles, implicitly, are ubiquitous in the theory of shifted symmetric functions.…”

Section: 13mentioning

confidence: 99%