The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories

I.M.Krichever

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The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories

I.M.Krichever

High Energy Physics - Theory · Submitted 29 May 1992

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The authors' main claim It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

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arXiv IDhep-th/9205110 (version 1)

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CitationI.M.Krichever. "The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories." arXiv:hep-th/9205110 (1992). https://pith.science/paper/PUE3W6QK

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The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The universal Witham hierarchy is considered from the point of view of topological field theories.	contribution 0.50	0.57	0.05	asserted 0.96
s1	The $\tau$-function for this hierarchy is defined.	contribution 0.97	0.84	0.06	asserted 1.00
s2	It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.	result 0.98	0.84	0.79	asserted 1.00

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Original abstract

The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

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References

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Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Seiberg-Witten Prepotential From Instanton Counting
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Equivariant localization on the instanton moduli space yields a closed combinatorial formula for the Seiberg-Witten prepotential, matching known instanton expansions up to five instantons.
Invariants of algebraic curves and topological expansion
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A single residue recursion at the branch points of an algebraic curve reproduces the topological (large-N) expansion of one-matrix, two-matrix, and external-field matrix models, and yields a new proof that the Kontsev...
Seiberg-Witten Theory and Random Partitions
hep-th 2003-06 accept novelty 8.0 of 10

The low-energy Seiberg-Witten prepotential of N=2 gauge theories is derived from the exact instanton partition function by a saddle-point analysis of random partitions.
Matrix eigenvalue model: Feynman graph technique for all genera
math-ph 2006-04 conditional novelty 7.0 of 10

A Feynman-graph method computes all orders of the 1/N free energy for the arbitrary-beta matrix eigenvalue model with multiple cuts, including a new F_{0,2} term.
Dispersionless Hirota equations of two-component BKP hierarchy
nlin.SI 2006-04 conditional novelty 7.0 of 10

The dispersionless limit of the two-component BKP hierarchy is shown to be governed by a system of Hirota-type equations equivalent to Hamilton-Jacobi equations, from which dispersionless Lax equations follow.

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Hodge Theory of Novikov Type Landau--Ginzburg Models
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A strong Hodge decomposition is established for L^2 forms on non-compact Kähler manifolds twisted by a closed holomorphic one-form, with harmonic space concentrated in top degree and isomorphic to Milnor rings.
Universal Correlators on Exponentially Ramified Spectral Curves
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Generalized topological recursion extends to spectral curves with essential singularities by replacing residues at the essential point with residues at ordinary meromorphic points.
$x-y$ swap for $(2,2p+1)$ minimal string
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An x-y swapped spectral curve is conjectured to reproduce (2,2p+1) minimal string tachyon correlators without resonance transformations; verified at low genus, with a ground-ring extension that does not match HEM.
On Painlev\'e/gauge theory correspondence
hep-th 2016-12 conditional novelty 6.0 of 10

The isomonodromic connection of Painlevé equations is shown to match the oper limit of the Hitchin connection, and the resulting long-distance tau-function expansions agree with magnetic and dyonic prepotentials of ra...
On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
hep-th 2010-11 conditional novelty 4.0 of 10

Inserting the simplest degenerate operator into a conformal block yields a Schrödinger-like equation whose Bohr-Sommerfeld periods generate the one-parameter deformed Seiberg-Witten prepotential.

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36 arXiv papers cite this in 72 sentences

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Papers citing it · 47 sentences without a confident label

Builds on it 2 papers

This is a new ingredient in the formulation of quasiclassical hierarchy, going beyond the original setup of [ref] .

Seiberg-Witten Theory and Extended Toda Hierarchy · Extended quasiclassical Toda hierarchy · 0.91
We made contact with Whitham hierarchies [ref] , generalizing and clarifying some of the earlier work on the subject [ref] .

Vershik-Kerov in higher times · Conclusions and further directions · 0.93

Uses its method or tool 3 papers

Following [ref] we are going to demonstrate how it can save efforts for the computation of the exact conformal blocks for the twist fields in theories with W-symmetry.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations · Introduction · 0.84
A direct way to prove the consistency of qder in the framework of [ref] is to use the RBR for the differentials

Tau-functions for Quiver Gauge Theories · Prepotentials and tau-functions · 0.91
Below we are going to use the techniques of the quasiclassical integrable systems [ref] to study the expansion of the prepotentials of quiver gauge theories in various regions of the moduli space, to be called as weak-coupling expansion, since it coincides with the perturbative and instanton analysis in all known cases, though goes itself beyond the scope, where such analysis is valid.

Residue Formulas for Prepotentials, Instanton Expansions and Conformal Blocks · Introduction · 1.00

Compares with it 1 paper

Ω= θ”(ξ)θ(ξ) = (logθ(ξ))” + ((logθ(ξ))')^2 ≡J'(ξ) + J^2(ξ) Ω(ξ_0+1)-Ω(ξ_0) = 0 Ω(ξ_0+τ)-Ω(ξ_0) = ((J(ξ_0+τ)-J(ξ_0))((J(ξ_0+τ)+J(ξ_0))= = -4πi(J(ξ_0)-iπ) = -4πi (θ(ξ_0)'θ(ξ_0)-iπ) so that the jump (cf., for example, with [ref] )

Non Abelian gauge theories, prepotentials and Abelian differentials · Elliptic U(1) theory · 0.90

Cites it as support 1 paper

The total system of equations SW, Bper is also integrable [ref] due to the Riemann bilinear relations.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations · Seiberg-Witten integrable system · 0.96

Cites it as background 14 papers

While considering finite-gap solutions to the integrable system one deals with a spectral curve and a tau-function [equation] $θ(z|τ)$ - is a conventional theta-function, [equation] If one introduces "slow"(Whitham) times $t_i=ϵT_i, ϵ0$, Whitham hierarchy equations tell us how moduli can be slowly varied provided ([ref]) still gives the solution to the leading order in $ϵ$ [ref] , [ref] .

RG-Whitham dynamics and complex Hamiltonian systems · Generalities · 0.94
Let us now turn directly to the tau-functions [ref] for the $SU(2)$ quiver gauge theories.

Tau-functions for Quiver Gauge Theories · Prepotentials and tau-functions · 0.81
These prepotentials obey remarkable properties, which can be shortly characterized by the fact, that they are quasiclassical tau-functions [ref] , and the low-energy effective theory [ref] can be formulated in terms of an integrable system [ref] .

On Microscopic Origin of Integrability in Seiberg-Witten Theory · Introduction · 0.83

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