The τ‐function of the universal whitham hierarchy, matrix models and topological field theories

Abstract

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

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

“…We thus recover the well known result: the two-points function is equal to the Bergmann kernel on the Riemann surface corresponding to the algebraic equation E(x, y) = 0 (cf [5,20,27,30] (y(r) − y(q))U 1 (q, y(r); p 1 ) dx(q) = − W 2 (q; p 1 )U 0 (q, y(r)) dx(q) 2 − P 1 (x(q), y(r); p 1 ) +d p 1 U 0 (p 1 , y(r)) (x(q) − x(p 1 )) dx(p 1 ) (B.7)…”

“…, x k ) are obtained by derivation of w 1 with respect to the potential In what follows, we assume that the leading resolvent, i.e. the function Y (x) is known, and we refer the reader to the existing literature on that topic, for instance [5,20,27,30].…”

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

“…In the A 1 case, for example, the qq-character modifies to In this way we get the realization of the W -algebra and its qq-deformation in gauge theory. We also get a new perspective on the rôle of Whitham hierarchies [46,63] and their quantum and qq-deformations in gauge theory. See also [13] for more applications of qq-characters in the U(1) case.…”

Geometry of 2D topological field theories

“…(The at coordinates t 1 , ..., t n can be expressed via Casimirs of the original Poisson bracket and action variables and wave n umbers along the invariant tori -see details in [53,54].) To de ne a tensor c (t) o n M (or, equivalently, the \primary free energy" F(t)) we need to use a semiclassical limit of the -function of the original hierarchy [87,88,44,45,133]. For the dispersionless limit the de nition of the semiclassical -function reads log semiclassical (T 0 ; T 1 ; : : : ) = lim !0 2 log ( t 0 ; t 1 ; : : : ) : (6:20) Then F = log semiclassical (6:21) for a particular -function of the hierarchy.…”