Mirror symmetry for extended affine Weyl groups

Abstract

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau-Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson-Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko-Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax-Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

Five-dimensional gauge theories and the local B-model

“…Its (non-trivial) existence and explicit construction are reduced to a problem in commutative algebra, which we solve for all G and for all different realisations of Seiberg-Witten geometries when more than one is available (such as for non-simply laced classical groups). By the mirror theorem of [20], for simply laced G this restricts to the structure constants of the affine-Weyl Frobenius algebras mentioned previously.…”

“…Suppose first that G is of type ADE. We propose that SW periods are, in the terminology of [30], the odd periods of the canonical Frobenius manifold structure on the orbits of the Dubrovin-Zhang extension of the affine Weyl group of type G in the reflection representation [31], which was recently explicitly constructed in [16,20]. This is a natural generalisation of an idea of Dubrovin [30] (see also [34,48] for previous work on this), where the polynomial Frobenius manifold of type ADE played a similar role in the reconstruction of the SW periods for four-dimensional N = 2 super Yang-Mills with simply laced gauge symmetry.…”

“…In (2), {q i } r i=1 are the α-basis coefficients of the highest element of the root system of type G, h g is the Coxeter number, the coordinates t i are a homogeneous choice of flat coordinates for the associated affine Weyl Frobenius manifold [20,31], and C k i j are the structure constants of the Frobenius product in those coordinates. From the engineering point of view, and as an application to local mirror symmetry/(orbifold) Gromov-Witten theory, the Picard-Fuchs ideals (2) specialise to the GKZ system for the associated Y N ,0 toric singularities considered in [19,33] for G = SU(N ).…”

“…The reasons are already well known from the study of the 4d set-up, where the auxiliary Frobenius manifold would arise from the spectral curves of the non-relativistic periodic Toda lattice. It was found shortly after the work of Seiberg-Witten [66] that the integrable system relevant for N = 2 super Yang-Mills theory with gauge group G is the Toda lattice associated with the twisted Kac-Moody algebra (G (1) ) ∨ , whereas the construction in [20,31] of Frobenius manifolds on orbits of extended affine Weyl groups pertains to its untwisted counterpart. Unfortunately there turns out to be no strict analogue of the Frobenius manifolds of [31] in the twisted Kac-Moody world, as the would-be Frobenius metric constructed from the spectral curve becomes either degenerate, or curved, in that setting.…”

“…We start by providing a very brief account of the construction of Dubrovin and Zhang [31] of a semisimple Frobenius manifold on the space of regular orbits of an extended affine Weyl group, referring the reader to Dubrovin and Zhang [31], Brini [16] and Brini and van Gemst [20] for a more extended treatment. Let g be a complex simple Lie algebra of rank r , and write h and W g for, respectively, its Cartan subalgebra and the Weyl group.…”