Results 11  20
of
75
Identification of the Givental formula with the spectral curve topological recursion procedure
 Comm. Math. Phys
"... ar ..."
(Show Context)
On almost duality for Frobenius manifolds
, 2004
"... We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theo ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg Witten duality.
Isomonodromic deformation of resonant rational connections
, 2005
"... We analyze isomonodromic deformations of rational connections on the Riemann sphere with Fuchsian and irregular singularities. The Fuchsian singularities are allowed to be of arbitrary resonant index; the irregular singularities are also allowed to be resonant in the sense that the leading coefficie ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
We analyze isomonodromic deformations of rational connections on the Riemann sphere with Fuchsian and irregular singularities. The Fuchsian singularities are allowed to be of arbitrary resonant index; the irregular singularities are also allowed to be resonant in the sense that the leading coefficient matrix at each singularity may have arbitrary Jordan canonical form, with a genericity condition on the Lidskii submatrix of the subleading term. We also give the relevant notion of isomonodromic tau function extending the one of nonresonant deformations introduced by MiwaJimboUeno. The tau function is
Symmetries and solutions of Getzler’s equation for Coxeter and extended affine Weyl Frobenius manifolds
"... Abstract. The Gfunction associated to the semisimple Frobenius manifold C n /W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Abstract. The Gfunction associated to the semisimple Frobenius manifold C n /W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the Gfunction corresponding to the Frobenius manifold C n / ˜ W (k) (An−1) , where ˜ W (k) (An−1) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space ˆ M0;k−1,n−k−1), together with the general form of the Gfunction in terms of data on caustics. Symmetries of the G function are also studied.
The curvature of a Hessian metric
 Internat. J. Math
"... Given a smooth function f on an open subset of a real vector space, one can define the associated “Hessian metric ” using the second derivatives of f, gij: = ∂ 2 f/∂xi∂xj. In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case wher ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
Given a smooth function f on an open subset of a real vector space, one can define the associated “Hessian metric ” using the second derivatives of f, gij: = ∂ 2 f/∂xi∂xj. In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case where f is a homogeneous polynomial (also called a “form”) of degree d at least 2. Following Okonek and van de Ven [23], Wilson considers the “index cone, ” the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1, ∗)) and f is positive. He restricts the indefinite metric −1/d(d − 1)∂2f/∂xi∂xj to the hypersurface M: = {f = 1} in the index cone, where it is a Riemannian metric, which he calls the Hodge metric. (In affine differential geometry, this metric is known as the “centroaffine metric ” of the hypersurface M, up to a constant factor.) Wilson considers two main questions about the Riemannian manifold M. First, when does M have nonpositive sectional curvature? (It does have nonpositive sectional curvature in many examples.) Next, when does M have constant negative curvature?
The symplectic and twistor geometry of the general isomonodromic deformation problem
, 2008
"... ..."
(Show Context)
Stokes matrices for the quantum cohomologies of Grassmannians, Int
 Math. Res. Not
"... GromovWitten invariants of homogeneous spaces contain enumerative information such as the number of nodal rational curves of a given degree passing through a given set of points in general position. The theory of Frobenius manifold allows a systematic treatment of these invariants. A Frobenius man ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
GromovWitten invariants of homogeneous spaces contain enumerative information such as the number of nodal rational curves of a given degree passing through a given set of points in general position. The theory of Frobenius manifold allows a systematic treatment of these invariants. A Frobenius manifold is a complex manifold whose
Milanov T., Simple singularities and integrable hierarchies, in The breadth of symplectic and Poisson geometry
 2005, 173–201, math.AG/0307176. (n, 1)Reduced DKP Hierarchy 19
"... Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobeniusmanifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the An−1 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobeniusmanifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the An−1singularity satisfies the modulon reduction of the KPhierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,Etype. Our description of the hierarchy is parallel to the vertex operator construction of Kac – Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac – Wakimoto theory are studied on a casebycase basis and remain, generally speaking, unknown. 1. The ADEhierarchies. According to Date–Jimbo–Kashiwara–Miwa [6] and I. Frenkel [10], the KdVhierarchy of integrable systems can be placed under the name A1 into the list of more general integrable hierarchies corresponding to the ADE Dynkin diagrams. These hierarchies are usually constructed (see [16]) using representation theory of the corresponding loop groups. V. Kac and M. Wakimoto [17] describe the hierarchies even more explicitly in the form of the so called Hirota quadratic equations expressed in terms of suitable vertex operators. One of the goals of the present paper is to show how the vertex operator description of the Hirota quadratic equations (certainly the same ones, even though we don’t quite prove this) emerges from the theory of vanishing cycles associated with the ADE singularities. Let f be a weightedhomogeneous polynomial inC3 with a simple critical point at the origin. According to V. Arnold [1] simple singularities of holomorphic functions are classified by simplylaced Dynkin diagrams: AN, N ≥ 1: f = x
Profiling the brane drain in a nonsupersymmetric orbifold
 JHEP 0601
, 2006
"... We study Dbranes in a nonsupersymmetric orbifold of type C 2 /Γ, perturbed by a tachyon condensate, using a gauged linear sigma model. The RG flow has both higgs and coulomb branches, and each branch supports different branes. The coulomb branch branes account for the “brane drain ” from the higgs ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
We study Dbranes in a nonsupersymmetric orbifold of type C 2 /Γ, perturbed by a tachyon condensate, using a gauged linear sigma model. The RG flow has both higgs and coulomb branches, and each branch supports different branes. The coulomb branch branes account for the “brane drain ” from the higgs branch, but their precise relation to fractional branes has hitherto been unknown. Building on the results of hepth/0403016 we construct, in detail, the map between fractional branes and the coulomb/higgs branch branes for two examples in the type 0 theory. This map depends on the phase of the tachyon condensate in a surprising and intricate way. In the mirror LandauGinzburg picture the dependence on the tachyon phase is manifested by discontinuous changes in the shape of the Dbrane. July 20, 2005 1. Introduction and