Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful normal trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group UM of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ‖xp_‖=τ(p|x|)^1/p, p?1. The main results include the following. The unitary group carries on a rectifiable distance dp induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance dO,p. We prove that the distances and dO,p coincide. Based on this fact, we show that the metric space is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of UM with the p-norm.     

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

On the Frobenius manifolds for cusp singularities

We show that the Frobenius manifold associated to the pair of a cusp singularity and its canonical primitive form is isomorphic to the one constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points. We also calculate the intersection form of the Frobenius manifold.     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy

Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.     

Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal (axiomatic) Gromov-Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction (due to van de Leur) describes the action of the same group on the space of Frobenius structures via the multi-component KP hierarchies. Our main theorem states that the genus zero restriction of the Y.-P. Lee formulas coincides with the tangent van de Leur action.     

Frobenius manifolds from regular classical W-algebras

We obtain polynomial Frobenius manifolds from classical W-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.     

Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures

The Drinfeld-Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of Miura-type transformations.     

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.     

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.     

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a K?hler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42