Polynomial Modular Frobenius Manifolds

The simplest family of Frobenius manifolds are those for which the prepotential -a solution of the underlying WDVV-equations -is a polynomial in the flat variables and the Saito construction [20] provides a construction of these polynomial solutions on the orbit space C N /W , starting from any finite Coxeter group W . It was conjectured by Dubrovin [7] and later proved by Hertling [12] that all polynomial solutions arise in this way, and hence polynomial Frobenius manifolds have now been completely classified. Beyond polynomial solutions there are trigonometric and elliptic solutions and variations of the original Saito construction, corresponding to extended affine Weyl and Jacobi groups, which provide large classes of examples. One may conjecture that these constructions provide all such solutions within their classes, but currently, unlike in the polynomial case, such conjectures remain unproved.

In [15] the authors introduced the idea of a modular Frobenius manifold. These sit at the fixed points of a natural involutive symmetry that exists on the moduli space of Frobenius manifolds and hence inherit very special properties, basically an invariance under the modular group.

The aim of this paper is to study such modular Frobenius manifolds further and in particular to begin a low-dimensional classification of polynomial examples, where this means that the prepotential is polynomial in the variables t 1 , . . . , t N -1 and transcendental in the last variable t N (throughout this paper N will denote the dimension of the Frobenius manifold). A number of explicit examples have appeared in the literature before:

• the A 1 Jacobi group example [7];

• the D (1,1) 4 solution [24]; • the simple elliptic singularities E 6,7,8 [13,16,21,23,30]. What is perhaps surprising is that the simplest family of examples -those based on the A N -2 -Jacobi group -do not fall into this class, apart from the simplest N = 3 case. In general these are polynomial in the variables t 1 , . . . t N -2 but have rational dependence in the variable t N -1 [3]. Thus polynomial modular Frobenius manifolds seem to be extremely special. Some other solutions may be found in [10] and in the work of Bertola [3] , and an aim of this paper is to provide a systematic analysis of the solution space of polynomial modular manifolds rather than just isolated examples. In fact a close examination of Satake’s construction [23] (itself based on the Saito construction [21]) for the so-called codimension one cases shows that the prepotential will be polynomial and modular. Since there are only a finite number of codimension one examples, a plausible conjecture would be that semi-simple polynomial modular manifolds are actually finite in number.

We recall the basic definitions and symmetries of a Frobenius manifold [7].

2.1. Frobenius Manifolds and the WDVV Equations. Frobenius manifolds were introduced as a way to give a geometric understanding to solutions of the Witten-Dijkraaf-Verlinde-Verlinde (WDVV) equations, (1)

∂t µ ∂t α ∂t γ for some quasihomogeneous function F (t). Throughout this paper η αβ will be defined via η αβ η βκ = δ α κ where

(2) η αβ = ∂ 3 F ∂t 1 ∂t α ∂t β is constant and non-degenerate. We recall briefly how to establish the correspondence between Frobenius manifolds and solutions of WDVV. With this one may define a Frobenius manifold.

Definition 2. Let M be a smooth manifold. M is called a Frobenius manifold if each tangent space T t M is equipped with the structure of a Frobenius algebra varying smoothly with t ∈ M, and further 1. The invariant inner product η defines a flat metric on M. 2. The unity vector field is covariantly constant with respect to the Levi-Civita connection for η,

then the (0, 4) tensor ∇ W c(X, Y, Z) is totally symmetric. 4. There exists a vector field E ∈ Γ(T M) such that ∇∇E = 0 and (5)

E is called the Euler vector field.

Condition 1 implies there exists a choice of coordinates (t 1 , …, t N ) such that the Gram matrix η αβ = (∂ α , ∂ β ) is constant. Furthermore, this may be done in such a way that e = ∂ 1 . In such a coordinate system, partial and covariant derivatives coincide, and condition 3 becomes c αβγ,κ = c αβκ,γ . Successive applications of the Poincaré lemma then implies local existence of a function F (t) called the free energy of the Frobenius manifold such that ( 6)

Defining (η αβ ) -1 = η αβ , the components of • are given by c α βγ = η αε c εβγ . Associativity of • is then equivalent to (1). We assume from now on that ∇E is diagonalizable and that η ij = δ i+j,N +1 . Condition 4 leads to the requirement that F is quasihomogeneous, that is, ( 8)

and, on using the freedom in the definition of flat coordinates, the Euler vector field may be taken to be ( 9)

We will, for the most part, restrict ourselves in this paper to those Frobenius manifolds with the property that r σ = 0 , σ = 1 , . . . , N . So

This excludes those examples coming from quantum cohomology and the extended

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