The differential geometry of the orbit space of extended affine Jacobi group \(A_1\) (Q2023425)

The paper is motivated by dualities between orbit spaces of certain groups and Hurwitz spaces, both taken as Dubrovin-Frobenius manifolds. For example, the orbit space of the finite Coxeter group \(A_1\) is dual to \(H_{0,1}\), of the Jacobi group \(\mathcal{J}(A_1)\) to \(H_{1,1}\), and of the extended affine Weyl group \(\tilde{A}_1\) to \(H_{0,0,0}\). In this vein, the author constructs an extension of both \(\mathcal{J}(A_1)\) and \(\tilde{A}_1\), named extended affine Jacobi group and denoted \(\mathcal{J}(\tilde{A}_1)\), whose orbit space under the action on \(\mathbb{C}\oplus\mathbb{C}^{2}\oplus\mathbb{H}\) carries the Dubrovin-Frobenius structure of \(H_{1,0,0}\). The WDVV solution is extracted from the data of \(\mathcal{J}(\tilde{A}_1)\) directly, without the Hurwitz space construction, and is written explicitly. More precisely, the main result states that a suitable covering of the orbit space \(\left(\mathbb{C}\oplus\mathbb{C}^{2}\oplus\mathbb{H}\right)/\mathcal{J}(\tilde{A}_1)\) has Dubrovin-Frobenius manifold structure, and is isomorphic to a suitable covering of the Hurwitz space \(H_{1,0,0}\). The orbit space construction of \(\mathcal{J}(\tilde{A}_1)\) can be extended to \(\mathcal{J}(\tilde{A}_n)\), and further to the \(B_n\) and \(D_n\) analogues, and may give rise to some new integrable hierarchies. To define the notion of invariant \(\mathcal{J}(\tilde{A}_1)\) sections the author generalizes the theory of Jacobi forms for \(\mathcal{J}(A_1)\) due to Eichler-Zagier and Wirthmüller to weak \(\tilde{A}_1\)-invariant Jacobi forms. Unlike their holomorphic prototypes, these are only meromorphic in one of the variables over a fixed divisor. Their ring gives the Euler vector field, and a Chevalley-type theorem allows to do differential geometry on the orbit space of \(\mathcal{J}(\tilde{A}_1)\) in their terms. The natural invariant metric on \(\mathcal{J}(\tilde{A}_1)\) gives the intersection form, which, in turn, gives rise to another invariant metric on \(\mathcal{J}(\tilde{A}_1)\), the Saito metric. Proving its flatness is a major technical hurdle dealt with in the paper.