[Paper Review] Integrable Structure behind WDVV Equations

H. Aratyn, Johan van de Leur|ArXiv.org|Nov 27, 2001

Nonlinear Waves and Solitons4 references4 citations

TL;DR

This paper identifies an integrable structure underlying the WDVV equations through a Riemann-Hilbert problem on the homogeneous $\widehat{GL}(N,\mathbb{C})$ loop group, with reductions via a loop group automorphism of order two yielding a sub-hierarchy of odd symmetry flows. The key result is that dressing matrices of the reduced model provide explicit solutions to the WDVV equations, while Virasoro constraints enforce the required homogeneity of the Darboux-Egoroff system.

ABSTRACT

An integrable structure behind Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations is identified with reduction of a Riemann-Hilbert problem for a homogeneous GL(N, C) loop group. Reduction requires the dressing matrices to be fixed points of a loop group automorphism of order two resulting in a sub-hierarchy of gl(N,C) hierarchy containing only odd symmetry flows. The model possesses Virasoro symmetry and imposing Virasoro constraints ensures homogeneity property of the Darboux-Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.

Motivation & Objective

• To uncover the underlying integrable structure of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.
• To establish a connection between the WDVV equations and the Riemann-Hilbert problem in the context of the $\widehat{GL}(N,\mathbb{C})$ loop group.
• To show that reductions via a loop group automorphism of order two yield a sub-hierarchy of odd symmetry flows that satisfy the WDVV system.
• To demonstrate that Virasoro symmetry and constraints ensure the homogeneity property of the Darboux-Egoroff structure.
• To construct explicit solutions of the WDVV equations using dressing matrices from the reduced integrable hierarchy.

Proposed method

• Formulate a Riemann-Hilbert factorization problem for the homogeneous $\widehat{GL}(N,\mathbb{C})$ loop group, decomposing the wave function into $G_-$ and $G_+$ components.
• Define symmetry flows via the multi-time Riemann-Hilbert problem, with time variables $u_j^{(n)}$ and associated wave functions $\Psi_0$, $\Theta$, and $M$.
• Apply a loop group automorphism of order two to reduce the hierarchy, restricting to odd symmetry flows and fixing the dressing matrices under the involution.
• Use the Cartan involution to enforce symmetry conditions on the rotation coefficients, ensuring the Darboux-Egoroff system is satisfied.
• Introduce Virasoro symmetry and impose Virasoro constraints to guarantee the homogeneity condition $\sum_k u_k \partial_{u_k} \beta_{ij} = -\beta_{ij}$.
• Construct the Frobenius manifold via the matrix $M(u)$, with flat coordinates and metric derived from eigenvectors and the $\eta$-matrix, ensuring the WDVV prepotential structure.

Experimental results

Research questions

• RQ1 How can the WDVV equations be derived from an integrable hierarchy based on the $\widehat{GL}(N,\mathbb{C})$ loop group?
• RQ2 What role does the loop group automorphism of order two play in reducing the hierarchy to a sub-system with only odd flows?
• RQ3 How do Virasoro constraints enforce the homogeneity property of the Darboux-Egoroff system?
• RQ4 In what way do the dressing matrices of the reduced model yield solutions to the WDVV equations?
• RQ5 How is the Frobenius manifold structure, including the metric and prepotential, reconstructed from the reduced Riemann-Hilbert problem?

Key findings

• The Riemann-Hilbert problem on $\widehat{GL}(N,\mathbb{C})$ provides a unified framework for generating solutions to the WDVV equations via dressing matrices.
• Reduction via an involution of order two yields a sub-hierarchy of $\widehat{gl}(N,\mathbb{C})$ containing only odd symmetry flows, preserving integrability.
• Virasoro symmetry and constraints ensure the homogeneity condition $\sum_k u_k \partial_{u_k} \beta_{ij} = -\beta_{ij}$, which is essential for the Darboux-Egoroff system.
• The matrix $M(u)$, constructed from the reduced wave function, provides eigenvectors for the Euler vector field, enabling the construction of a local semisimple Frobenius manifold.
• The metric $ds^2 = \sum_i h_i^2 (du_i)^2$ and structure constants $c_{\alpha\beta\gamma} = \sum_i \frac{m_{i\alpha} m_{i\beta} m_{i\gamma}}{m_{i1}}$ are explicitly derived from the dressing matrix $M(u)$, confirming the WDVV prepotential structure.
• The matrix $\mu = S^{-1} \mathcal{V} S$ is diagonal with $\mu_i = -\mu_{N+1-i}$, ensuring the correct scaling dimensions $\mu_\alpha - \mu_1$ for the Frobenius manifold.