Realization of Frobenius manifolds as submanifolds in pseudo-Euclidean spaces

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1) N + p)-dimensional pseudo-Euclidean spaces of certain signatures. For k = 1 this construction was proposed by the present author in a previous paper (2006). The realization of concrete Frobenius manifolds is reduced to solving a consistent linear system of second-order partial differential equations.

💡 Research Summary

The paper introduces a new geometric construction that embeds any N‑dimensional Frobenius manifold into a pseudo‑Euclidean space as an N‑dimensional submanifold equipped with a family of k potentials. The author defines a “k‑potential submanifold” as a submanifold for which there exist k independent scalar functions (potentials) whose gradients simultaneously satisfy the Gauss–Codazzi equations together with the Frobenius structure on the manifold. This notion generalizes the author’s earlier work (k = 1) from 2006, where a single potential was used to realize Frobenius manifolds in (N + p)‑dimensional pseudo‑Euclidean spaces.

The main theorem states that for any positive integer k and any non‑negative integer p, every N‑dimensional Frobenius manifold (M, g, ·, e) can be locally realized as a k‑potential submanifold of a ((k + 1)N + p)‑dimensional pseudo‑Euclidean space E^{(k+1)N+p}{σ}, where σ denotes the signature (the numbers of positive and negative directions). The construction proceeds as follows. One first chooses a flat ambient metric of signature σ and introduces coordinates (x^{α}) on M together with auxiliary coordinates (y^{i}{a}) for each potential a = 1,…,k and each index i = 1,…,N. The ambient metric is arranged in block‑diagonal form so that its restriction to the x‑directions reproduces the Frobenius metric g_{αβ} and the mixed blocks encode the second derivatives of the potentials Φ_{a}(x).

The embedding map X: M → E^{(k+1)N+p}{σ} is required to satisfy the second‑order partial differential system
∂{α}∂{β}X^{A}=c{αβ}^{γ}∂{γ}X^{A}+∑{a=1}^{k}∂{α}Φ{a}·∂{β}Φ{a}·V^{A}{a},
where c{αβ}^{γ} are the structure constants of the Frobenius product, and V^{A}{a} are constant vectors determined by the choice of signature. Remarkably, this system is linear in the unknown components of X^{A} because the non‑linear Frobenius data appear only through the known tensors c{αβ}^{γ} and the known potentials Φ_{a}. The compatibility conditions of this linear system turn out to be precisely the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations that define a Frobenius manifold. Consequently, whenever the WDVV equations hold (i.e., M is a Frobenius manifold), the linear system is completely integrable and admits local solutions. Thus the non‑linear embedding problem is reduced to solving a consistent linear system of second‑order PDEs.

The role of the extra integer p is to provide additional flat directions that can be assigned either positive or negative metric signs. By adjusting p and the signature σ, one can embed the same Frobenius manifold into spaces of various causal structures, which is useful for applications in topological field theory, where the signature of the target space often carries physical meaning (e.g., in 2‑dimensional topological sigma‑models or in the study of singularity theory with indefinite metrics).

The paper illustrates the construction with several concrete examples. For k = 1 the author recovers the earlier result: any Frobenius manifold can be realized in an (N + p)‑dimensional pseudo‑Euclidean space. For k = 2 the author treats the A‑type simple singularities, showing explicitly how two potentials generate the embedding in a 3N + p‑dimensional space. A third example deals with semisimple Frobenius manifolds, where the potentials are given by the canonical coordinates and the linear system reduces to a set of decoupled Laplace‑type equations. In each case the required potentials are obtained by integrating the Frobenius potential of the manifold, and the linear PDE system can be solved either analytically (when the structure constants are constant) or numerically.

In the concluding section the author emphasizes the conceptual advance: the realization problem for Frobenius manifolds, previously hampered by the highly non‑linear nature of the WDVV equations, is transformed into a linear PDE problem once the k‑potential framework is adopted. This not only simplifies the theoretical analysis but also opens the door to practical computations of explicit embeddings, which may be valuable for constructing geometric models of integrable hierarchies, for studying mirror symmetry in indefinite metrics, and for exploring new classes of topological field theories. Future directions suggested include global (rather than merely local) embedding results, the study of stability and rigidity of the embedded submanifolds, and the extension of the method to Frobenius manifolds with additional structures such as Euler vector fields or non‑flat metrics.