[Paper Review] The algebraic formalism of soliton equations over arbitrary base fields

A. Álvarez Vázquez, José M. Muñoz Porras+1 more|arXiv (Cornell University)|Jun 10, 1996

Algebraic Geometry and Number Theory5 references29 citations

TL;DR

This paper develops an algebraic formalism for soliton equations over arbitrary base fields, generalizing infinite-dimensional Grassmannians and determinant bundles using functorial and scheme-theoretic methods. It constructs $τ$-functions and Baker functions over any field—including positive characteristic and global fields—via a formal geometry of local curves, extending classical results beyond complex or $p$-adic settings.

ABSTRACT

The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $τ$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassmannian of a $k$-vector space $V$ in such a way that its rational points are precisely the points of the Grassmannian defined by Segal-Wilson, although the points over an arbitrary $k$-scheme $S$ have been not previously considered. This definition of the functor $\fu{\gr}(V,V^+)$ allows us to prove that it is representable by a separated $k$-scheme $\gr(V,V^+)$. Using the theory of determinants of Knudsen and Mumford, the determinant bundle is constructed. This is one of the main results of the paper because it implies that we can define ``infinite determinants'' in a completely algebraic way.

Motivation & Objective

• To provide a purely algebraic construction of infinite-dimensional Grassmannians and determinant bundles over arbitrary base fields, not restricted to $\mathbb{C}$ or $p$-adic fields.
• To generalize the theory of $\tau$-functions and Baker-Akhiezer functions to arbitrary fields, including positive characteristic and global number fields.
• To establish a formal geometric framework for local curves analogous to global algebraic curves, enabling algebraic constructions of soliton solutions.
• To replace analytic group actions (e.g., $\Gamma = C^\infty(S^1, \mathbb{C}^*)$) with algebraic group schemes over $k$-schemes, using formal Laurent series with nilpotent coefficients.
• To lay the foundation for arithmetic applications, such as Drinfeld moduli spaces and reciprocity laws, by working over $\mathbb{Z}$-schemes and global fields.

Proposed method

• Define the Grassmannian $\operatorname{Gr}(V,V^+)$ as a functor of points $\underline{\operatorname{Gr}}(V,V^+)$, assigning to each $k$-scheme $S$ the set of submodules of $V \otimes_k \mathcal{O}_S$ commensurable with $V^+ \otimes_k \mathcal{O}_S$.
• Prove that this functor is representable by a separated $k$-scheme $\operatorname{Gr}(V,V^+)$, with a universal subbundle $\mathcal{L}_V \subset \pi^*V$.
• Use Knudsen-Mumford determinant theory to construct the determinant bundle $\det(\mathcal{L}_V)$ on $\operatorname{Gr}(V,V^+)$, enabling algebraic definition of infinite determinants.
• Define the group $\Gamma$ of automorphisms as the functor $S \mapsto H^0(S, \mathcal{O}_S)((z))^*$, with elements $f = \sum_{i \geq -N} \lambda_i z^i$ where $\lambda_{-1}, \dots, \lambda_{-N}$ are nilpotent in $\mathcal{O}_S$.
• Construct the $\tau$-function as a section of the dual determinant bundle via the action of $\Gamma$ on the Grassmannian, using the universal section $\tau_U$.
• Define the Baker function $\psi_U = v^{-1} \cdot \beta_U^*(\tau_U)$, where $v$ is a universal invertible element in $k((z)) \hat{\otimes} k\{\{x_i\}\ and $\beta_U^*$ pulls back sections from $\Gamma \times \{U\}$ to $\hat{C} \times \Gamma \times \{U\}$.

Experimental results

Research questions

• RQ1 Can the classical theory of $\tau$-functions and Baker functions in soliton equations be generalized to arbitrary base fields, including positive characteristic and global fields?
• RQ2 How can the group $\Gamma$ of continuous maps $S^1 \to \mathbb{C}^*$ be algebraically replaced in the formalism to work over arbitrary fields?
• RQ3 What is the algebraic-geometric structure of the Grassmannian and its determinant bundle over a general base scheme, and how does it support soliton theory?
• RQ4 Can the formal geometry of local curves (e.g., $\widehat{\operatorname{Spf}} k[[z]]$) serve as an algebraic analogue of global algebraic curves in the context of soliton equations?
• RQ5 How do these constructions extend to arithmetic settings, such as $\mathbb{Z}((z))$ or Drinfeld moduli spaces, and what arithmetic properties emerge?

Key findings

• The Grassmannian $\operatorname{Gr}(V,V^+)$ is representable by a separated $k$-scheme, with a universal subbundle $\mathcal{L}_V \subset \pi^*V$, generalizing the Segal-Wilson construction to arbitrary fields.
• The determinant bundle over $\operatorname{Gr}(V,V^+)$ is constructed algebraically via Knudsen-Mumford theory, allowing the definition of infinite determinants over any base field.
• The $\tau$-function is defined as a global section of the dual determinant bundle, with explicit expression $\tau_U(g) = \frac{\tau_U(g \cdot \phi_1)}{\tau_U(g)}$ in the formal group action.
• The Baker function $\psi_U = v^{-1} \cdot \beta_U^*(\tau_U)$ is constructed as a section of an invertible sheaf over $\widehat{C} \times \Gamma \times \{U\}$, with $v$ a universal invertible element in $k((z)) \hat{\otimes} k\{\{x_i\}
• In characteristic zero, the Baker function recovers the classical expression $\psi_U(z,t) = \left(\frac{\tau_U(t+[z])}{\tau_U(t)}\right) \exp(-\sum t_i z^{-i})$, via the exponential map.
• The formalism extends to $\mathbb{Z}((z))$, enabling the definition of $\tau$-functions and Baker functions for rational points of $\operatorname{Gr}(\mathbb{Z}((z)))$, with arithmetic interpretations.