Connections on the total Picard sheaf and the KP hierarchy (Q1909379)

The paper is inspired by \textit{D. Mumford} [Proc. Int. Symp. Alg. Geom. Kyoto 1977, 115-153 (1977; Zbl 0423.14007)] and analysis of the Krichever approach to KP equations [see also \textit{G. Segal} and \textit{G. Wilson}, Publ. Math., Inst. Hautes Étud. Sci. 65, 5-65 (1985; Zbl 0592.35112)]. The construction relies on certain special flat connections \(\nabla\) on the total Picard sheaf \(\mathcal E\) (a direct limit of Poincaré sheaves of a curve) defined on the Jacobian of the curve. To get the KP hierarchy, \(\mathcal E\) is embedded in a formal completion \(\widehat {\mathcal E}\) equipped with the sheaf of pseudodifferential operators. Then \(\mathcal E\) and \(\widehat {\mathcal E}\) naturally trivialize by a connection which is not \(\nabla\), and the difference yields the sought hierarchy.