Supercurves, their Jacobians, and super KP equations

Abstract

We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves ("generic SKP curves") we define a period matrix. We show that the odd part of the period matrix controls the cohomology of the dual curve. The Jacobian of a generic SKP curve is a smooth supermanifold; it is principally polarized, hence projective, if the even part of the period matrix is symmetric. In general symmetry is not guaranteed by the Riemann bilinear equations for our contour integration, so it remains open whether Jacobians are always projective or carry theta functions.These results on generic SKP curves are applied to the study of algebro-geometric solutions of the super KP hierarchy. The tau function is shown to be, essentially, a meromorphic section of a line bundle with trivial Chern class on the Jacobian, rationally expressible in terms of super theta functions when these exist. Also we relate the tau function and the Baker function for this hierarchy, using a generalization of Cramer's rule to the supercase.

Berezinians, Exterior Powers and Recurrent Sequences

“…Using the Gelfand-Retakh theory of quasi-determinants, Bergvelt and Rabin in [4] found an analog of Cramer's formula in the supercase. The situation with Cramer's rule, i.e., calculating the inverse of a supermatrix, is a bit peculiar.…”

“…Earlier such a generalization was obtained by Bergveldt and Rabin in [4], who used the 'hard tools' of the Gelfand-Retakh quasi-determinants theory (see [10,9]). Our approach does not use anything but the main properties of the Berezinian.…”

Super KP equations and Darboux transformations: another perspective on the Jacobian Super KP hierarchy

“…It is by doing this that the super universal Grassmannian arises. Let us first of all recall its definition [34,2,38]. Denote by V := Λ((z −1 )) ⊕ Λ((z −1 )) · θ the quotient ring of the ring of formal power series in z −1 and θ over 2 Λ, and let…”

“…the space of orbits of ∂ 2 . Essentially, we have to consider k = 1 and interpret the first two families of equations of motion reported in Table 2 as recursive definitions of the currents, as differential polynomials (in the space variable x = t 2 ) of the generators H (1) and H (2) . With respect to the bosonic case, there is a subtlety, connected with the relation of the first time t 1 of SCS with the fermionic partner ϕ of x.…”

“…Observe that since W is generated by the action of the operator ∂ x +ĥ (2) on the pair (ĥ (0) = 1,ĥ (1) = θa + ϕh), W will be generated by the action of A(∂ x +ĥ (2) ) A −1 on the pair (l (0) ,l (1) ). We notice that…”