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… Show more
“…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.…”
Section: Motivation and Backgroundmentioning
confidence: 99%
“…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.…”
Section: Cramer's Rule In Supermathematicsmentioning
Abstract. We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V . We show that traces of exterior powers of A satisfy universal recurrence relations of period q. 'Underlying' recurrence relations hold in the Grothendieck ring of representations of GL(V ). They are expressed by vanishing of certain Hankel determinants of order q + 1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to explicitly express the Berezinian of an operator as a rational function of traces. We analyze the Cayley-Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer's rule.
“…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.…”
Section: Motivation and Backgroundmentioning
confidence: 99%
“…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.…”
Section: Cramer's Rule In Supermathematicsmentioning
Abstract. We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V . We show that traces of exterior powers of A satisfy universal recurrence relations of period q. 'Underlying' recurrence relations hold in the Grothendieck ring of representations of GL(V ). They are expressed by vanishing of certain Hankel determinants of order q + 1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to explicitly express the Berezinian of an operator as a rational function of traces. We analyze the Cayley-Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer's rule.
“…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…”
Section: The Super Central Systemmentioning
confidence: 99%
“…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.…”
Section: Hskp As a "Reduction" Of Scsmentioning
confidence: 99%
“…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…”
Abstract. We generalize to the supersymmetric case the representation of the KP hierarchy as a set of conservation laws for the generating series of the conserved densities. We show that the hierarchy so obtained is isomorphic to the JSKP of Mulase and Rabin. We identify its "bosonic content" with the so-called Darboux-KP hierarchy, which geometrically encompasses the theory of Darboux-Bäcklund transformations, and is an extension both of the KP theory and of the modified KP theory. Finally, we show how the hierarchy can be linearized and how the supersymmetric counterpart of a wide class of rational solution can be quite explicitly worked out.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.