Second N=1 Superanalog of Complex Structure (original) (raw)

Noninvertible N=1 superanalog of complex structure

Journal of Mathematical Physics, 1997

We found another N = 1 odd superanalog of complex structure (the even one is widely used in the theory of super Riemann surfaces). New N = 1 superconformal-like transformations are similar to anti-holomorphic ones of nonsupersymmetric complex function theory. They are dual to the ordinary superconformal transformations subject to the Berezinian addition formula presented, noninvertible, highly degenerated and twist parity of the tangent space in the standard basis. They also lead to the "mixed cocycle condition" which can be used in building noninvertible objects analogous to super Riemann surfaces. A new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations.

Superconformal-like transformations and nonlinear realizations

1998

We consider various properties of N = 1 superconformal-like transformations which generalize conformal transformations to supersymmetric and noninvertible case. Alternative tangent space reduction in N = 1 superspace leads to some new transformations which are similar to the anti-holomorphic ones of the complex function theory, which g i v es new odd N = 1 superanalog of complex structure. They are dual to the ordinary superconformal transformations subject to the Berezinian addition formula presented, noninvertible, highly degenerated and twist parity of the tangent space in the standard basis, and they also lead to some "mixed cocycle condition". A new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations. The nonlinear realization of invertible and noninvertible N = 1 superconformal-like transformations is studied by means of the odd curve motion technique and introduced clear diagrammatic method. 1991 A.M.S. Subject Classi cation Codes.

The Superconformal Structures of Super Riemann Surfaces

Progress of Theoretical Physics, 1991

It is shown that every super Riemann surface has a unique superconformal structure in a framework which provides a proof of the fact that super Riemann surfaces in the sense of Kostant are equivalent to Riemann surfaces with spin structures.

The geometry of super Riemann surfaces

1988

We define super Riemann surfaces as smooth 2{2-dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2 x 2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the torsion constraints of 2d supergravity. We show that they are both necessary and sufficient for a frame to admit local superconformal coordinates. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Civita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface cannot be regarded as having only one complex dimension. Nevertheless, in certain important aspects super Riemann surfaces behave as nicely as if they had only one dimension. In particular they posses an analog ~ of the Cauchy-Riemann operator on ordinary Riemann surfaces, a differential operator taking values in the bundle of half-volume forms. This operator furnishes a short resolution of the structure sheaf, making possible a Quillen theory of determinant line bundles. Finally we show that the moduli space of super Riemann surfaces is embedded in the larger space of complex curves of dimension 111.

Conformal Field Theory on Super Riemann Surfaces

... the fermionic case, one must also include a jacobian for fixing 6, O. The (super)jacobian for ... action is thus 1 Sconf 47 f d2zd2e [ZDXf'DXG,,(X)+BDC+BDC], (3) and its ... For the closed superstring of Green and Schwarz [29], both halves are representations of the superconformal ...

Generalized Complex Manifolds and Supersymmetry

Communications in Mathematical Physics, 2005

We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two-dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in context of deformation quantization. the formalism can be seen as the two complementary pictures of string theory -from the world-sheet and from the supergravity point of view. Pure spinors naturally emerge in the low-energy context, in which, in particular, the above mentioned mirror symmetry proposal was formulated. In this paper, we are going to see how generalized complex structures emerge from the world-sheet point of view.

Geometry of superconformal manifolds

Communications in Mathematical Physics, 1988

The main facts about complex curves are generalized to superconformal manifolds. The results thus obtained are relevant to the fermion string theory and, in particular, they are useful for computation of determinants of super laplacians which enter the string partition function.

(Super)conformal algebra on the (super)torus

A generalization of the Virasoro algebra has recently been introduced by Krichever and Novikov (KN). The KN algebra describes the algebra of general conformal transformations in a basis appropriate to a genus-g Riemann surface. We examine in detail the genus-one KN algebra, and find explicit expressions for the central extension. We, further, construct explicitly the superconformal algebra of the supertorus, which yields supersymmetric generalizations of the genus-one KN algebra. A novel feature of the odd-spin-structure case is that the algebra includes a central element which is anticommuting. We comment on possible applications to string theory.

The dual superconformal surface

Annals of Global Analysis and Geometry, 2015

It is shown that a superconformal surface with arbitrary codimension in flat Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is S-Willmore, the latter a well-known necessary condition to allow a dual as shown by Ma [12]. Duality means that both surfaces envelope the same central sphere congruence and are conformal with the induced metric. Our main result is that the dual surface to a superconformal surface can easily be described in parametric form in terms of a parametrization of the latter. Moreover, it is shown that the starting surface is conformally equivalent, up to stereographic projection in the nonflat case, to a minimal surface in a space form (hence, S-Willmore) if and only if either the dual degenerates to a point (flat case) or the two surfaces are conformally equivalent (nonflat case).