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Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on…
We extend our previous construction of global solutions to Type IIB supergravity that are invariant under the superalgebra $F(4)$ and are realized on a spacetime of the form $AdS_6 \times S^2$ warped over a Riemann surface $\Sigma$ by…
We construct polylogarithms on families of pointed Riemann surfaces of any genus which describe monodromies of meromorphic connections with simple poles. Furthermore, we show that the polylogaritms are computable as power series in…
In this paper we are investigated the monodromy group for linearly polymorphic functions on compact Riemann surface of genus $g \geq 2,$ in connection with standard uniformization of these surfaces by Kleinian groups, and are found a…
We define analytic maps between super Riemann surfaces which extend the notion of branched covering maps to a supersymmetric setting. We show that these super covering maps appear naturally both in symmetric product orbifolds of…
A superconnection is a supermatrix whose even part contains the gauge-potential one-forms of a local gauge group, while the odd parts contain the (0-form) Higgs fields; the combined grading is thus odd everywhere. We demonstrate that the…
An effective action is obtained for the $N=1$, $2D-$induced supergravity on a compact super Riemann surface (without boundary) $\hat\Sigma$ of genus $g>1$, as the general solution of the corresponding superconformal Ward identity. This is…
We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples,…
We propose a new lattice superfield formalism in momentum representation which accommodates species doublers of the lattice fermions and their bosonic counterparts as super multiplets. We explicitly show that one dimensional $N=2$ model…
We construct supergravity theories in twelve and thirteen dimensions with the respective signatures (10,2) and (11,2) with some technical details. Starting with N=1 supergravity in 10+2 dimensions coupled to Green-Schwarz superstring, we…
We investigate a supersymmetric generalisation of topological recursion from two perspectives: algebraic and geometric. The algebraic side concerns a recursive structure encoded in modules of a super Virasoro algebra, and the geometric…
We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice-punctured torus times a 9D Minkowski space-time. It corresponds to a M2-brane formulated in 11D space with ten non-compact…
A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a…
We extend the notion of super-Minkowski space-time to include $\mathbb{Z}_2^n$-graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical…
(Talk given at Strings '93, Berkeley, and at XXVII. Internationales Symposium \"uber Elementarteilchentheorie, Wendisch-Rietz, 1993) We review the superconformal properties of matter coupled to $2d$ gravity, and $W$-extensions thereof. We…
We study Type IIB supergravity solutions with spacetime of the form $AdS_6\times S^2$ warped over a Riemann surface $\Sigma$, where $\Sigma$ includes punctures around which the supergravity fields have non-trivial $SL(2,R)$ monodromy.…
A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables…
We develop an abstract framework for coupled period--realization of meromorphic $1$--forms on punctured Riemann surfaces. A configuration datum $C$ gives the combinatorics and determines a restricted character domain…
We study surface defects in 4d $\mathcal{N}=1$ $SU(N)$ superconformal gauge theories of class $\mathcal{S}_k$ obtained from the 6d (1,0) theories of type $A_{N-1}$, which are worldvolume theories on $N$ M5-branes at…
When a globally supersymmetric theory is scale invariant, it must possess a Virial supercurrent supermultiplet. The multiplet structure is analogous to the R-current supermultiplet in globally R-symmetric theories but we put extra "$i$"s in…