The purpose of this paper is to describe the projective linear supergroup, its relation with the automorphisms of the projective superspace and to determine the supergroup of SUSY preserving automorphisms of ${mathbf P}^{1|1}$
We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $mathbb{P}^1$ of all degrees in full genera.
We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $rho(g-1)$ for some integer $rhoge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This exte
nds results of Belolipetzky and the second author for $rho>6$, and of the first and third authors for $rho=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; this extends results of Conder, v Sirav n and Tucker for maps.
We consider the KZ differential equations over $mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the s
pace of polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic equations for that subbundle and argue that the equations correspond to highest weight vectors of the associated $hat{sl}_2$ Verma modules over the field $mathbb F_p$.
In this essay we study various notions of projective space (and other schemes) over $mathbb{F}_{1^ell}$, with $mathbb{F}_1$ denoting the field with one element. Our leading motivation is the Hiden Points Principle, which shows a huge deviation betwee
n the set of rational points as closed points defined over $mathbb{F}_{1^ell}$, and the set of rational points defined as morphisms $texttt{Spec}(mathbb{F}_{1^ell}) mapsto mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $mathbb{F}_{1^ell}$-type, and consider their zeta functions.
The problem of fermions in 1+1 dimensions in the presence of a pseudoscalar Coulomb potential plus a mixing of vector and scalar Coulomb potentials which have equal or opposite signs is investigated. We explore all the possible signs of the potential
s and discuss their bound-state solutions for fermions and antifermions. We show the relation between spin and pseudospin symmetries by means of charge-conjugation and $gamma^{5}$ chiral transformations. The cases of pure pseudoscalar and mixed vector-scalar potentials, already analyzed in previous works, are obtained as particular cases. The results presented can be extended to 3+1 dimensions.