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Added by
Alexander Alldridge PD Dr.
Publication date
2013
fields
Physics
and research's language is
English
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We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.
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We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted Kahler-Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms, we prove the existence of singular gravitating vortices whose Kahler metric has conical singularities and Hermitian metric has parabolic singularities. In the Bogomolnyi phase, we establish an existence result for singular Einstein-Bogomolnyi equations, which represents cosmic strings with singularities.
In this paper, we develop results in the direction of an analogue of Sjamaar and Lermans singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kahler manifolds.
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $omega$ in dimension $2n$, i.e. differential 1-forms $gamma$ which vanish at a point and such that $dgamma = omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin cite{Lychagin-1stOrder1975}, Webster cite{Webster-1stOrder1987} and Zhitomirskii cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.
In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in the tautological ring of the moduli space of 2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.
In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the Virasoro conjecture.
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