Abstract
Moduli spaces of a large set of 3d mathcal{N}=4 effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa’s theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80’s mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently [1] showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as collapse. Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.
Highlights
In the previous work [1] a new relation was found between brane dynamics in Type IIB superstring theory [2] and the geometry of nilpotent orbits in the sl(n) algebra over the field C
This is an interesting link between quantum field theory and the geometry of Lie algebras: the Higgs mechanism can be utilized to reproduce the slicing, in the BrieskornSlodowy sense [7, 8], of the moduli space of the 3d N = 4 theory
This Higgs mechanism has a clear interpretation in terms of the brane dynamics of the superstring embedding of the quantum field theory
Summary
In the previous work [1] a new relation was found between brane dynamics in Type IIB superstring theory [2] and the geometry of nilpotent orbits in the sl(n) algebra over the field C. This paper aims to do the same, to expand the analysis in [1] to the other classical cases. One of the main goals of this paper is to bring attention to the Brieskorn-Slodowy program [7, 8]. This a very interesting way to understand the geometry of a variety and its singularities. We say that locally the variety looks like the direct product S×(G·x) In this way we can learn a great deal about the geometry of a variety and the nature of its singularities by finding subvarieties which are orbits of its isometry group and computing their transverse slices. In the relevant cases of our study, these slices are always singular varieties
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