82,660 research outputs found

    On the 5d instanton index as a Hilbert series

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    The superconformal index for N=2 5d theories contains a non-perturbative part arising from 5d instantonic operators which coincides with the Nekrasov instanton partition function. In this note, for pure gauge theories, we elaborate on the relation between such instanton index and the Hilbert series of the instanton moduli space. We propose a non-trivial identification of fugacities allowing the computation of the instanton index through the Hilbert series. We show the agreement of our proposal with existing results in the literature, as well as use it to compute the exact index for a pure U(1) gauge theory.Comment: 13 pages, 2 figure

    Hilbert regularity of Z-graded modules over polynomial rings

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    Let M be a finitely generated Z-graded module over the standard graded polynomial ring R=K[X1,…,Xd] with K a field, and let HM(t)=QM(t)/(1−t)d be the Hilbert series of~M. We introduce the Hilbert regularity of~M as the lowest possible value of the Castelnuovo-Mumford regularity for an R-module with Hilbert series HM. Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of~M to the smallest~k such that the power series QM(1−t)/(1−t)k has no negative coefficients. Finally, we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an R-module.The second author was partially supported by the Spanish GovernmentMinisterio de Educaci´on y Ciencia (MEC), grants MTM2007-64704 and MTM2012–36917–C03–03 in cooperation with the European Union in the framework of the founds “FEDER”

    Hilbert Series for Moduli Spaces of Two Instantons

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    The Hilbert Series (HS) of the moduli space of two G instantons on C^2, where G is a simple gauge group, is studied in detail. For a given G, the moduli space is a singular hyperKahler cone with a symmetry group U(2) \times G, where U(2) is the natural symmetry group of C^2. Holomorphic functions on the moduli space transform in irreducible representations of the symmetry group and hence the Hilbert series admits a character expansion. For cases that G is a classical group (of type A, B, C, or D), there is an ADHM construction which allows us to compute the HS explicitly using a contour integral. For cases that G is of E-type, recent index results allow for an explicit computation of the HS. The character expansion can be expressed as an infinite sum which lives on a Cartesian lattice that is generated by a small number of representations. This structure persists for all G and allows for an explicit expressions of the HS to all simple groups. For cases that G is of type G_2 or F_4, discrete symmetries are enough to evaluate the HS exactly, even though neither ADHM construction nor index is known for these cases.Comment: 53 pages, 9 tables, 24 figure

    Constructing equivariant vector bundles via the BGG correspondence

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    We describe a strategy for the construction of finitely generated GG-equivariant Z\mathbb{Z}-graded modules MM over the exterior algebra for a finite group GG. By an equivariant version of the BGG correspondence, MM defines an object F\mathcal{F} in the bounded derived category of GG-equivariant coherent sheaves on projective space. We develop a necessary condition for F\mathcal{F} being isomorphic to a vector bundle that can be simply read off from the Hilbert series of MM. Combining this necessary condition with the computation of finite excerpts of the cohomology table of F\mathcal{F} makes it possible to enlist a class of equivariant vector bundles on P4\mathbb{P}^4 that we call strongly determined in the case where GG is the alternating group on 55 points

    Comments on twisted indices in 3d supersymmetric gauge theories

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    We study three-dimensional N=2{\mathcal N}=2 supersymmetric gauge theories on Σg×S1{\Sigma_g \times S^1} with a topological twist along Σg\Sigma_g, a genus-gg Riemann surface. The twisted supersymmetric index at genus gg and the correlation functions of half-BPS loop operators on S1S^1 can be computed exactly by supersymmetric localization. For g=1g=1, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on R2×S1{\mathbb R}^2 \times S^1. This also provides a powerful and simple tool to study 3d N=2{\mathcal N}=2 Seiberg dualities. Finally, we study A- and B-twisted indices for N=4{\mathcal N}=4 supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the S2×S1S^2 \times S^1 twisted indices and the Hilbert series of N=4{\mathcal N}=4 moduli spaces.Comment: 66 pages plus appendix; v2: corrected typos and added reference
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