82,660 research outputs found
On the 5d instanton index as a Hilbert series
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
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
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
We describe a strategy for the construction of finitely generated
-equivariant -graded modules over the exterior algebra for a
finite group . By an equivariant version of the BGG correspondence,
defines an object in the bounded derived category of
-equivariant coherent sheaves on projective space. We develop a necessary
condition for being isomorphic to a vector bundle that can be
simply read off from the Hilbert series of . Combining this necessary
condition with the computation of finite excerpts of the cohomology table of
makes it possible to enlist a class of equivariant vector bundles
on that we call strongly determined in the case where is the
alternating group on points
Comments on twisted indices in 3d supersymmetric gauge theories
We study three-dimensional supersymmetric gauge theories on
with a topological twist along , a genus-
Riemann surface. The twisted supersymmetric index at genus and the
correlation functions of half-BPS loop operators on can be computed
exactly by supersymmetric localization. For , 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 . This also provides a
powerful and simple tool to study 3d Seiberg dualities.
Finally, we study A- and B-twisted indices for 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 twisted indices and the Hilbert series of
moduli spaces.Comment: 66 pages plus appendix; v2: corrected typos and added reference
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