5,323 research outputs found
The spectral problem of the ABJ Fermi gas
The partition function on the three-sphere of ABJ theory can be rewritten
into a partition function of a non-interacting Fermi gas, with an accompanying
one-particle Hamiltonian. We study the spectral problem defined by this
Hamiltonian. We determine the exact WKB quantization condition, which involves
quantities from refined topological string theory, and test it successfully
against numerical calculations of the spectrum.Comment: 31 page
Extrapolating an Euler class
Let be a noetherian ring of dimension and let be an integer so
that . Let be a unimodular row so that
the ideal has height . Jean Fasel has associated to this
row an element in the Euler class group , with
given by . If
contains an infinite field then we show that the rule of Fasel defines a
homomorphism from to . The main
problem is to get a well defined map on all of . Similar results
have been obtained by Mrinal Kanti Das and MD Ali Zinna, with a different
proof. Our proof uses that every Zariski open subset of is path
connected for walks made up of elementary matrices.Comment: 7 pages, reference update
From Mennicke symbols to Euler class groups
Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian
groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class
group. We suggest that this is the last map in a longer exact sequence of
abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the
sequence is an orbit set of unimodular two by n matrices over the ring A. If n
is at least four we describe a partially defined operation on two by n
matrices. We conjecture that this operation describes a group structure on G if
A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup
of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.Comment: 11 pages, to appear in the Proceedings of the International
Colloquium on Algebra, Arithmetic and Geometry. TIFR, Mumbai, January 4-12,
200
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