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 $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n \leq d\leq 2n-3$. Let $(a_1,...,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,...,a_n)$ has height $n$. Jean Fasel has associated to this row an element $[(J,\omega_J)]$ in the Euler class group $E^n(R)$, with $\omega_J:(R/J)^n\to J/J^2$ given by $(a_1,...,a_{n-1},a_n a_{n+1})$. If $R$ contains an infinite field $F$ then we show that the rule of Fasel defines a homomorphism from $WMS_{n+1}(R)=Um_{n+1}(R)/E_{n+1}(R)$ to $E^n(R)$. The main problem is to get a well defined map on all of $Um_{n+1}(R)$. 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 $SL_{n+1}(F)$ 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