29,925 research outputs found
On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function
We derive new results about properties of the Witten zeta function associated
with the group SU(3), and use them to prove an asymptotic formula for the
number of n-dimensional representations of SU(3) counted up to equivalence. Our
analysis also relates the Witten zeta function of SU(3) to a summation identity
for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new
proof of that identity and show that it is a special case of a stronger
identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic
Basic hypergeometry of supersymmetric dualities
We introduce several new identities combining basic hypergeometric sums and
integrals. Such identities appear in the context of superconformal index
computations for three-dimensional supersymmetric dual theories. We give both
analytic proofs and physical interpretations of the presented identities.Comment: 25 pages, v2: minor corrections and comment
Simple derivation of general Fierz-type identities
General Fierz-type identities are examined and their well known connection
with completeness relations in matrix vector spaces is shown. In particular, I
derive the chiral Fierz identities in a simple and systematic way by using a
chiral basis for the complex matrices. Other completeness relations
for the fundamental representations of SU(N) algebras can be extracted using
the same reasoning.Comment: 9pages. Few sentences modified in introduction and in conclusion.
Typos corrected. An example added in introduction. Title modifie
The elliptic gamma function and SL(3,Z) x Z^3
The elliptic gamma function is a generalization of the Euler gamma function
and is associated to an elliptic curve. Its trigonometric and rational
degenerations are the Jackson q-gamma function and the Euler gamma function,
respectively. The elliptic gamma function appears in Baxter's formula for the
free energy of the eight-vertex model and in the hypergeometric solutions of
the elliptic qKZB equations. In this paper, the properties of this function are
studied. In particular we show that elliptic gamma functions are
generalizations of automorphic forms of G=SL(3,Z) x Z^3 associated to a
non-trivial class in H^3(G,Z).Comment: 27 pages, LaTeX References added, minor correction
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