656 research outputs found
Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs
We study the representation theory of a certain finite group for which
Kloosterman sums appear as character values. This leads us to consider a
concrete family of commuting hermitian matrices which have Kloosterman sums as
eigenvalues. These matrices satisfy a number of "magical" combinatorial
properties and they encode various arithmetic properties of Kloosterman sums.
These matrices can also be regarded as adjacency matrices for multigraphs which
display Ramanujan-like behavior.Comment: 20 page
The balanced Voronoi formulas for GL(n)
In this paper we show how the GL(N) Voronoi summation formula of [MiSc2] can
be rewritten to incorporate hyper-Kloosterman sums of various dimensions on
both sides. This generalizes a formula for GL(4) with ordinary Kloosterman sums
on both sides that was considered by Xiaoqing Li and the first-named author,
and later by the second-named author in [Zho]
Nonperturbative black hole entropy and Kloosterman sums
Non-perturbative quantum corrections to supersymmetric black hole entropy
often involve nontrivial number-theoretic phases called Kloosterman sums. We
show how these sums can be obtained naturally from the functional integral of
supergravity in asymptotically AdS_2 space for a class of black holes. They are
essentially topological in origin and correspond to charge-dependent phases
arising from the various gauge and gravitational Chern-Simons terms and
boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections
are essential to obtain an integer from supergravity in agreement with the
quantum degeneracies, and reveal an intriguing connection between topology,
number theory, and quantum gravity. We give an assessment of the current
understanding of quantum entropy of black holes.Comment: 35 pages; minor changes, JHEP versio
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