138 research outputs found
One-Loop BPS amplitudes as BPS-state sums
Recently, we introduced a new procedure for computing a class of one-loop
BPS-saturated amplitudes in String Theory, which expresses them as a sum of
one-loop contributions of all perturbative BPS states in a manifestly T-duality
invariant fashion. In this paper, we extend this procedure to all BPS-saturated
amplitudes of the form \int_F \Gamma_{d+k,d} {\Phi}, with {\Phi} being a weak
(almost) holomorphic modular form of weight -k/2. We use the fact that any such
{\Phi} can be expressed as a linear combination of certain absolutely
convergent Poincar\'e series, against which the fundamental domain F can be
unfolded. The resulting BPS-state sum neatly exhibits the singularities of the
amplitude at points of gauge symmetry enhancement, in a chamber-independent
fashion. We illustrate our method with concrete examples of interest in
heterotic string compactifications.Comment: 42 pages; v4: a few misprints correcte
Rankin-Selberg periods for spherical principal series
By the unfolding method, Rankin-Selberg L-functions for can be expressed in terms of period integrals. These
period integrals actually define invariant forms on tensor products of the
relevant automorphic representations. By the multiplicity-one theorems due to
Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and
can therefore be related to invariant forms on equivalent principal series
representations. We construct meromorphic families of such invariant forms for
spherical principal series representations of and
conjecture that their special values at the spherical vectors agree in absolute
value with the archimedean local L-factors of the corresponding L-functions. We
verify this conjecture in several cases.
This work can be viewed as the first of two steps in a technique due to
Bernstein-Reznikov for estimating L-functions using their period integral
expressions.Comment: 25 pages. v3 replaces the previous versions which have a gap in Lemma
4.1 due to the non-co-compactness of the lattic
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction
On Sums of SL(3,Z) Kloosterman Sums
We show that sums of the SL(3,Z) long element Kloosterman sum against a
smooth weight function have cancellation due to the variation in argument of
the Kloosterman sums, when each modulus is at least the square root of the
other. Our main tool is Li's generalization of the Kuznetsov formula on
SL(3,R), which has to date been prohibitively difficult to apply. We first
obtain analytic expressions for the weight functions on the Kloosterman sum
side by converting them to Mellin-Barnes integral form. This allows us to relax
the conditions on the test function and to produce a partial inversion formula
suitable for studying sums of the long-element SL(3,Z) Kloosterman sums.Comment: 44 pages, 1 figure, Revised version accepted by the Ramanujan Journa
Rapid computation of -functions attached to Maass forms
Let be a degree- -function associated to a Maass cusp form. We
explore an algorithm that evaluates values of on the critical line in
time . We use this algorithm to rigorously compute an
abundance of consecutive zeros and investigate their distribution
Period integrals and Rankin-Selberg L-functions on GL(n)
We compute the second moment of a certain family of Rankin-Selberg
-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n).
Our bound is as strong as the Lindel\"of hypothesis on average, and recovers
individually the convexity bound. This result is new even in the classical case
n=2.Comment: accepted version with minor change
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