41 research outputs found
Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
Motivated by string theory scattering amplitudes that are invariant under a
discrete U-duality, we study Fourier coefficients of Eisenstein series on
Kac-Moody groups. In particular, we analyse the Eisenstein series on ,
and corresponding to certain degenerate principal
series at the values s=3/2 and s=5/2 that were studied in 1204.3043. We show
that these Eisenstein series have very simple Fourier coefficients as expected
for their role as supersymmetric contributions to the higher derivative
couplings and coming from 1/2-BPS and 1/4-BPS
instantons, respectively. This suggests that there exist minimal and
next-to-minimal unipotent automorphic representations of the associated
Kac-Moody groups to which these special Eisenstein series are attached. We also
provide complete explicit expressions for degenerate Whittaker vectors of
minimal Eisenstein series on , and that have not
appeared in the literature before.Comment: 62 pages. Journal versio
Playing it safe: information constrains collective betting strategies
Every interaction of a living organism with its environment involves the
placement of a bet. Armed with partial knowledge about a stochastic world, the
organism must decide its next step or near-term strategy, an act that
implicitly or explicitly involves the assumption of a model of the world.
Better information about environmental statistics can improve the bet quality,
but in practice resources for information gathering are always limited. We
argue that theories of optimal inference dictate that ``complex'' models are
harder to infer with bounded information and lead to larger prediction errors.
Thus, we propose a principle of ``playing it safe'' where, given finite
information gathering capacity, biological systems should be biased towards
simpler models of the world, and thereby to less risky betting strategies. In
the framework of Bayesian inference, we show that there is an optimally safe
adaptation strategy determined by the Bayesian prior. We then demonstrate that,
in the context of stochastic phenotypic switching by bacteria, implementation
of our principle of ``playing it safe'' increases fitness (population growth
rate) of the bacterial collective. We suggest that the principle applies
broadly to problems of adaptation, learning and evolution, and illuminates the
types of environments in which organisms are able to thrive.Comment: 23 pages, 10 figure
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 fundamental domains and volumes of hyperbolic Coxeter-Weyl groups
We present a simple method for determining the shape of fundamental domains
of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody
algebras. These domains are given as subsets of certain generalized upper half
planes, on which the Weyl groups act via generalized modular transformations.
Our construction only requires the Cartan matrix of the underlying
finite-dimensional Lie algebra and the associated Coxeter labels as input
information. We present a simple formula for determining the volume of these
fundamental domains. This allows us to re-produce in a simple manner the known
values for these volumes previously obtained by other methods.Comment: v2: to be published in Lett Math Phys (reference added, typo
corrected
COST 733 – WG4: Applications of weather type classifications
Presentación realizada para: European Geosciences Union General Assembly celebrado del 19-24 de abril de 2009 en Viena
Eisenstein series for infinite-dimensional U-duality groups
We consider Eisenstein series appearing as coefficients of curvature
corrections in the low-energy expansion of type II string theory four-graviton
scattering amplitudes. We define these Eisenstein series over all groups in the
E_n series of string duality groups, and in particular for the
infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that,
remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains
only a finite number of terms for particular choices of a parameter appearing
in the definition of the series. This resonates with the idea that the constant
term of the Eisenstein series encodes perturbative string corrections in
BPS-protected sectors allowing only a finite number of corrections. We underpin
our findings with an extensive discussion of physical degeneration limits in
D<3 space-time dimensions.Comment: 69 pages. v2: Added references and small additions, to be published
in JHE
Electric dipole moments and the search for new physics
Static electric dipole moments of nondegenerate systems probe mass scales for
physics beyond the Standard Model well beyond those reached directly at high
energy colliders. Discrimination between different physics models, however,
requires complementary searches in atomic-molecular-and-optical, nuclear and
particle physics. In this report, we discuss the current status and prospects
in the near future for a compelling suite of such experiments, along with
developments needed in the encompassing theoretical framework.Comment: Contribution to Snowmass 2021; updated with community edits and
endorsement
Bronze IHES Logo Plaque
A bronze dedication plaque from the Institut Hautes des Etudes Scientifique (IHES) documenting the name of the sculptorhttps://orb.binghamton.edu/mathematical_sculptures/1384/thumbnail.jp