11 research outputs found
Automorphic forms: a physicist's survey
Motivated by issues in string theory and M-theory, we provide a pedestrian
introduction to automorphic forms and theta series, emphasizing examples rather
than generality.Comment: 22 pages, to appear in the Proceedings of Les Houches Winter School
``Frontiers in Number Theory, Physics and Geometry'', March 9-21, 2003; v2:
minor changes and clarifications, section 3.5 on pure spinors has been
rewritte
Minimal representations, spherical vectors, and exceptional theta series I
Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantized BPS membrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group is simply laced. Specifically, we review and construct explicitly the minimal representation of which generalizes the Schr\"odinger representation of symplectic groups. The real spherical vector invariant under the maximal compact subgroup is computed in this representation and yields the action appearing in the summand of the automorphic theta series. The summation measure can be obtained from the p-adic form of the spherical vector and is left to the sequel of this paper. The simplicity of our result is suggestive of a new Born-Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with hidden non-compact symmetries
couplings, the fundamental membrane and exceptional theta correspondences
This letter is an attempt to carry out a first-principle computation in M-theory using the point of view that the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS couplings in M-theory compactified on a torus from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an Sl(3,\Zint) modular invariance hidden in the light-cone gauge (but obvious in the Polyakov approach), and recover the correct classical spectrum and membrane instantons; the summation measure however is incorrect. It is argued that the correct membrane amplitude should be given by an exceptional theta correspondence lifting Sl(3,\Zint) modular forms to \exc(\Zint) automorphic forms, generalizing the usual theta lift between Sl(2,\Zint) and SO(d,d,\Zint) in string theory. The exceptional correspondence offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes
Quantum Cosmology and Conformal Invariance
According to Belinsky, Khalatnikov and Lifshitz, gravity near a space-like
singularity reduces to a set of decoupled one-dimensional mechanical models at
each point in space. We point out that these models fall into a class of
conformal mechanical models first introduced by de Alfaro, Fubini and Furlan
(DFF). The deformation used by DFF to render the spectrum discrete corresponds
to a negative cosmological constant. The wave function of the universe is the
zero-energy eigenmode of the Hamiltonian, also known as the spherical vector of
the representation of the conformal group SO(1,2). A new class of conformal
quantum mechanical models is constructed, based on the quantization of
nilpotent coadjoint orbits, where the conformal group is enhanced to an ADE
non-compact group for which the spherical vector is known.Comment: 4 pages, latex2e, uses revtex
BRST Detour Quantization
We present the BRST cohomologies of a class of constraint (super) Lie
algebras as detour complexes. By giving physical interpretations to the
components of detour complexes as gauge invariances, Bianchi identities and
equations of motion we obtain a large class of new gauge theories. The pivotal
new machinery is a treatment of the ghost Hilbert space designed to manifest
the detour structure. Along with general results, we give details for three of
these theories which correspond to gauge invariant spinning particle models of
totally symmetric, antisymmetric and K\"ahler antisymmetric forms. In
particular, we give details of our recent announcement of a (p,q)-form K\"ahler
electromagnetism. We also discuss how our results generalize to other special
geometries.Comment: 43 pages, LaTeX, added reference
BPS black holes, quantum attractor flows and automorphic forms
We propose a program for counting microstates of four-dimensional BPS black
holes in N >= 2 supergravities with symmetric-space valued scalars by
exploiting the symmetries of timelike reduction to three dimensions. Inspired
by the equivalence between the four dimensional attractor flow and geodesic
flow on the three-dimensional scalar manifold, we radially quantize stationary,
spherically symmetric BPS geometries. Connections between the topological
string amplitude, attractor wave function, the Ooguri-Strominger-Vafa
conjecture and the theory of automorphic forms suggest that black hole
degeneracies are counted by Fourier coefficients of modular forms for the
three-dimensional U-duality group, associated to special "unipotent"
representations which appear in the supersymmetric Hilbert space of the quantum
attractor flow.Comment: 9 pages, revtex; v2: references added and typos correcte
Quantum Attractor Flows
Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a
holographic correspondence in the mini-superspace approximation, we study the
radial quantization of stationary, spherically symmetric black holes in four
dimensions. A key ingredient is the classical equivalence between the radial
evolution equation and geodesic motion of a fiducial particle on the moduli
space M^*_3 of the three-dimensional theory after reduction along the time
direction. In the case of N=2 supergravity, M^*_3 is a para-quaternionic-Kahler
manifold; in this case, we show that BPS black holes correspond to a particular
class of geodesics which lift holomorphically to the twistor space Z of M^*_3,
and identify Z as the BPS phase space. We give a natural quantization of the
BPS phase space in terms of the sheaf cohomology of Z, and compute the exact
wave function of a BPS black hole with fixed electric and magnetic charges in
this framework. We comment on the relation to the topological string amplitude,
extensions to N>2 supergravity theories, and applications to automorphic black
hole partition functions.Comment: 43 pages, 6 figures; v2: typos and references added; v3: published
version, minor change
Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
In these lecture notes, we review some recent developments on the relation
between the macroscopic entropy of four-dimensional BPS black holes and the
microscopic counting of states, beyond the thermodynamical, large charge limit.
After a brief overview of charged black holes in supergravity and string
theory, we give an extensive introduction to special and very special geometry,
attractor flows and topological string theory, including holomorphic anomalies.
We then expose the Ooguri-Strominger-Vafa (OSV) conjecture which relates
microscopic degeneracies to the topological string amplitude, and review
precision tests of this formula on ``small'' black holes. Finally, motivated by
a holographic interpretation of the OSV conjecture, we give a systematic
approach to the radial quantization of BPS black holes (i.e. quantum
attractors). This suggests the existence of a one-parameter generalization of
the topological string amplitude, and provides a general framework for
constructing automorphic partition functions for black hole degeneracies in
theories with sufficient degree of symmetry.Comment: 103 pages, 8 figures, 21 exercises, uses JHEP3.cls; v5: important
upgrade, prepared for the proceedings of Frascati School on Attractor
Mechanism; Sec 7 was largely rewritten to incorporate recent progress; more
figures, more refs, and minor changes in abstract and introductio