122 research outputs found
Casimir eigenvalues for universal Lie algebra
For two different natural definitions of Casimir operators for simple Lie
algebras we show that their eigenvalues in the adjoint representation can be
expressed polynomially in the universal Vogel's parameters and give explicit formulae for the generating functions of these
eigenvalues.Comment: Slightly revised versio
Minkowski superspaces and superstrings as almost real-complex supermanifolds
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that
mathematicians study are real or (almost) complex ones, while Minkowski
superspaces are completely different objects. They are what we call almost
real-complex supermanifolds, i.e., real supermanifolds with a non-integrable
distribution, the collection of subspaces of the tangent space, and in every
subspace a complex structure is given.
An almost complex structure on a real supermanifold can be given by an even
or odd operator; it is complex (without "always") if the suitable superization
of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we
define the circumcised analog of the Nijenhuis tensor. We compute it for the
Minkowski superspaces and superstrings. The space of values of the circumcised
Nijenhuis tensor splits into (indecomposable, generally) components whose
irreducible constituents are similar to those of Riemann or Penrose tensors.
The Nijenhuis tensor vanishes identically only on superstrings of
superdimension 1|1 and, besides, the superstring is endowed with a contact
structure. We also prove that all real forms of complex Grassmann algebras are
isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to
related recent work by Witten is adde
Uniformizing the Stacks of Abelian Sheaves
Elliptic sheaves (which are related to Drinfeld modules) were introduced by
Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can
be viewed as function field analogues of elliptic curves and hence are objects
"of dimension 1". Their higher dimensional generalisations are called abelian
sheaves. In the analogy between function fields and number fields, abelian
sheaves are counterparts of abelian varieties. In this article we study the
moduli spaces of abelian sheaves and prove that they are algebraic stacks. We
further transfer results of Cerednik--Drinfeld and Rapoport--Zink on the
uniformization of Shimura varieties to the setting of abelian sheaves. Actually
the analogy of the Cerednik--Drinfeld uniformization is nothing but the
uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper
half space. Our results generalise this uniformization. The proof closely
follows the ideas of Rapoport--Zink. In particular, analogies of -divisible
groups play an important role. As a crucial intermediate step we prove that in
a family of abelian sheaves with good reduction at infinity, the set of points
where the abelian sheaf is uniformizable in the sense of Anderson, is formally
closed.Comment: Final version, appears in "Number Fields and Function Fields - Two
Parallel Worlds", Papers from the 4th Conference held on Texel Island, April
2004, edited by G. van der Geer, B. Moonen, R. Schoo
The hypertoric intersection cohomology ring
We present a functorial computation of the equivariant intersection
cohomology of a hypertoric variety, and endow it with a natural ring structure.
When the hyperplane arrangement associated with the hypertoric variety is
unimodular, we show that this ring structure is induced by a ring structure on
the equivariant intersection cohomology sheaf in the equivariant derived
category. The computation is given in terms of a localization functor which
takes equivariant sheaves on a sufficiently nice stratified space to sheaves on
a poset.Comment: Significant revisions in Section 5, with several corrected proof
Twisting gauged non-linear sigma-models
We consider gauged sigma-models from a Riemann surface into a Kaehler and
hamiltonian G-manifold X. The supersymmetric N=2 theory can always be twisted
to produce a gauged A-model. This model localizes to the moduli space of
solutions of the vortex equations and computes the Hamiltonian Gromov-Witten
invariants. When the target is equivariantly Calabi-Yau, i.e. when its first
G-equivariant Chern class vanishes, the supersymmetric theory can also be
twisted into a gauged B-model. This model localizes to the Kaehler quotient
X//G.Comment: 33 pages; v2: small additions, published versio
Mixed Hodge polynomials of character varieties
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties
M_n of Riemann surfaces by counting points over finite fields using the
character table of the finite group of Lie-type GL(n,F_q) and a theorem proved
in the appendix by N. Katz. We deduce from this calculation several geometric
results, for example, the value of the topological Euler characteristic of the
associated PGL(n,C)-character variety. The calculation also leads to several
conjectures about the cohomology of M_n: an explicit conjecture for its mixed
Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture
relating the pure part to absolutely indecomposable representations of a
certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New
definition for homogeneous weight in Definition 4.1.6, subsequent arguments
modified. Some other minor changes. To appear in Invent. Mat
Ramification theory for varieties over a local field
We define generalizations of classical invariants of wild ramification for
coverings on a variety of arbitrary dimension over a local field. For an l-adic
sheaf, we define its Swan class as a 0-cycle class supported on the wild
ramification locus. We prove a formula of Riemann-Roch type for the Swan
conductor of cohomology together with its relative version, assuming that the
local field is of mixed characteristic.
We also prove the integrality of the Swan class for curves over a local field
as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture
of Serre on the Artin character for a group action with an isolated fixed point
on a regular local ring, assuming the dimension is 2.Comment: 159 pages, some corrections are mad
Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations
In this paper we show a local Jacquet-Langlands correspondence for all
unitary irreducible representations. We prove the global Jacquet-Langlands
correspondence in characteristic zero. As consequences we obtain the
multiplicity one and strong multiplicity one theorems for inner forms of GL(n)
as well as a classification of the residual spectrum and automorphic
representations in analogy with results proved by Moeglin-Waldspurger and
Jacquet-Shalika for GL(n).Comment: 49 pages; Appendix by N. Grba
Lectures on F-theory compactifications and model building
These lecture notes are devoted to formal and phenomenological aspects of
F-theory. We begin with a pedagogical introduction to the general concepts of
F-theory, covering classic topics such as the connection to Type IIB
orientifolds, the geometry of elliptic fibrations and the emergence of gauge
groups, matter and Yukawa couplings. As a suitable framework for the
construction of compact F-theory vacua we describe a special class of
Weierstrass models called Tate models, whose local properties are captured by
the spectral cover construction. Armed with this technology we proceed with a
survey of F-theory GUT models, aiming at an overview of basic conceptual and
phenomenological aspects, in particular in connection with GUT breaking via
hypercharge flux.Comment: Invited contribution to the proceedings of the CERN Winter School on
Supergravity, Strings and Gauge Theory 2010, to appear in Classical and
Quantum Gravity; 63 pages; v2: references added, typos correcte
Fuchs versus Painlev\'e
We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e
VI. We then show that the polynomiality of the expressions of the correlation
functions (and form factors) in terms of the complete elliptic integral of the
first and second kind,
and , is a straight consequence of the fact that the differential
operators corresponding to the entries of Toeplitz-like determinants, are
equivalent to the second order operator which has as solution (or,
for off-diagonal correlations to the direct sum of and ). We show
that this can be generalized, mutatis mutandis, to the anisotropic Ising model.
The singled-out second order linear differential operator being replaced
by an isomonodromic system of two third-order linear partial differential
operators associated with , the Jacobi's form of the complete elliptic
integral of the third kind (or equivalently two second order linear partial
differential operators associated with Appell functions, where one of these
operators can be seen as a deformation of ). We finally explore the
generalizations, to the anisotropic Ising models, of the links we made, in two
previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and
elliptic curves. In particular the elliptic representation of Painlev\'e VI has
to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of
Difference Equations, SIDE VII meeting held in Melbourne during July 200
- …