71,947 research outputs found
Eisenstein Series in String Theory
We discuss the relevance of Eisenstein series for representing certain
G(Z)-invariant string theory amplitudes which receive corrections from BPS
states only. The Eisenstein series are constructed using G(Z)-invariant mass
formulae and are manifestly invariant modular functions on the symmetric space
K\G(R) of non-compact type, with K the maximal compact subgroup of G(R). In
particular, we show how Eisenstein series of the T-duality group SO(d,d,Z) can
be used to represent one- and g-loop amplitudes in compactified string theory.
We also obtain their non-perturbative extensions in terms of the Eisenstein
series of the U-duality group E_{d+1(d+1)}(Z).Comment: 11 pages, Latex, submitted to Proceedings of Strings '99, published
versio
Central extensions of mapping class groups from characteristic classes
Tangential structures on smooth manifolds, and the extension of mapping class groups they induce, admit a natural formulation in terms of higher (stacky) differential geometry. This is the literal translation of a classical construction in differential topology to a sophisticated language, but it has the advantage of emphasizing how the whole construction naturally emerges from the basic idea of working in slice categories. We characterize, for every higher smooth stack equipped with tangential structure, the induced higher group extension of the geometric realization of its higher automor- phism stack. We show that when restricted to smooth manifolds equipped with higher degree topological structures, this produces higher extensions of homotopy types of diffeomorphism groups. Passing to the groups of connected components, we obtain abelian extensions of mapping class groups and we derive sufficient conditions for these being central. We show as a special case that this provides an elegant re-construction of Segal’s approach to -extensions of mapping class groups of surfaces that provides the anomaly cancellation of the modular functor in Chern-Simons theory. Our construction generalizes Segal’s approach to higher central extensions of mapping class groups of higher dimensional manifolds with higher tangential structures, expected to provide the analogous anomaly cancellation for higher dimensional TQFTs
A multi-Frey approach to Fermat equations of signature
In this paper, we give a resolution of the generalized Fermat equations for all integers , and all integers which are not a multiple of , respectively,
using the modular method with Frey elliptic curves over totally real fields.
The results require a refined application of the multi-Frey technique, which we
show to be effective in new ways to reduce the bounds on the exponents .
We also give a number of results for the equations , where
, under additional local conditions on the solutions. This includes a
result which is reminiscent of the second case of Fermat's Last Theorem, and
which uses a new application of level raising at modulo .Comment: Includes more details regarding the connection of this paper with its
sequel 'Some extensions of the modular method and Fermat-equations of
signature (13,13,n)'. More precisely: extended Remark 7.4; added details on
the computational parts of the proofs of Proposition 9 and Theorem 2;
included new comments and polished the auxiliary Magma files for Proposition
9 and Theorem
Modular Invariants from Subfactors: Type I Coupling Matrices and Intermediate Subfactors
A braided subfactor determines a coupling matrix Z which commutes with the S-
and T-matrices arising from the braiding. Such a coupling matrix is not
necessarily of "type I", i.e. in general it does not have a block-diagonal
structure which can be reinterpreted as the diagonal coupling matrix with
respect to a suitable extension. We show that there are always two intermediate
subfactors which correspond to left and right maximal extensions and which
determine "parent" coupling matrices Z^\pm of type I. Moreover it is shown that
if the intermediate subfactors coincide, so that Z^+=Z^-, then Z is related to
Z^+ by an automorphism of the extended fusion rules. The intertwining relations
of chiral branching coefficients between original and extended S- and
T-matrices are also clarified. None of our results depends on non-degeneracy of
the braiding, i.e. the S- and T-matrices need not be modular. Examples from
SO(n) current algebra models illustrate that the parents can be different,
Z^+\neq Z^-, and that Z need not be related to a type I invariant by such an
automorphism.Comment: 25 pages, latex, a new Lemma 6.2 added to complete an argument in the
proof of the following lemma, minor changes otherwis
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