90,229 research outputs found
Elliptic Genera of Symmetric Products and Second Quantized Strings
In this note we prove an identity that equates the elliptic genus partition
function of a supersymmetric sigma model on the N-fold symmetric product
of a manifold M to the partition function of a second quantized
string theory on the space . The generating function of these
elliptic genera is shown to be (almost) an automorphic form for O(3,2,Z). In
the context of D-brane dynamics, this result gives a precise computation of the
free energy of a gas of D-strings inside a higher-dimensional brane.Comment: 17 pages, latex, 1 figure, to appear in Commun. Math. Phy
A new construction of Eisenstein's completion of the Weierstrass zeta function
In the theory of elliptic functions and elliptic curves, the Weierstrass
function (which is essentially an antiderivative of the Weierstrass
function) plays a prominent role. Although it is not an elliptic
function, Eisenstein constructed a simple (non-holomorphic) completion of this
form which is doubly periodic. This theorem has begun to play an important role
in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as
well as Alfes, Griffin, Ono, and the author. In particular, this simple
completion of provides a powerful method to construct harmonic Maass
forms of weight zero which serve as canonical lifts under the differential
operator of weight 2 cusp forms, and this has been shown in to have
deep applications to determining vanishing criteria for central values and
derivatives of twisted Hasse-Weil -functions for elliptic curves.
Here we offer a new and motivated proof of Eisenstein's theorem, relying on
the basic theory of differential operators for Jacobi forms together with a
classical identity for the first quasi-period of a lattice. A quick inspection
of the proof shows that it also allows one to easily construct more general
non-holomorphic elliptic functions.Comment: 3 pages, minor additions and correction
Classification of totally real elliptic Lefschetz fibrations via necklace diagrams
We show that totally real elliptic Lefschetz fibrations that admit a real
section are classified by their "real loci" which is nothing but an
-valued Morse function on the real part of the total space. We assign to
each such real locus a certain combinatorial object that we call a
\emph{necklace diagram}. On the one hand, each necklace diagram corresponds to
an isomorphism class of a totally real elliptic Lefschetz fibration that admits
a real section, and on the other hand, it refers to a decomposition of the
identity into a product of certain matrices in . Using an algorithm
to find such decompositions, we obtain an explicit list of necklace diagrams
associated with certain classes of totally real elliptic Lefschetz fibrations.
Moreover, we introduce refinements of necklace diagrams and show that refined
necklace diagrams determine uniquely the isomorphism classes of the totally
real elliptic Lefschetz fibrations which may not have a real section. By means
of necklace diagrams we observe some interesting phenomena underlying special
feature of real fibrations.Comment: 25 pages, 30 figure
Black Hole Entropy Associated with Supersymmetric Sigma Model
By means of an identity that equates elliptic genus partition function of a
supersymmetric sigma model on the -fold symmetric product of
(, is the symmetric group of elements) to the
partition function of a second quantized string theory, we derive the
asymptotic expansion of the partition function as well as the asymptotic for
the degeneracy of spectrum in string theory. The asymptotic expansion for the
state counting reproduces the logarithmic correction to the black hole entropy.Comment: 11 pages, no figures, version to appear in the Phys. Rev. D (2003
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