4,280 research outputs found
The Genus Two Free Boson in Arakelov Geometry
Using Arakelov geometry, we compute the partition function of the noncompact
free boson at genus two. We begin by compiling a list of modular invariants
which appear in the Arakelov theory of Riemann surfaces. Using these
quantities, we express the genus two partition function as a product of modular
forms, as in the well-known genus one case. We check that our result has the
expected obstruction to holomorphic factorization and behavior under
degeneration.Comment: 21 pages; added subleading order degeneration calculation
Asymptotic expansions, -values and a new Quantum Modular Form
In 2010 Zagier introduced the notion of a quantum modular form. One of his
first examples was the "strange" function of Kontsevich. Here we produce
a new example of a quantum modular form by making use of some of Ramanujan's
mock theta functions. Using these functions and their transformation behaviour,
we also compute asymptotic expansions similar to expansions of .Comment: 7 page
Nonstandard Parafermions and String Compactification
Nonstandard parafermions are built and their central charges and dimensions
are calculated. We then construct new N=2 superconformal field theories by
tensoring the parafermions with a free boson. We study the spectrum and modular
transformations of these theories. Superstring and heterotic strings in four
dimensions are then obtained by tensoring the new superconformal field theories
along with some minimal models. The generations and antigenerations are
studied. We give an example of the theory which is shown to have
three net generations.Comment: 29 pages; typos corrected and some remarks adde
A Framework for Modular Properties of False Theta Functions
False theta functions closely resemble ordinary theta functions, however they
do not have the modular transformation properties that theta functions have. In
this paper, we find modular completions for false theta functions, which among
other things gives an efficient way to compute their obstruction to modularity.
This has potential applications for a variety of contexts where false and
partial theta series appear. To exemplify the utility of this derivation, we
discuss the details of its use on two cases. First, we derive a convergent
Rademacher-type exact formula for the number of unimodal sequences via the
Circle Method and extend earlier work on their asymptotic properties. Secondly,
we show how quantum modular properties of the limits of false theta functions
can be rederived directly from the modular completion of false theta functions
proposed in this paper.Comment: 20 page
N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds
We study the elliptic genera of hyperKahler manifolds using the
representation theory of N=4 superconformal algebra. We consider the
decomposition of the elliptic genera in terms of N=4 irreducible characters,
and derive the rate of increase of the multiplicities of half-BPS
representations making use of Rademacher expansion. Exponential increase of the
multiplicity suggests that we can associate the notion of an entropy to the
geometry of hyperKahler manifolds. In the case of symmetric products of K3
surfaces our entropy agrees with the black hole entropy of D5-D1 system.Comment: 25 pages, 1 figur
Regularized inner products of meromorphic modular forms and higher Green's functions
In this paper we study generalizations of quadratic form Poincar\'e series,
which naturally occur as outputs of theta lifts. Integrating against them
yields evaluations of higher Green's functions. For this we require a new
regularized inner product, which is of independent interest
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