24 research outputs found
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
Covariant Symplectic Structure and Conserved Charges of Topologically Massive Gravity
We present the covariant symplectic structure of the Topologically Massive
Gravity and find a compact expression for the conserved charges of generic
spacetimes with Killing symmetries.Comment: 10 pages, Dedicated to the memory of Yavuz Nutku (1943-2010),
References added, Conserved charges of non-Einstein solutions of TMG are
added, To appear in Phys. Rev.
Quantum Modular Forms from Real Quadratic Double Sums
In 2015, Lovejoy and Osburn discovered twelve -hypergeometric series and
proved that their Fourier coefficients can be understood as counting functions
of ideals in certain quadratic fields. In this paper, we study their modular
and quantum modular properties and show that they yield three vector-valued
quantum modular forms on the group .Comment: 26 pages; v2: Brief comments added. To appear in the Quarterly
Journal of Mathematic
Wireless Network Simplification: the Gaussian N-Relay Diamond Network
We consider the Gaussian N-relay diamond network, where a source wants to
communicate to a destination node through a layer of N-relay nodes. We
investigate the following question: what fraction of the capacity can we
maintain by using only k out of the N available relays? We show that
independent of the channel configurations and the operating SNR, we can always
find a subset of k relays which alone provide a rate (kC/(k+1))-G, where C is
the information theoretic cutset upper bound on the capacity of the whole
network and G is a constant that depends only on N and k (logarithmic in N and
linear in k). In particular, for k = 1, this means that half of the capacity of
any N-relay diamond network can be approximately achieved by routing
information over a single relay. We also show that this fraction is tight:
there are configurations of the N-relay diamond network where every subset of k
relays alone can at most provide approximately a fraction k/(k+1) of the total
capacity. These high-capacity k-relay subnetworks can be also discovered
efficiently. We propose an algorithm that computes a constant gap approximation
to the capacity of the Gaussian N-relay diamond network in O(N log N) running
time and discovers a high-capacity k-relay subnetwork in O(kN) running time.
This result also provides a new approximation to the capacity of the Gaussian
N-relay diamond network which is hybrid in nature: it has both multiplicative
and additive gaps. In the intermediate SNR regime, this hybrid approximation is
tighter than existing purely additive or purely multiplicative approximations
to the capacity of this network.Comment: Submitted to Transactions on Information Theory in October 2012. The
new version includes discussions on the algorithmic complexity of discovering
a high-capacity subnetwork and on the performance of amplify-and-forwar
ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine
We consider double scaled little string theory on . These theories are
labelled by a positive integer and an root lattice with Coxeter
number . We count BPS fundamental string states in the holographic dual of
this theory using the superconformal field theory . We show that the BPS fundamental string states that are counted
by the second helicity supertrace of this theory give rise to weight two mixed
mock modular forms. We compute the helicity supertraces using two separate
techniques: a path integral analysis that leads to a modular invariant but
non-holomorphic answer, and a Hamiltonian analysis of the contribution from
discrete states which leads to a holomorphic but not modular invariant answer.
From a mathematical point of view the Hamiltonian analysis leads to a mixed
mock modular form while the path integral gives the completion of this mixed
mock modular form. We also compare these weight two mixed mock modular forms to
those that appear in instances of Umbral Moonshine labelled by Niemeier root
lattices that are powers of root lattices and find that they are
equal up to a constant factor that we determine. In the course of the analysis
we encounter an interesting generalization of Appell-Lerch sums and
generalizations of the Riemann relations of Jacobi theta functions that they
obey.Comment: 1+56 page
On the asymptotic behavior for partitions separated by parity
The study of partitions with parts separated by parity was initiated by
Andrews in connection with Ramanujan's mock theta functions, and his variations
on this theme have produced generating functions with a large variety of
different modular properties. In this paper, we use Ingham's Tauberian theorem
to compute the asymptotic main term for each of the eight functions studied by
Andrews
Integral Representations of Rank Two False Theta Functions and Their Modularity Properties
False theta functions form a family of functions with intriguing modular
properties and connections to mock modular forms. In this paper, we take the
first step towards investigating modular transformations of higher rank false
theta functions, following the example of higher depth mock modular forms. In
particular, we prove that under quite general conditions, a rank two false
theta function is determined in terms of iterated, holomorphic, Eichler-type
integrals. This provides a new method for examining their modular properties
and we apply it in a variety of situations where rank two false theta functions
arise. We first consider generic parafermion characters of vertex algebras of
type and . This requires a fairly non-trivial analysis of Fourier
coefficients of meromorphic Jacobi forms of negative index, which is of
independent interest. Then we discuss modularity of rank two false theta
functions coming from superconformal Schur indices. Lastly, we analyze
-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing -graphs. Along the way, our method clarifies previous results on depth two
quantum modularity.Comment: 26 page
Network Simplification: The Gaussian diamond network with multiple antennas
We consider the N-relay Gaussian diamond network when the source and the destination have n(s) >= 2 and n(d) >= 2 antennas respectively. We show that when n(s) = n(d) = 2 and when the individual MISO channels from the source to each relay and the SIMO channels from each relay to the destination have the same capacity, there exists a two relay sub-network that achieves approximately all the capacity of the network. To prove this result, we establish a simple relation between the joint entropies of three Gaussian random variables, which is not implied by standard Shannon-type entropy inequalities.(1