140,898 research outputs found
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
Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
In this paper, we make the case that Clifford algebra is the natural
framework for root systems and reflection groups, as well as related groups
such as the conformal and modular groups: The metric that exists on these
spaces can always be used to construct the corresponding Clifford algebra. Via
the Cartan-Dieudonn\'e theorem all the transformations of interest can be
written as products of reflections and thus via `sandwiching' with Clifford
algebra multivectors. These multivector groups can be used to perform concrete
calculations in different groups, e.g. the various types of polyhedral groups,
and we treat the example of the tetrahedral group in detail. As an aside,
this gives a constructive result that induces from every 3D root system a root
system in dimension four, which hinges on the facts that the group of spinors
provides a double cover of the rotations, the space of 3D spinors has a 4D
euclidean inner product, and with respect to this inner product the group of
spinors can be shown to be closed under reflections. In particular the 4D root
systems/Coxeter groups induced in this way are precisely the exceptional ones,
with the 3D spinorial point of view also explaining their unusual automorphism
groups. This construction simplifies Arnold's trinities and puts the McKay
correspondence into a wider framework. We finally discuss extending the
conformal geometric algebra approach to the 2D conformal and modular groups,
which could have interesting novel applications in conformal field theory,
string theory and modular form theory.Comment: 14 pages, 1 figure, 5 table
Hadamard matrices modulo p and small modular Hadamard matrices
We use modular symmetric designs to study the existence of Hadamard matrices
modulo certain primes. We solve the -modular and -modular versions of
the Hadamard conjecture for all but a finite number of cases. In doing so, we
state a conjecture for a sufficient condition for the existence of a
-modular Hadamard matrix for all but finitely many cases. When is a
primitive root of a prime , we conditionally solve this conjecture and
therefore the -modular version of the Hadamard conjecture for all but
finitely many cases when , and prove a weaker result for
. Finally, we look at constraints on the existence of
-modular Hadamard matrices when the size of the matrix is small compared to
.Comment: 14 pages; to appear in the Journal of Combinatorial Designs; proofs
of Lemma 4.7 and Theorem 5.2 altered in response to referees' comment
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