15,720 research outputs found
Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I
We define the partition and -point correlation functions for a vertex
operator superalgebra on a genus two Riemann surface formed by sewing two tori
together. For the free fermion vertex operator superalgebra we obtain a closed
formula for the genus two continuous orbifold partition function in terms of an
infinite dimensional determinant with entries arising from torus Szeg\"o
kernels. We prove that the partition function is holomorphic in the sewing
parameters on a given suitable domain and describe its modular properties.
Using the bosonized formalism, a new genus two Jacobi product identity is
described for the Riemann theta series. We compute and discuss the modular
properties of the generating function for all -point functions in terms of a
genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one
point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun.
Math. Phy
What Does an Exemplary Middle School Mathematics Teacher Look Like? The Use of a Professional Development Rubric
A School University Research Network (SURN) committee composed of current mathematics teachers, central office math supervisors, building administrators, mathematicians, and mathematics educators researched numerous sources regarding best practices in mathematics instruction. The resulting professional development rubric synthesizes their findings and can serve a professional development role by providing teachers and administrators with a tool to develop clarity and consensus on best mathematics instructional practices, and how these practices are implemented in the classroom. It is also being used as a tool for cooperating teachers in their supervision of student teachers and as a reflective method for self-evaluation
Superconductor-Insulator Transition in a Capacitively Coupled Dissipative Environment
We present results on disordered amorphous films which are expected to
undergo a field-tuned Superconductor-Insulator Transition.The addition of a
parallel ground plane in proximity to the film changes the character of the
transition.Although the screening effects expected from "dirty-boson" theories
are not evident,there is evidence that the ground plane couples a certain type
of dissipation into the system,causing a dissipation-induced phase
transition.The dissipation due to the phase transition couples similarly into
quantum phase transition systems such as superconductor-insulator transitions
and Josephson junction arrays.Comment: 4 pages, 4 figure
Crossover and scaling in a two-dimensional field-tuned superconductor
Using an analysis similar to that of Imry and Wortis, it is shown that the
apparent first order superconductor to metal transition, which has been claimed
to exist at low values of the magnetic field in a two-dimensional field-tuned
system at zero temperature,can be consistentlyinterpreted as a sharp crossover
from a strong superconductor to an inhomogeneous state, which is a weak
superconductor. The true zero-temperature superconductor to insulator
transition within the inhomogenous state is conjectured to be that of randomly
diluted XY model. An explaination of the observed finite temperature
approximate scaling of resistivity close to the critical point is speculated
within this model.Comment: 5 pages, 2 figures, corrected and modified according to referee
Report
Intrinsic Energy Localization through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals
We present a systematic study of the existence and stability of discrete
breathers that are spatially localized in the bulk of a one-dimensional chain
of compressed elastic beads that interact via Hertzian contact. The chain is
diatomic, consisting of a periodic arrangement of heavy and light spherical
particles. We examine two families of discrete gap breathers: (1) an unstable
discrete gap breather that is centered on a heavy particle and characterized by
a symmetric spatial energy profile and (2) a potentially stable discrete gap
breather that is centered on a light particle and is characterized by an
asymmetric spatial energy profile. We investigate their existence, structure,
and stability throughout the band gap of the linear spectrum and classify them
into four regimes: a regime near the lower optical band edge of the linear
spectrum, a moderately discrete regime, a strongly discrete regime that lies
deep within the band gap of the linearized version of the system, and a regime
near the upper acoustic band edge. We contrast discrete breathers in anharmonic
FPU-type diatomic chains with those in diatomic granular crystals, which have a
tensionless interaction potential between adjacent particles, and highlight in
that the asymmetric nature of the latter interaction potential may lead to a
form of hybrid bulk-surface localized solutions
Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
We extend our method of partner symmetries to the hyperbolic complex
Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We
show the existence of partner symmetries and derive the relations between them
for both equations. For certain simple choices of partner symmetries the
resulting differential constraints together with the original heavenly
equations are transformed to systems of linear equations by an appropriate
Legendre transformation. The solutions of these linear equations are
generically non-invariant. As a consequence we obtain explicitly new classes of
heavenly metrics without Killing vectors.Comment: 20 pages, 1 table, corrected typo
The ADHM Construction of Instantons on Noncommutative Spaces
We present an account of the ADHM construction of instantons on Euclidean
space-time from the point of view of noncommutative geometry. We
recall the main ingredients of the classical construction in a coordinate
algebra format, which we then deform using a cocycle twisting procedure to
obtain a method for constructing families of instantons on noncommutative
space-time, parameterised by solutions to an appropriate set of ADHM equations.
We illustrate the noncommutative construction in two special cases: the
Moyal-Groenewold plane and the Connes-Landi plane
.Comment: Latex, 40 page
Modular Invariance for Twisted Modules over a Vertex Operator Superalgebra
The purpose of this paper is to generalize Zhu's theorem about characters of
modules over a vertex operator algebra graded by integer conformal weights, to
the setting of a vertex operator superalgebra graded by rational conformal
weights. To recover SL_2(Z)-invariance of the characters it turns out to be
necessary to consider twisted modules alongside ordinary ones. It also turns
out to be necessary, in describing the space of conformal blocks in the
supersymmetric case, to include certain `odd traces' on modules alongside
traces and supertraces. We prove that the set of supertrace functions, thus
supplemented, spans a finite dimensional SL_2(Z)-invariant space. We close the
paper with several examples.Comment: 42 pages. Published versio
Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere
equation so that this equation itself emerges as an algebraic consequence. We
regard the function in the extended Lax equations as a complex potential. We
identify the real and imaginary parts of the potential, which we call partner
symmetries, with the translational and dilatational symmetry characteristics
respectively. Then we choose the dilatational symmetry characteristic as the
new unknown replacing the K\"ahler potential which directly leads to a Legendre
transformation and to a set of linear equations satisfied by a single real
potential. This enables us to construct non-invariant solutions of the Legendre
transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler
metrics with anti-self-dual Riemann curvature 2-form that admit no Killing
vectors.Comment: submitted to J. Phys.
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