63 research outputs found
On transversally elliptic operators and the quantization of manifolds with -structure
An -structure on a manifold is an endomorphism field
\phi\in\Gamma(M,\End(TM)) such that . Any -structure
determines an almost CR structure E_{1,0}\subset T_\C M given by the
-eigenbundle of . Using a compatible metric and connection
on , we construct an odd first-order differential operator ,
acting on sections of , whose principal symbol is of the
type considered in arXiv:0810.0338. In the special case of a CR-integrable
almost -structure, we show that when is the generalized
Tanaka-Webster connection of Lotta and Pastore, the operator is given by D
= \sqrt{2}(\dbbar+\dbbar^*), where \dbbar is the tangential Cauchy-Riemann
operator.
We then describe two "quantizations" of manifolds with -structure that
reduce to familiar methods in symplectic geometry in the case that is a
compatible almost complex structure, and to the contact quantization defined in
\cite{F4} when comes from a contact metric structure. The first is an
index-theoretic approach involving the operator ; for certain group actions
will be transversally elliptic, and using the results in arXiv:0810.0338,
we can give a Riemann-Roch type formula for its index. The second approach uses
an analogue of the polarized sections of a prequantum line bundle, with a CR
structure playing the role of a complex polarization.Comment: 31 page
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
Special biconformal changes of K\"ahler surface metrics
The term "special biconformal change" refers, basically, to the situation
where a given nontrivial real-holomorphic vector field on a complex manifold is
a gradient relative to two K\"ahler metrics, and, simultaneously, an
eigenvector of one of the metrics treated, with the aid of the other, as an
endomorphism of the tangent bundle. A special biconformal change is called
nontrivial if the two metrics are not each other's constant multiples. For
instance, according to a 1995 result of LeBrun, a nontrivial special
biconformal change exists for the conformally-Einstein K\"ahler metric on the
two-point blow-up of the complex projective plane, recently discovered by Chen,
LeBrun and Weber; the real-holomorphic vector field involved is the gradient of
its scalar curvature. The present paper establishes the existence of nontrivial
special biconformal changes for some canonical metrics on Del Pezzo surfaces,
viz. K\"ahler-Einstein metrics (when a nontrivial holomorphic vector field
exists), non-Einstein K\"ahler-Ricci solitons, and K\"ahler metrics admitting
nonconstant Killing potentials with geodesic gradients.Comment: 16 page
Overtwisted energy-minimizing curl eigenfields
We consider energy-minimizing divergence-free eigenfields of the curl
operator in dimension three from the perspective of contact topology. We give a
negative answer to a question of Etnyre and the first author by constructing
curl eigenfields which minimize energy on their co-adjoint orbit, yet are
orthogonal to an overtwisted contact structure. We conjecture that -contact
structures on -bundles always define tight minimizers, and prove a partial
result in this direction.Comment: published versio
Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds
We extend classical Euclidean stability theorems corresponding to the
nonrelativistic Hamiltonians of ions with one electron to the setting of non
parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar
Tight Beltrami fields with symmetry
Let be a compact orientable Seifered fibered 3-manifold without a
boundary, and an -invariant contact form on . In a suitable
adapted Riemannian metric to , we provide a bound for the volume
and the curvature, which implies the universal tightness of the
contact structure .Comment: 26 page
Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type
Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization
Once more on the Witten index of 3d supersymmetric YM-CS theory
The problem of counting the vacuum states in the supersymmetric 3d
Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy
between its original calculation by Witten at large volumes and the calculation
based on the evaluation of the effective Lagrangian in the small volume limit.
We show that the latter calculation suffers from uncertainties associated with
the singularities in the moduli space of classical vacua where the
Born-Oppenheimer approximation breaks down. We also show that these
singularities can be accurately treated in the Hamiltonian Born-Oppenheimer
method, where one has to match carefully the effective wave functions on the
Abelian valley and the wave functions of reduced non-Abelian QM theory near the
singularities. This gives the same result as original Witten's calculation.Comment: 27 page
Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes
We construct a class of solutions to the Cauchy problem of the Klein-Gordon
equation on any standard static spacetime. Specifically, we have constructed
solutions to the Cauchy problem based on any self-adjoint extension (satisfying
a technical condition: "acceptability") of (some variant of) the
Laplace-Beltrami operator defined on test functions in an -space of the
static hypersurface. The proof of the existence of this construction completes
and extends work originally done by Wald. Further results include the
uniqueness of these solutions, their support properties, the construction of
the space of solutions and the energy and symplectic form on this space, an
analysis of certain symmetries on the space of solutions and of various
examples of this method, including the construction of a non-bounded below
acceptable self-adjoint extension generating the dynamics
Spherical nucleic acids as an infectious disease vaccine platform
Despite recent efforts demonstrating that organization and presentation of vaccine components are just as important as composition in dictating vaccine efficacy, antiviral vaccines have long focused solely on the identification of the immunological target. Herein, we describe a study aimed at exploring how vaccine component presentation in the context of spherical nucleic acids (SNAs) can be used to elicit and maximize an antiviral response. Using COVID-19 as a topical example of an infectious disease with an urgent need for rapid vaccine development, we designed an antiviral SNA vaccine, encapsulating the receptor-binding domain (RBD) subunit into a liposome and decorating the core with a dense shell of CpG motif toll-like receptor 9 agonist oligonucleotides. This vaccine induces memory B cell formation in human cells, and in vivo administration into mice generates robust binding and neutralizing antibody titers. Moreover, the SNA vaccine outperforms multiple simple mixtures incorporating clinically employed adjuvants. Through modular changes to SNA structure, we uncover key relationships and proteomic insights between adjuvant and antigen ratios, concepts potentially translatable across vaccine platforms and disease models. Importantly, when humanized ACE2 transgenic mice were challenged in vivo against a lethal live virus, only mice that received the SNA vaccine had a 100% survival rate and lungs that were clear of virus by plaque analysis. This work underscores the potential for SNAs to be implemented as an easily adaptable and generalizable platform to fight infectious disease and demonstrates the importance of structure and presentation in the design of next-generation antiviral vaccines
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