An arena for model building in the Cohen—Glashow very special relativity

The Cohen—Glashow Very Special Relativity (VSR) algebra is defined as the part of the Lorentz algebra which upon addition of CP or T invariance enhances to the full Lorentz group, plus the space—time translations. We show that noncommutative space—time, in particular noncommutative Moyal plane, with light- like noncommutativity provides a robust mathematical setting for quantum field theories which are VSR invariant and hence set the stage for building VSR invariant particle physics models. In our setting the VSR invariant theories are specified with a single deformation parameter, the noncommutativity scale ╕_NC. Preliminary analysis with the available data leads to ╕_NC ≳ 1–10 TeV.     

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A_∞-algebras) by closed strings (L_∞-algebras). H. K is supported by JSPS Research Fellowships for Young Scientists. J. S. is supported in part by NSF grant FRG DMS-0139799 and US-Czech Republic grant INT-0203119.     

Strong connections on quantum principal bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS ²→RP ². A certain class of strongU _q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU _q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq. This work was in part supported by the NSF grant 1-443964-21858-2. Writing up the revised version was partially supported by the KBN grant 2 P301 020 07 and by a visiting fellowship at the International Centre for Theoretical Physics in Trieste.     

Curvature corrections and Kac–Moody compatibility conditions

We study possible restrictions on the structure of curvature corrections to gravitational theories in the context of their corresponding Kac–Moody algebras, following the initial work on E ₁₀ in Damour and Nicolai [Class Quant Grav 22:2849 (2005)]. We first emphasize that the leading quantum corrections of M-theory can be naturally interpreted in terms of (non-gravity) fundamental weights of E ₁₀. We then heuristically explore the extent to which this remark can be generalized to all over-extended algebras by determining which curvature corrections are compatible with their weight structure, and by comparing these curvature terms with known results on the quantum corrections for the corresponding gravitational theories.     

Order parameter quantum fluctuations in a two-dimensional system of mesoscopic Josephson junctions

The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n ₀ (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY-model—and approaches the latter as n ₀ increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n ₀<10, while quantum fluctuations in the phase are significant when n ₀<8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY-model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/n ₀ using the method of functional integration. Zh. éksp. Teor. Fiz. 113, 261–277 (January 1998)     

On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain

We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain. The boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed. The aim is to weaken the conditions sufficient for justifying the classical asymptotic formula for the eigenvalues. We show that it suffices to assume the boundary to be C ¹ in a neighborhood of the support of the weight outside a closed subset of zero measure. The work of the author is supported by the RFBR grant no. 04-01-00914.     

The classification of monopoles for the classical groups

By studying a construction of Nahm, we compute the moduli spaces of monopoles with maximal symmetry breaking at infinity forSU(N),SO(N) andSp(N); these are found to be equivalent to spaces of holomorphic maps from ₁ into flag manifolds.Research supported in part by NSERC grant A8361 and FCAR grant EQ3518     

Differential geometry on linear quantum groups

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group GL _q (N). SL _q (N) is then found by constructing matrices with determnant unity, and the induced calculus is found.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139.     

Topological quantum field theories on manifolds with a boundary

We consider a class of exactly soluble topological quantum field theories on manifolds with a boundary that are invariant on-shell under diffeomorphisms which preserve the boundary. After showing that the functional integral of the two-point function with boundary conditions yields precisely the linking number, we use it to derive topological properties of the linking number. Considering gauge fixing, we obtain exact results of the partition function (Ray-Singer torsion of manifolds with a boundary) and theN-point functions in closed expressions.     

On the equivalence of the two most favoured Calabi-Yau compactifications

We discuss the two known multiply connected Calabi-Yau manifolds which give rise to three generations of elementary particles when chosen as the classical vacuum configuration of theE ₈×E ₈ heterotic superstring. It is shown that these two manifolds are diffeomorphic.Part of this work carried out at and supported by the IBM T.J. Watson Research Center, Yorktown Heights, NYOn leave from Lyman Laboratory of Physics, Harvard University     

Calabi-Yau manifolds — Motivations and constructions

The possible ways of compacitification of theE ₈ E ₈ Superstring theory to four dimensions are reviewed. The phenomenological need forN=1 supersymmetry is argued (on quite general grounds) to favour the choice of a Calabi-Yau manifold for the compact internal manifold. The massless spectrum after compactification is derived in full detail revealing, beside the usual particles, others that may have great phenomenological impact. The technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given. A big family of such constructions, giving rise to many new Calabi-Yau manifolds, is presented and its relevance to the search of a phenomenologically acceptable solution is discussed.This work was supported by the National Science Foundation     

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ,1/b ₁}, where ρ is the bound on the convergent ones, the instanton radius, and b ₁ the first coefficient of the β-function, while in general it is bounded by the above. Research supported by grant NSF-DMS/0603781. Supported by CNRS.     

Eu2+ and Eu3+ based “concentrated phosphors” as converters for UV LED light: two approaches and two new examples

The absolute majority of phosphors are composed of a host lattice and some percentage of an activator. At higher activator concentrations the concentration quenching occurs. However, there are phosphors in which only minor quenching of the emission occurs with increasing of the activator content. Based on the existence of two different valence states of the Eu ion (2+ and 3+), two approaches for the development of “concentrated phosphors”, i.e. light emitting materials in which the activator ion is a main part of the crystal lattice, are discussed. In both approaches, reduced energy migration leading to the luminescence quenching is considered as a main condition to reach a high quantum efficiency of a concentrated phosphor. Two kinds of phosphors—Eu²⁺-doped alumosilicate and Eu³⁺-doped oxyfluoride—are used as an experimental basis for this discussion. Starting from the stoichiometric Ca_1-xEu_x²⁺Al₂Si₂O₈\mathrm{Ca}_{1-x}\mathrm{Eu}_{x}^{2+}\mathrm{Al}_{2}\mathrm{Si}_{2}\mathrm{O}_{8} anorthite and Eu³⁺OF oxyfluorides, the non-stoichiometric powders with Eu²⁺_0.92Al_1.76Si_2.24O₈\mathrm{Eu}^{2+}_{0.92}\mathrm{Al}_{1.76}\mathrm{Si}_{2.24}\mathrm{O}_{8}, Eu³⁺(O, F)_2,35 and Eu³⁺(O, F)_2,16 compositions were synthesized by a solid state reaction and investigated. It was shown that—in spite of the almost 100% Eu concentration—light converters with high quantum efficiency of more than 45% can be realized. A possible application of these materials as UV LED light converters for white light emitting diodes are discussed as well.     

Realizations of causal manifolds by quantum fields

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e., by the homogenous spacesD(n)=GL(C _R ⁿ )/U(n) withn=1 for mechanics andn=2 for relativistic fields. The rankn gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e., two characteristic masses in the relativistic case. ‘Canonical’ field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifoldD(2). Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable—in addition to representations with eigenvectors (states, particle), they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. In theorthogonal andunitary groupsO(N ₊,N ₋), respectively, thepositive orthogonal and unitary ones areO(N) andU(N), respectively.     

Spin Gromov-Witten Invariants

We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r^th Gelfand-Dickey hierarchy (or, equivalently, the A_r−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397     

Geometry of Batalin-Vilkovisky quantization

The geometry ofP-manifolds (odd symplectic manifolds) andSP-manifolds (P-manifolds provided with a volume element) is studied. A complete classification of these manifolds is given. This classification is used to prove some results about Batalin-Vilkovisky procedure of quantization, in particular to obtain a very general result about gauge independence of this procedure.Research supported in part by NSF grant No. DMS-9201366     

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the papers style. This research was partially supported by NSF grant DMS-0305048.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

Quantized Knizhnik-Zamolodchikov equations,quantum Yang-Baxter equation,and difference equations forq-hypergeometric functions

Thes₂ quantized Knizhnik-Zamolodchikov equations are solved inq-hypergeometric functions. New difference equations are derived for generalq-hypergeometric functions. The equations are given in terms of quantum Yang-Baxter matrices and have the form similar to quantum Knizhnik-Zamolodchikov equations for quantum affine algebras introduced by Frenkel and Reshetikhin.This work was supported by NSF grant DMS-9203929.     

Quantum sets and clifford algebras

The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton. This material is based upon work supported in part by NSF Grant No. PHY8007921.