On mod 2 and higher elliptic genera

In the first part of this paper, we construct mod 2 elliptic genera on manifolds of dimensions 8k+1, 8k+2 by mod 2 index formulas of Dirac operators. They are given by mod 2 modular forms or mod 2 automorphic functions. We also obtain an integral formula for the mod 2 index of the Dirac operator. As a by-product we find topological obstructions to group actions. In the second part, we construct higher elliptic genera and prove some of their rigidity properties under group actions. In the third part we write down characteristic series for all Witten genera by Jacobi theta-functions. The modular property and transformation formulas of elliptic genera then follow easily. We shall also prove that Krichever's genera, which come from integrable systems, can be written as indices of twisted Dirac operators forSU-manifolds. Some general discussions about elliptic genera are given.     

M‐strings,elliptic genera and N = 4 string amplitudes

We study mass‐deformed N = 2 gauge theories from various points of view. Their partition functions can be computed via three dual approaches: firstly, (p,q)‐brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M‐strings configurations. We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail. To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of ℂ² through a (singular) theta‐transform. This form appears naturally as a specific class of one‐loop scattering amplitudes in type II string theory on T², which we calculate explicitly.     

Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies

There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS₃, which also describe the three-dimensional Euclidean black holes, the pure N=1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.     

Elliptic genera and quantum field theory

It is shown that in elliptic cohomology — as recently formulated in the mathematical literature — the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the Dirac operator inK-theory. This leads to several insights concerning both elliptic cohomology and string theory. Some of the relevant calculations have been done previously by Schellekens and Warner in a different context.Supported in part by NSF grants PHY 80-19754 and PHY 86-16129     

Curvature of super diff(S1)/S1

Motivated by the work of Bowick and Rajeev, we calculate the curvature of the infinite-dimensional flag manifolds Diff(S¹)/S¹ and Super Diff(S¹)/S¹ using standard finite-dimensional coset space techniques. We regularize the infinity by ζ-function regularization and recover the conformal and superconformal anomalies respectively for a specific choice of the torsion.     

Constraints and actions in two and three dimensionalN=1 conformal supergravity

We investigate closure of the gauge algebra and constraints inN=1 conformal supergravity in 2 and 3 dimensions. In the 2 dimensional case, contrary to 3 or higher dimensions, some parts of the gauge fields are algebraically unsolvable in the constraint equations on group curvatures. It will be shown that these unsolvable parts are decoupled from the transformation law as well as from the kinetic multiplets. Hence they are absent in the invariant action for matter multiplets coupled to conformal supergravity which is relevant to the old superstrings. Explicit construction of the invariant actions are illustrated for the case of spinning strings and locally supersymmetric σ-models with the Wess-Zumino term.     

Quantum-algebras and elliptic algebras

We define a quantum-algebra associated to as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary-algebra of , or theq-deformed classical-algebra algebra of . We construct free field realizations of the quantum-algebra and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in.The research of the second author was partially supported by NSF grant DMS-9501414.     

Electron energy spectra in an elliptic quantum wire and elliptic nanotube

In the effective mass approximation, the electron spectra in an elliptic quantum wire and elliptic semiconductor nanotube are investigated. An exact electron energy spectrum in the GaAs elliptic quantum wire and elliptic semiconductor nanotube with impenetrable walls is obtained and an approximate solution of the Schrödinger equation is derived for a finite potential barrier height in the GaAs/Al _x Ga _1?x As elliptic quantum wire. It is demonstrated that the ellipticity of the quantum wire and nanotube results in removal of degeneration of the quasiparticle energy spectrum. Dependences of energies of even and odd states on the ratio of the ellipse semiaxes are nonmonotonic in character. In the limiting case of degeneration of elliptic quantum wires and nanotubes into circular ones, the quasi-particle energy spectrum coincides with the corresponding spectrum in cylindrical nanosystems.     

Confined states in two-dimensional flat elliptic quantum dots and elliptic quantum wires

The energy spectrum and corresponding wave functions of a flat quantum dot with elliptic symmetry are obtained exactly. A detailed study is made of the effect of ellipticity on the energy levels and the corresponding wave functions. The analytical behavior of the energy levels in certain limiting cases is obtained.     

The conformal factor and DiffS1/S1

Two-dimensional gravity coupled with an open string is considered to geometrically construct a string field theory following Bowick and Rajeev. The phase space of this system can be thought of as an extended loop space, and is in fact a Kähler manifold. Each loop carries a conformal factor. Bowick and Rajeev's proposal is modified so that the ghost fields stand on the same footing as the x^μ-fields. The closed string field is modified to be the Kähler potential of the extended loop space. The canonical line bundle is not needed for the equation of motion for a closed string field. A covariant (free) open string field theory is constructed as a by-product in this approach.     

Quantum cosmology inR 1×S 1×S n space time

Classical and quantum cosmological aspects for (n + 2) dimensional anisotropic spherically symmetric space-time with topology of (n + 1) spaceS ¹×S ⁿ have been studied. The Lorentzian field equations are reduced to an autonomous system by a change of field variables and are discussed near the critical points. The path integral expression for propagation amplitude is converted to a single ordinary integration over the lapse function by the usual technique and is evaluated in terms of Bessel functions.     

Jet quenching and elliptic flow

In jet quenching, a hard QCD parton, before fragmenting into a jet of hadrons, deposits a fraction of its energy in the medium, leading to suppressed production of high-pT$p_T$ hadrons. Assuming that the deposited energy quickly thermalizes, we simulate the subsequent hydrodynamic evolution of the QGP fluid. Explicit simulation of Au + Au collision with and without a quenching jet indicate that elliptic flow is greatly reduced in a jet event. The result can be used to identify the jet events in heavy ion collisions.     

q-deformed W-algebras and elliptic algebras

The elliptic algebra A _q,p {sl}(N _c) at the critical level c = –N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the W _N algebra, are constructed. The operators t(z) also close an exchange algebra when (–p ^1/2) ^NM = q ^–c–N for M . It becomes Abelian when in addition p = q ^Nh where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W _N algebras depending on the parity of h, characterizing the exchange structures at p q ^Nh as new W _q,p(sl(N)) algebras.     

S1 and S2 excited States of gas-phase Schiff-base retinal chromophores

Photoabsorption studies of 11-cis and all-trans Schiff-base retinal chromophore cations in the gas phase have been performed at the electrostatic ion storage ring in Aarhus. A broad absorption band due to the optically allowed excitation to the first electronically excited singlet state (S1) is observed at around 600 nm. A second "dark" excited state (S2) just below 400 nm is reported for the first time. It is located approximately 1.2 eV above S1 for both chromophores. The S2 state was not visible in a solution measurement where only one highly blueshifted absorption band corresponding to the first excited state was visible. Knowledge of the position of the excited states in retinal is essential for the understanding of the fast photoisomerization in, for example, visual pigments.     

Super elliptic curves

A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces with odd spin structure are not freely generated modules. The divisor theory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group Pic, but this map is a projection, not an isomorphism as it is for ordinaty tori. The geometric realization of the addition law on Pic via intersections of the supertorus with superlines in projective space is described. The isomorphisms of Pic with the Jacobian and the divisor class group are verified. All possible isogenies, or surjective holomorphic maps between supertori, are determined and shown to induce homomorphisms of the Picard groups. Finally, the solutions to the new super Kadomtsev-Petviashvili hierarchy of Mulase-Rabin which arise from super elliptic curves via the Krichever construction are exhibited.     

Quiver elliptic W-algebras

We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction (Kimura and Pestun in Quiver W-algebras, 2015. arXiv:1512.08533 [hep-th]) with six-dimensional gauge theory.