A. A. Bytsenko

String partition functions, Hilbert schemes and affine Lie algebra representations on homology groups

This review paper contains a concise introduction to highest weight representations of infinite dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera of superconformal quantum mechanics and superstring models. The common link of all these concepts and of the many examples considered in the paper is to be found in a very important feature of the theory of infinite dimensional Lie algebras: the modular properties of the characters (generating functions) of certain representations. The characters of the highest weight modules represent the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. We discuss the role of the unimodular (and modular) groups and the (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated q-series. For mathematicians, elliptic genera are commonly associated to new mathematical invariants for spaces, while for physicists elliptic genera are one-loop string partition function (therefore they are applicable, for instance, to topological Casimir effect calculations). We show that elliptic genera can be conveniently transformed into product expressions which can then inherit the homology properties of appropriate polygraded Lie algebras.

Spacetime topology, temperature and the vanishing of vacuum energies in dimensionally reduced supersymmetric theories

In dimensionally reduced field theories the large vacuum energies (interpreted as the cosmological constant) can appear when taking quantum effects into account. In a cosmological context one may achieve the vanishing of such vacuum energies in the framework of both Friedmann cosmology and the inflationary universe scenario. This vanishing will essentially depend on the topological properties of physical 4-dimensional spacetime and also on the topology of the (4 + q)-dimensional spacetime under reduction (q is a natural number). The possibility of such a vanishing is considered using the example of supersymmetric Wess-Zumino and N = 1 Yang-Mills models at finite temperature in manifolds with topologies Mr × (S1)q, where the physical 4-dimensional spacetime Mr possesses either the topology of flat homogeneous Clifford-Klein spacetimes, i.e., Mr = R4 − r × (S1)r, r = 1, 2, 3 or M0 = R4 is the Minkowski spacetime.

The supersymmetric Casimir effect and quantum creation of the universe with nontrivial topology

Abstract We estimate the probability of quantum creation of the universe, having the spatial topology (S 1 ) 3 , and filled with the fields of minimal N = 1 supergravity, in the semiclassical approximation. After creation, inflation of the universe occurs due to the topological Casimir effect. Creation of the universe with an isotropic topology is found to be the most preferable.

Effective finite temperature partition function for fields on noncommutative flat manifolds

The first quantum correction to the finite temperature partition function for a self-interacting massless scalar field on a D-dimensional flat manifold with p noncommutative extra dimensions is evaluated by means of dimensional regularization, supplemented with zeta-function techniques. It is found that the zeta function associated with the effective one-loop operator may be nonregular at the origin. The important issue of the determination of the regularized vacuum energy, namely the first quantum correction to the energy in such a case, is discussed.

Truncated heat kernel and one-loop determinants for the BTZ geometry

There is a special relation between the spectrum and the truncated heat kernel of the Euclidean BTZ black hole which entails the Patterson–Selberg zeta function. Using an orbifold description of this relation we calculate the on-shell corrections of the gravitational quantum fluctuations.

Estimates of the gluon concentrations in the confining SU(3)-Yang–Mills field for the first three states of charmonium

Abstract The estimates of the gluon concentrations in the classical SU(3)-Yang–Mills field modelling confinement are given for the first three states of charmonium whose spectrum is tuned by calculating electromagnetic transitions among the mentioned levels in dipole approximation. For comparison the corresponding estimates for the photon concentration in the ground state of positronium (parapositronium and orthopositronium) are also adduced.

Casimir effect in supergravity theories and the quantum birth of the Universe with non-trivial topology

The authors discuss the possibility of obtaining the non-zero one-loop topological Casimir effect in supergravity theories and evaluate it for a number of supergravity models in the flat homogeneous Clifford-Klein spacetimes with topologies Mq=(S1)q*R4-q, q=1, 2, 3. These models are pure N=1 supergravity, N=1 supergravity coupled to matter and gauge fields, N=2 supergravity coupled to matter fields and N=8 supergravity. The results of calculations permit us to estimate the probability of the quantum birth of the Universe, possessing the effective Mq topology and filled with the fields of any of the above theories in the semiclassical approximation. In all cases it turns out that the birth of the Universe with more isotropic topology is more plausible. They also outline the perspectives of an extension of the results obtained.

Topological Casimir effect for a class of hyperbolic three-dimensional Clifford-Klein spacetimes

The topological Casimir energy for all topologically inequivalent configurations of a massless real scalar field on the hyperbolic three-dimensional Clifford-Klein spacetimes R*H2/ Gamma is evaluated for the case when H2/ Gamma is a compact Riemannian surface of genus g>1. Calculations are performed at both zero and finite temperature with the help of the Selberg trace formula for compact Riemannian surfaces of genus g>1.

Topological violation of supersymmetry

Abstract The mechanism of supersymmetry violation via a nontrivial spacetime topology is considered by the example of the supersymmetric Wess-Zumino model in the flat homogeneous Clifford-Klein spacetimes with topologies M q =(S 1 ) q ×R 4− q , q =1,2,3. The nontrivial spacetime topology leads to a number of supersymmetry violation possibilities. The described mechanism may be interesting for building cosmological scenarios.

Quantum black holes, elliptic genera and spectral partition functions

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.