Dmitri V. Vassilevich

Heat kernel expansion: User's manual

Abstract The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local boundary conditions, in the presence of various types of singularities (e.g., domain walls). In each case the heat kernel coefficients are given in terms of several geometric invariants. These invariants are derived for scalar and spinor theories with various interactions, Yang–Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding anomalous actions, and covariant perturbation expansions of the effective action (both “low-” and “high-energy” ones).

Heat kernels and zeta-function regularization for the mass of the supersymmetric kink

We apply zeta-function regularization to the kink and susy kink and compute its quantum mass. We fix ambiguities by the renormalization condition that the quantum mass vanishes as one lets the mass gap tend to infinity while keeping scattering data fixed. As an alternative we write the regulated sum over zero point energies in terms of the heat kernel and apply standard heat kernel subtractions. Finally we discuss to what extent these procedures are equivalent to the usual renormalization conditions that tadpoles vanish.

Finite temperature Casimir effect for graphene

We adopt the Dirac model for quasiparticles in graphene and calculate the finite temperature Casimir interaction between a suspended graphene layer and a parallel conducting surface. We find that at high temperature the Casimir interaction in such system is just one half of that for two ideal conductors separated by the same distance. In this limit single graphene layer behaves exactly as a Drude metal. In particular, the contribution of the TE mode is suppressed, while one of the TM mode saturates the ideal metal value. Behaviour of the Casimir interaction for intermediate temperatures and separations accessible for an experiment is studied in some detail. We also find an interesting interplay between two fundamental constants of graphene physics: the fine structure constant and the Fermi velocity.

Heat kernel asymptotics with mixed boundary conditions

Abstract We calculate the coefficient a 5 of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.

Noncommutative Heat Kernel

We consider a natural generalisation of the Laplace type operators for the case of noncommutative (Groenewold—Moyal star) product. We demonstrate existence of a power law asymptotic expansion for the heat trace of such operators on Tn. First four coefficients of this expansion are calculated explicitly. We also find an analog of the UV/IR mixing phenomenon when analysing the localised heat kernel

Area spectrum in Lorentz covariant loop gravity

We use the manifestly Lorentz covariant canonical formalism to evaluate eigenvalues of the area operator acting on Wilson lines. To this end we modify the standard definition of the loop states to make it applicable to the present case of noncommutative connections. The area operator is diagonalized by using the usual shift ambiguity in the definition of the connection. The eigenvalues are then expressed through quadratic Casimir operators. No dependence on the Immirzi parameter appears.

Heat kernel, effective action and anomalies in noncommutative theories

Being motivated by physical applications (as the 4 model) we calculate the heat kernel coefficients for generalised laplacians on the Moyal plane containing both left and right multiplications. We found both star-local and star-nonlocal terms. By using these results we calculate the large mass and strong noncommutativity expansion of the effective action and of the vacuum energy. We also study the axial anomaly in the models with gauge fields acting on fermions from the left and from the right.

Heat trace asymptotics with transmittal boundary conditions and quantum brane-world scenario

Abstract We study the spectral geometry of an operator of Laplace type on a manifold with a singular surface. We calculate the first several coefficients of the heat kernel expansion. These coefficients are responsible for divergences and conformal anomaly in quantum brane-world scenario.

Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary

Let M be a compact Riemannian manifold with smooth boundary. We study the vacuum expectation value of an operator Q by studying TrL2Qe−tD, where D is an operator of Laplace type on M, and where Q is a second order operator with scalar leading symbol; we impose Dirichlet or modified Neumann boundary conditions.

Consistent boundary conditions for supergravity

We derive the complete orbit of boundary conditions for supergravity models which is closed under the action of all local symmetries of these models, and which eliminates spurious field equations on the boundary. We show that the Gibbons–Hawking boundary conditions break local supersymmetry if one imposes local boundary conditions on all fields. Nonlocal boundary conditions are not ruled out. We extend our analysis to BRST symmetry and to the Hamiltonian formulation of these models.