Heat Kernel Coefficients Research Articles

Heat Kernel Coefficients on Kahler Manifolds

Heat Kernel Coefficients on Kahler Manifolds

Thermal Casimir effect for a Dirac field on flat space with a nontrivial circular boundary condition

This work investigates the thermal Casimir effect associated with a massive spinor field defined on a four-dimensional flat space with a circularly compactified spatial dimension whose periodicity is oriented along a vector in the xy plane. We employ the generalized zeta function method to establish a finite definition for the vacuum free energy density. This definition conveniently separates into the zero-temperature Casimir energy density and additional terms accounting for temperature corrections. The structure of existing divergences is analyzed from the asymptotic behavior of the spinor heat kernel function and removed in the renormalization by subtracting scheme. The only non-null heat coefficient is the one associated with the Euclidean divergence. We also address the need for a finite renormalization to treat the ambiguity in the zeta function regularization prescription associated with this Euclidean heat kernel coefficient and ensure that the renormalization procedure is unique. The high- and low-temperature asymptotic limits are also explored. In particular, we explicitly show that free energy density lacks a classical limit at high temperatures, and the entropy density agrees with the Nernst heat theorem at low temperatures. Published by the American Physical Society 2024

Heat kernel coefficients for massive gravity

Heat kernel coefficients for massive gravity

Vacuum energy, temperature corrections and heat kernel coefficients in (D + 1)-dimensional spacetimes with nontrivial topology

Vacuum energy, temperature corrections and heat kernel coefficients in (<i>D</i> + 1)-dimensional spacetimes with nontrivial topology

Worldline approach for spinor fields in manifolds with boundaries

The worldline formalism is a useful scheme in Quantum Field Theory which has also become a powerful tool for numerical computations. It is based on the first quantisation of a point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field theory in a bounded manifold one needs to restrict the path integration domain of the point-particle to a specific subset of worldlines enclosed by those boundaries. In the present article it is shown how to implement this restriction for the case of a spinor field in a two-dimensional curved half-plane under MIT bag boundary conditions, and compute the first few heat-kernel coefficients as a verification of the proposed construction. This construction admits several generalisations.

One-loop effective action up to dimension eight: Integrating out heavy fermion(s)

We present the universal one-loop effective action up to dimension eight after integrating out heavy fermion(s) using the Heat-Kernel method. We have discussed how the Dirac operator being a weak elliptic operator, the fermionic operator still can be written in the form of a strong elliptic one such that the Heat-Kernel coefficients can be used to compute the fermionic effective action. This action captures the footprint of both the CP conserving as well as violating UV interactions. As it does not rely on the specific forms of either UV or low energy theories, can be applicable for a very generic action. Our result encapsulates the effects of heavy fermion loop through Yukawa couplings only.

One-loop effective action in chiral Einstein–Cartan gravity

In chiral Einstein–Cartan gravity, a new gauge fixing procedure is implemented recently, leading to a very economical perturbation expansion of the action. Using this formulation and the relevant gauge fixing, we develop the ghost Lagrangian on an arbitrary Einstein background using the Becchi-Rouet-Stora-Tyutin (BRST) formalism. The novelty is the appearance of a new term quadratic in the tetrad field. We next compute the heat kernel coefficients and understand the divergences arising in the gravitational one-loop effective action. In our computation, the arising heat kernel coefficients depend only on the self-dual part of the Weyl curvature. We make a comparison between our results and what has been obtained for metric general relativity.

Local momentum space: scalar field and gravity

We use the local momentum space technique to obtain an expansion of the Feynman propagators for scalar field and graviton up to first order in the background curvature. The expressions for the propagators are cross-checked with the past literature as well as with the expressions for the traced heat kernel coefficients. The propagators so obtained are used to compute one-loop divergences in the Vilkovisky-Dewitt’s (VD’s) effective action for a scalar field non-minimally coupled with gravity for an arbitrary spacetime metric background. The VD effective action is then compared with the standard effective action in the limit κ = 0, where in terms of the Planck mass. The comparison yields the important result that taking the limit κ = 0 after computing the VD effective action is not equivalent to computing the VD effective action for the same theory in the absence of gravity.

New covariant Feynman rules for effective field theories

We provide a new and completely general formalism to compute the effective field theory matching contributions from integrating out massive fields in a manifestly gauge covariant way, at any desired loop order. The formalism is based on old ideas such as the background field method and the heat kernel, however we add some crucial new ingredients that greatly improve the simplicity and general applicability of the approach. We formulate our method in terms of Feynman rules, the resulting effective action is expressed in terms of local heat kernel coefficients. We also provide as supplementary material a mathematica code that facilitates the computation of these coefficients.

Casimir self-energy of a δ−δ′ sphere

We extend previous work on the vacuum energy of a massless scalar field in the presence of singular potentials. We consider a single sphere defined by the so-called $\ensuremath{\delta}\text{\ensuremath{-}}{\ensuremath{\delta}}^{\ensuremath{'}}$ interaction. Contrary to the Dirac $\ensuremath{\delta}$-potential, we find a nontrivial one-parameter family of potentials such that the regularization procedure gives an unambiguous result for the Casimir self-energy. The procedure employed is based on the zeta function regularization and the cancellation of the heat kernel coefficient ${a}_{2}$. The results obtained are in agreement with particular cases, such as the Dirac $\ensuremath{\delta}$ or Robin and Dirichlet boundary conditions.

Gauge-invariant coefficients in perturbative quantum gravity

Heat kernel methods are useful for studying properties of quantum gravity. We recompute the first three heat kernel coefficients in perturbative quantum gravity with cosmological constant to ascertain which ones are correctly reported in the literature. They correspond to the counterterms needed to renormalize the one-loop effective action in four dimensions. They may be evaluated at arbitrary dimensions D, in which case they identify only a subset of the divergences appearing in the effective action for Dge 6. Generically, these coefficients depend on the gauge-fixing choice adopted in quantizing the Einstein–Hilbert action. However, they become gauge-invariant once evaluated on-shell, i.e. using Einstein’s equations with cosmological constant. Thus, we identify these gauge invariant coefficients and use them as a benchmark for testing alternative approaches to perturbative quantum gravity. One of these approaches describes the graviton in first-quantization through the {{mathcal {N}}}=4 spinning particle, characterized by four supersymmetries on the worldline and a set of worldline gauge invariances. This description has been used for computing the gauge-invariant coefficients as well. We verify their correctness at D=4, but find a mismatch at arbitrary D when comparing with the benchmark fixed previously. We interpret this result as signaling that the path integral quantization of the {{mathcal {N}}}=4 spinning particle should be amended. We perform this task by fixing the correct counterterm that must be used in the worldline path integral quantization of the {{mathcal {N}}}=4 spinning particle to make it consistent in arbitrary dimensions.

On holography in general background and the boundary effective action from AdS to dS

We study quantum fields on an arbitrary, rigid background with boundary. We derive the action for a scalar in the holographic basis that separates the boundary and bulk degrees of freedom. A relation between Dirichlet and Neumann propagators valid for any background is obtained from this holographic action. As a simple application, we derive an exact formula for the flux of bulk modes emitted from the boundary in a warped background. We also derive a formula for the Casimir pressure on a (d − 1)-brane depending only on the boundary-to-bulk propagators, and apply it in AdS. Turning on couplings and using the holographic basis, we evaluate the one-loop boundary effective action in AdS by means of the heat kernel expansion. We extract anomalous dimensions of single and double trace CFT operators generated by loops of heavy scalars and nonabelian vectors, up to third order in the large squared mass expansion. From the boundary heat kernel coefficients we identify CFT operator mixing and corrections to OPE data, in addition to the radiative generation of local operators. We integrate out nonabelian vector fluctuations in AdS4,5,6 and obtain the associated holographic Yang-Mills β functions. Turning to the expanding patch of dS, following recent proposals, we provide a boundary effective action generating the perturbative cosmological correlators using analytical continuation from dS to EAdS. We obtain the “cosmological” heat kernel coefficients in the scalar case and work out the divergent part of the dS4 effective action which renormalizes the cosmological correlators. We find that bulk masses and wavefunction can logarithmically run as a result of the dS4 curvature, and that operators on the late time boundary are radiatively generated. More developments are needed to extract all one-loop information from the cosmological effective action.

A formula for the heat kernel coefficients of the Dirac Laplacians on spin manifolds

Based on Getzler's rescaling transformation, we obtain a formula for the heat kernel coefficients of the Dirac Laplacian on a spin manifold. One can compute them explicitly up to an arbitrarily high order by using only a basic knowledge of calculus added to the formula.

A calculation of the Weyl anomaly for 6D conformal higher spins

In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein space-filling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

Boundary conformal invariants and the conformal anomaly in five dimensions

In odd dimensions the integrated conformal anomaly is entirely due to the boundary terms [1]. In this paper we present a detailed analysis of the anomaly in five dimensions. We give the complete list of the boundary conformal invariants that exist in five dimensions. Additionally to 8 invariants known before we find a new conformal invariant that contains the derivatives of the extrinsic curvature along the boundary. Then, for a conformal scalar field satisfying either the Dirichlet or the conformal invariant Robin boundary conditions we use the available general results for the heat kernel coefficient a5, compute the conformal anomaly and identify the corresponding values of all boundary conformal charges.

Hearing the shape of inequivalent spin structures and exotic Dirac operators

Exotic spinor fields arise from inequivalent spin structures on non-trivial topological manifolds, M. This induces an additional term in the Dirac operator, defined by the cohomology group that rules a ech cohomology class. This formalism is extended for manifolds of any finite dimension, endowed with a metric of arbitrary signature. The exotic corrections to heat kernel coefficients, relating spectral properties of exotic Dirac operators to the geometric invariants of M, are derived and scrutinized.

Interface conformal anomalies

We consider two d ≥ 2 conformal field theories (CFTs) glued together along a codimension one conformal interface. The conformal anomaly of such a system contains both bulk and interface contributions. In a curved-space setup, we compute the heat kernel coefficients and interface central charges in free theories. The results are consistent with the known boundary CFT data via the folding trick. In d = 4, two interface invariants generally allowed as anomalies turn out to have vanishing interface charges. These missing invariants are constructed from components with odd parity with respect to flipping the orientation of the defect. We conjecture that all invariants constructed from components with odd parity may have vanishing coefficient for symmetric interfaces, even in the case of interacting interface CFT.

Path integral calculation of heat kernel traces with first order operator insertions

We study generalized heat kernel coefficients, which appear in the trace of the heat kernel with an insertion of a first-order differential operator, by using a path integral representation. These coefficients may be used to study gravitational anomalies, i.e. anomalies in the conservation of the stress tensor. We use the path integral method to compute the coefficients related to the gravitational anomalies of theories in a non-abelian gauge background and flat space of dimensions 2, 4, and 6. In 4 dimensions one does not expect to have genuine gravitational anomalies. However, they may be induced at intermediate stages by regularization schemes that fail to preserve the corresponding symmetry. A case of interest has recently appeared in the study of the trace anomalies of Weyl fermions.

A formula for the heat kernel coefficients on Riemannian manifolds

Based on the idea of adiabatic expansion theory, we will present a new formula for the asymptotic expansion coefficients of every derivative of the heat kernel on a compact Riemannian manifold. It will be very useful for having systematic understanding of the coefficients, and, furthermore, by using only a basic knowledge of calculus added to the formula, one can describe them explicitly up to an arbitrarily high order.

Heat kernel coefficients on the sphere in any dimension

We derive all heat kernel coefficients for Laplacians acting on scalars, vectors, and tensors on fully symmetric spaces, in any dimension. Final expressions are easy to evaluate and implement, and confirmed independently using spectral sums and the Euler–Maclaurin formula. We also obtain the Green’s function for Laplacians acting on transverse traceless tensors in any dimension, and new integral representations for heat kernels using known eigenvalue spectra of Laplacians. Applications to quantum gravity and the functional renormalisation group, and other, are indicated.