Heat Kernel Expansion Research Articles

Commutator technique for the heat kernel of minimal higher derivative operators

We suggest a new technique of the asymptotic heat kernel expansion for minimal higher derivative operators of a generic 2Mth order, F(∇)=(−□)M+⋯, in the background field formalism of gauge theories and quantum gravity. This technique represents the conversion of the recently suggested Fourier integral method of generalized exponential functions [A. O. Barvinsky and W. Wachowski, Heat kernel expansion for higher order minimal and nonminimal operators, ] into the commutator algebra of special differential operators, which allows one to express expansion coefficients for F(∇) in terms of the Schwinger-DeWitt coefficients of a minimal second-order operator H(∇). This procedure is based on special functorial properties of the formalism including the Mellin-Barnes representation of the complex operator power HM(∇) and naturally leads to the origin of generalized exponential functions without directly appealing to the Fourier integral method. The algorithm is essentially more straightforward than the Fourier method and consists of three steps ready for a computer codification by symbolic manipulation programs. They begin with the decomposition of the operator into a power of some minimal second-order operator H(∇) and its lower derivative perturbation part W(∇), F(∇)=HM(∇)+W(∇), followed by considering their multiple nested commutators. The second step is the construction of special local differential operators—the perturbation theory in powers of the lower derivative part W(∇). The final step is the so-called procedure of their “Syngification,” consisting of a special modification of the covariant derivative monomials in these operators by the Synge world function σ(x,x′) with their subsequent action on the Hadamard-Minakshisundaram-DeWitt coefficients of H(∇). Published by the American Physical Society 2024

Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase Model

In a recent contribution, we identified possible points of contact between the asymptotically safe and canonical approaches to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity, which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantized, and one can construct the generating functional of time-ordered Wightman (i.e., Feynman) or Schwinger distributions, respectively, from the corresponding time-translation unitary group or contraction semigroup, respectively, as a path integral. For the unitary choice, that path integral can be rewritten in terms of the Lorentzian Einstein–Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantization and a reduction term that encodes the reduction of the full phase space to the phase space of observables. This path integral can then be treated with the methods of asymptotically safe quantum gravity in its Lorentzian version. We also exemplified the procedure using a concrete, minimalistic example, namely Einstein–Klein–Gordon theory, with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper, we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature, which has a strong impact on the convergence of “heat” kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.

Erratum: Heat kernel expansion for higher order minimal and nonminimal operators [Phys. Rev. D 105 , 065013 (2022)

Erratum: Heat kernel expansion for higher order minimal and nonminimal operators [Phys. Rev. D <b>105</b> , 065013 (2022)

On torsion contribution to chiral anomaly via Nieh–Yan term

In this note we present a solution to the question of whether or not, in the presence of torsion, the topological Nieh–Yan term contributes to chiral anomaly. The integral of Nieh–Yan term is non-zero if topology is non-trivial; the manifold has a boundary or vierbeins have singularities. Noting that singular Nieh–Yan term could be written as a sum of delta functions, we argue that the heat kernel expansion cannot end at finite steps. This leads to a sinusoidal dependence on the Nieh–Yan term and the UV cut-off of the theory (or alternatively the minimum length of spacetime). We show this ill-behaved dependence can be removed if a quantization condition on length scales is applied. It is expected as the Nieh–Yan term can be derived as the difference of two Chern class integrals (i.e. Pontryagin terms). On the other hand, in the presence of a cosmological constant, we find that indeed the Nieh–Yan term contributes to the index with a dimensionful anomaly coefficient that depends on the de Sitter length or equivalently inverse Hubble rate. We find similar result in thermal field theory where the anomaly coefficient depends on temperature. In both examples, the anomaly coefficient depends on IR cut-off of the theory. Without singularities, the Nieh–Yan term can be smoothly rotated away, does not contribute to topological structure and consequently does not contribute to chiral anomaly.

A baby–Skyrme model with anisotropic DM interaction: Compact skyrmions revisited

We consider a baby–Skyrme model with Dzyaloshinskii–Moriya interaction (DMI) and two types of potential terms. The model has a close connection with the vacuum functional of fermions coupled with O(3) nonlinear n-fields and with a constant SU(2) gauge background. The energy functional is derived from the heat-kernel expansion for the fermion determinant. The model possesses normal skyrmions with topological charge Q=1. The restricted version of the model also includes both the weak-compacton case (at the boundary, not continuously differentiable) and genuine-compacton case (continuously differentiable). The model consists of only the Skyrme term, and the DMI provides soliton solutions that are known as skyrmions without any potential. The BPS equation in the supersymmetric soliton models implies that the impurity coupling is closely related to the DMI. Therefore, the effect of an exponentially localized DMI is also studied in the present model.

Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as {mathbb {C}}^n with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li–Yau’s famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. This is the second of our papers in understanding Landau–Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray–Singer torsion for Witten deformation in the non-compact setting.

The thermodynamic limit of an ideal Bose gas by asymptotic expansions and spectral ζ-functions

We analyze the thermodynamic limit—modeled as the open-trap limit of an isotropic harmonic potential—of an ideal, non-relativistic Bose gas with a special emphasis on the phenomenon of Bose–Einstein condensation. This is accomplished by the use of an asymptotic expansion of the grand potential, which is derived by ζ-regularization techniques. Herewith, we can show that the singularity structure of this expansion is directly interwoven with the phase structure of the system: In the non-condensation phase, the expansion has a form that resembles usual heat kernel expansions. By this, thermodynamic observables are directly calculable. In contrast, the expansion exhibits a singularity of infinite order above a critical density, and a renormalization of the chemical potential is needed to ensure well-defined thermodynamic observables. Furthermore, the renormalization procedure forces the system to exhibit condensation. In addition, we show that characteristic features of the thermodynamic limit, such as the critical density or the internal energy, are entirely encoded in the coefficients of the asymptotic expansion.

Special functions for heat kernel expansion

In this paper, we study an asymptotic expansion of the heat kernel for a Laplace operator on a smooth Riemannian manifold without a boundary at enough small values of the proper time. The Seeley-DeWitt coefficients of this decomposition satisfy a set of recurrence relations, which we use to construct two function families of a special kind. Using these functions, we study the expansion of a local heat kernel for the inverse Laplace operator. We show that the new functions have some important properties. For example, we can consider the Laplace operator on the function set as a shift one. Also we describe various applications useful in theoretical physics and, in particular, we find a decomposition of Green's functions near the diagonal in terms of new functions.

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.

Green’s Function of a Subgraph of a Complete Graph

Abstract By using heat kernel expansions for the Laplacian on a subgraph of a complete graph, we obtain formal expansions for the Green function of the Laplacian on a subgraph of a complete graph. We study this formal series for both the cases that the subgraph is complete and not complete, and in both cases, we obtain precise information about the convergence of the series.

Heat kernel expansion for higher order minimal and nonminimal operators

We build a systematic calculational method for the covariant expansion of the two-point heat kernel $\hat K(\tau|x,x')$ for generic minimal and non-minimal differential operators of any order. This is the expansion in powers of dimensional background field objects -- the coefficients of the operator and the corresponding spacetime and vector bundle curvatures, suitable in renormalization and effective field theory applications. For minimal operators whose principal symbol is given by an arbitrary power of the covariant Laplacian $(-\Box)^M$, $M>1$, this result generalizes the well-known Schwinger--DeWitt (or Seeley--Gilkey) expansion to the infinite series of positive and negative fractional powers of the proper time $\tau^{1/M}$, weighted by the generalized exponential functions of the dimensionless argument $-\sigma(x,x')/2\tau^{1/M}$ depending on the Synge world function $\sigma(x,x')$. The coefficients of this series are determined by the chain of auxiliary differential operators acting on the two-point parallel transport tensor, which in their turn follow from the solution of special recursive equations. The derivation of these operators and their recursive equations are based on the covariant Fourier transform in curved spacetime. The series of negative fractional powers of $\tau$ vanishes in the coincidence limit $x'=x$, which makes the proposed method consistent with the heat kernel theory of Seeley--Gilkey and generalizes it beyond the heat kernel diagonal in the form of the asymptotic expansion in the domain $\sigma(x,x')\ll\tau^{1/M}$, $\tau\to 0$.$\nabla^a\sigma(x,x')\ll\tau^{1/2M}$, $\tau\to 0$.

Torsion Type Invariants of Singularities

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on \(\mathbb {C}^{n}\) by a quasi-homogeneous polynomial f. Under some mild assumption to f, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integration. The heat kernel expansion provides the spectral invariants of f. Furthermore, we can define the torsion type invariants associated to a homogeneous singularity. The spectral invariants provide another way to classify the singularity.

Logarithmic correction to the entropy of extremal black holes in mathcal{N} = 1 Einstein-Maxwell supergravity

We study one-loop covariant effective action of “non-minimally coupled” mathcal{N} = 1, d = 4 Einstein-Maxwell supergravity theory by heat kernel tool. By fluctuating the fields around the classical background, we study the functional determinant of Laplacian differential operator following Seeley-DeWitt technique of heat kernel expansion in proper time. We then compute the Seeley-DeWitt coefficients obtained through the expansion. A particular Seeley-DeWitt coefficient is used for determining the logarithmic correction to Bekenstein-Hawking entropy of extremal black holes using quantum entropy function formalism. We thus determine the logarithmic correction to the entropy of Kerr-Newman, Kerr and Reissner-Nordström black holes in “non-minimally coupled” mathcal{N} = 1, d = 4 Einstein-Maxwell supergravity theory.

Heat kernel expansion for the Heston-Hull-White model

Heat kernel expansion for the Heston-Hull-White model

A physicist’s guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as [Formula: see text]-dimensional balls (with [Formula: see text]), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

Heat Kernel Approximation on Kendall Shape Space

The heat kernel on Kendall shape subspaces is approximated by an expansion. The Kendall space is useful for representing the shapes associated to collections of landmarks’positions. The Minakshisundaram-Pleijel recursion formulas are used in order to calculate the closed-form approximations of the first and second coefficients of the heat kernel expansion. Prior to the exploitation of the recursion scheme, the expression of the Laplace-Beltrami operator is adapted to the targeted space using geodesic spherical and angular coordinates.

The Ricci curvature for noncommutative three tori

We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analysed. Using Connes’ pseudodifferential calculus for the noncommutative tori, we explicitly compute the second term of the short time heat kernel expansion for the perturbed Laplacians on functions and on 1-forms. The Ricci curvature is defined by localizing heat traces suitably. Equivalently, it can be defined through special values of localized spectral zeta functions. We also compute the scalar curvatures and compare our results with previous calculations in the conformal case. Finally we compute the classical limit of our formulas and show that they coincide with classical formulas in the commutative case.

Chiral effective theories with a light scalar at one loop

There are indications that some theories with spontaneous symmetry breaking also feature a light scalar in their spectrum, with a mass comparable to the one of the Goldstone modes. In this paper, we perform the one-loop renormalization of a theory of Goldstone modes invariant under a chiral SU(n)×SU(n) symmetry group coupled to a generic scalar singlet. We employ the background field method, together with the heat kernel expansion, to get an expression for the effective action at one loop and single out the anomalous dimensions, which can be read off from the second Seeley-DeWitt coefficient. As a relevant application, we use our master formula to renormalize chiral-scale perturbation theory, an alternative to SU(3) chiral perturbation theory where the f0(500) meson is interpreted as a dilaton. Based on our results, we briefly discuss strategies to test and discern both effective field theories using lattice simulations.

On finite temperature Casimir effect for Dirac lattices

We consider polarizable sheets modeled by a lattice of delta function potentials. The Casimir interaction of two such lattices is calculated at nonzero temperature. The heat kernel expansion for periodic singular background is discussed in relation with the high temperature asymptote of the free energy.

Conditions for Bose–Einstein condensation in periodic background

We investigate Bose–Einstein condensation of a noninteracting gas of Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, this property persists also in the presence of the lattice for all examples which are considered in the present paper and we could only formulate some conditions which are necessary for condensation. We also considered the three-dimensional case and showed that the lattice does not destroy condensation. We calculated, for small coupling, the change in the critical temperature, which is lowered by the lattice. Finally, we took another, more general view on the problem using heat kernel expansion, and discuss BEC for Casimir effect related configurations.