Seeley-DeWitt Coefficients Research Articles

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.

Proper-time evaluation of the effective action: Unequal masses in the loop

The Fock-Schwinger proper-time method is used to derive the effective action in the field theory with the chiral $U(3)\times U(3)$ symmetry explicitly broken by unequal masses of heavy particles. The one-loop effective action is presented as a series in inverse powers of heavy masses. The first two Seeley-DeWitt coefficients of this expansion are explicitly calculated. This powerful technique opens a promising avenue for studying explicit flavor symmetry breaking effects in the effective field theories.

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

We reviewed the field redefinition approach of Seeley-DeWitt expansion for the determination of Seeley-DeWitt coefficients from arXiv:1505.01156. We apply this approach to compute the first three Seeley-DeWitt coefficients for “non-minimal” mathcal{N} = 1 Einstein-Maxwell supergravity in four dimensions. Finally, we use the third coefficient for the computation of the logarithmic corrections to the Bekenstein-Hawking entropy of non-extremal black holes following arXiv:1205.0971. We determine the logarithmic corrections for non-extremal Kerr-Newman, Kerr, Reissner-Nordström and Schwarzschild black holes in “non-minimal” mathcal{N} = 1, d = 4 Einstein-Maxwell supergravity.

Generalized Einstein-Maxwell theory: Seeley-DeWitt coefficients and logarithmic corrections to the entropy of extremal and nonextremal black holes

We present a consolidated manual of Euclidean gravity approaches for finding the logarithmic corrections to the entropy of the full Kerr-Newman family of black holes in both extremal and nonextremal limits. Seeley-DeWitt coeffcients for the quadratic fluctuations of a concern gravity theory appear to be the key ingredients in this manual. Following the manual, we calculate the first three Seeley-DeWitt coefficients and logarithmic corrections to the entropy of extremal and nonextremal black holes in a generalized Einstein-Maxwell theory minimally coupled to additional massless scalar, vector, spin-$1/2$ Dirac and spin-$3/2$ Rarita-Schwinger fields. We finally employ the Seeley-DeWitt data to reproduce the logarithmic entropy corrections for extremal black holes in all $\mathcal{N}\ensuremath{\ge}2$ Einstein-Maxwell supergravity via an alternative local supersymmetrization method.

Logarithmic corrections to black hole entropy in matter coupled mathcal{N} \u2265 1 Einstein-Maxwell supergravity

We calculate the first three Seeley-DeWitt coefficients for fluctuation of the massless fields of a mathcal{N} = 2 Einstein-Maxwell supergravity theory (EMSGT) distributed into different multiplets in d = 4 space-time dimensions. By utilizing the Seeley-DeWitt data in the quantum entropy function formalism, we then obtain the logarithmic correction contribution of individual multiplets to the entropy of extremal Kerr-Newman family of black holes. Our results allow us to find the logarithmic entropy corrections for the extremal black holes in a fully matter coupled mathcal{N} = 2, d = 4 EMSGT, in a particular class of mathcal{N} = 1, d = 4 EMSGT as consistent decomposition of mathcal{N} = 2 multiplets ( mathcal{N} = 2 → mathcal{N} = 1) and in mathcal{N} ≥ 3, d = 4 EMSGTs by decomposing them into mathcal{N} = 2 multiplets ( mathcal{N} ≥ 3 → mathcal{N} = 2). For completeness, we also obtain logarithmic entropy correction results for the non-extremal Kerr-Newman black holes in the matter coupled mathcal{N} ≥ 1, d = 4 EMSGTs by employing the same Seeley-DeWitt data into a different Euclidean gravity approach developed in [17].

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: Proper-time method, Fock–Schwinger gauge, path integral, and Wilson line

The proper time method plays an important role in modern mathematics and physics. It includes many approaches, each of which has its pros and cons. This work is devoted to the description of one model case, which reflects the subtleties of construction and can be extended to a more general cases (curved space, manifold with boundary), and contains two interrelated parts: asymptotic expansion and path intergal representation. The paper discusses in details the importance of gauge conditions and role of the ordered exponentials, gives the proof of a new non-recursive formula for the Seeley-DeWitt coefficients on the diagonal, as well as the equivalence of the two main approaches using the exponential formula.

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.

Seeley-DeWitt coefficients in mathcal{N} = 2 Einstein-Maxwell supergravity theory and logarithmic corrections to mathcal{N} = 2 extremal black hole entropy

We investigate the heat kernel method for one-loop effective action following the Seeley-DeWitt expansion technique of heat kernel with Seeley-DeWitt coefficients. We also review a general approach of computing the Seeley-DeWitt coefficients in terms of background or geometric invariants. We, then consider the Einstein-Maxwell theory em-bedded in minimal mathcal{N} = 2 supergravity in four dimensions and compute the first three Seeley-DeWitt coefficients of the kinetic operator of the bosonic and the fermionic fields in an arbitrary background field configuration. We find the applications of these results in the computation of logarithmic corrections to Bekenstein-Hawking entropy of the extremal Kerr-Newman, Kerr and Reissner-Nordström black holes in minimal mathcal{N} = 2 Einstein-Maxwell supergravity theory following the quantum entropy function formalism.

One-loop \u03b2-functions in 4-derivative gauge theory in 6 dimensions

A classically scale-invariant 6d analog of the 4d Yang-Mills theory is the 4-derivative (∇F )2 + F 3 gauge theory with two independent couplings. Motivated by a search for a perturbatively conformal but possibly non-unitary 6d models we compute the one-loop β-functions in this theory. A systematic way of doing this using the back-ground field method requires the (previously unknown) expression for the b6 Seeley-DeWitt coefficient for a generic 4-derivative operator; we derive it here. As an application, we also compute the one-loop β-function in the (1,0) supersymmetric (∇F )2 6d gauge theory con-structed in hep-th/0505082.

Modular forms in the spectral action of Bianchi IX gravitational instantons

We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.

Diagram Technique for the Heat Kernel of the Covariant Laplace Operator

We present a diagram technique used to calculate the Seeley-DeWitt coefficients for a covariant Laplace operator. We use the combinatorial properties of the coefficients to construct a matrix formalism and derive a formula for an arbitrary coefficient.

Regularization versus Renormalization: Why Are Casimir Energy Differences So Often Finite?

One of the very first applications of the quantum field theoretic vacuum state was in the development of the notion of Casimir energy. Now, field theoretic Casimir energies, considered individually, are always infinite. However, differences in Casimir energies (at worst regularized, not renormalized) are quite often finite—a fortunate circumstance which luckily made some of the early calculations, (for instance, for parallel plates and hollow spheres), tolerably tractable. We will explore the extent to which this observation can be made systematic. For instance: What are necessary and sufficient conditions for Casimir energy differences to be finite (with regularization but without renormalization)? Additionally, when the Casimir energy differences are not formally finite, can anything useful nevertheless be said by invoking renormalization? We will see that it is the difference in the first few Seeley–DeWitt coefficients that is central to answering these questions. In particular, for any collection of conductors (be they perfect or imperfect) and/or dielectrics, as long as one merely moves them around without changing their shape or volume, then physically the Casimir energy difference (and so also the physically interesting Casimir forces) is guaranteed to be finite without invoking any renormalization.

General heat kernel coefficients for massless free spin-3/2 Rarita–Schwinger field

We review the general heat kernel method for the Dirac spinor field as an elementary example in any arbitrary background. We, then compute the first three Seeley–DeWitt coefficients for the massless free spin-3/2 Rarita–Schwinger field without imposing any limitations on the background geometry.

CT for higher derivative conformal fields and anomalies of (1, 0) superconformal 6d theories

In [8] we proposed the linear relations between the Weyl anomaly c1, c2, c3 coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories. These relations were determined up to one free parameter ξ and its value was then conjectured using some additional assumptions. A different value for ξ was recently suggested in arXiv:1702.03518 using an alternative method. Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient c3 for the higher derivative (1,0) superconformal 6d vector multiplet. This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar. We find the corresponding value of c3 which is proportional to the coefficient CT in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space. We present some general results of the computation of the Rényi entropy and CT from the partition function on S1 × ℍd − 1 for higher derivative conformal scalars, spinors and vectors in even dimensions. We also give an independent derivation of the conformal anomaly coefficients of the 6d higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.

The minimum supersymmetric standard model on noncommutative geometry

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space |$\mathcal {H}$| consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra |$\mathcal {A}$| in the triple. So, we obtain not only the vector supermultiplets that mediate |${\rm SU}(3)\otimes {\rm SU}(2)\otimes {\rm U}(1)_Y$| gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley–DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of |${\rm SU}(3)$|⁠, |${\rm SU}(2)$|⁠, and |${\rm U}(1)_Y$| is same as that of SU(5) unification theory.

TOWARD A SUPERGRAVITY SPECTRAL ACTION

A spectral action for a generalized bosonic sector corresponding to the Dirac operator of Euclidean supergravity is proposed. We calculate, up to a4, the Seeley–DeWitt coefficients in the expansion of the spectral action. It is in general not known how to construct a "matter fermionic" supersymmetric partner to the spectral action. The action we propose provides the effective action to be completed to get, at any order of the expansion, the corresponding supersymmetric action.

Worldline approach to noncommutative field theory

The study of the heat-trace expansion in non-commutative field theory has shown the existence of Moyal non-local Seeley–DeWitt coefficients which are related to the UV/IR mixing and manifest, in some cases, the non-renormalizability of the theory. We show that these models can be studied in a worldline approach implemented in phase space and arrive at a master formula for the n-point contribution to the heat-trace expansion. This formulation could be useful in understanding some open problems in this area, as the heat-trace expansion for the non-commutative torus or the introduction of renormalizing terms in the action, as well as for generalizations to other non-local operators.

Heat kernel expansion and induced action for the matrix model Dirac operator

We compute the quantum effective action induced by integrating out fermions in Yang-Mills matrix models on a 4-dimensional background, expanded in powers of a gauge-invariant UV cutoff. The resulting action is recast into the form of generalized matrix models, manifestly preserving the SO(D) symmetry of the bare action. This provides non-commutative (NC) analogs of the Seeley-de Witt coefficients for the emergent gravity which arises on NC branes, such as curvature terms. From the gauge theory point of view, this provides strong evidence that the non-commutative \( \mathcal{N} = 4 \) SYM has a hidden SO(10) symmetry even at the quantum level, which is spontaneously broken by the space-time background. The geometrical view proves to be very powerful, and allows to predict nontrivial loop computations in the gauge theory.

Supersymmetry of noncompact MQCD-like membrane instantons and heat kernel asymptotics

We perform a heat kernel asymptotics analysis of the nonperturbative superpotential obtained from wrapping of an M2-brane around a supersymmetric noncompact three-fold embedded in a (noncompact) G_2-manifold as obtained in [1], the three-fold being the one relevant to domain walls in Witten's MQCD [2], in the limit of small "zeta", a complex constant that appears in the Riemann surfaces relevant to defining the boundary conditions for the domain wall in MQCD. The MQCD-like configuration is interpretable, for small but non-zero zeta as a noncompact/"large" open membrane instanton, and for vanishing zeta, as the type IIA D0-brane (for vanishing M-theory cicle radius). We find that the eta-function Seeley de-Witt coefficients vanish, and we get a perfect match between the zeta-function Seeley de-Witt coefficients (up to terms quadratic in zeta) between the Dirac-type operator and one of the two Laplace-type operators figuring in the superpotential. This is an extremely strong signature of residual supersymmetry for the nonperturbative configurations in M-theory considered in this work.