arXiv subjects

Markus B. Fröb

Publications and source records attributed to Markus B. Fröb.

At least 37 records · Page 2Linked to original sources

arXiv2024

Modular Hamiltonian and modular flow of massless fermions on a cylinder

We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.

math-ph↗

arXiv2024

Universal definition of the non-conformal trace anomaly

We show that there exists a generalized, universal notion of the trace anomaly for theories which are not conformally invariant at the classical level. The definition is suitable for any regularization scheme and clearly states to what extent the classical equations of motion should be used, thus resolving existing controversies surrounding previous proposals. Additionally, we exhibit the link between our definition of the anomaly and the functional Jacobian arising from a Weyl transformation.

hep-th↗

arXiv2024

Entropy-area law and temperature of de Sitter horizons from modular theory

We derive an entropy-area law for the future horizon of an observer in diamonds inside the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita--Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature $H/(2π)$.

hep-th↗

arXiv2023

Modular Hamiltonian for de Sitter diamonds

We determine the Tomita-Takesaki modular data for CFTs in double cone and light cone regions in conformally flat spacetimes. This includes in particular the modular Hamiltonian for diamonds in the de Sitter spacetime. In the limit where the diamonds become large, we show that the modular automorphisms become time translations in the static patch. As preparation, we also provide a pedagogical rederivation of the known results for Minkowski spacetime. With our results and using the Araki formula, it becomes possible to compute relative entanglement entropies for CFTs in these regions.

hep-th↗

arXiv2023

Synchronous coordinates and gauge-invariant observables in cosmological spacetimes

We consider the relational approach to construct gauge-invariant observables in cosmological perturbation theory using synchronous coordinates. We construct dynamical synchronous coordinates as non-local scalar functionals of the metric perturbation in the fully non-linear theory in an arbitrary gauge. We show that the observables defined in this dynamical coordinate system are gauge-independent, and that the full perturbed metric has the expected form in these coordinates. Our construction generalises the familiar synchronous gauge in linearised gravity, widely used in cosmological perturbation theory, to the non-linear theory. We also work out the expressions for the gauge-invariant Einstein equation, sourced either by an ideal fluid or a scalar field up to second order in perturbation theory, and give explicit expressions for the Hubble rate -- as measured by synchronous observers or by observers co-moving with the matter field -- up to that order. Finally, we consider quantised linear perturbations around Minkowski and de~Sitter backgrounds, and compute the two-point function of the gauge-invariant metric perturbation in synchronous coordinates, starting with two-point function in a general linear covariant gauge. Although the gauge-fixed two-point function contains gauge modes, we show that the resulting gauge-invariant two-point function only contains the physical tensor modes and it is thus positive, \ie, it has a spectral representation.

gr-qc↗

arXiv2023

A quantum energy inequality in the Sine--Gordon model

We consider the stress tensor in the massless Sine--Gordon model in the finite regime $β^2 < 4 π$ of the theory. We prove convergence of the renormalised perturbative series for the interacting stress tensor defined using the Bogoliubov formula in an arbitrary Hadamard state, even for the case that the smearing is only along a one-dimensional time-like worldline and not in space-time. We then show that the interacting energy density, as seen by an observer following this worldline, satisfies an absolute lower bound, that is a bound independent of the quantum state. Our proof employs and generalises existing techniques developed for free theories by Flanagan, Fewster and Smith.

math-ph↗

arXiv2023

Non-commutative coordinates from quantum gravity

Local observables in (perturbative) quantum gravity are notoriously hard to define, since the gauge symmetry of gravity -- diffeomorphisms -- moves points on the manifold. In particular, this is a problem for backgrounds of high symmetry such as Minkowski space or de Sitter spacetime which describes the early inflationary phase of our universe. Only recently this obstacle has been overcome, and a field-dependent coordinate system has been constructed to all orders in perturbation theory, in which observables are fully gauge-invariant. We show that these field-dependent coordinates are non-commutative, and compute their commutator to second order in the Planck length. This provides the first systematic derivation of non-commutativity that arises due to quantum gravity effects.

gr-qc↗

arXiv2022

Constructing CFTs from AdS flows

We study the renormalization group flow equations for correlation functions of weakly coupled quantum field theories in AdS. Taking the limit where the external points approach the conformal boundary, we obtain a flow of conformally invariant correlation functions. We solve the flow for one- and two-point functions and show that the corrections to the conformal dimensions can be obtained as an integral over the Mellin amplitude of the four-point function. We also derive the flow of the Mellin amplitude for higher $n$-point functions. We then consider the flows at tree level and one loop (in AdS), and show that one obtains exactly the recursion relations for the corresponding Mellin amplitudes derived earlier by Fitzpatrick et al. [arXiv:1107.1499] at tree level and Yuan [arXiv:1710.01361,arXiv:1801.07283] at one loop. As an application, we furthermore compute one-loop corrections to the conformal dimensions for some operators in the CFT dual to an $\mathrm{O}(N)$ scalar model in AdS.

hep-th↗

arXiv2022

On well-posedness and algebraic type of the five-dimensional charged rotating black hole with two equal-magnitude angular momenta

We study various mathematical aspects of the charged rotating black hole with two equal-magnitude angular momenta in five dimensions. We introduce a coordinate system that is regular on the horizon and in which Einstein-Maxwell equations reduce to an autonomous system of ODEs. Employing Bondi and Kruskal-like coordinates, we analyze the geometric regularity of the black hole metric at infinity and the horizon, respectively, and the well-posedness of the corresponding boundary value problem. We also study the algebraic types of the electromagnetic and curvature tensors. While outside the horizon the electromagnetic and Ricci tensors are of type D, the Weyl tensor is algebraically general. The Weyl tensor simplifies to type~II on the horizon and type~D on the bifurcation sphere. These results imply inconsistency of the metric with the Kerr--Schild form with a geodesic Kerr-Schild vector. This feature is shared by the four-dimensional Kerr-Newman metric and the vacuum Myers-Perry or charged Schwarzschild-Tangherlini geometries in arbitrary dimension, but hence not by the black hole we have considered here.

gr-qc↗

arXiv2022

Cosmological Perturbations and Invariant Observables in Geodesic Lightcone Coordinates

We consider a recent approach to the construction of gauge-invariant relational observables in gravity in the context of cosmological perturbation theory. These observables are constructed using a field-dependent coordinate system, which we take to be geodesic lightcone coordinates. We show that the observables are gauge-independent in the fully non-linear theory, and that they have the expected form when one adopts the geodesic lightcone gauge for the metric. We give explicit expressions for the Sasaki-Mukhanov variable at linear order, and the Hubble rate -- as measured both by geodesic observers and by observers co-moving with the inflaton -- to second order. Moreover, we show that the well-known linearised equations of motion for the Sasaki-Mukhanov variable and the scalar constraint variables follow from the gauge-invariant Einstein's equations.

gr-qc↗

arXiv2021

Deformations of Supergravity and Supersymmetry Anomalies

We present a BRST analysis of supersymmetry anomalies of $\mathcal{N} = 1$ supersymmetric quantum field theories with anomalous R symmetry. To this end, we consider the coupling of the matter theory to classical $\mathcal{N} = 1$ new minimal supergravity. We point out that a supersymmetry anomaly cocycle associated to the $\mathrm{U}(1){}_\mathrm{R}$ field does exist for this theory. It is non-trivial in the space of supergravity fields (and ghosts), but it becomes BRST-exact in the functional space that includes antifields. Equivalently, the $\mathrm{U}(1){}_\mathrm{R}$ supersymmetry anomaly cocycle vanishes "on-shell". It is therefore removable. However, to remove it -- precisely because it is not trivial in the smaller space of fields -- one needs to deform the supergravity BRST operator. This deformation is triggered, at first order in the anomaly coefficient, by a local operator $S_1$ of ghost number 1. We give a cohomological characterization of $S_1$ and compute it in full detail. At higher orders in the anomaly coefficient, we expect a priori that further deformations of the BRST rules are necessary.

hep-th↗

arXiv2021

Kerr-Schild metrics in teleparallel gravity

We show that the Kerr-Schild ansatz can be extended from the metric to the tetrad, and then to teleparallel gravity where curvature vanishes but torsion does not. We derive the equations of motion for the Kerr-Schild null vector, and describe the solution for a rotating black hole in this framework. It is shown that the solution depends on the chosen tetrad in a non-trivial way if the spin connection is fixed to be the one of the flat background spacetime. We show furthermore that any Kerr-Schild solution with a flat background is also a solution of $f(\mathcal{T})$ gravity.

gr-qc↗

arXiv2021

FieldsX -- An extension package for the xAct tensor computer algebra suite to include fermions, gauge fields and BRST cohomology

I present the tensor computer algebra package FieldsX, which extends the xAct suite of tensor algebra packages to perform computations in field theory with fermions and gauge fields. This includes the standard tools of curved-space $γ$ matrices, Fierz identities, invariant tensors on Lie algebras, arbitrary gradings and left and right variational derivatives, as well as the decomposition of spinor products into irreducible components following the approach of d'Auria, Fré, Maina and Regge [Annals Phys. 139 (1982) 93]. Lastly, it also includes functions to work with nilpotent differentials such as the BV-BRST differential for (supersymmetric) gauge theories and to compute their (relative) cohomologies, from which anomalies and gauge-invariant operators can be determined. I illustrate the use of the package with the example of $\mathcal{N}=1$ Super-Yang--Mills theory.

hep-th↗

arXiv2021

All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory

We prove the existence of the operator product expansion (OPE) in Euclidean Yang-Mills theories as a short-distance expansion, to all orders in perturbation theory. We furthermore show that the Ward identities of the underlying gauge theory are reflected in the OPE; especially, the OPE of an arbitrary number of gauge-invariant composite operators only involves gauge-invariant composite operators. Moreover, we derive recursion relations which allow to construct the OPE coefficients, the quantum BRST differential and the quantum antibracket order by order in perturbation theory, starting from the known free-theory objects. These relations are completely finite from the start, and do not need any further renormalisation as is usually the case in other approaches. Our results underline the importance of the OPE as a general structure underlying quantum field theories. The proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, and generalise similar results recently obtained for scalar field theories [J. Holland and S. Hollands, Commun. Math. Phys. 336 (2015) 1555; J. Holland and S. Hollands, J. Math. Phys. 56 (2015) 122303]. Combining their results with our recursion formula, we also obtain associativity of the OPE coefficients.

math-ph↗

arXiv2019

Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories

We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul-Tate and the full BRST differential in the BV-BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an $L_\infty$ algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras.

math-ph↗

arXiv2019

Approaches to linear local gauge-invariant observables in inflationary cosmologies

We review and relate two recent complementary constructions of linear local gauge-invariant observables for cosmological perturbations in generic spatially flat single-field inflationary cosmologies. After briefly discussing their physical significance, we give explicit, covariant and mutually invertible transformations between the two sets of observables, thus resolving any doubts about their equivalence. In this way, we get a geometric interpretation and show the completeness of both sets of observables, while previously each of these properties was available only for one of them.

gr-qc↗

arXiv2019

Trace anomaly for chiral fermions via Hadamard subtraction

We calculate the trace (conformal) anomaly for chiral fermions in a general curved background using Hadamard subtraction. While in intermediate steps of the calculation imaginary terms proportional to the Pontryagin density appear, imposing a vanishing divergence of the stress tensor these terms completely cancel, and we recover the well-known result equal to half the trace anomaly of a Dirac fermion. We elaborate in detail on the advantages of the Hadamard method for the general definition of composite operators in general curved spacetimes, and speculate on possible causes for the appearance of the Pontryagin density in other calculations.

hep-th↗

arXiv2019

One-loop quantum gravitational backreaction on the local Hubble rate

We determine corrections to the Hubble rate due to graviton loops in a cosmological background spacetime of constant deceleration parameter. The corrections are gauge-invariant, based on a recent proposal for all-order gauge-invariant observables in perturbative quantum gravity. We find explicit expressions for the cases of matter- and radiation-dominated eras and slow-roll inflation with vanishing second slow-roll parameter. Interestingly, in the latter case the corrections can be described by a quantum-corrected first slow-roll parameter, which brings the spacetime closer to de Sitter space.

gr-qc↗