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53 pages, 560 KiB

Open AccessReview

Notes on Derived Deformation Theory for Field Theories and Their Symmetries

by Ingmar Saberi

Symmetry 2025, 17(8), 1172; https://doi.org/10.3390/sym17081172 - 22 Jul 2025

Viewed by 222

These notes are an informal overview of techniques related to deformation theory in the context of physics. Beginning from motivation for the concept of a sheaf, they build up through derived functors, resolutions, and the functor of points to the notion of a moduli problem, emphasizing physical motivation and the principles of locality and general covariance at each step. They are primarily aimed at those who have some prior exposure to quantum field theory and are interested in acquiring some intuition or orientation regarding modern mathematical methods. A couple of small things are new, including a discussion of the twist of

N = 1

conformal supergravity in generic backgrounds at the level of the component fields and a computation relating the two-dimensional local cocycle for the Weyl anomaly to the one for the Virasoro anomaly. I hope they will serve as a useful appetizer for the more careful and complete treatments of this material that are already available. Full article

(This article belongs to the Special Issue Symmetries, and Symmetry Breaking in String Theory)

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21 pages, 462 KiB

Open AccessArticle

Inflaton Decay in No-Scale Supergravity and Starobinsky-like Models

by Yohei Ema, Marcos A. G. Garcia, Wenqi Ke, Keith A. Olive and Sarunas Verner

Universe 2024, 10(6), 239; https://doi.org/10.3390/universe10060239 - 30 May 2024

Cited by 7 | Viewed by 984

Abstract

We consider the decay of the inflaton in Starobinsky-like models arising from either an

R + R^{2}

theory of gravity or

N = 1

no-scale supergravity models. If Standard Model matter is simply introduced to the

R + R^{2}

theory, the [...] Read more.

We consider the decay of the inflaton in Starobinsky-like models arising from either an

R + R^{2}

theory of gravity or

N = 1

no-scale supergravity models. If Standard Model matter is simply introduced to the

R + R^{2}

theory, the inflaton (which appears when the theory is conformally transformed into the Einstein frame) couples to matter predominantly in Standard Model Higgs kinetic terms. This will typically lead to a reheating temperature of ∼3 ×

10^{9}

GeV. However, if the Standard Model Higgs is conformally coupled to curvature, the decay rate may be suppressed and vanishes for conformal coupling

ξ = 1 / 6

. Nevertheless, the inflaton decays through the conformal anomaly, leading to a reheating temperature of the order of

10^{8}

GeV. The Starobinsky potential may also arise in no-scale supergravity. In this case, the inflaton decays if there is a direct coupling of the inflaton to matter in the superpotential or to gauge fields through the gauge kinetic function. We also discuss the relation between the theories and demonstrate the correspondence between the no-scale models and the conformally coupled

R + R^{2}

theory (with

ξ = 1 / 6

). Full article

(This article belongs to the Special Issue Particle Physics and Cosmology: A Themed Issue in Honor of Professor Dimitri Nanopoulos)

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15 pages, 319 KiB

Open AccessArticle

On the Vacuum Structure of the

N = 4

Conformal Supergravity

by Ioannis Dalianis, Alex Kehagias, Ioannis Taskas and George Tringas

Universe 2021, 7(11), 409; https://doi.org/10.3390/universe7110409 - 28 Oct 2021

Viewed by 1868

Abstract

We consider

N = 4

conformal supergravity with an arbitrary holomorphic function of the complex scalar S which parametrizes the

S U (1, 1) / U (1)

coset. Assuming non-vanishings vevs for S and the scalars in a [...] Read more.

We consider

N = 4

conformal supergravity with an arbitrary holomorphic function of the complex scalar S which parametrizes the

S U (1, 1) / U (1)

coset. Assuming non-vanishings vevs for S and the scalars in a symmetric matrix

E_{i j}

of the

\bar{10}

S U (4)

R-symmetry group, we determine the vacuum structure of the theory. We find that the possible vacua are classified by the number of zero eigenvalues of the scalar matrix and the spacetime is either Minkowski, de Sitter, or anti-de Sitter. We determine the spectrum of the scalar fluctuations and we find that it contains tachyonic states which, however, can be removed by appropriate choice of the unspecified at the supergravity level holomorphic function. Finally, we also establish that S-supersymmetry is always broken whereas Q-supersymmetry exists only on flat Minkowski spacetime. Full article

(This article belongs to the Special Issue Gauge Theory, Strings and Supergravity)

21 pages, 309 KiB

Open AccessArticle

Conformal Symmetry and Supersymmetry in Rindler Space

by Jan-Willem van Holten

Universe 2020, 6(9), 144; https://doi.org/10.3390/universe6090144 - 4 Sep 2020

Cited by 1 | Viewed by 2728

Abstract

This paper addresses the fate of extended space-time symmetries, in particular conformal symmetry and supersymmetry, in two-dimensional Rindler space-time appropriate to a uniformly accelerated non-inertial frame in flat 1+1-dimensional space-time. Generically, in addition to a conformal co-ordinate transformation, the transformation of fields from Minkowski to Rindler space is accompanied by local conformal and Lorentz transformations of the components, which also affect the Bogoliubov transformations between the associated Fock spaces. I construct these transformations for massless scalars and spinors, as well as for the ghost and super-ghost fields necessary in theories with local conformal and supersymmetries, as arising from coupling to two-dimensional (2-D) gravity or supergravity. Cancellation of the anomalies in Minkowski and in Rindler space requires theories with the well-known critical spectrum of particles that arise in string theory in the limit of infinite strings, and it is relevant for the equivalence of Minkowski and Rindler frame theories. Full article

(This article belongs to the Special Issue Gauge Theory, Strings and Supergravity)

44 pages, 416 KiB

Open AccessReview

Conformal Symmetry in Field Theory and in Quantum Gravity

by Lesław Rachwał

Universe 2018, 4(11), 125; https://doi.org/10.3390/universe4110125 - 15 Nov 2018

Cited by 26 | Viewed by 3509

Abstract

Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Full article

(This article belongs to the Special Issue Gravity, Black Holes and Cosmology XXI)

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