Supersymmetric invariant theories

We study field models for which a quantum action (i.e. the action appearing in the generating functional of Green functions) is invariant under supersymmetric transformations. We derive the Ward identity which is a direct consequence of this invariance. We consider a change of variables in functional integral connected with supersymmetric transformations when its parameter is replaced by a nilpotent functional of fields. Exact form of the corresponding Jacobian is found. We find restrictions on generators of supersymmetric transformations when a consistent quantum description of given field theories exists.

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Field-dependent BRST–anti-BRST Lagrangian transformations

International Journal of Modern Physics A ◽

10.1142/s0217751x15500219 ◽

2015 ◽

Vol 30 (04n05) ◽

pp. 1550021 ◽

Cited By ~ 13

Author(s):

Pavel Yu. Moshin ◽

Alexander A. Reshetnyak

Keyword(s):

Symmetry Breaking ◽

Ward Identity ◽

Brst Symmetry ◽

Lagrangian Formalism ◽

Yang Mills ◽

Gauge Dependence ◽

Change Of Variables ◽

Field Dependent ◽

Renormalization Group Approach ◽

Quantum Action

We continue our study of finite BRST–anti-BRST transformations for general gauge theories in Lagrangian formalism, initiated in [arXiv:1405.0790 [hep-th] and arXiv:1406.0179 [hep-th]], with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters, and prove the correctness of the explicit Jacobian in the partition function announced in [arXiv:1406.0179 [hep-th]], which corresponds to a change of variables with functionally dependent parameters λa = UaΛ induced by a finite Bosonic functional Λ(ϕ, π, λ) and by the anticommuting generators Ua of BRST–anti-BRST transformations in the space of fields ϕ and auxiliary variables πa, λ. We obtain a Ward identity depending on the field-dependent parameters λa and study the problem of gauge dependence, including the case of Yang–Mills theories. We examine a formulation with BRST–anti-BRST symmetry breaking terms, additively introduced into the quantum action constructed by the Sp(2)-covariant Lagrangian rules, obtain the Ward identity and investigate the gauge independence of the corresponding generating functional of Green's functions. A formulation with BRST symmetry breaking terms is developed. It is argued that the gauge independence of the above generating functionals is fulfilled in the BRST and BRST–anti-BRST settings. These concepts are applied to the average effective action in Yang–Mills theories within the functional renormalization group approach.

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Renormalisation II: Improving perturbation theory

10.1093/oso/9780198802877.003.0012 ◽

2018 ◽

Author(s):

Michael Kachelriess

Keyword(s):

Perturbation Theory ◽

Critical Phenomena ◽

Renormalisation Group ◽

Green Functions ◽

Ward Identities ◽

Generating Functional ◽

Vacuum Polarisation ◽

Quantum Action ◽

Effective Theories

This chapter introduces the quantum action as the generating functional of 1PI Green functions. The Ward identities of QED are derived and the vacuum polarisation is calculated. The renormalisation group equations are introduced. Critical phenomena and Wilsonian effective theories are discussed.

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Boundary conditions in topological AdS4/CFT3

Journal of High Energy Physics ◽

10.1007/jhep02(2021)156 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Pietro Benetti Genolini ◽

Matan Grinberg ◽

Paul Richmond

Keyword(s):

Field Theory ◽

Free Energy ◽

Boundary Conditions ◽

Smooth Solution ◽

Three Dimensional ◽

Field Theories ◽

Legendre Transformation ◽

Generating Functional ◽

Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

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SHEAF COHOMOLOGY AND FUNCTIONAL INTEGRATION

International Journal of Modern Physics A ◽

10.1142/s0217751x89002089 ◽

1989 ◽

Vol 04 (18) ◽

pp. 4919-4928

Author(s):

CHARLES NASH

Keyword(s):

Functional Integration ◽

Functional Integral ◽

Field Theories ◽

Quantum Field Theories ◽

Quantum Field ◽

Topological Properties ◽

Sheaf Cohomology

Various analytic and topological properties of the spaces of functions arising in the functional integral are derived. It is shown that these spaces can possess attractive properties such as continuity, smoothness, and complex analyticity. We provide illustrations of the results with examples taken from several quantum field theories in varying dimensions.

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Some comments on infinities on quantum field theory: A functional integral approach

Random Operators and Stochastic Equations ◽

10.1515/rose-2016-0006 ◽

2016 ◽

Vol 24 (2) ◽

Author(s):

Luiz C. L. Botelho

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Function Space ◽

Functional Integral ◽

Field Theories ◽

Integral Approach ◽

Quantum Field ◽

Functional Integrals ◽

Euclidean Field ◽

The Subject

AbstractWe analyze on the formalism of probabilities measures-functional integrals on function space the problem of infinities on Euclidean field theories. We also clarify and generalize our previous published studies on the subject.

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Interacting Fields

10.1093/oso/9780192844200.003.0011 ◽

2021 ◽

pp. 237-252

Author(s):

J. Iliopoulos ◽

T.N. Tomaras

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Perturbation Expansion ◽

Field Theories ◽

Green Functions ◽

Quantum Field ◽

Minkowski Space Time ◽

Dimensional Minkowski Space ◽

Feynman Rules ◽

Wightman Axioms

We present a simple form of the Wightman axioms in a four-dimensional Minkowski space-time which are supposed to define a physically interesting interacting quantum field theory. Two important consequences follow from these axioms. The first is the invariance under CPT which implies, in particular, the equality of masses and lifetimes for particles and anti-particles. The second is the connection between spin and statistics. We give examples of interacting field theories and develop the perturbation expansion for Green functions. We derive the Feynman rules, both in configuration and in momentum space, for some simple interacting theories. The rules are unambiguous and allow, in principle, to compute any Green function at any order in perturbation.

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The Principles of Renormalisation

10.1093/oso/9780192844200.003.0015 ◽

2021 ◽

pp. 304-328

Author(s):

J. Iliopoulos ◽

T.N. Tomaras

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Perturbation Expansion ◽

Power Counting ◽

Field Theories ◽

Asymptotic Freedom ◽

Green Functions ◽

Quantum Field Theories ◽

Quantum Field

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.

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