We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory (i) at the classical level within the ambit of Lagrangian formulation, and (ii) at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles.

India

Bhagya. R

Harsha Sreekumar

E. Harikumar

R.P. Malik

Constraints, conserved charges and extended BRST algebra for a 3D field-theoretic example for Hodge theory

We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too.

B. Chauhan

A.K. Rao

Constraints, symmetry transformations and conserved charges for massless Abelian 3-form theory

We derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations for any arbitrary D-dimensional Stückelberg-modified massive Abelian 3-form theory within the framework of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism where, in addition to the horizontality condition (HC), we exploit the theoretical strength of the gauge invariant restriction (GIR) to deduce the proper transformations for the gauge, associated (anti-)ghost fields, auxiliary fields, Stückelberg compensating field, etc. In fact, it is an elegant and delicate combination of HC and GIR (within the ambit of AVSA) that is crucial for all our discussions and derivations. One of the highlights of our present endeavour is the deduction of a new set of (anti-)BRST invariant Curci-Ferrari (CF)-type restrictions which are not found in the massless version of our present theory where only the HC plays an important role in the derivations of all the (anti-)BRST transformations and a very specific set of CF-type restrictions. The alternative ways of the derivation of the full set of the latter, from various theoretical considerations, are also interesting results of our present investigation.

Nilpotent symmetries of a modified massive Abelian 3-form theory: Augmented superfield approach

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two- <math id="M1"><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfenced></math> dimensional (2D) diffeomorphism symmetry transformations on the worldsheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari- (CF-) type restrictions (using MBTSA) which turn out to be the root cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already-known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

A. Tripathi

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

Within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the full set of proper BRST and anti-BRST transformations for a 2D diffeomorphism invariant theory which is described by the Lagrangian density of a standard bosonic string. The above (anti-)BRST transformations are off-shell nilpotent and absolutely anticommuting. The latter property is valid on a submanifold of the space of quantum fields where the 2D version of the universal (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is satisfied. We derive the precise forms of the BRST and anti-BRST invariant Lagrangian densities as well as the exact expressions for the conserved (anti-)BRST and ghost charges. The lucid derivation of the proper anti-BRST symmetry transformations and the emergence of the CF-type restrictions are completely novel results for our present bosonic string which has already been discussed earlier in literature where only the BRST symmetry transformations have been pointed out. We briefly mention the derivation of the CF-type restrictions from the modified version of the Bonora-Tonin superfield approach, too.

Nilpotent Symmetries of a Model of 2D Diffeomorphism Invariant Theory: BRST Approach

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one <math id="M1"><mfenced><mrow><mn>0</mn><mo>+</mo><mn>1</mn></mrow></mfenced></math>-dimensional (1D) model of a free massive spinning relativistic particle (i.e., a supersymmetric system) by exploiting its classical infinitesimal and continuous reparameterization symmetry transformations. We use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the target space variables and the (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction for the 1D model of our supersymmetric (SUSY) system. The nilpotent (anti-)BRST symmetry transformations for other variables of our model are derived by using the (anti-)chiral supervariable approach (ACSA) to BRST formalism. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e., a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e., 1D diffeomorphism) invariant models of the (non-)relativistic particles.

Reparameterization Invariant Model of a Supersymmetric System: BRST and Supervariable Approaches

We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose space <math id="M1"><mfenced><mrow><mi>x</mi></mrow></mfenced></math> and time <math id="M2"><mfenced><mrow><mi>t</mi></mrow></mfenced></math> variables are a function of an evolution parameter <math id="M3"><mfenced><mrow><mi>τ</mi></mrow></mfenced></math>. The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter <math id="M4"><mfenced><mrow><mi>τ</mi></mrow></mfenced></math>. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by <math id="M5"><mi>τ</mi></math>) is generalized onto a <math id="M6"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math>-dimensional supermanifold which is characterized by the superspace coordinates <math id="M7"><msup><mrow><mi>Z</mi></mrow><mrow><mi>M</mi></mrow></msup><mo>=</mo><mfenced><mrow><mi>τ</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mover><mi>θ</mi><mo>¯</mo></mover></mrow></mfenced></math> where a pair of the Grassmannian variables satisfy the fermionic relationships: <math id="M8"><msup><mrow><mi>θ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mover><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn></math>, <math id="M9"><mi>θ</mi><mtext> </mtext><mover><mi>θ</mi><mo>¯</mo></mover><mo>+</mo><mover><mi>θ</mi><mo>¯</mo></mover><mtext> </mtext><mi>θ</mi><mo>=</mo><mn>0</mn></math>, and <math id="M10"><mi>τ</mi></math> is the bosonic evolution parameter. In the context of ACSA, we take into account only the <math id="M11"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math>-dimensional (anti)chiral super submanifolds of the general<math id="M12"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math>-dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.

Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

We discuss the continuous and infinitesimal gauge, supergauge, reparameterization, nilpotent Becchi-Rouet-Stora-Tyutin (BRST), and anti-BRST symmetries and derive corresponding nilpotent charges for the one <math id="M1"><mfenced><mrow><mn>0</mn><mo>+</mo><mn>1</mn></mrow></mfenced></math>-dimensional (1D) massive model of a spinning relativistic particle. We exploit the theoretical potential and power of the BRST and supervariable approaches to derive the (anti-)BRST symmetries and coupled (but equivalent) Lagrangians for this system. In particular, we capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of the newly proposed (anti-)chiral supervariable approach (ACSA) to BRST formalism where only the (anti-)chiral supervariables (and their suitable super expansions) are taken into account along the Grassmannian direction(s). One of the novel observations of our present investigation is the derivation of the Curci-Ferrari- (CF-) type restriction by the requirement of the absolute anticommutativity of the (anti-)BRST charges in the ordinary space. We obtain the same restriction within the framework of ACSA to BRST formalism by (i) the symmetry invariance of the coupled Lagrangians and (ii) the proof of the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges. The observation of the anticommutativity property of the (anti-)BRST charges is a novel result in view of the fact that we have taken into account only the (anti-)chiral super expansions.

Massive Spinning Relativistic Particle: Revisited under BRST and Supervariable Approaches

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) approach, we discuss mainly the fermionic (i.e., off-shell nilpotent) (anti-)BRST, (anti-)co-BRST, and some discrete dual symmetries of the appropriate Lagrangian densities for a two (<math id="M1"><mn>1</mn><mo>+</mo><mn>1</mn></math>)-dimensional (2D) modified Proca (i.e., a massive Abelian 1-form) theory without any interaction with matter fields. One of the novel observations of our present investigation is the existence of some kinds of restrictions in the case of our present Stückelberg-modified version of the 2D Proca theory which is not like the standard Curci-Ferrari (CF) condition of a non-Abelian 1-form gauge theory. Some kinds of similarities and a few differences between them have been pointed out in our present investigation. To establish the sanctity of the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries, we derive them by using our newly proposed (anti-)chiral superfield formalism where a few specific and appropriate sets of invariant quantities play a decisive role. We express the (anti-)BRST and (anti-)co-BRST conserved charges in terms of the superfields that are obtained after the applications of (anti-)BRST and (anti-)co-BRST invariant restrictions and prove their off-shell nilpotency and absolute anticommutativity properties, too. Finally, we make some comments on (i) the novelty of our restrictions/obstructions and (ii) the physics behind the negative kinetic term associated with the pseudoscalar field of our present theory.

S. Kumar

Modified 2D Proca Theory: Revisited under BRST and (Anti-)chiral Superfield Formalisms

We propose a novel mechanism for the production of gravitational waves in the early Universe that originates from the relaxation processes induced by the QCD phase transition. While the energy density of the quark-gluon mean-field is monotonously decaying in real time, its pressure undergoes a series of violent oscillations at the characteristic QCD time scales that generate a primordial multi-peaked gravitational waves signal in the radio frequencies’ domain. The signal is an echo of the QCD phase transition that is accessible by planned measurements at the FAST and SKA telescopes.

China

Andrea Addazi

Antonino Marcianò

Sweden

Roman Pasechnik

Time-crystal ground state and production of gravitational waves from QCD phase transition A.A and A.M. acknowledge support by the NSFC(11875113), the Shanghai Municipality(KBH1512299), and by Fudan University(JJH1512105). R.P. was partially supported by the Swedish Research Council, contract numbers 621-2013-428 and 2016-05996, by CONICYT grant MEC80170112 (Chile), as well as by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme(668679). This work was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic, project LT17018. The work has been performed in the framework of COST Action CA15213 “Theory of hot matter and relativistic heavy-ion collisions” (THOR)