We show that for classical Liouville field theory, diffeomorphism invariance, Weyl invariance and locality cannot hold together. This is due to a genuine Virasoro center, present in the theory, that leads to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and with a non-vanishing trace, in curved space. Our focus is on a field-independent term, proportional to the square of the Weyl gauge field, <math><msub><mrow><mi>W</mi></mrow><mrow><mi>μ</mi></mrow></msub><msup><mrow><mi>W</mi></mrow><mrow><mi>μ</mi></mrow></msup></math>, that makes the action Weyl-invariant and was disregarded in previous investigations of Weyl and conformal symmetry. We show this term to be related to the classical center of the Virasoro algebra. The mechanism uncovered here is a classical version of the quantum anomalous phenomenon: the generalization to curved space only allows to keep one of the two symmetries enjoyed by the flat space theory, either Lorentz (diffeomorphism) or conformal invariance.

Czechia

Pavel Haman

Alfredo Iorio

Classical gravitational anomalies of Liouville theory

It was recently shown that, taking into account the granular structure of graphene lattice, the Dirac-like dynamics of its quasiparticles resists beyond the lowest energy approximation. This can be described in terms of new phase-space variables, <math><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>,</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>)</mo></math>, that enjoy generalized Heisenberg algebras. In this letter, we add to that picture the important case of noncommuting <math><mover><mrow><mi>X</mi></mrow><mrow><mo>→</mo></mrow></mover></math>, for which <math><mo>[</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>j</mi></mrow></msup><mo>]</mo><mo>=</mo><mi>i</mi><mspace width="0.2em"></mspace><msup><mrow><mi>θ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup></math> and we find that <math><msup><mrow><mi>θ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace width="0.2em"></mspace><msup><mrow><mi>ϵ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup></math>, with ℓ the lattice spacing. We close by giving both the general recipe and a possible specific kinematic setup for the practical implementation of this approach to test noncommutative theories in tabletop analog experiments on graphene.

Austria

Boris Ivetić

Chile

Pablo Pais

Turning graphene into a lab for noncommutativity

To fully clarify the invariance of the classical Liouville field theory under the Virasoro algebra, we first elucidate in detail the concept of classical anomaly, discuss the occurrence of two symmetry algebras associated to this theory, and provide some new formulae to compute the classical center in a general fashion. We apply this to the study of the symmetries of the free boson in two dimensions. Moving to Liouville, we see how this gives rise to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and a non-vanishing trace, in curved space. We provide a variety of improvements of the (local) theory, that restore Weyl invariance. With explicit computations, we show that the covariant conservation of the Weyl-invariance-improved energy-momentum tensor is lost, in general, and relate the chosen improvement with the corresponding subset of preserved diffeomorphisms. The non-tensorial transformation rule of the Weyl-invariance-improved energy-momentum tensor in curved space is explicitly back-traced to the Virasoro center.

Trace and diffeomorphism anomalies of the classical Liouville theory, Virasoro algebras, Weyl-gauging and all that

We study the thermodynamics of spherically symmetric, neutral and non-rotating black holes in conformal (Weyl) gravity. To this end, we apply different methods: (i) the evaluation of the specific heat; (ii) the study of the entropy concavity; (iii) the geometrical approach to thermodynamics known as thermodynamic geometry; (iv) the Poincaré method that relates equilibrium and out-of-equilibrium thermodynamics. We show that the thermodynamic geometry approach can be applied to conformal gravity too, because all the key thermodynamic variables are insensitive to Weyl scaling. The first two methods, (i) and (ii), indicate that the entropy of a de Sitter black hole is always in the interval  $$2/3\le S\le 1$$  <math> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> <mo>≤</mo> <mi>S</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math>  , whereas thermodynamic geometry suggests that, at  $$S=1$$  <math> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>  , there is a second order phase transition to an Anti de Sitter black hole. On the other hand, we obtain from the Poincaré method (iv) that black holes whose entropy is  $$S &lt; 4/3$$  <math> <mrow> <mi>S</mi> <mo>&lt;</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> </mrow> </math>   are stable or in a saddle-point, whereas when  $$S&gt;4/3$$  <math> <mrow> <mi>S</mi> <mo>&gt;</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> </mrow> </math>   they are always unstable, hence there is no definite answer on whether such transition occurs. Since thermodynamics geometry takes the view that the entropy is an extensive quantity, while the Poincaré method does not require extensiveness, it is valuable to present here the analysis based on both approaches, and so we do.

Italy

Daniele Lanteri

China

Shen-Song Wan

Paolo Castorina

Stability of Schwarzschild (Anti)de Sitter black holes in conformal gravity

We investigate the structure of the gravity-induced generalized uncertainty principle in three dimensions. The subtleties of lower-dimensional gravity, and its important differences concerning four and higher dimensions, are duly taken into account, by considering different possible candidates for the gravitational radius, <math><msub><mi>R</mi><mi>g</mi></msub></math>, that is the minimal length/maximal resolution of the quantum mechanical localization process. We find that the event horizon of the <math><mi>M</mi><mo>≠</mo><mn>0</mn></math> Bañados-Teitelboim-Zanelli micro-black-hole furnishes the most consistent <math><msub><mi>R</mi><mi>g</mi></msub></math>. This allows us to obtain a suitable formula for the generalized uncertainty principle in three dimensions, and also to estimate the corrections induced by the latter on the Hawking temperature and Bekenstein entropy. We also point to the extremal <math><mi>M</mi><mo>=</mo><mn>0</mn></math> case, and its natural unit of length introduced by the cosmological constant, <math><mo>ℓ</mo><mo>=</mo><mn>1</mn><mo>/</mo><msqrt><mrow><mo>−</mo><mi>Λ</mi></mrow></msqrt></math>, as a possible alternative to <math><msub><mi>R</mi><mi>g</mi></msub></math>, and present a condensed matter analog realization of this scenario.

Gaetano Lambiase

Netherlands

Fabio Scardigli

Generalized uncertainty principle in three-dimensional gravity and the BTZ black hole

Assuming dislocations could be meaningfully described by torsion, we propose here a scenario based on the role of time in the low-energy regime of two-dimensional Dirac materials, for which coupling of the fully antisymmetric component of the torsion with the emergent spinor is not necessarily zero. Appropriate inclusion of time is our proposal to overcome well-known geometrical obstructions to such a program, that stopped further research of this kind. In particular, our approach is based on the realization of an exotic time-loop, that could be seen as oscillating particle-hole pairs. Although this is a theoretical paper, we moved the first steps toward testing the realization of these scenarios, by envisaging Gedankenexperiments on the interplay between an external electromagnetic field (to excite the pair particle-hole and realize the time-loops), and a suitable distribution of dislocations described as torsion (responsible for the measurable holonomy in the time-loop, hence a current). Our general analysis here establishes that we need to move to a nonlinear response regime. We then conclude by pointing to recent results from the interaction laser-graphene that could be used to look for manifestations of the torsion-induced holonomy of the time-loop, e.g., as specific patterns of suppression/generation of higher harmonics.

Spain

Marcelo F. Ciappina

Adamantia Zampeli

Torsion in quantum field theory through time-loops on Dirac materials