Standard Model in Weyl conformal geometry

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal with no new fields beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry D(1) (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of D(1) by a geometric Stueckelberg mechanism in which the Weyl gauge field ($$\omega _\mu $$ ω μ ) acquires mass by “absorbing” the spin-zero mode of the $${\tilde{R}}^2$$ R ~ 2 term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field. The Higgs field ($$\sigma $$ σ ) has direct couplings to the Weyl gauge field ($$\sigma ^2\omega _\mu \omega ^\mu $$ σ 2 ω μ ω μ ). The SM fermions only acquire such couplings for non-vanishing kinetic mixing of the gauge fields of $$D(1)\times U(1)_Y$$ D ( 1 ) × U ( 1 ) Y . If this mixing is present, part of the mass of Z boson is not due to the usual Higgs mechanism, but to its mixing with massive $$\omega _\mu $$ ω μ . Precision measurements of Z mass then set lower bounds on the mass of $$\omega _\mu $$ ω μ which can be light (few TeV). In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio r on the spectral index $$n_s$$ n s is similar to that in Starobinsky inflation but mildly shifted to lower r by the Higgs non-minimal coupling to Weyl geometry.

More on the SW-QNM correspondence

We exploit the recently proposed correspondence between gravitational perturbations and quantum Seiberg-Witten curves to compute the spectrum of quasi-normal modes of asymptotically flat Kerr Newman black holes and establish detailed gauge/gravity dictionaries for a large class of black holes, D-branes and fuzzballs in diverse dimensions. QNM frequencies obtained from the quantum periods of SU(2) $$ \mathcal{N} $$ N = 2 SYM with Nf = 3 flavours are compared against numerical results, WKB (eikonal) approximation and geodetic motion showing remarkable agreement. Starting from the master example relating quasi-normal modes of Kerr-Newman black holes in AdS4 to SU(2) gauge theory with Nf = 4, we illustrate the procedure for some simple toy-models that allow analytic solutions. We also argue that the AGT version of the gauge/gravity correspondence may give precious hints as to the physical/geometric origin of the quasi-normal modes/Seiberg-Witten connection and further elucidate interesting properties (such as tidal Love numbers and grey-body factors) that can help discriminating black holes from fuzzballs.

Gauging scale symmetry and inflation: Weyl versus Palatini gravity

We present a comparative study of inflation in two theories of quadratic gravity with gauged scale symmetry: (1) the original Weyl quadratic gravity and (2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ($$w_\mu $$ w μ ) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of $$w_\mu $$ w μ ), Planck scale and metricity emerge in the broken phase after $$w_\mu $$ w μ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ($$\phi _1$$ ϕ 1 ), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their $$R^2$$ R 2 term, both theories have a small tensor-to-scalar ratio ($$r\sim 10^{-3}$$ r ∼ 10 - 3 ), larger in Palatini case. For a fixed spectral index $$n_s$$ n s , reducing the non-minimal coupling ($$\xi _1$$ ξ 1 ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $$\xi _1\le 10^{-3}$$ ξ 1 ≤ 10 - 3 , unlike the Palatini version, Weyl theory gives a dependence $$r(n_s)$$ r ( n s ) similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.

Langevin field equations: Properties and renormalization

Langevin equations for fields have been proposed to describe the dynamics of critical phenomena, or as an alternative method of quantization, which could be useful in cases where ordinary methods lead to difficulties, like in gauge theories. Some of their general properties will be described here. For a number of problems, in particular related to perturbation theory, it is more convenient to work with an action and a field integral than with the equation directly, because standard methods of quantum field theory (QFT) then become available. For this purpose, one can associate a field integral representation, involving a dynamic action to the Langevin equation. The dynamic action can be interpreted as generated by the Langevin equation, considered as a constraint equation. Quite generally, the integral representation of constraint equations, including stochastic equations, leads to an action that has a Slavnov–Taylor (ST) symmetry and, in a different form, has an anticommuting type Becchi–Rouet–Stora–Tyutin (BRST) symmetry, a symmetry that involves anticommuting parameters. This symmetry has no geometric origin, but is merely a consequence of associating a specific form of integral representations to the constraint equations. This symmetry is used in a number of different examples to prove the renormalizability of non-Abelian gauge theories, or to prove the geometric stability of models defined on homogeneous spaces under renormalization. In this chapter, it is used to prove Ward-Takahashi (WT) identities, and to determine how the Langevin equation renormalizes.

Bio-inspired origami metamaterials with metastable phases through mechanical phase transitions

Structural instability, once a catastrophic phenomenon to be avoided in engineering applications, is being harnessed to improve functionality of structures and materials, and has catalyzed a substantial research in the field. One important application is to create functional metamaterials that deform their internal structure to adjust performance, resembling phase transformations in natural materials. In this paper, we propose a novel origami pattern, named the Shrimp pattern, with application to multi-phase architected metamaterials whose phase transition is achieve mechanically by snap-through. The Shrimp pattern consists of units that can be easily tessellated in two dimensions, either periodically with homogeneous local geometry, or non-periodically with heterogeneous local geometries. We can use a few design parameters to program the unit cell to become either monostable or bistable, and tune the energy barrier between the bistable states. By tessellating these unit cells into an architected metamaterial, we can create complex yet navigable energy landscape, leading to multiple metastable phases of the material. As each phase has different geometry, the metamaterial can switch between different mechanical properties and shapes. The geometric origin of the multi-stable behavior implies that our designs are scale-independent, making them candidates for a variety of innovative applications, including reprogrammable materials, reconfigurable acoustic wave guides, and microelectronic mechanical systems and energy storage systems.

Four-dimensional Brane–Chern–Simons Gravity and Cosmology

From the field equations corresponding to a four-dimensional brane embedded in the five-dimensional spacetime of the Einstein–Chern–Simons theory for gravity, we find cosmological solutions that describe an accelerated expansion for a flat universe. Apart from a quintessence-type evolution scheme, we obtain a transient phantom evolution, which is not ruled out by the current observational data. Additionally, a bouncing solution is shown. The introduction of a kinetic term in the action shows a de Sitter behavior although the energy density is not constant. A quintessence behavior is also found. We conjecture on a possible geometric origin of dark energy coming from this action.

Einstein-aether theory in Weyl integrable geometry

We study the Einstein-aether theory in Weyl integrable geometry. The scalar field which defines the Weyl affine connection is introduced in the gravitational field equation. We end up with an Einstein-aether scalar field model where the interaction between the scalar field and the aether field has a geometric origin. The scalar field plays a significant role in the evolution of the gravitational field equations. We focus our study on the case of homogeneous and isotropic background spacetimes and study their dynamical evolution for various cosmological models.

Gnoseological research of geometric (by origin) method of analysis and synthesis

The subject of this research in the method of finding proof of theoretical hypotheses based on their content. It demonstrated that historically this method emerged in geometry, and thus is called geometric (by origin) method of analysis and synthesis. Then it was shifted to mechanics by Galileo, and later to other sciences, never obtaining the status of the universal method for confirming theoretical hypotheses. The author conducts gnoseological analysis of this method. First, gnoseological analysis of the three initial forms is carried out on the basis of the three elementary geometric theorems. Then description is given to qualitative transformations that followed its transfer to other sciences. The author discusses the correlation of this method with the classical method of analysis and synthesis, as well as with experimental and hypothetical-deductive method. The main conclusion consists in the thesis that in accordance with its gnoseological status, the method of geometric (by origin) analysis and synthesis is on the same level as the classical method of analysis and synthesis, as well as experimental and hypothetical-deductive method. The author’s special contribution lies in substantiation of the thesis that with the development of science, this method of geometric origin was generalized to the method of finding proof of any theoretical hypotheses. Determination of qualitative changes that occurred in terms of its transfer from geometry to special sciences has an important methodological meaning. The author notes the role of idealization in the process of shifting into theoretical description of researched objects the results of practical actions with them.

Higher form symmetries and M-theory

We discuss the geometric origin of discrete higher form symmetries of quantum field theories in terms of defect groups from geometric engineering in M-theory. The flux non-commutativity in M-theory gives rise to (mixed) ’t Hooft anomalies for the defect group which constrains the corresponding global structures of the associated quantum fields. We analyze the example of 4d $$ \mathcal{N} $$ N = 1 SYM gauge theory in four dimensions, and we reproduce the well-known classification of global structures from reading between its lines. We extend this analysis to the case of 7d $$ \mathcal{N} $$ N = 1 SYM theory, where we recover it from a mixed ’t Hooft anomaly among the electric 1-form center symmetry and the magnetic 4-form center symmetry in the defect group. The case of five-dimensional SCFTs from M-theory on toric singularities is discussed in detail. In that context we determine the corresponding 1-form and 2-form defect groups and we explain how to determine the corresponding mixed ’t Hooft anomalies from flux non-commutativity. Several predictions for non-conventional 5d SCFTs are obtained. The matching of discrete higher-form symmetries and their anomalies provides an interesting consistency check for 5d dualities.