Narrow Results

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension $\mathrm{\Delta }$ from the generalized ${ℛ}(p,q)$-deformed quantum algebra and investigate the ${ℛ}(p,q)$-deformed super Virasoro algebra with the particular conformal dimension $\mathrm{\Delta }=1.$ Furthermore, we perform the ${ℛ}(p,q)$-conformal Virasoro $n$-algebra, the ${ℛ}(p,q)$-conformal super Virasoro $n$-algebra ($n$-even) and discuss a toy model for the ${ℛ}(p,q)$-conformal Virasoro constraints and ${ℛ}(p,q)$-conformal super Virasoro constraints. Besides, we generalized the notion of the ${ℛ}(p,q)$-elliptic Hermitian matrix model with an arbitrary conformal dimension $\mathrm{\Delta }$. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.

The model of a point particle in the background of external symmetric tensor fields is analyzed from the higher spin theory perspective. It is proposed that the gauge transformations of the infinite collection of symmetric tensor fields may be read off from the covariance properties of the point particle action w.r.t. general canonical transformations. The gauge group turns out to be a semidirect product of all phase space canonical transformations to an Abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. A general configuration of external fields includes rank-0,1,2 symmetric tensors, so the whole system may be truncated to ordinary particle in Einstein–Maxwell backgrounds by switching off the higher-rank symmetric tensors. When otherwise all the higher rank tensors are switched on, the full gauge group provides a huge gauge symmetry acting on the whole infinite collection of symmetric tensors. We analyze this gauge symmetry and show that the symmetric tensors which couple to the point particle should not be interpreted as Fronsdal gauge fields, but rather as gauge fields of some conformal higher spin theories. It is shown that the Fronsdal fields system possesses twice as many symmetric tensor fields as is contained in the general background of the point particle. Besides, the particle action in general backgrounds is shown to reproduce De Wit–Freedman point particle–symmetric tensors first order interaction suggested many years ago, and extends their result to all orders in interaction, while the generalized equivalence principle completes the first order covariance transformations found in their paper, in all orders.

We show that logarithmic conformal field theories may be derived using nilpotent scale transformation. Using such nilpotent weights we derive properties of LCFT's, such as two and three point correlation functions solely from symmetry arguments. Singular vectors and the Kac determinant may also be obtained using these nilpotent variables, hence the structure of the four point functions can also be derived. This leads to non homogeneous hypergeometric functions. Also we consider LCFT's near a boundary. Constructing "superfields" using a nilpotent variable, we show that the superfield of conformal weight zero, composed of the identity and the pseudo identity is related to a superfield of conformal dimension two, which comprises of energy momentum tensor and its logarithmic partner. This device also allows us to derive the operator product expansion for logarithmic operators. Finally we discuss the AdS/LCFT correspondence and derive some correlation functions and a BRST symmetry.

The trace anomaly for a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary is considered. In the context of a perturbative evaluation of the theory's effective action explicit calculations are given for those contributions to the conformal anomaly which emerge as a result of free scalar propagation as well as from scalar self-interactions up to second order in the scalar self-coupling. The renormalization-group behavior of the theory is, subsequently, exploited in order to advance the evaluation of the conformal anomaly to third order in the scalar self-coupling. As a direct consequence the effective action is evaluated to the same order. In effect, complete contributions to the theory's conformal anomaly and effective action are evaluated up to fourth-loop order.

We discuss tachyon-free examples of (Type IIB on) noncompact nonsupersymmetric orbifolds. Tachyons are projected out by discrete torsion between orbifold twists, while supersymmetry is broken by a Scherk–Schwarz phase (+1/-1 when acting on space–time bosons/fermions) accompanying some even order twists. The absence of tachyons is encouraging for constructing nonsupersymmetric D3-brane gauge theories with stable infrared fixed points. The D3-brane gauge theories in our orbifold backgrounds have chiral supersymmetric spectra, but nonsupersymmetric interactions.

In this brief talk, I will try to focus on the things that happened at the conference that seemed very important, but that I didn’t understand.

We study knots in 𝕊³ obtained by the intersection of a minimal surface in ℝ⁴ with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold $M$:

Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schrödinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures — the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity.

Conformal, concircular, quasi-conformal and conharmonic curvature tensors play an important role in Riemannian geometry. In this paper, we study on normal complex contact metric manifolds under flatness conditions of these tensors.

The complete structure of the Casimir ${{𝒲}{𝒜}}_N$ algebras is shown to exist in such a way that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. We can conclude that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra, so it is clear from any construction of ${{𝒲}}_{\infty }$ algebra that by putting infinite number of fields $W_s$ with $s>N$ to zero, we arrive at the Casimir ${{𝒲}{𝒜}}_N$ algebra. We concentrated in this work only for the particular case of ${{𝒲}{𝒜}}_5$ algebra since this example gives us explicitly a method on how to deal with the general case $N$.

With the extended navigation data, we consider the generalized Zermelo navigation on Riemannian manifolds, admitting a space-dependent ship’s speed in the presence of perturbation determined by a weak velocity vector field, with application of Finsler metric of Randers type. The approach is shown via indicatrix and inner product. We also compare our findings in the context of conformality for the cases of weak and critical winds. The study is illustrated with the example in dimension 2.