The generalization of squeezing is realized in terms of the Virasoro algebra. The higher-order squeezing can be introduced through the higher-order time-dependent potential, in which the standard squeezing operator is generalized to higher-order Virasoro operators. We give a formula that describes the number of particles generated by the higher-order squeezing when a parameter specifying the degree of squeezing is small. The formula (18) shows that the higher the order of squeezing becomes the larger the number of generated particles grows.
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
In this lecture we discuss `beyond CFT from symmetry point of view. After reviewing the Virasoro algebra, we introduce deformed Virasoro algebras and elliptic algebras. These algebras appear in solvable lattice models and we study them by free field approach.
Using the {it off-shell} Noether current and potential we compute the entropy for the AdS black holes in new massive gravity. For the non-extremal BTZ black holes by implementing the so-called stretched horizon approach we reproduce the correct expression for the horizon entropy. For the extremal case, we adopt standard formalism in the AdS/CFT correspondence and reproduce the corresponding entropy by computing the central extension term on the asymptotic boundary of the near horizon geometry. We explicitly show the invariance of the angular momentum along the radial direction for extremal as well as non-extremal BTZ black holes in our model. Furthermore, we extend this invariance for the black holes in new massive gravity coupled with a scalar field, which correspond to the holographic renormalization group flow trajectory of the dual field theory. This provides another realization for the holographic c-theorem.
We show that the diffeomorphism anomaly together with the trace anomaly reveal a chiral Virasoro algebra near the event horizon of a black hole. This algebra is the same irrespective of whether the anomaly is covariant or consistent, thereby manifesting its universal character and the fact that only the outgoing modes are relevant near the horizon. Our analysis therefore clarifies the role of the trace anomaly in the diffeomorphism anomaly approach cite{wilczek, isowilczek, shailesh, shailesh2, sunandan, sunandan10, rabin10} to the Hawking radiation.
In this paper, we study large $c$ Virasoro blocks by using the Zamolodchikov monodromy method beyond its known limits. We give an analytic proof of our recent conjectur, which implied that the asymptotics of the large $c$ conformal blocks can be expressed in very simple forms, even if outside its known limits, namely the semiclassical limit or the heavy-light limit. In particular, we analytically discuss the fact that the asymptotic behavior of large $c$ conformal blocks drastically changes when the dimensions of external primary states reach the value $frac{c}{32}$, which is conjectured by our numerical studies. The results presented in this work imply that the general solutions to the Zamolodchikov recursion relation are given by Cardy-like formula, which is an important conclusion that can be numerically drawn from our recent works. Mathematical derivations and analytical results imply that, in the bulk, the collision behavior between two heavy particles may undergo a remarkable transition associated with their masses.