Holographic renormalization for z = 2 Lifshitz spacetimes from AdS

Lifshitz spacetimes with the critical exponent z = 2 can be obtained by the dimensional reduction of Schrödinger spacetimes with the critical exponent z = 0. The latter spacetimes are asymptotically AdS solutions of AdS gravity coupled to an axion–dilaton system and can be uplifted to solutions of type IIB supergravity. This basic observation is used to perform holographic renormalization for four-dimensional asymptotically z = 2 locally Lifshitz spacetimes by the Scherk–Schwarz dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically locally AdS spacetimes coupled to an axion–dilaton system. We can thus define and characterize a four-dimensional asymptotically locally z = 2 Lifshitz spacetime in terms of five-dimensional AdS boundary data. In this setup the four-dimensional structure of the Fefferman–Graham expansion and the structure of the counterterm action, including the scale anomaly, will be discussed. We find that for asymptotically locally z = 2 Lifshitz spacetimes obtained in this way, there are two anomalies each with their own associated nonzero central charge. Both anomalies follow from the Scherk–Schwarz dimensional reduction of the five-dimensional conformal anomaly of AdS gravity coupled to an axion–dilaton system. Together, they make up an action that is of the Horava–Lifshitz type with a nonzero potential term for z = 2 conformal gravity.

Lifshitz holographic renormalization from anti-de Sitter 1

Lifshitz space–times with critical exponent z = 2 can be obtained by dimensional reduction of Schrödinger space–times with critical exponent z = 0. The latter space–times are asymptotically anti-de Sitter (AdS) solutions of AdS gravity coupled to an axion–dilaton system (or even just a massless scalar field). This basic observation is used to perform holographic renormalization for four-dimensional asymptotically locally Lifshitz space–times by dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically AdS space–times coupled to an axion–dilaton system. In this setup the four-dimensional structure of the Lifshitz – Fefferman-Graham expansion and the structure of the counterterm action, including the scale anomaly, will be summarized.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

From quantum to classical dynamics: The relativistic O ( N ) model in the framework of the real-time functional renormalization group

We investigate the transition from unitary to dissipative dynamics in the relativistic O(N) vector model with the λ(φ2)2 interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent z in the limit of large temperatures and in 2≤d≤4 spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.