Towards a CC-function in 4D quantum gravity

We develop a generally applicable method for constructing functions, $C$, which have properties similar to Zamolodchikov's $C$-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations. Employing the Euclidean framework of the Effective Average Action (EAA), we propose a $C$-function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime dimensions. It becomes stationary both at critical points and in classical regimes, and decreases monotonically along RG trajectories provided the breaking of the split-symmetry which relates background and quantum fields is sufficiently weak. Within the Asymptotic Safety approach we test the proposal for Quantum Einstein Gravity in $d>2$ dimensions, performing detailed numerical investigations in $d=4$. We find that the bi-metric Einstein-Hilbert truncation of theory space introduced recently is general enough to yield perfect monotonicity along the RG trajectories, while its more familiar single-metric analog fails to achieve this behavior which we expect on general grounds. Investigating generalized crossover trajectories connecting a fixed point in the ultraviolet to a classical regime with positive cosmological constant in the infrared, the $C$-function is shown to depend on the choice of the gravitational instanton which constitutes the background spacetime. For de Sitter space in 4 dimensions, the Bekenstein-Hawking entropy is found to play a role analogous to the central charge in conformal field theory. We also comment on the idea of a `$\Lambda$-$N$ connection' and the `$N$-bound' discussed earlier.Comment: 15 figures; additional comment

Quantum theory as a statistical theory under symmetry

Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory. A total parameter space $\Phi$, equipped with a group $G$ of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter $\phi$ can be estimated. Choose an experiment/ question $a$, and get from this a parameter space $\Lambda^{a}$, perhaps after some model reduction compatible with the group structure. The essentially statistical construction of this paper leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born's formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions. The theory is illustrated by a simple macroscopic example, and by the example of a spin 1/2 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry.Comment: The paper has been withdrawn because it is outdate

Condensates and instanton - torus knot duality. Hidden Physics at UV scale

We establish the duality between the torus knot superpolynomials or the Poincar\'e polynomials of the Khovanov homology and particular condensates in $\Omega$-deformed 5D supersymmetric QED compactified on a circle with 5d Chern-Simons(CS) term. It is explicitly shown that $n$-instanton contribution to the condensate of the massless flavor in the background of four-observable, exactly coincides with the superpolynomial of the $T(n,nk+1)$ torus knot where $k$ - is the level of CS term. In contrast to the previously known results, the particular torus knot corresponds not to the partition function of the gauge theory but to the particular instanton contribution and summation over the knots has to be performed in order to obtain the complete answer. The instantons are sitting almost at the top of each other and the physics of the "fat point" where the UV degrees of freedom are slaved with point-like instantons turns out to be quite rich. Also also see knot polynomials in the quantum mechanics on the instanton moduli space. We consider the different limits of this correspondence focusing at their physical interpretation and compare the algebraic structures at the both sides of the correspondence. Using the AGT correspondence, we establish a connection between superpolynomials for unknots and q-deformed DOZZ factors.Comment: v2: text substantially improve

Three-point function of semiclassical states at weak coupling

We give the derivation of the previously announced analytic expression for the correlation function of three heavy non-BPS operators in N=4 super-Yang-Mills theory at weak coupling. The three operators belong to three different su(2) sectors and are dual to three classical strings moving on the sphere. Our computation is based on the reformulation of the problem in terms of the Bethe Ansatz for periodic XXX spin-1/2 chains. In these terms the three operators are described by long-wave-length excitations over the ferromagnetic vacuum, for which the number of the overturned spins is a finite fraction of the length of the chain, and the classical limit is known as the Sutherland limit. Technically our main result is a factorized operator expression for the scalar product of two Bethe states. The derivation is based on a fermionic representation of Slavnov's determinant formula, and a subsequent bosonisation.Comment: 28 pages, 5 figures, cosmetic changes and more typos corrected in v

Coupled quantum kicked rotors: a study about dynamical localization, slow heating and thermalization

Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating

Hilbert modular forms and the theory of complex multiplication

En aquesta tesi presentem les propietats principals de les superfícies modulars de Hilbert i les formes modulars associades. La més remarcable és que poden ser vistes com a varietats modulars associades al grup ortogonal d'un espai quadràtic de tipus (2,2). Aquesta propietat dona una font de formes modulars, que estudiarem, posant un especial èmfasi al Borcherds lift i el Doi-Naganuma lift. Una vegada els fonaments per les superfícies modulars de Hilbert hagin estat establerts, introduirem la teoria de la Multiplicació Complexa, començant per alguns fets bàsics en el cas de corbes el·líptiques que servirà com a introducció per al cas de Multiplicació Complexa per a superfícies modulars de Hilbert. Mostrarem com obtenir els anomenats punts CM a la superfície modular de Hilbert i com avaluar el Borcherds lift en aquests punts. També veurem que aquests valors són nombres algebraics que pertanyen a cossos concrets i que quan avaluem una funció modular en tot un cicle CM obtenim nombres racionals amb múltiples factors primers. Donem diversos exemples de càlculs numèrics fets amb SageMath per confirmar els resultats teòrics.En esta tesis presentamos las propiedades principales de las superficies modulares de Hilbert y las formas modulares asociadas. La más remarcable es que pueden ser vistas como variedades modulares asociadas al grupo ortogonal de un espacio cuadrático de tipo (2,2). Esta propiedad nos da una fuente de formas modulares, que estudiaremos, poniendo especial énfasis en el Borcherds lift y el Doi-Naganuma lift. Una vez hayamos establecido los fundamentos de las superficies modulares de Hilbert, introduciremos la teoría de la Multiplicación Compleja, empezando por algunos hechos básicos en el caso de curvas elípticas que nos servirá como introducción para el caso de Multiplicación Compleja para superficies modulares de Hilbert. Mostraremos cómo obtener los llamados puntos CM en la superficie modular de Hilbert y cómo evaluar el Borcherds lift en esos puntos. También veremos que esos valores son números algebraicos pertenecientes a unos cuerpos concretos y que cuando evaluamos una función modular en todo el ciclo CM obtenemos números racionales con muchos factores primos. Damos varios ejemplos de los cálculos numéricos realizados con SageMath para respaldar los resultados teóricos.In this thesis we present the main properties of Hilbert modular surfaces and their associated modular forms. The most remarkable one is that they can be viewed as modular varieties associated to the orthogonal group of a quadratic space of type (2,2). This property provides a source of modular forms, which we will study, with a special focus on the so-called Borcherds lift and the Doi-Naganuma lift. Once the foundations of Hilbert modular surfaces and modular forms are established, we introduce the theory of Complex Multiplication, starting with some basic facts for elliptic curves that will serve as an introduction to the Theory of Complex Multiplication for Hilbert modular surfaces. We will show how to obtain the so-called CM points on the Hilbert Modular surface and how to evaluate Borcherds lifts on them. We will also see that those values are nice algebraic numbers in some concrete fields and that when we evaluate our modular function on a full CM cycle we get rational numbers with several prime factors. We provide several examples of those numerical computations on SageMath to support the theoretical results.Outgoin