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

Multifractal Characterization of Protein Contact Networks

The multifractal detrended fluctuation analysis of time series is able to reveal the presence of long-range correlations and, at the same time, to characterize the self-similarity of the series. The rich information derivable from the characteristic exponents and the multifractal spectrum can be further analyzed to discover important insights about the underlying dynamical process. In this paper, we employ multifractal analysis techniques in the study of protein contact networks. To this end, initially a network is mapped to three different time series, each of which is generated by a stationary unbiased random walk. To capture the peculiarities of the networks at different levels, we accordingly consider three observables at each vertex: the degree, the clustering coefficient, and the closeness centrality. To compare the results with suitable references, we consider also instances of three well-known network models and two typical time series with pure monofractal and multifractal properties. The first result of notable interest is that time series associated to proteins contact networks exhibit long-range correlations (strong persistence), which are consistent with signals in-between the typical monofractal and multifractal behavior. Successively, a suitable embedding of the multifractal spectra allows to focus on ensemble properties, which in turn gives us the possibility to make further observations regarding the considered networks. In particular, we highlight the different role that small and large fluctuations of the considered observables play in the characterization of the network topology

Komplexe Algebraische Geometrie

[no abstract available

Investigations of topological phases for quasi-1D systems

For a long time, quantum states of matter have been successfully characterized by the Ginzburg-Landau formalism that was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. The latter are materials that host metallic edge states in an insulating bulk, some of which are protected by the existing symmetries. Complementary to the search of topological phases in condensed matter, great efforts have been made in quantum simulations based on cold atomic gases. Sophisticated laser schemes provide optical lattices with different geometries and allow to tune interactions and the realization of artificial gauge fields. At the same time, new concepts coming from quantum information, based on entanglement, are pushing the frontier of our understanding of quantum phases as a whole. The concept of entanglement has revolutionized the description of quantum many-body states by describing wave functions with tensor networks (TN) that are exploited for numerical simulations based on the variational principle. This thesis falls within the framework of the studies in condensed matter physics: it focuses indeed on the so-called synthetic realization of quantum states of matter, more specifically, of topological ones, which may have on the long-run outfalls towards robust quantum computers. We propose a theoretical investigation of cold atoms in optical lattice pierced by effective (magnetic) gauge fields and subjected to experimentally relevant interactions, by adding a modern numerical approach based on TN algorithms. More specifically, this work will focus on (i) interacting topological phases in quasi-1D systems and, in particular, the Creutz-Hubbard model, (ii) the connection between condensed matter and high energy physics studying the Gross-Neveu model and the discretization of Wilson-Hubbard model, (iii) implementing tensor network-based algorithms.Durante mucho tiempo, los estados cuánticos de la materia se han caracterizado con éxito por el formalismo de Ginzburg-Landau que permitió de clasificar todos los diferentes tipos de transiciones de fase. Esta visión cambió con el descubrimiento del efecto Hall cuántico y los aislantes topológicos. Estos últimos son materiales que albergan estados de borde metálicos en una masa aislante, algunos de los cuales están protegidos por las simetrías existentes. Conjuntamente a la búsqueda de fases topológicas en materia condensada, se han hecho grandes esfuerzos en simulaciones cuánticas basadas en gases atómicos fríos. Los sofisticados esquemas láser proporcionan redes ópticas con diferentes geometrías y permiten ajustar las interacciones y la realización de campos de gauge artificial. Al mismo tiempo, los nuevos conceptos que provienen de la información cuántica, basados en el entanglement, están empujando la frontera de nuestra comprensión de las fases cuánticas en su conjunto. El concepto de entanglement ha revolucionado la descripción de los estados cuánticos de muchos cuerpos al describir las funciones de onda con redes tensoras (TN) que se explotan para simulaciones numéricas basadas en el principio de variación. Esta tesis se enmarca en los estudios de física de la materia condensada: en particular, se centra en la llamada realización sintética de los estados cuánticos de la materia, más específicamente, de los topológicos, que pueden tener en las salidas a largo plazo hacia computadoras cuánticas robustas. Se propone una investigación teórica de los átomos fríos en la red óptica con campos de gauge efectivos y sometidos a interacciones relevantes experimentalmente, agregando un enfoque numérico moderno basado en algoritmos TN. Más específicamente, este trabajo se centrará en (i) fases topológicas en los sistemas cuasi-1D y, en particular, el modelo Creutz-Hubbard, (ii) la conexión entre la materia condensada y la física de alta energía estudiando el modelo Gross-Neveu y el discretización del modelo Wilson-Hubbard, (iii) implementación de algoritmos basados en redes tensoras