Algebraic Structures and Eigenstates for Integrable Collective Field Theories

Conditions for the construction of polynomial eigen--operators for the Hamiltonian of collective string field theories are explored. Such eigen--operators arise for only one monomial potential $v(x) = \mu x^2$ in the collective field theory. They form a $w_{\infty}$--algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non--zero--energy polynomial eigen--operators. This analysis leads us to consider a particular potential $v(x)= \mu x^2 + g/x^2$. A Lie algebra of polynomial eigen--operators is then constructed for this potential. It is a symmetric 2--index Lie algebra, also represented as a sub--algebra of $U (s\ell (2)).$Comment: 27 page

Torsion and anomalies in the warped limit of Lifschitz theories

We describe the physics of fermionic Lifschitz theories once the anisotropic scaling exponent is made arbitrarily small. In this limit the system acquires an enhanced (Carrollian) boost symmetry. We show, both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous. This turns out to be a mixed anomaly between boosts and translations. In a Newton-Cartan formulation of the space-time geometry such anomaly is sourced by torsion. We use these results to give an effective field theory description of the anomalous transport coefficients, which were originally computed through Kubo formulas in [1]. Along the way we provide a link with warped CFTsThis work is supported by FPA2015-65480-P and by the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severo Ochoa SEV2016-0597. The work of C.C. is funded by Fundación La Caixa under “La Caixa-Severo Ochoa” international predoctoral grant. the author would like to thank Karl Landsteiner and Eric Bergshoeff for discussions and comments on the draft. He also would like to thank the organizers of the “Effective Theories of Quantum Phases of Matter” workshop at NORDITA, where part of this work was presente

Theory and algorithms for swept manifold intersections

Recent developments in such fields as computer aided geometric design, geometric modeling, and computational topology have generated a spate of interest towards geometric objects called swept volumes. Besides their great applicability in various practical areas, the mere geometry and topology of these entities make them a perfect testbed for novel approaches aimed at analyzing and representing geometric objects. The structure of swept volumes reveals that it is also important to focus on a little simpler, although a very similar type of objects - swept manifolds. In particular, effective computability of swept manifold intersections is of major concern. The main goal of this dissertation is to conduct a study of swept manifolds and, based on the findings, develop methods for computing swept surface intersections. The twofold nature of this goal prompted a division of the work into two distinct parts. At first, a theoretical analysis of swept manifolds is performed, providing a better insight into the topological structure of swept manifolds and unveiling several important properties. In the course of the investigation, several subclasses of swept manifolds are introduced; in particular, attention is focused on regular and critical swept manifolds. Because of the high applicability, additional effort is put into analysis of two-dimensional swept manifolds - swept surfaces. Some of the valuable properties exhibited by such surfaces are generalized to higher dimensions by introducing yet another class of swept manifolds - recursive swept manifolds. In the second part of this work, algorithms for finding swept surface intersections are developed. The need for such algorithms is necessitated by a specific structure of swept surfaces that precludes direct employment of existing intersection methods. The new algorithms are designed by utilizing the underlying ideas of existing intersection techniques and making necessary technical modifications. Such modifications are achieved by employing properties of swept surfaces obtained in the course of the theoretical study. The intersection problems is also considered from a little different prospective. A novel, homology based approach to local characterization of intersections of submanifolds and s-subvarieties of a Euclidean space is presented. It provides a method for distinguishing between transverse and tangential intersection points and determining, in some cases, whether the intersection point belongs to a boundary. At the end, several possible applications of the obtained results are described, including virtual sculpting and modeling of heterogeneous materials

Theories of Class F and Anomalies

We consider the 6d (2,0) theory on a fibration by genus g curves, and dimensionally reduce along the fiber to 4d theories with duality defects. This generalizes class S theories, for which the fibration is trivial. The non-trivial fibration in the present setup implies that the gauge couplings of the 4d theory, which are encoded in the complex structures of the curve, vary and can undergo S-duality transformations. These monodromies occur around 2d loci in space-time, the duality defects, above which the fiber is singular. The key role that the fibration plays here motivates refering to this setup as theories of class F. In the simplest instance this gives rise to 4d N=4 Super-Yang-Mills with space-time dependent coupling that undergoes SL(2, Z) monodromies. We determine the anomaly polynomial for these theories by pushing forward the anomaly polynomial of the 6d (2,0) theory along the fiber. This gives rise to corrections to the anomaly polynomials of 4d N=4 SYM and theories of class S. For the torus case, this analysis is complemented with a field theoretic derivation of a U(1) anomaly in 4d N=4 SYM. The corresponding anomaly polynomial is tested against known expressions of anomalies for wrapped D3-branes with varying coupling, which are known field theoretically and from holography. Extensions of the construction to 4d N = 0 and 1, and 2d theories with varying coupling, are also discussed.Comment: 54 pages, 1 figure, v2: added discussion of non-supersymmetric extension, v3: version as appears in JHE

ADE Spectral Networks

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.Comment: 68 pages plus appendices; visit http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that generates spectral networks; v2: version published in JHEP plus minor correction

Model independent constraints on geometrical extended theories of gravity

155 p.Desde los primeros estudios sobre el uso de Supernovas Tipo Ia a modo de candelas estándar, sabemosque el universo se expande aceleradamente. La descripción físico-matemática de este fenómeno puedeabordarse desde perspectivas muy diferentes. Por un lado, si se quiere mantener un universo descrito porla teoría de la Relatividad General de Einstein, es necesario acudir a la idea de que un tipo de fluidodiferente de la materia bariónica debe estar presente. Dentro de este escenario, una posibilidad es incluirla denominada constante cosmológica, que puede interpretarse como un fluido con ecuación de estadoconstante. De hecho, este modelo es el que actualmente está más aceptado y es considerado el modeloestándar de la Cosmología. Otra opción bastante más general es considerar un fluido dinámico, lo quenormalmente se conoce como energía oscura. Por último, una tercera perspectiva que puede adoptarse,conlleva dejar de lado el contexto de la Relatividad General y establecer otro marco teórico partiendo deuna definición diferente de la acción. Dentro de este grupo se encuentran las teorías de gravedadmodificada. Otra alternativa ampliamente estudiada dentro del marco de la Relatividad General es ladenominada energía oscura unificada. Dado que la presencia de energía oscura y de materia oscura en eluniverso es aproximadamente del mismo orden, y teniendo en cuenta que desconocemos la naturaleza deambas, parece lógico considerar un fluido capaz de describir el comportamiendo de ambos componentes.En esta tesis analizamos cada una de las perspectivas expuestas en lo anterior. Más concretamenteestudiamos las teorías f( R), f( Q) y un fluido unificado perteneciente a la familia de los Chaplygin

Model independent constraints on geometrical extended theories of gravity

155 p.Desde los primeros estudios sobre el uso de Supernovas Tipo Ia a modo de candelas estándar, sabemosque el universo se expande aceleradamente. La descripción físico-matemática de este fenómeno puedeabordarse desde perspectivas muy diferentes. Por un lado, si se quiere mantener un universo descrito porla teoría de la Relatividad General de Einstein, es necesario acudir a la idea de que un tipo de fluidodiferente de la materia bariónica debe estar presente. Dentro de este escenario, una posibilidad es incluirla denominada constante cosmológica, que puede interpretarse como un fluido con ecuación de estadoconstante. De hecho, este modelo es el que actualmente está más aceptado y es considerado el modeloestándar de la Cosmología. Otra opción bastante más general es considerar un fluido dinámico, lo quenormalmente se conoce como energía oscura. Por último, una tercera perspectiva que puede adoptarse,conlleva dejar de lado el contexto de la Relatividad General y establecer otro marco teórico partiendo deuna definición diferente de la acción. Dentro de este grupo se encuentran las teorías de gravedadmodificada. Otra alternativa ampliamente estudiada dentro del marco de la Relatividad General es ladenominada energía oscura unificada. Dado que la presencia de energía oscura y de materia oscura en eluniverso es aproximadamente del mismo orden, y teniendo en cuenta que desconocemos la naturaleza deambas, parece lógico considerar un fluido capaz de describir el comportamiendo de ambos componentes.En esta tesis analizamos cada una de las perspectivas expuestas en lo anterior. Más concretamenteestudiamos las teorías f( R), f( Q) y un fluido unificado perteneciente a la familia de los Chaplygin

The Analysis of Matched Layers

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of B\'erenger. We prove that the B\'erenger and closely related layers define well posed transmission problems in great generality. When the B\'erenger method or one of its close relatives is well posed, perfect matching is proved. The proofs use the energy method, Fourier-Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the B\'erenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in B\'erenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of B\'erenger's algorithm for Maxwell's equations our favorite choice leads to a strongly well posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence

Distribution of Resonances in Scattering by Thin Barriers

We study high energy resonances for the operators $-\Delta +\delta_{\partial\Omega}\otimes V$ and $-\Delta+\delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega$ is strictly convex with smooth boundary, $V:L^2(\partial\Omega)\to L^2(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$. These operators are model Hamiltonians for quantum corrals and leaky quantum graphs. We give a quantum version of the Sabine Law from the study of acoustics for both the $\delta$ and $\delta'$ interactions. It characterizes the decay rates (imaginary parts of resonances) in terms of the system's ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For the $\delta$ interaction we show that generically there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of the $\delta'$ interaction, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate and is optimal in the case of the unit disk in the plane. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose--Taylor parametrix for complex energies. We use these constructions to give a complete microlocal description of boundary layer operators and to prove sharp high energy estimates on the boundary layer operators in the case that $\partial\Omega$ is smooth and strictly convex.Comment: A portion of the semiclassical preliminaries section is taken from arXiv:1204.1305 with the authors' permission. This update includes the material from the previous versio

A Monte Carlo approach to statics and dynamics of quantum fluids

The main objective of the thesis is to study static and/or dynamic properties of a set of quantum fluids by means of quantum Monte Carlo techniques, mainly using the path integral formalism to obtain results both at zero temperature and finite temperature. First, we present briefly some of the more important quantum Monte Carlo methods, and introduce the Path Integral Monte Carlo (PIMC) method, which has been used during all this thesis, as well as the Path Integral Ground State (PIGS), which is an extension of the first at ground state. After introducing the basic formalism, we comment on the approximations needed and provide a comparison between different actions. We also comment on parallelization schemes and advanced sampling techniques. The first results shown in this thesis are for the phase diagram of a one-dimensional Coulomb gas, which have been obtained using the PIMC method. The phase diagram have been constructed mainly by calculating energetic and structural properties. The obtained results extend previous knowledge of different phases in the one-dimensional Coulomb gas at zero temperature. Our results show the existence of a quantum Wigner crystal regime and a Ideal Fermi gas regime at low temperatures. As temperature increases, we reach a classic Wigner crystal regime and a classical gas. In the following chapter we show the results of a quasi-one-dimensional para-H2. The aim of this work is to see how the quasi-one-dimensionality affects the Luttinger parameter when comparing it with the pure one-dimensional case. This is done at zero temperature using PIGS. As para-hydrogen is an important candidate to superfluidity, the main idea behind study a quasi-one-dimensional system is to reduce dimensionality in order to soften intermolecular interaction. For that, we try different external potentials to control the opening of the system in two dimensions. Despite an increase in the Luttinger parameter in the various quasi-one-dimensional cases, it still does not reach the values displaying superfluidity. The next work shown in the thesis is our extensive study of the dynamic structure factor for the 4He. Using Path Integral Monte Carlo, we compute the intermediate scattering function at different temperatures and perform an inversion in order to gain access at the dynamics of the system. Despite the ill-posed problem of this inversion, we obtain results in a qualitative agreement with the experiments and prove that our method of inversion, despite having to yield with inversion problems achieves to obtain better numerical results for 4He at finite temperature than the ones previously reported. In this sense, we provide comparisons with the Maximum Entropy method and with experimental results. The study at different temperatures shows us the dissappearance of the roton peak when we cross T=2.17K from the superfluid regime to the normal fluid. We also observe a kink in the momentum distribution at the superfluid regime that dissappears at higher temperatures, for which does not exist an explanation in the theory. In the final chapter of the thesis we provide a method to sample complex-time correlation functions whose aim is to obtain better dynamic structure factor functions than the ones obtained via pure imaginary-time correlation functions. This model has already been tested for single-particle systems. Our aim is to test it for multi-particle systems, and to see if we can still recover good results at a reasonable high complex-time when the number of particles is closer to the typical simulation values of real systems. We tested it with particles interacting with an harmonic potential. Despite an increased variance compared with the one-particle case, we obtain good results that allow us to obtain the dynamic structure factor. Comparing the results with ones obtained at pure-imaginary time, we show how the complex-time inversion is superior and provides results closer to the exact ones.L'objectiu principal d'aquesta tesi es l'estudi de propietats estàtiques i dinàmiques de diferents fluids quàntics utilitzant tècniques de Monte Carlo quàntiques, principalment emprant el formalisme de path integrals per obtenir resultats tan a temperatura zero com a temperatura finita. Primer de tot, presentem els mètodes de Monte Carlo quàntics més importants, i introduïm el mètode de Path Integral Monte Carlo (PIMC), que fem servir al llarg de tota la tesi, i el mètode de Path Integral Ground State (PIGS), que es una extensió del primer però a temperatura zero. Després d'introduir el formalisme bàsic, comentem les diferents aproximacions necessàries i aportem una comparació entre elles. També expliquem un possible mètode de paral·lelització i tècniques de mostreig avançat. Els primers resultats que mostrem en aquesta tesi son pel diagrama de fases d'un gas de Coulomb unidimensional, que hem obtingut emprant PIMC. Hem construït el diagrama de fases mitjançant el càlcul de propietats energètiques i estructurals. Els nostres resultats amplien estudis previs que s'havien realitzat pel mateix sistema a temperatura zero. Els nostres resultats mostren l'existència d'un règim de cristall de Wigner quàntic i un d'un gas de Fermi ideal a temperatures baixes. Incrementant la temperatura obtenim un cristall de Wigner clàssic i un gas clàssic. En el següent capítol ensenyem els resultats per un sistema quasi-unidimensional de parahidrogen. L'objectiu d'aquest estudi es veure si la quasi-unidimensionalitat afecta al paràmetre de Luttinger quan el comparem pel cas purament unidimensional. Això ho fem a temperatura zero utilitzant PIGS. Sent el parahidrogen un fort candidat a ser superfluid, la idea principal es veure si reduint la dimensionalitat del sistema podem alleugerir suficient la interacció intermolecular. Per fer-ho, provem diferents potencials externs per controlar l'obertura del sistema en dues de les dimensions. Tot i l'increment del paràmetre de Luttinger respecte al cas unidimensional, aquest no arriba als valors esperats per mostrar superfluïdesa. El següents resultats són del nostre estudi sobre el factor d'estructura dinàmic per 4He. Utilitzant PIMC, calculem la funció de dispersió a diferents temperatures i fem una inversió per tal d'accedir a les propietats dinàmiques del sistema. Tot i la naturalesa de problema mal posat d'aquesta inversió, obtenim resultats qualitativament bons en comparació amb els experimentals, i provem que el nostre mètode d'inversió obté resultats superiors per 4He a temperatura finita que els obtinguts prèviament utilitzant altres mètodes. En aquest sentit, aportem una comparació amb el mètode de màxima entropia i amb resultats experimentals. L'estudi a diferents temperatures ens deixa veure la desaparició del pic del rotó quan creuem T=2.17K des de el règim superfluid al fluid normal. També observem una curvatura estranya en la distribució de moments en el règim de superfluïdesa que desapareix a temperatures més elevades, i pel qual no existeix cap explicació teòrica. Finalment, mostrem un mètode per calcular funcions de correlació en temps complex, l'objectiu del qual es obtenir factors d'estructura dinàmic superiors als obtinguts en temps purament imaginari. Aquest model ha sigut provat amb èxit en sistemes d'una sola partícula. El nostre objectiu es veure si obtenim resultats bons en sistemes amb més partícules, i si el temps complex màxim al que podem accedir no es redueix amb aquest increment. Tot i l'increment en la variança, obtenim bons resultats pel factor dinàmic i, comparant-los amb els obtinguts amb temps imaginari, podem veure com el temps complex ofereix resultats més pròxims als exactes