<div><div>In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying</div><div>definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.</div></div>

We consider integrable boundary states in the Heisenberg model. We begin by reviewing the algebraic Bethe Ansatz as well as integrable boundary states in spin chains. Then a new class of integrable states that was introduced last year by Caetano and Komatsu is described and expanded. We call these states the crosscap states. In these states each spin is entangled with its antipodal spin. We present a novel proof of the integrability of both a crosscap state that is known in the literature and one that is not previously known. We then use the machinery of the algebraic Bethe Ansatz to derive the overlaps between the crosscap states and off-shell Bethe states in terms of scalar products and other known overlaps. / Vi undersöker integrable gränstillstånd i Heisenbergmodellen. Vi börjar med att gå igenom den algebraiska Betheansatsen och integrabla gränstillstånd i spinnkedjor. Sedan beskrivs och expanderas en ny klass av integrabla tillstånd som introducerades förra året av Caetano och Komatsu. Vi kallar dessa tillstånd crosscap-tillstånd. I dessa tillstånd är varje spinn intrasslat med sin antipodala motsvarighet. Vidare presenterar vi ett nytt bevis av integrerbarheten hos både ett tidigare känt och ett nytt crosscap-tillstånd. Sedan använder vi den algebraiska Betheansatsens maskineri för att härleda överlappen mellan crosscap-tillstånden och off-shell Bethe tillstånd i termer av skalärprodukter och andra kända överlapp.

O objetivo deste trabalho foi o estudo do Ansatz de Bethe Algébrico (ABA), que é uma técnica utilizada na obtenção dos auto-estados do hamiltoniano de inúmeros modelos da Mecânica Estatística e da Teoria Quântica de Campos. Aplicamos este procedimento na diagonalização de três modelos de spins: o modelo de Heisenberg, o modelo de Heisenberg-Sklyanin e o modelo de Heisenberg-Cherednik. Na diagonalização do primeiro modelo, não foi possível encontrar todos os auto-estados do hamiltoniano através do ABA e, durante o procedimento de obtenção das expressões analíticas, nos deparamos com um conjunto de identidades inédito na literatura. A matriz de borda do modelo de Heisenberg-Sklyanin acopla o último e o primeiro sítios, generalizando o modelo anterior, e permite estabelecer uma relação limite com outros modelos integráveis. Neste caso também não conseguimos obter todos os auto-estados utilizando a técnica do ABA. Diferentemente do que ocorreu para os primeiros modelos, o de Heisenberg-Cherednik, com acoplamentos que alternam a intensidade ao longo da cadeia de spin, apresentou um conjunto completo de auto-estados quando diagonalizado pelo ABA. / The goal of this work was to study the Algebraic Bethe ansatz (ABA), which is a technique used to obtain the eigenstates of Hamiltonian of many models of Statistical Mechanics and Quantum Field Theory. We apply this procedure to diagonalize three types of spin models: the Heisenberg model, the Heisenberg-Sklyanin model and the Heisenberg-Cherednik model. On diagonalization of the …rst model, we could not …nd all the eigenstates of Hamiltonian through ABA, and during the procedure for obtaining the analytical expressions, we face an unprecedented set of identities in literature. The Sklyanin´s boundary matrix couples the fi…rst and last sites, generalizing the previous model, and provides a limit for other integrable models. In this case also did not get all eigenstates using the technique of ABA. Unlike what happened with the …rst models, the Heisenberg-Cherednik model, with alternating couplings the intensity along the spin chain, presented a complete set of eigenstates when diagonalized by ABA.

Nesta tese estudamos as propriedades críticas dos modelos anisotrópicos (isotrópicos) de Heisenberg com spin s arbitrário. O espectro das Hamiltonianas, com condições periódicas de contorno, foi calculado para redes finitas, resolvendo-se as equações do Bethe ansatz associadas. Nossos resultados indicam que a anomalia conforme destes modelos tem o valor c=3s/(1+s), independente da anisotropia, e os expoentes críticos variam continuamente com a anisotropia assim como no modelo de 8-vértices. O conteúdo de operadores destes modelos indica que a teoria de campos que governa a criticalidade destes modelos de spin é descrita por operadores formados pelo produto de um operador Gaussiano por outro com simetria Z(2s). Estudando estes modelos, com certas condições especiais de contorno, mostramos que eles são relacionados com uma nova classe de teorias unitárias recentemente propostas / This thesis is concerned with the critical properties of anisotropic (isotropic) Heisenberg chain,with arbitrary spin-s. The eigenspectrum of these Hamiltoniana, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The results indicate that the conformal anomaly hás the value c=3s/1+s, independently of the anisotropy, and the exponentes vary continuously with the anisotropy like in the 8-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-s models are described by composite fields formed by the product of Gaussian and Z(2s) fields. Studying these models, with some special boundary conditions, we show that they are related with a large class of unitary conformal field theories recntly introduced

C'est l'histoire d'un mathématicien qui est allé à la recherche des tores perdus<br />dans la jungle des systèmes complètement intégrables. Il a trouvé des feuilles<br />particulières et des tores pour construire une petite cabane qui donne une vue<br />topologique sur la jungle.

Using Sympelctic Field Theory as a computational tool, we compute Gromov-Witten theory of target curves using gluing formulas and quantum integrable systems. In the smooth case this leads to a relation of the results of Okounkov and Pandharipande with the quantum dispersionless KdV hierarchy, while in the orbifold case we prove triple mirror symmetry between GW theory of target P^1 orbifolds of positive Euler characteristic, singularity theory of a class of polynomials in three variables and extended affine Weyl groups of type ADE.

One of the main topics in the modern String Theory are the conjectured string/gauge (AdS/CFT) dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, i.e. the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality.The first part of this thesis is focused on the gravity side of the AdS5/CFT4 duality: we investigate the quantum integrability of the type IIB superstring on AdS5 x S5. In the pure spinor formulation we analyze the operator algebra by computing the operator product expansion of the Maurer-Cartan currents at the leading order in perturbation theory. With the same approach at one loop order, we show the path-independence of the monodromy matrix which implies the charge conservation law, strongly supporting the quantum integrability of the string sigma-model. We also verify that the Lax pair field strength remains well-defined at one-loop order being free from UV divergences. The same string sigma-model is analyzed in the Green-Schwarz formalism in the near-flat-space (NFS) limit. Such a limit remarkably simplifies the string world-sheet action but still leaving interesting physics. We use the NFS truncation to show the factorization of the world-sheet S-matrix at one-loop order. This property defines a two-dimensional field theory as integrable: it is the manifestation of the higher conserved charges. Hence, we have explicitly checked their presence at quantum level. The second part is dedicated to the AdS4/CFT3 duality: in particular the type IIA superstring on AdS4 x CP3. We compute the leading quantum corrections to the string energies for string configurations with a large but yet finite angular momentum on CP3 and show that they match the conjectured all-loop Bethe Ansatz equations.

Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.