Dimers in a bottle

We revisit D3-branes at toric CY3 singularities with orientifolds and their description in terms of dimer models. We classify orientifold actions on the dimer through smooth involutions of the torus. In particular, we describe new orientifold projections related to maps on the dimer without fixed points, leading to Klein bottles. These new orientifolds lead to novel N = 1 SCFT's that resemble, in many aspects, non-orientifolded theories. For instance, we recover the presence of fractional branes and some of them trigger a cascading RG-flow a la Klebanov-Strassler. The remaining involutions lead to non-supersymmetric setups, thus exhausting the possible orientifolds on dimers

Dimers in a Bottle

We revisit D3-branes at toric CY$_3$ singularities with orientifolds and their description in terms of dimer models. We classify orientifold actions on the dimer through smooth involutions of the torus. In particular, we describe new orientifold projections related to maps on the dimer without fixed points, leading to Klein bottles. These new orientifolds lead to novel $\mathcal{N}=1$ SCFT's that resemble, in many aspects, non-orientifolded theories. For instance, we recover the presence of fractional branes and some of them trigger a cascading RG-flow \`a la Klebanov-Strassler. The remaining involutions lead to non-supersymmetric setups, thus exhausting the possible orientifolds on dimers.Comment: 42 pages, 42 figures and 1 Tikz drawin

Dimers, orientifolds and stability of supersymmetry breaking vacua

We study (orientifolded) toric Calabi-Yau singularities in search for D-brane configurations which lead to dynamical supersymmetry breaking at low energy. By exploiting dimer techniques we are able to determine that while most realizations lead to a Coulomb branch instability, a rather specific construction admits a fully stable supersymmetry breaking vacuum. We describe the geometric structure that a singularity should have in order to host such a construction, and present its simplest example, the Octagon

Cusps, congruence groups and Monstrous dessins

We study general properties of the dessins d’enfants associated with the Hecke congruence subgroups Γ0(N) of the modular group PSL2(Z). The definition of the Γ0(N) as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set Γ0(N)∖PSL2(Z) as the projective lattices N-hyperdistant from a reference one, and hence as the projective line over the ring Z∕NZ. The natural action of PSL2(Z) on the lattices defines a dessin d’enfant structure, allowing for a combinatorial approach to features of the classical modular curves, such as the torsion points and the cusps. We tabulate the dessins d’enfants associated with the 15 Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group

A short introduction to Monstrous Moonshine

This paper is an introduction to the Monstrous Moonshine correspondence aiming at an undergraduate level. We review first the classification of finite simple groups and some properties of the monster $\mathbb{M}$, and then the theory of classical modular functions and modular forms, in order to define Klein's $J$-invariant. Eventually we turn to the correspondence itself, the historical framework in which it appeared, the ideas that were developped in its proof, and its status nowadays

Structures amassées, orientifolds de modèles de dimères et laminations supérieures

Les algèbres et variétés amassées se manifestent naturellement dans divers champs de la physique mathématique, comme la théorie de Teichmüller de rang supérieur et l'étude des modèles de dimères. D'une part, on étudie la généralisation des laminations de Thurston aux espaces de Teichmüller de rang supérieur correspondant à des groupes réels déployés. Cela conduit notamment à l'introduction de théories topologiques des champs quantiques liées aux algèbres de Iwahori-Hecke des groupes de Coxeter finis. Celles-là associent un polynôme de Laurent entier à chaque surface épointée de type fini. D'autre part, les modèles de dimères nous permettent de prouver l'existence d'une complétion ultraviolette stable du modèle SU(5) de brisure dynamique de supersymétrie. On dérive de plus des résultats généraux quant à l'existence d'anomalies de jauge sur des D-branes transverses à des orientifolds de singularités Calabi-Yau affines toriques. Enfin, on donne un sens physique aux modèles de dimères sur la bouteille de Klein. Par ailleurs, les deux parties introductives de ce manuscrit présentent de manière pédagogique la théorie de Teichmüller de rang supérieur de Fock et Goncharov, puis les modèles de dimères en théorie des cordes ainsi que leur emploi dans l'étude des correspondances holographiques.Cluster algebras and varieties naturally appear in various fields of mathematical physics, such as higher Teichmüller theory and dimer models - known as brane tilings in the context of string theory. On the first hand, we study the generalisation of Thurston's laminations to higher Teichmüller spaces in the real split case. This guides us towards introducing topological quantum field theories associated with the Iwahori-Hecke algebras of finite Coxeter groups. Those assign a Laurent polynomial with integer coefficients to each punctured surface of finite type. On the other hand, we use dimer models to prove the existence of a stable ultraviolet completion of the dynamical supersymmetry breaking SU(5) model. Moreover, we derive general results on the existence of gauge anomalies in the worldvolume of D-branes at orientifolds of affine toric Calabi-Yau singularities. Lastly, we provide a physical interpretation of brane tilings on the Klein bottle. Besides, the two preliminary parts of this dissertation are pedogogical invitations first to Fock and Goncharov's higher Teichmüller theory and then to the use of dimer models in string theory and in holography

Structures Amassées, Orientifolds de Modèles de Dimères et Laminations Supérieures.

Cluster algebras and varieties naturally appear in various fields of mathematical physics, such as higher Teichmüller theory and dimer models -- known as brane tilings in the context of string theory.On the first hand, we study the generalisation of Thurston's laminations to higher Teichmüller spaces in the real split case. This guides us towards introducing topological quantum field theories associated with the Iwahori--Hecke algebras of finite Coxeter groups. Those assign a Laurent polynomial with integer coefficients to each punctured surface of finite type.On the other hand, we use dimer models to prove the existence of a stable ultraviolet completion of the dynamical supersymmetry breaking $\mathrm{SU}(5)$ model. Moreover, we derive general results on the existence of gauge anomalies in the worldvolume of D-branes at orientifolds of affine toric Calabi--Yau singularities. Lastly, we provide a physical interpretation of brane tilings on the Klein bottle.Besides, the two preliminary parts of this dissertation are pedogogical invitations first to Fock and Goncharov's higher Teichmüller theory and then to the use of dimer models in string theory and in holography.Les algèbres et variétés amassées se manifestent naturellement dans divers champs de la physique mathématique, comme la théorie de Teichmüller de rang supérieur et l'étude des modèles de dimères. D'une part, on étudie la généralisation des laminations de Thurston aux espaces de Teichmüller de rang supérieur correspondant à des groupes réels déployés. Cela conduit notamment à l'introduction de théories topologiques des champs quantiques liées aux algèbres de Iwahori--Hecke des groupes de Coxeter finis. Celles-là associent un polynôme de Laurent entier à chaque surface épointée de type fini.D'autre part, les modèles de dimères nous permettent de prouver l'existence d'une complétion ultraviolette stable du modèle $\mathrm{SU}(5)$ de brisure dynamique de supersymétrie. On dérive de plus des résultats généraux quant à l'existence d'anomalies de jauge sur des D-branes transverses à des orientifolds de singularités Calabi--Yau affines toriques. Enfin, on donne un sens physique aux modèles de dimères sur la bouteille de Klein.Par ailleurs, les deux parties introductives de ce manuscrit présentent de manière pédagogique la théorie de Teichmüller de rang supérieur de Fock et Goncharov, puis les modèles de dimères en théorie des cordes ainsi que leur emploi dans l'étude des correspondances holographiques