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A bstract We consider the double scaling limit of β -deformed planar N = 4 supersymmetric Yang-Mills theory (SYM), which has been argued to be conformal and integrable. It is a special point in the three-parameter space of double-scaled γ i -deformed N = 4 SYM, preserving N = 1 supersymmetry. The Feynman diagrams of the general three-parameter models form a “dynamical fishnet” that is much harder to analyze than the original one-parameter fishnet, where major progress in uncovering the model’s integrable structure has been made in recent years. Here we show that by applying N = 1 superspace techniques to the β -deformed model the dynamical nature of its Feynman graph expansion disappears, and we recover a regular lattice structure of brick wall (honeycomb) type. As a first application, we compute the zero-mode-fixed thermodynamic free energy of this model by applying Zamolodchikov’s method of inversion to the supersymmetric brick wall diagrams.

We study the free energy of an integrable, planar, chiral and non-unitary four-dimensional Yukawa theory, the bi-fermion fishnet theory discovered by Pittelli and Preti. The typical Feynman-diagrams of this model are of regular “brick-wall”-type, replacing the regular square lattices of standard fishnet theory. We adapt A. B. Zamolodchikov’s powerful classic computation of the thermodynamic free energy of fishnet graphs to the brick-wall case in a transparent fashion, and find the result in closed form. Finally, we briefly discuss two further candidate integrable models in three and six dimensions related to the brick wall model.

A bstract Modular symmetries naturally combine with traditional flavor symmetries and CP , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ 2 orbifold, which is equipped with two modular symmetries SL(2 , ℤ) T and SL(2 , ℤ) U associated with two moduli: the Kähler modulus T and the complex structure modulus U . The resulting finite modular group is (( S 3 × S 3 ) ⋊ ℤ 4 ) × ℤ 2 including mirror symmetry (that exchanges T and U ) and a generalized CP -transformation. Together with the traditional flavor symmetry ( D 8 × D 8 )/ℤ 2 , this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and T = U = exp π i 3 . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.

A bstract We present a detailed analysis of the eclectic flavor structure of the two-dimensional ℤ 2 orbifold with its two unconstrained moduli T and U as well as SL(2 , ℤ) T × SL(2 , ℤ) U modular symmetry. This provides a thorough understanding of mirror symmetry as well as the R -symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the ( T , U ) modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP and R -symmetries on the superpotential and Kähler potential of the theory.

A bstract We propose a double-scaling limit of β -deformed ABJM theory in three-dimensional N = 2 superspace, and a non-local deformation thereof. Due to the regular appearance of the theory’s Feynman supergraphs, we refer to this superconformal and integrable theory as the superfishnet theory. We use techniques inspired by the integrability of bi-scalar fishnet theory and adapted to superspace to calculate the zero-mode-fixed thermodynamic free energy, the corresponding critical coupling, and the exact all-loop scaling dimensions of various operators. Furthermore, we confirm the results of the supersymmetric dynamical fishnet theory by applying our methods to four-dimensional N = 1 superspace.

We propose a double-scaling limit of β -deformed ABJM theory in three-dimensional$$ \\mathcal{N} $$N = 2 superspace, and a non-local deformation thereof. Due to the regular appearance of the theory’s Feynman supergraphs, we refer to this superconformal and integrable theory as the superfishnet theory. We use techniques inspired by the integrability of bi-scalar fishnet theory and adapted to superspace to calculate the zero-mode-fixed thermodynamic free energy, the corresponding critical coupling, and the exact all-loop scaling dimensions of various operators. Furthermore, we confirm the results of the supersymmetric dynamical fishnet theory by applying our methods to four-dimensional$$ \\mathcal{N} $$N = 1 superspace.

Abstract We propose a double-scaling limit of β-deformed ABJM theory in three-dimensional N $$ \\mathcal{N} $$ = 2 superspace, and a non-local deformation thereof. Due to the regular appearance of the theory’s Feynman supergraphs, we refer to this superconformal and integrable theory as the superfishnet theory. We use techniques inspired by the integrability of bi-scalar fishnet theory and adapted to superspace to calculate the zero-mode-fixed thermodynamic free energy, the corresponding critical coupling, and the exact all-loop scaling dimensions of various operators. Furthermore, we confirm the results of the supersymmetric dynamical fishnet theory by applying our methods to four-dimensional N $$ \\mathcal{N} $$ = 1 superspace.

This thesis examines the correspondence between models of statistical physics and Feynman graphs of quantum field theories by a common property: integrability. We review integrable structures for periodic boundary conditions on both sides, while focusing on the eightand six-vertex model and the bi-scalar fishnet theory. The latter is a double-scaled γdeformations of N = 4 super Yang-Mills theory. Interesting applications of integrability existing in the literature that we reconsider are the computation of the free energy in the thermodynamic limit and its quantum field theory (QFT) counterpart, the critical coupling. In addition, we provide a detailed overview of the calculation of exact anomalous dimensions and operator product expansion (OPE) coefficients in the conformal bi-scalar fishnet theory.The original contributions of this work comprise the results of the critical coupling for models with fermions, the brick wall theory, and the fermionic fishnet theory. Additionally, we extend the study of integrable Feynman graphs to supersymmetric diagrams in superspace. By establishing an efficient graphical formalism, we obtain the critical coupling of double-scaled β-deformations of N = 4 super Yang-Mills theory and Aharony-BergmanJafferis-Maldacena theory, the super brick wall and superfishnet theory, respectively. Moreover, we apply superspace methods to the superfishnet theory and find results for anomalous dimensions and an OPE coefficient, which are all-loop exact in the coupling. In addition, we study boundary integrability in the six-vertex model and for Feynman diagrams. We present new box-shaped boundary conditions for the six-vertex model and conjecture a closed form for its partition function at any lattice size. On the QFT side, we find integrable boundary scattering matrices in the form of generalized Feynman diagrams by graphical methods.

We propose a double-scaling limit of \\(\\beta\\)-deformed ABJM theory in three-dimensional \\(\\mathcal{N} = 2\\) superspace, and a non-local deformation thereof. Due to the regular appearance of the theory's Feynman supergraphs, we refer to this superconformal and integrable theory as the superfishnet theory. We use techniques inspired by the integrability of bi-scalar fishnet theory and adapted to superspace to calculate the zero-mode-fixed thermodynamic free energy, the corresponding critical coupling, and the exact all-loop scaling dimensions of various operators. Furthermore, we confirm the results of the supersymmetric dynamical fishnet theory by applying our methods to four-dimensional \\(\\mathcal{N} = 1\\) superspace.

We consider the double scaling limit of \\(\\beta\\)-deformed planar N = 4 supersymmetric Yang-Mills theory (SYM), which has been argued to be conformal and integrable. It is a special point in the three-parameter space of double-scaled \\(\\gamma_i\\)-deformed N = 4 SYM, preserving N = 1 supersymmetry. The Feynman diagrams of the general three-parameter models form a \"dynamical fishnet\" that is much harder to analyze than the original one-parameter fishnet, where major progress in uncovering the model's integrable structure has been made in recent years. Here we show that by applying N = 1 superspace techniques to the \\(\\beta\\)-deformed model the dynamical nature of its Feynman graph expansion disappears, and we recover a regular lattice structure of brick wall (honeycomb) type. As a first application, we compute the zero-mode-fixed thermodynamic free energy of this model by applying Zamolodchikov's method of inversion to the supersymmetric brick wall diagrams.