Antoine Bourget

We show that all 5D balanced (ADE-shaped) special unitary quivers with no Chern-Simons level admit a UV completion which is a 5D conformal matter superconformal field theory (SCFT). We give explicit local Calabi-Yau threefolds realizing each of these models in M-theory. This unified description enables a systematic exploration of their physical properties, such as their Higgs branch, as well as connections to class- S constructions and the affine Grassmannian.

Vides et modularité dans les théories de jauge supersymétriques N = 1*: Nous explorons la structure des vides dans une déformation massive de la théorie de Yang-Mills maximalement supersymétrique en quatre dimensions. Sur un espace-temps topologiquement trivial, la théorie des orbites nilpotentes dans les algèbres de Lie rend possible le calcul exact de l'indice de Witten. Nous en donnons les fonctions génératrices pour les algèbres classiques, et recourons à un calcul explicite pour les exceptionnelles. Après compactification sur un cercle, un lien entre les théories de jauge supersymétriques et les systèmes intégrables est exploitable pour réduire la chasse aux vides à une extrémisation du hamiltonien de Calogero-Moser elliptique twisté. Une analyse soigneuse des propriétés globales du groupe de jauge et des opérateurs de ligne est nécessaire pour obtenir un accord parfait. En combinant exploration numérique sur ordinateur et contrôle analytique grâce à la théorie des formes modulaires, nous exhibons la structure des vides massifs pour des algèbres de rang petit, et mettons en évidence de nouvelles propriétés modulaires. Nous montrons que des branches de vides de masse nulle existent, et nous en donnons la structure exacte pour les algèbres de rang deux.

Obtaining the classification of three-dimensional (3D) N = 4 quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with eight supercharges in any dimension. In this work, we derive a full classification for such Abelian quivers with arbitrary charges and identify all possible Coulomb branch geometries as quotients of H n by U(1) or a finite cyclic group. We give two proofs, one which uses the “decay and fission” algorithm and another one relying only on explicit computations involving 3D mirror symmetry. In the process, we put forward a method for computing the 3D mirror of any U ( 1 ) r gauge theory, which is sensitive to discrete gauge factors in the mirror theory. This constitutes a confirmation for the decay and fission algorithm.

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this paper, we extend this connection and provide a geometric realization of brane configurations including 7-branes. A web with 7-branes defines a so-called generalized toric polygon (GTP), which corresponds to combinatorial data that is obtained by removing vertices along external edges of a toric polygon. We identify the geometries associated to GTPs as non-toric deformations of toric Calabi-Yau threefolds and provide a precise, algebraic description of the geometry, when 7-branes are introduced along a single edge. The key ingredients in our analysis are T-branes in a type IIA frame, which includes D6-branes. We show that performing Hanany-Witten moves for the 7-branes on the type IIB side corresponds to switching on semisimple vacuum expectation values on the worldvolume of D6-branes, which in turn uplifts to complex structure deformations of the Calabi-Yau geometries. We test the proposal by computing the crepant resolutions of the deformed geometries, thereby checking consistency with the expected properties of the SCFTs.

Magnetic quivers have been an instrumental technique for advancing our understanding of Higgs branches of supersymmetric theories with eight supercharges. In this work, we present the “decay and fission” algorithm for unitary magnetic quivers. It enables the derivation of the complete phase (Hasse) diagram and is characterized by the following key attributes: First and foremost, the algorithm is inherently simple, just relying on convex linear algebra. Second, any magnetic quiver can only undergo decay or fission processes; these reflect the possible Higgs branch RG flows (Higgsings), and the quivers thereby generated are the magnetic quivers of the new RG fixed points. Third, the geometry of the decay or fission transition (i.e., the transverse slice) is simply read off. As a consequence, the algorithm does not rely on a complete list of minimal transitions, but rather outputs the transverse slice geometry automatically. As a proof of concept, its efficacy is showcased across various scenarios, encompassing superconformal field theories from dimensions 3 to 6, instanton moduli spaces, and little string theories. Published by the American Physical Society 2024

In exploring supersymmetric theories with eight supercharges, the Higgs branches present an intriguing window into strong coupling dynamics. Magnetic quivers serve as crucial tools for understanding these branches. Here, we introduce the decay and fission algorithm for unitary magnetic quivers. It efficiently derives complete phase diagrams (Hasse diagrams) through convex linear algebra. It allows magnetic quivers to undergo decay or fission, reflecting Higgs branch RG flows in the theory. Importantly, the algorithm generates magnetic quivers for the RG fixed points and simplifies the understanding of transverse slice geometry with no need for a list of minimal transitions. In contrast, the algorithm hints to the existence of a new minimal transition, whose geometry and physics need to be explored.