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Two highly successful approaches to constructing 5d SCFTs are geometric engineering using M-theory on a Calabi-Yau 3-fold and the use of 5-brane webs suspended from 7-branes in Type IIB string theory. In the brane web realization, the extended Coulomb branch of the 5d SCFT can be studied by opening the web using rigid triple intersections of branes — i.e. configurations with no deformations. In this paper, we argue that the geometric engineering counterpart of these rigid triple intersections are the T-cones introduced in the mathematical literature. We extend the class of rigid brane webs to include locked superpositions of the minimal ones. These rigid brane webs serve as fundamental building blocks for supersymmetrically tessellating Generalized Toric Polygons (GTPs) from first principles. Interestingly, we find that the extended Coulomb branch generally exhibits a structure consisting of multiple cones intersecting at a single point. Hanany-Witten (HW) transitions in the web have been conjectured to correspond geometrically to flat fibrations over a line, where the central and generic fibers represent the geometries dual to the webs before and after the transition. We demonstrate this explicitly in an example, showing that for GTPs reducing to standard toric diagrams, the HW transition corresponds to a deformation of the BPS quiver that we map to the geometric deformation. 

We discuss the realization of 2d (0,2) gauge theories in terms of branes focusing on Brane Brick Models, which are T-dual to D1-branes probing toric Calabi-Yau 4-folds. These brane setups fully encode the infinite class of 2d (0,2) quiver gauge theories on the worldvolume of the D1-branes and substantially streamline their connection to the probed geometries. We review various methods for efficiently generating Brane Brick Models. These algorithms are then used to construct 2d (0,2) gauge theories for the cones over all the smooth Fano 3-folds and two infinite families of Sasaki-Einstein 7-manifolds with known metrics. This note is based on the author’s talk at the Gauged Linear Sigma Models @ 30 conference at the Simons Center for Geometry and Physics. 

Generalized global symmetries, in particular non-invertible and categorical symmetries, have become a focal point in the recent study of quantum field theory (QFT). In this paper, we investigate aspects of symmetry topological field theories (SymTFTs) and anomalies of non-invertible symmetries for 2D QFTs from a string theory perspective. Our primary focus is on an infinite class of 2D QFTs engineered on D1-branes probing toric Calabi-Yau 4-fold singularities. We derive 3D SymTFTs from the topological sector of IIB supergravity and discuss the resulting 2D QFTs, which can be intrinsically relative or absolute. For intrinsically relative QFTs, we propose a sufficient condition for them to exist. For absolute QFTs, we show that they exhibit non-invertible symmetries with an elegant brane origin. Furthermore, we find that these non-invertible symmetries can suffer from anomalies, which we discuss from a top-down perspective. Explicit examples are provided, including theories for including theories for Y(p,k)(ℙ2), Y(2,0)(ℙ1×ℙ1), and ℂ4/ℤ4 geometries. 

A new type of quiver theories, denoted twin quivers, was recently introduced for studying 5d SCFTs engineered by webs of 5-branes ending on 7-branes. Twin quivers provide an alternative perspective on various aspects of such webs, including Hanany-Witten moves and the s-rule. More ambitiously, they can be regarded as a first step towards the construction of combinatorial objects, generalizing brane tilings, encoding the corresponding BPS quivers. This paper continues the investigation of twin quivers, focusing on their non-uniqueness, which stems from the multiplicity of toric phases for a given toric Calabi-Yau 3-fold. We find that the different twin quivers are necessary for describing what we call quiver tails, which in turn correspond to certain sub-configurations in the webs. More generally, the multiplicity of twin quivers captures the roots of the Higgs branch in the extended Coulomb branch of 5d theories. 

A<sc>bstract</sc> We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over ℙ¹interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers. 

A<sc>bstract</sc> We introduce a class of 4-dimensional crystal melting models that count the BPS bound state of branes on toric Calabi-Yau 4-folds. The crystalline structure is determined by the brane brick model associated to the Calabi-Yau 4-fold under consideration or, equivalently, its dual periodic quiver. The crystals provide a discretized version of the underlying toric geometries. We introduce various techniques to visualize crystals and their melting configurations, including 3-dimensional slicing and Hasse diagrams. We illustrate the construction with the D0-D8 system on$${\mathbb{C}}$$⁴. Finally, we outline how our proposal generalizes to arbitrary toric CY 4-folds and general brane configurations. 

A bstract We propose a unified perspective on two sets of objects that usually arise in the study of bipartite field theories. Each of the sets consists of a polytope, or equivalently a toric Calabi-Yau, and a quiver theory. We refer to the two sets of objects as original and twin. In the simplest cases, the two sides of the correspondence are connected by the graph operation known as untwisting. The democratic treatment that we advocate raises new questions regarding the connections between these objects, some of which we explore. With this motivation in mind, we establish a correspondence between the mutations of the original polytope and the twin quiver. This leads us to propose that non-toric twin quivers are naturally associated to generalized toric polygons (GTPs) and we explore various aspects of this idea. Supporting evidence includes global symmetries, the ability of twin quivers to encode the generalized s -rule, and the connection between the mutations of polytopes and of configurations of webs of 5-branes suspended from 7-branes. We introduce three methods for constructing twin quivers for GTPs. We also investigate the connection between twin quivers obtained using different toric phases. Twin quivers provide a powerful new perspective on GTPs. The ideas presented in this paper may represent a step towards the generalization of brane tilings to GTPs. 

A<sc>bstract</sc> We investigate a class of mass deformations that connect pairs of 2d(0,2) gauge theories associated to different toric Calabi-Yau 4-folds. These deformations are generalizations to 2dof the well-known Klebanov-Witten deformation relating the 4dgauge theories for the ℂ²/ℤ₂× ℂ orbifold and the conifold. We investigate various aspects of these deformations, including their connection to brane brick models and the relation between the change in the geometry and the pattern of symmetry breaking triggered by the deformation. We also explore how the volume of the Sasaki-Einstein 7-manifold at the base of the Calabi-Yau 4-fold varies under deformation, which leads us to conjecture that it quantifies the number of degrees of freedom of the gauge theory and its dependence on the RG scale. 

A bstract The 2 d (0 , 2) supersymmetric gauge theories corresponding to the classes of Y p,k (ℂℙ 1 × ℂℙ 1 ) and Y p,k (ℂℙ 2 ) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2 d (0 , 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. 

A bstract We introduce and initiate the study of a general class of 2 d $$ \mathcal{N} $$ N = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of theories as BFT 2 ’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their master and moduli spaces. Invariance of the moduli space is indeed a powerful criterion for identifying theories in the same triality class. We also investigate the reducibility of these theories.