4d Crystal Melting, Toric Calabi-Yau 4-Folds and Brane Brick Models

We initiate a systematic study of 2d (0,2) quiver gauge theories on the worldvolume of D1-branes probing singular toric Calabi-Yau 4-folds. We present an algorithm for efficiently calculating the classical mesonic moduli spaces of these theories, which correspond to the probed geometries. We also introduce a systematic procedure for constructing the gauge theories for arbitrary toric singularities by means of partial resolution, which translates to higgsing in the field theor...

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions ...

An infinite class of $4d$ $\mathcal{N}=1$ gauge theories can be engineered on the worldvolume of D3-branes probing toric Calabi-Yau 3-folds. This kind of setup has multiple applications, ranging from the gauge/gravity correspondence to local model building in string phenomenology. Brane tilings fully encode the gauge theories on the D3-branes and have substantially simplified their connection to the probed geometries. The purpose of this paper is to push the boundaries of com...

It has recently been shown that the statistical mechanics of crystal melting maps to A-model topological string amplitudes on non-compact Calabi-Yau spaces. In this note we establish a one to one correspondence between two and three dimensional crystal melting configurations and certain BPS black holes given by branes wrapping collapsed cycles on the orbifolds C^2/Z_n and C^3/Z_n x Z_n in the large n limit. The ranks of gauge groups in the associated gauged quiver quantum mec...

We survey geometrical and especially combinatorial aspects of generalized Donaldson-Thomas invariants (also called BPS invariants) for toric Calabi-Yau manifolds, emphasizing the role of plane partitions and their generalizations in the recently proposed crystal melting model. We also comment on equivalence with a vicious walker model and the matrix model representation of the partition function.

We provide a brane realization of 2d (0,2) Gadde-Gukov-Putrov triality in terms of brane brick models. These are Type IIA brane configurations that are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. Triality translates into a local transformation of brane brick models, whose simplest representative is a cube move. We present explicit examples and construct their triality networks. We also argue that the classical mesonic moduli space of brane brick model theories...

These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant coho...

We study BPS spectra of D-branes on local Calabi-Yau threefolds $\mathcal{O}(-p)\oplus\mathcal{O}(p-2)\to \mathbb{P}^1$ with $p=0,1$, corresponding to $\mathbb{C}^3/\mathbb{Z}_{2}$ and the resolved conifold. Nonabelianization for exponential networks is applied to compute directly unframed BPS indices counting states with D2 and D0 brane charges. Known results on these BPS spectra are correctly reproduced by computing new types of BPS invariants of 3d-5d BPS states, encoded b...

We construct a statistical model that reproduces the BPS partition function of D4-D2-D0 bound states on a class of toric Calabi-Yau three-folds. The Calabi-Yau three-folds we consider are obtained by adding a compact two-cycle to $A_{N-1}$-ALE $\times \mathbb{C}$. We show that in the small radii limit of the Calabi-Yau the D4-D2-D0 partition function is correctly reproduced by counting the number of triangles and parallelograms.

Solid partitions are the 4D generalization of the plane partitions in 3D and Young diagrams in 2D, and they can be visualized as stacking of 4D unit-size boxes in the positive corner of a 4D room. Physically, solid partitions arise naturally as 4D molten crystals that count equivariant D-brane BPS states on the simplest toric Calabi-Yau fourfold, $\mathbb{C}^4$, generalizing the 3D statement that plane partitions count equivariant D-brane BPS states on $\mathbb{C}^3$. In the ...