Cohomological Hall Algebras, Vertex Algebras and Instantons

200

We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are “bootstrapped” from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles.

Section: Jhep11(2020)035mentioning

confidence: 99%

“…Related problems were posed recently in[28] 3. The quiver Yangian is conjectured to be the Drinfeld double of CoHA; or inversely, the CoHA captures the positive part of the quiver Yangian, i.e.…”

mentioning

confidence: 99%

Quiver Yangian from crystal melting

Yamazaki

2020

200

“…If x creates a in the asymptotic Young diagram on the un-hatted side, it creates a in the asymptotic Young diagram on the hatted side. Adopting conventions of[23], the transverse coordinates of the 's must be negative 25. If the in the asymptotic Young diagram on the un-hatted side is displaced along the x 1 direction, there are two natural options for the displacement of in the Young diagram on the hatted side: either −ĥ 1 or −ĥ 3 .…”

mentioning

confidence: 99%

“…At least, in the case when ρ = 0 they precisely coincide with (framed) Donaldson-Thomas partitions functions in maximal chambers of the moduli space of stability conditions. For other values of ρ we discuss their interpretation in section 8.2.8 For more recent study of related ideas see e.g [25]. and references therein.…”

mentioning

confidence: 99%

Gluing two affine Yangians of 𝔤𝔩1

Longhi

2019

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of gl 1. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the N = 2 affine Yangian, which is isomorphic to the universal enveloping algebra of u(1)⊕W N =2 ∞ [λ]. All algebras that we construct have natural representations in terms of "twin plane partitions", a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.

“…One can associate to this counting problem a Hall algebra and it would be interesting to see if there is any connection between these algebraic structures and the structures that we were studying. Another analogous natural algebraic structure is the cohomological Hall algebra [91] which in the context of W 1+∞ and its truncations was discussed in [92].…”

Section: Jhep09(2020)150mentioning

confidence: 99%

The Grassmannian VOA

Eberhardt

Procházka²

2020

We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT $$ \mathfrak{u} $$ u (M + N )k /($$ \mathfrak{u} $$ u (M )k×$$ \mathfrak{u} $$ u (N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the $$ {\mathcal{W}}_{\infty } $$ W ∞ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $$ \mathcal{N} $$ N = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.