arXiv subjects

Dmitry Galakhov

Publications and source records attributed to Dmitry Galakhov.

27 records · Page 2Linked to original sources

arXiv2022

Gauge/Bethe correspondence from quiver BPS algebras

We study the Gauge/Bethe correspondence for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the $R$-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional $\mathcal{N}=2$ gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature.

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arXiv2022

Toroidal and Elliptic Quiver BPS Algebras and Beyond

The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $\mathcal{N}=2$ supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

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arXiv2021

Shifted Quiver Yangians and Representations from BPS Crystals

We introduce a class of new algebras, the shifted quiver Yangians, as the BPS algebras for type IIA string theory on general toric Calabi-Yau three-folds. We construct representations of the shifted quiver Yangian from general subcrystals of the canonical crystal. We derive our results via equivariant localization for supersymmetric quiver quantum mechanics for various framed quivers, where the framings are determined by the shape of the subcrystals. Our results unify many known BPS state counting problems, including open BPS counting, non-compact D4-branes, and wall crossing phenomena, simply as different representations of the shifted quiver Yangians. Furthermore, most of our representations seem to be new, and this suggests the existence of a zoo of BPS state counting problems yet to be studied in detail.

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arXiv2020

Quiver Yangian and Supersymmetric Quantum Mechanics

The statistical model of crystal melting represents BPS configurations of D-branes on a toric Calabi-Yau three-fold. Recently it has been noticed that an infinite-dimensional algebra, the quiver Yangian, acts consistently on the crystal-melting configurations. We physically derive the algebra and its action on the BPS states, starting with the effective supersymmetric quiver quantum mechanics on the D-brane worldvolume. This leads to remarkable combinatorial identities involving equivariant integrations on the moduli space of the quantum mechanics, which can be checked by numerical computations.

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arXiv2019

BPS Hall Algebra of Scattering Hall States

Starting with a very pedestrian point of view we compare two different at the first glance definitions for an algebra associated to BPS states in supersymmetric fields theories. One proposed by Harvey and Moore exploits $S$-matrices of BPS states as structure constants of a new algebra. Another one proposed by Kontsevich and Soibelman gives a construction according to the structure of cohomological Hall algebras. We show these two constructions give equivalent algebras.

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arXiv2017

Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we associate instantons in LG model to specific WKB line configurations we call null-webs.

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arXiv2016

Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

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arXiv2014

Spectral Networks with Spin

The BPS spectrum of d=4 N=2 field theories in general contains not only hyper- and vector-multipelts but also short multiplets of particles with arbitrarily high spin. This paper extends the method of spectral networks to give an algorithm for computing the spin content of the BPS spectrum of d=4 N=2 field theories of class S. The key new ingredient is an identification of the spin of states with the writhe of paths on the Seiberg-Witten curve. Connections to quiver representation theory and to Chern-Simons theory are briefly discussed.

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arXiv2013

Wild Wall Crossing and BPS Giants

We show that the BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of the BPS degeneracies with the charge is exponential. We show this using spectral networks and independently using wall-crossing formulae and quiver methods. The computations using spectral networks provide a very nontrivial example of how these networks determine the four-dimensional BPS spectrum. We comment on some physical implications of the wild spectrum: for example, exponentially many field-theoretic BPS states with large charge are gigantic. Finally, we exhibit some surprising, thus far unexplained, regularities of the BPS spectrum.

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