arXiv subjects

Mina Aganagic

Publications and source records attributed to Mina Aganagic.

At least 19 recordsLinked to original sources

arXiv2025

Two-parameter Langlands Correspondence

In our paper arXiv:1701.03146 we established, for every simply-laced Lie algebra g, a canonical isomorphism between the spaces of deformed conformal blocks of the deformed W-algebra and the quantum affine algebra corresponding to g, which we view as a q-deformation of the quantum Langlands correspondence. This was done by realizing the deformed conformal blocks of these algebras via the quantum K-theory of the Nakajima quiver varieties. We also linked this isomorphism to a duality emerging from the 6d little string theory. Here, we give a brief survey of these results and propose an extension to the non-simply laced case, which exhibits a Langlands-type duality.

math.QA↗

arXiv2024

Quiver Hecke algebras from Floer homology in Couloumb branches

Homology theories categorifying quantum group link invariants are known to be governed by the representation theory of quiver Hecke algebras, also called KLRW algebras. Here we show that certain cylindrical KLRW algebras, relevant in particular for cylindrical generalizations of link homology theories, can be realized by Lagrangian Floer homology in multiplicative Coulomb branches. This confirms a homological mirror symmetry prediction of the first author.

math.SG↗

arXiv2023

Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves

We derive two geometric approaches to categorification of quantum invariants of links associated to an arbitrary compact simple Lie group $^L{G}$. In part I, we describe the first approach, based on an equivariant derived category of coherent sheaves on ${\cal X}$, the moduli space of singular $G$-monopoles, where $G$ is related to $^LG$ by Langlands duality. In part II, we describe the second approach, based on the derived category of a Fukaya-Seidel category of a Calabi-Yau $Y$ with potential $W$. The two approaches are related by a version of mirror symmetry, which plays a crucial role in the story. In part III, we explain the string theory origin of these results, and the relation to an approach due to Witten.

hep-th↗

arXiv2023

Knot Categorification from Mirror Symmetry, Part II: Lagrangians

I provide two solutions to the problem of categorifying quantum link invariants, which work uniformly for all gauge groups and originate in geometry and string theory. The first is based on a category of equivariant B-type branes on ${\cal X}$ which is a moduli space of singular $G$-monopoles on ${\mathbb R}^3$. In this paper, I give the second approach, which is based on a category of equivariant A-type branes on $Y$ with potential $W$. The first and the second approaches are related by equivariant homological mirror symmetry: $Y$ is homological mirror to $X$, a core locus of ${\cal X}$ preserved by an equivariant action related to $\mathfrak{q}$. The theory of equivariant A-branes on $Y$ is the same as the derived category of modules of an algebra $A$, which is a cousin of the algebra considered by Khovanov, Lauda, Rouquier and Webster, but simpler. The result is a new, geometric formulation of Khovanov homology, which generalizes to all groups. In part III, I will explain the string theory origin of the two approaches, and the relation to an approach being developed by Witten. The three parts may be read independently.

hep-th↗

arXiv2023

Homological Link Invariants from Floer Theory

There is a generalization of Heegaard-Floer theory from ${\mathfrak{gl}}_{1|1}$ to other Lie (super)algebras $^L{\mathfrak{g}}$. The corresponding category of A-branes is solvable explicitly and categorifies quantum $U_q(^L{\mathfrak{g}})$ link invariants. The theory was discovered in \cite{A1,A2}, using homological mirror symmetry. It has novel features, including equivariance and, if $^L{\mathfrak{g}} \neq {\mathfrak{gl}}_{1|1}$, coefficients in categories. In this paper, we describe the theory and how it is solved in detail in the two simplest cases: the ${\mathfrak{gl}}_{1|1}$ theory itself, categorifying the Alexander polynomial, and the ${\mathfrak{su}}_{2}$ theory, categorifying the Jones polynomial. Our approach to solving the theory is new, even in the familiar ${\mathfrak{gl}}_{1|1}$ case.

hep-th↗

arXiv2022

Homological Knot Invariants from Mirror Symmetry

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning -- what are they homologies of? Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand. I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.

math.GT↗

arXiv2020

Elliptic stable envelopes

We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of $q$-difference equations arising the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik-Zamolodchikov equations.

math.AG↗

arXiv2018

Quantum q-Langlands Correspondence

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. The physical origin of the correspondence is the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d string theory becomes the 6d conformal field theory with (2,0) supersymmetry.

hep-th↗

arXiv2017

Topological Chern-Simons/Matter Theories

We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topological. We argue that this theory arises in topological A-model string theory on Lagrangian 3-branes in the presence of additional parallel coisotropic 5-branes. The theory obtained in this way is equivalent to an N=2 supersymmetric Chern-Simons matter theory on the same 3-manifold, which also only depends on the THF structure. The theory is a realization of a topological theory of class H, which allows splitting of a temporal direction from spatial directions. We briefly discuss potential condensed matter applications.

hep-th↗

arXiv2017

Quasimap counts and Bethe eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

math-ph↗

arXiv2015

String Theory and Math: Why This Marriage May Last

String theory is changing the relationship between mathematics and physics. The central role is played by the phenomenon of duality, which is intrinsic to quantum physics and abundant in string theory.

hep-th↗

arXiv2015

ADE Little String Theory on a Riemann Surface (and Triality)

We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.

hep-th↗

arXiv2014

Gauge/Vortex duality and AGT

This is the 12th article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories" ed. J. Teschner. This article describes one way to understand an important part of the AGT-correspondence in terms of a triality between four-dimensional gauge theory, the two-dimensional theory of its vortices, and conformal field theory. This triality is related to, and inspired by known large $N$ dualities of the topological string. It leads to a proof of some cases of the AGT-correspondence, and most importantly, of a generalisation of this correspondence to certain five-dimensional gauge theories.

hep-th↗

arXiv2014

$A_n$-Triality

A_n-type AGT correspondence anticipates that conformal blocks of A_n Toda CFT are related to partition functions of a family of 4d N=2 SCFTs. We use gauge/vortex duality to both give a precise form of the correspondence, and to prove it. Gauge/vortex duality relates the 4d theories and the 2d theories living on its vortices. Partition functions of the 2d theories on vortices provide Coulomb-gas representation of A_n Toda conformal blocks with discrete internal momenta. This gives a triality of relations between the gauge theory, its vortices and the Toda CFT. We prove that A_n triality holds for conformal blocks of A_n Toda on a sphere with all full punctures. The lift to one higher dimensional theories, compactified on a circle of arbitrary radius, and q-deformation of the Toda CFT, play a key role.

hep-th↗

arXiv2013

Topological Strings, D-Model, and Knot Contact Homology

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model." In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Q-deformation of the extension of A-polynomials associated with the link complement.

hep-th↗

arXiv2013

Gauge/Liouville Triality

Conformal blocks of Liouville theory have a Coulomb-gas representation as Dotsenko-Fateev (DF) integrals over the positions of screening charges. For q-deformed Liouville, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in DF representation, as the partition functions of a class of 3d U(N) gauge theories with N=2 supersymmetry, in the Omega-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Liouville vertex operator; the two real mass parameters determine the momentum and position of the puncture. The DF integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1 gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.

hep-th↗

arXiv2012

Refined Chern-Simons Theory and Topological String

We show that refined Chern-Simons theory and large N duality can be used to study the refined topological string with and without branes. We derive the refined topological vertex of hep-th/0701156 and hep-th/0502061 from a link invariant of the refined SU(N) Chern-Simons theory on S^3, at infinite N. Quiver-like Chern-Simons theories, arising from Calabi-Yau manifolds with branes wrapped on several minimal S^3's, give a dual description of a large class of toric Calabi-Yau. We use this to derive the refined topological string amplitudes on a toric Calabi-Yau containing a shrinking P^2 surface. The result is suggestive of the refined topological vertex formalism for arbitrary toric Calabi-Yau manifolds in terms of a pair of vertices and a choice of a Morse flow on the toric graph, determining the vertex decomposition. The dependence on the flow is reminiscent of the approach to the refined topological string in upcoming work of Nekrasov and Okounkov. As a byproduct, we show that large N duality of the refined topological string explains the ``mirror symmetry`` of the refined colored HOMFLY invariants of knots.

hep-th↗

arXiv2012

Refined Black Hole Ensembles and Topological Strings

We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that Z_{BH} = |Z_{top}|^2 relates the BPS black hole partition function to the topological string partition function Z_{top}. In the refined conjecture, Z_{BH} is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. Z_{top} becomes the partition function of the refined topological string, which is itself an index. Both the original and the refined conjecture are examples of large N duality in the 't Hooft sense. The refined conjecture applies to non-compact Calabi-Yau manifolds only, so the black holes are really BPS particles with large entropy, of order N^2. The refined OSV conjecture states that the refined BPS partition function has a large N dual which is captured by the refined topological string. We provide evidence that the conjecture holds by studying local Calabi-Yau threefolds consisting of line bundles over a genus g Riemann surface. We show that the refined topological string partition function on these geometries is computed by a two-dimensional TQFT. We also study the refined black hole partition function arising from N D4 branes on the Calabi-Yau, and argue that it reduces to a (q,t)-deformed version of two-dimensional SU(N) Yang-Mills. Finally, we show that in the large N limit this theory factorizes to the square of the refined topological string in accordance with the refined OSV conjecture.

hep-th↗