Research

A powerful invariant of superconformal field theories is the superconformal index, which is a generating function for the $R$-charges of BPS operators in the theory. Recently, it has been suggested in four dimensional examples that the superconformal index counts exactly those operators that survive the holomorphic twist.

We investigate categorifications of superconformal indices of ABJM theories and the 6d $\mathcal{N}=(2,0)$ theories in terms of topological-holomorphic factorization algebras of observables. As a first step, we construct these factorization algebras classically. The main technique is koszul duality, in its physical incarnation as a twisted form of holography, applied to the minimal twist of eleven dimensional supergravity (see below). Strikingly, the koszul descriptions suggest that the superconformal indices naturally arise as characters of modules for certain exceptional simple lie superalgebras $E(1|6)$ and $E(3|6)$ first studied by Kac.

Twisted supergravity refers to a particular class of supergravity backgrounds that are particularly useful for studying BPS physics. The background is defined to be one in which the bosonic ghost for the local supersymmetries of supergravity take a nonzero, nilpotent VEV. Coupling branes to such a background has the effect of twisting the worldvolume theory.

We study an interacting holomorphic-topological field theory in eleven dimensions defined on products of one-manifolds with Calabi-Yau five-folds whose phase space describes a certain deformation of the cotangent bundle to the moduli of Calabi-Yau deformations of the five-fold. We conjecture that this describes the minimal twist of eleven-dimensional supergravity. We establish several consistency checks for our proposed interaction, checking that dimensional reductions recover expected descriptions of twists of type IIA and type I supergravity, and that the theory deforms to the $G2\times SU(2)$ twist of 11-dimensional supergravity introduced by Costello. Strikingly, the global symmetry algebra of the interacting theory is a central extension of an exceptional simple lie superalgebra called $E(5|10)$ initially studied by Kac. We define the twisted analogues of $\mathrm{AdS}_4\times S^7$ and $\mathrm{AdS}_7\times S^4$ by backreacting $M2$ and $M5$ branes respectively and analyze the symmetries of such backgrounds.

The Gauge-Bethe correspondence of Nekrasov-Shatashvilli posits a surprising connection between quantum integrable systems and supersymmetric gauge theories. The correspondence identifies the ground states of quantum spin chains with supersymmetric vacuua of a family of quiver gauge theories. A hallmark of integrability is that the Hilbert space of a quantum spin chain carries the action of quantum groups with spectral R-matrices such as the Yangian and its cousins. In this way, a realization of a spin chain Hilbert space in terms of gauge theory vacuua gives geometric constructions of quantum groups and its representations, as codified in work of Nakajima-Varagnolo and Maulik-Okounkov

Our work deals with a realization of this correspondence in string theory. We propose a realization of rational $\mathfrak {gl}(m|n)$ spin chains with spins in Verma modules in terms of a family of D2-D4-NS5 configurations in type IIA string theory. From comparing with the vacuua of the worldvolume theories on D2 branes, we obtain a version of the Gauge-Bethe correspondence for $\mathfrak{gl}(m|n)$ spin chains, generalizing the recent work of Nekrasov in the finite dimensional case. A key ingredient in our stringy realization of spin chains is to use dualities to map the brane configurations in consideration to 4d Chern-Simons theory with gauge group $\operatorname {GL}(m|n)$.

As a consequence of our considerations, we conjecture that the Yangian of $\mathfrak{gl}(m|n)$ acts on the equivariant cohomology of certain varieties that describe the vacuua of the theories on the D2 branes, generalizing a recent construction of Rimanyi-Rozansky of finite dimensional modules of the Yangian of $\mathfrak{gl}(1|1)$.

This project aims to illustrate a framework for the systematic and rigorous investigation of some mathematical implications of string dualities. The key ingredients are some amazing conjectures of Costello-Li giving descriptions of certain supersymmetry protected sectors of type II superstrings in terms of topological strings. This allows one to recover many calculational maneuvers familiar to string theorists in terms of data attached to a Calabi-Yau category. A key feature of these protected sectors is that the worldvolume theories of D-branes are twists of the worldvolume theories one normally finds. Since twists of supersymmetric field theories now sit on relatively firm mathematical foundations, these conjectures afford a useful framework for making mathematical conjectures about the effects of string dualities on various homotopical algebraic/derived geometric data attached to supersymmetric field theories.

As a first step in this direction, we derive the action of S-duality on a certain supersymmetry protected sector of type IIB string theory. In mathematical terms, this amounts to constructing an action of $SL_2(\mathbb{Z})$ on a variant of the cyclic cochains of a Calabi-Yau 3-fold. We then provide evidence that S-duality in this protected sector is responsible for the Geometric Langlands correspondence for $GL_n$, and for a description of the quantized Coulomb branch ring of A-twisted 3d $\mathcal{N}=4$ quiver gauge theories in terms of shifted truncated Yangians. We conclude with some conjectures about other S-dual deformations of 4d $\mathcal{N}=4$ that our framework suggests. We are currently writing a second version of this paper where some constructions are stated more model-independently.

Seminal work of Kapustin-Witten shows that the Geometric Langlands correspondence can be understood as a consequence of S-duality of 4d $\mathcal{N}=4$ Super Yang-Mills. Namely, the theory has a family of twists labeled by $\mathbb{CP}^1$. The points at zero and infinity of this $\mathbb{CP}^1$ are the so-called A and B twists; compactifying these twisted theories on Riemann Surfaces yields two 2d TQFTs whose categories of boundary conditions are the categories appearing in Geometric Langlands. In my project with Philsang Yoo (discussed above) we show that the A and B twists can be viewed as deformations of the theory on a D3 brane gotten by turning on certain closed string fields in a certain protected sector of IIB string theory. We further show that these two closed string fields are in fact exchanged by S-duality. Curiously, the A and B twists sit in an infinite family of S-dual pairs of closed string fields.

This (in progress) project aims to establish a compatibility between twisted S-duality, and the so-called dolbeault Geomtric Langlands conjecture, in a way that hints at Langlands-like correspondences for other S-dual pairs of closed string fields. Mathematically, this is expressed as follows: let $\Sigma$ be a complex projective curve. A theorem of Donagi-Pantev establishes a nontrivial self equivalence of the category $\mathrm{Coh}(\mathrm{Higgs}_{GL_n}\Sigma)$ away from the discriminant locus. We show this induces an action of $\mathbb{Z}/4\mathbb{Z}$ on $\mathcal{O}(T^*[1]\mathrm{Higgs}_{GL_n}\Sigma)$ viewed as Hamiltonian deformations of the shifted symplectic stack $T^*[1]\operatorname{Higgs}_{GL_n}(\Sigma)$. Further, we construct a map from the cyclic cochains $\mathrm{HC}^\bullet(T^*\Sigma\times\mathbb{C})\to\mathcal{O}(T^*[1]\mathrm{Higgs}_{GL_n}\Sigma)$ intertwining the action of twisted S-duality on the source with the action of $\mathbb{Z}/4\mathbb{Z}$ on the target.

In my master's thesis I began a project to try and relate two constructions of Khovanov Homology. One is due to Witten and realizes Khovanov Homology of a link $L$ as the Hilbert space for the theory living on a surface defect supported on $L\times\mathbb{R}$ in an A-type twist of 5d $\mathcal{N}=2$ gauge theory. Another is due to Cautis-Kamnitzer and uses a certain 2-category built out of coherent sheaves on convolution products of orbits in the affine Grassmannian. My master's work substantiated a claim that the 2-category of Cautis-Kamnitzer can be understood as a specific subcategory of the 2-category of surface defects in a holomorphic-topological twist of 5d $\mathcal{N}=2$ gauge theory.

Since then, I have given a brane realization for the construction in my master's work in a particular twist of type IIA string theory. This brane realization can be lifted to a twist of M-theory studied by Costello, and upon doing so, matches the M-theoretic lift of the brane construction originally studied by Witten. Further work of Mykhailov-Witten uses a slight modification of the original brane construction to construct a knot homology they call "Khovanov homology for supergroups". The M-theoretic lift of this modification suggests a variation of the construction of my master's work that yields a certain geometrically defined 2-category from which Mykhailov-Witten's knot homology can conjecturally be computed as a certain Ext. I hope to further explore these ideas in the future.