Publications

Preprint

  1. Nafiz Ishtiaque, Seyed Faroogh Moosavian, Yehao Zhou. Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical R-Matrices for Superspin Chains from The Bethe/Gauge Correspondence. 2308.12333.

    Synopsis. We generalize Aganagic-Okounkov’s theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov’s and Bullimore-Zhang’s works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/Gauge Correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the R-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the stable envelopes, and in turn the geometric R-matrix, of the $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the R-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric R-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational R-matrix. The latter recovers the results of Rimányi and Rozansky..

  2. Nafiz Ishtiaque and Yehao Zhou. Line Operators in 4d Chern-Simons Theory and Cherkis Bows. 2211.00049.

    Synopsis. We show that the phase spaces of a large family of line operators in 4d Chern-Simons theory with $\text{GL}_n$ gauge group are given by Cherkis bow varieties with $n$ crosses. These line operators are characterized by Hanany-Witten type brane constructions involving D3, D5, and NS5 branes in an $\Omega$-background. Linking numbers of the five-branes and mass parameters for the D3 brane theories determine the phase spaces and in special cases they correspond to vacuum moduli spaces of 3d $\mathcal N=4$ quiver theories. Examples include line operators that conjecturally create T, Q, and L-operators in integrable spin chains.

Published

  1. Nafiz Ishtiaque, Seyed Faroogh Moosavian, Surya Raghavendran, and Junya Yagi. Superspin chains from superstring theory. SciPost Phys. 13, 083 (2022), 2110.15112.

    Synopsis. The main construction presented in the paper is an embedding of a rational $\mathfrak{gl}(m|n)$ spin chains into string theory which we use to derive the Bethe/gauge correspondence for rational superspin chains. We propose a version of the correspondence where spins are valued in Verma modules, extending the correspondence proposed by Nekrasov involving finite-dimensional representations [NEK19]. The key is the brane construction of 4d Chern-Simons theory with $\mathrm{GL}(m|n)$ gauge group. This is the twisted and $\Omega$-deformed world-volume theory on a stack of $m$ D5 branes and $n$ rotated D5 branes. The two stacks of D5 branes share four directions which is the support of the 4d Chern-Simons theory. The topological-holomorphic twist and $\Omega$-deformation are induced by certain closed string backgrounds including importantly an RR 2-form field. Line operators in the 4d Chern-Simons theory valued in Verma modules are created using semi-infinite D3 branes ending on the D5 branes. A set of parallel line operators in the 4d Chern-Simons theory represents an integrable spin chain [CWY18]. States in the spin chain correspond to supersymmetric configurations of fundamental strings suspended between the D5 branes. A sequence of string dualities converts this D5-D3-F1 brane configuration into an NS5-D4-D2 configuration. The D2 brane world-volume theory is a 2d $\mathcal N=(2,2)$ quiver gauge theory. Supersymmetric configurations of F1 strings are mapped by these duality transformations to supersymmetric configurations of D2 branes which correspond to vacua of the 2d theory — realizing the Bethe/gauge correspondence.

    Based on the physics we make some mathematical conjectures. In particular, the quivers defining the 2d gauge theories further define some Kähler varieties — the Higgs branches of the 2d theories carrying the actions of the flavor symmetry groups. We conjecture, roughly speaking, that the equivariant cohomology of these Higgs branches with respect to the maximal tori of the flavor symmetry gorups furnishes representations of the Yangian of $\mathfrak{gl}(m|n)$.

    We also show that Hanay-Witten transitions in the brane configurations lead to fermionic dualities of the superspin chains.

  2. Nafiz Ishtiaque, Seyed Faroogh Moosavian, and Yehao Zhou. Topological Holography: The Example of The D2-D4 Brane System. SciPost Phys. 9, 017 (2020), 1809.00372.

    Synopsis. We look at a topological version of $\mathrm{AdS}_3/\mathrm{CFT}_2$ holographic duality. The gravitational side is a topological-holomorphic version of $\mathrm{AdS}_3 \times S^3$ including a 4d defect supported on $\mathrm{AdS}_2 \times S^2$. The defect is described by 4d Chern-Simons theory with $\mathrm{GL}_K$ gauge group. The gauge theory side of the duality is a 2d BF theory with $\mathrm{GL}_N$ gauge group and a line defect with $\mathrm{GL}_N \times \mathrm{GL}_K$ symmetry. At large $N$ we compare the $\mathrm{GL}_N$-invariant operator algebra on this defect and the operator algebra at the asymptotic line in the $\mathrm{AdS}_2$ of the 4d Chern-Simons theory. We compute these algebras using Feynman and Witten diagrams and find that they are both isomorphic to the Yangian of $\mathfrak{gl}(K)$. The computations are all loop exact, giving an exactly solvable toy model of holography. Furthermore, from earlier results of Costello [COS13] deriving the local operator algebra on a line in 4d Chern-Simons to be the Koszul dual of the Yangian, our result gives a concrete example of the hypothesis by Costello and Li [CL16, COS17] that in a gravitational theory with holographic dual, algebra at infinity is Koszul dual to the local operator algebra. We also argue that this simple model is a supersymmetric subsector of the more famous $\mathrm{AdS}_5/\mathrm{CFT}_4$ duality.

  3. Nafiz Ishtiaque, Junya Yagi. Disk, interval, point: on constructions of quantum field theories with holomorphic action functionals. JHEP, 180 (2020), 2002.10488.
  4. Nafiz Ishtiaque. 2D BPS Rings from Sphere Partition Functions. JHEP, 124 (2018), 1712.02551.
  5. Efrat Gerchkovitz, Jaume Gomis, Nafiz Ishtiaque, Avner Karasik, Zohar Komargodski, and Silviu S. Pufu. Correlation Functions of Coulomb Branch Operators. JHEP, 103 (2017), 1602.05971.
  6. Jaume Gomis and Nafiz Ishtiaque. Kähler potential and ambiguities in 4d $ \mathcal N = 2 $ SCFTs. JHEP, 169 (2015), 1409.5325

Proceedings

Thesis