Correlators on the wall and $mathfrak{sl}_n$ spin chain

We study algebras and correlation functions of local operators at half-BPS interfaces engineered by the stacks of D5 or NS5 branes in the 4d $mathcal{N}=4$ super Yang-Mills. The operator algebra in this sector is isomorphic to a truncation of the Yangian $mathcal{Y}(mathfrak{gl}_n)$. The correlators, encoded in a trace on the Yangian, are controlled by the inhomogeneous $mathfrak{sl}_n$ spin chain, where $n$ is the number of fivebranes: they are given in terms of matrix elements of transfer matrices associated to Verma modules, or equivalently of products of Baxters Q-operators. This can be viewed as a novel connection between the $mathcal{N}=4$ super Yang-Mills and integrable spin chains. We also remark on analogous constructions involving half-BPS Wilson lines.

Algebras, traces, and boundary correlators in $mathcal{N}=4$ SYM

We study supersymmetric sectors at half-BPS boundaries and interfaces in the 4d $mathcal{N}=4$ super Yang-Mills with the gauge group $G$, which are described by associative algebras equipped with twisted traces. Such data are in one-to-one correspondence with an infinite set of defect correlation functions. We identify algebras and traces for known boundary conditions. Ward identities expressing the (twisted) periodicity of the trace highly constrain its structure, in many cases allowing for the complete solution. Our main examples in this paper are: the universal enveloping algebra $U(mathfrak{g})$ with the trace describing the Dirichlet boundary conditions; and the finite W-algebra $mathcal{W}(mathfrak{g},t_+)$ with the trace describing the Nahm pole boundary conditions.

4d/2d $rightarrow $ 3d/1d: A song of protected operator algebras

Superconformal field theories (SCFT) are known to possess solvable yet nontrivial sectors in their full operator algebras. Two prime examples are the chiral algebra sector on a two dimensional plane in four dimensional $mathcal{N}=2$ SCFTs, and the topological quantum mechanics (TQM) sector on a line in three dimensional $mathcal{N}=4$ SCFTs. Under Weyl transformation, they respectively map to operator algebras on a great torus in $S^1times S^3$ and a great circle in $S^3$, and are naturally related by reduction along the $S^1$ factor, which amounts to taking the Cardy (high-temperature) limit of the four dimensional theory on $S^1times S^3$. We elaborate on this relation by explicit examples that involve both Lagrangian and non-Lagrangian theories in four dimensions, where the chiral algebra sector is generally described by a certain W-algebra, while the three dimensional descendant SCFT always has a (mirror) Lagrangian description. By taking into account a subtle R-symmetry mixing, we provide explicit dictionaries between selected operator product expansion (OPE) data in the four and three dimensional SCFTs, which we verify in the examples using recent localization results in four and three dimensions. Our methods thus provide nontrivial support for various chiral algebra proposals in the literature. Along the way, we also identify three dimensional mirrors for Argyres-Douglas theories of type $(A_1, D_{2n+1})$ reduced on $S^1$, and find more evidence for earlier proposals in the case of $(A_1, A_{2n-2})$, which both realize certain superconformal boundary conditions for the four dimensional $mathcal{N}=4$ super-Yang-Mills. This is a companion paper to arXiv:1911.05741.

From VOAs to short star products in SCFT

We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d $mathcal{N}=2$ theories and an associative algebra in the Higgs sector of 3d $mathcal{N}=4$. The natural setting is a 4d $mathcal{N}=2$ SCFT placed on $S^3times S^1$: by sending the radius of $S^1$ to zero, we recover the 3d $mathcal{N}=4$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the $S^1$; 3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient $mathcal{A}_H = {rm Zhu}_{s}(V)/N$, where ${rm Zhu}_{s}(V)$ is the non-commutative Zhu algebra of the VOA $V$ (for ${s}in{rm Aut}(V)$), and $N$ is a certain ideal. This ideal is the null space of the (${s}$-twisted) trace map $T_{s}: {rm Zhu}_{s}(V) to mathbb{C}$ determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips $mathcal{A}_H$ with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map $T_{s}$ is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non-$C_2$-cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.

Chiral Algebra, Localization, Modularity, Surface defects, And All That

We study the 2D vertex operator algebra (VOA) construction in 4D $mathcal{N}=2$ superconformal field theories (SCFT) on $S^3 times S^1$, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus $mathbb{T}^2subset S^3times S^1$, it is $frac12mathbb{Z}$-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $mathbb{Z} +frac12$-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserve the VOA, we find that it admits discrete deformations, which allow access to the remaining spin structures (1,1), (0,1) and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle, the two-point functions, and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the $T$-transformation acts on our four partition functions and lifts to a large diffeomorphism on $S^3times S^1$. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled both by the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the $S$-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the $Omega$-background.

Coulomb Branch Quantization and Abelianized Monopole Bubbling

We develop an approach to the study of Coulomb branch operators in 3D $mathcal{N}=4$ gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For `good and `ugly theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

3d TQFTs from Argyres-Douglas theories

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres-Douglas theories on $S^1 times M_3$ with a non-trivial holonomy of a discrete global symmetry along the $S^1$. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern-Simons and Rozansky-Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the $(G_2)_1$ and $(F_4)_1$ Chern-Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d $mathcal{N}=2$ theory. We test our proposal by applying localization techniques to the conjectural $mathcal{N}=1$ UV Lagrangian descriptions of the $(A_1,A_2)$, $(A_1,A_3)$ and $(A_1,D_3)$ theories.

Gluing II: Boundary Localization and Gluing Formulas

We describe applications of the gluing formalism discussed in the companion paper. When a $d$-dimensional local theory $text{QFT}_d$ is supersymmetric, and if we can find a supersymmetric polarization for $text{QFT}_d$ quantized on a $(d-1)$-manifold $W$, gluing along $W$ is described by a non-local $text{QFT}_{d-1}$ that has an induced supersymmetry. Applying supersymmetric localization to $text{QFT}_{d-1}$, which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of `gluing formulas in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D $mathcal{N}=4$ theories on spheres: one providing the Coulomb branch representation of gluing, and another providing the Higgs branch representation. This allows to study various properties of their $(2,2)$-preserving boundary conditions in relation to Mirror Symmetry. After that we derive a gluing formula in 4D $mathcal{N}=2$ theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D $mathcal{N}=2$ theories.

Gluing I: Integrals and Symmetries

We review some aspects of the cutting and gluing law in local quantum field theory. In particular, we emphasize the description of gluing by a path integral over a space of polarized boundary conditions, which are given by leaves of some Lagrangian foliation in the phase space. We think of this path integral as a non-local $(d-1)$-dimensional gluing theory associated to the parent local $d$-dimensional theory. We describe various properties of this procedure and spell out conditions under which symmetries of the parent theory lead to symmetries of the gluing theory. The purpose of this paper is to set up a playground for the companion paper where these techniques are applied to obtain new results in supersymmetric theories.

Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional $mathcal{N} = 4$ abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the ($nleq 3$)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere $HS^3$ with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the $HS^3$ wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on $S^3$ with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D $mathcal{N} = 2$ theories decorated by BPS t Hooft-Wilson loops.