4D Operations Research Articles

Semi-chiral operators in 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 gauge theories

We discuss the properties of quarter-BPS local operators in four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric Yang-Mills theory using the formalism of holomorphic twists. We study loop corrections both to the space of local operators and to algebraic operations which endow the twisted theory with an infinite symmetry algebra. We classify all single-trace quarter-BPS operators in the planar approximation for SU(N) gauge theory and propose a holographic dual description for the twisted theory. We classify perturbative quarter-BPS operators in SU(2) and SU(3) gauge theories with sufficiently small quantum numbers and discuss possible non-perturbative corrections to the answer. We set up analogous calculations for some theories with matter.

Superconformal two-point functions of the Nahm pole defect in [formula omitted] super-Yang-Mills theory

We derive a manifestly superconformal expression for the leading-order two-point functions of all single trace chiral primary operators in 4d N=4 super-Yang-Mills theory with a co-dimension one Nahm pole defect. Notably, our result involves only a finite number of superblocks that correspond to 1/2-BPS representations in the defect channel. The derivation builds on a non-trivial exact result for traces of fuzzy spherical harmonics in combination with powers of su(2) generators, which leads to a remarkably compact expression for the bulk-to-defect couplings. Apart from being interesting in its own right as a closed-form expression at finite N, the final result is an essential prerequisite for applying the defect superconformal bootstrap programme to this model.

Webs, Nijenhuis operators, and heavenly PDEs

In 1989 Mason and Newman (Commun. Math. Phys. 121 659–68) proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields which are divergence free with respect to some volume form. Earlier Plebański (1975 J. Math. Phys. 16 2395–402) showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason–Newman vector fields are also known in the literature: Park (1992 Int. J. Mod. Phys. A 7 1415), Husain (1994 Phys. Rev. Lett. 72 800), Grant (1993 Phys. Rev. D 48 2606–12), Schief (Phys. Lett. A 223 55–62). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new ‘heavenly’ PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko–Schief–Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the ‘heavenly’ PDEs and the existence of related vacuum Einstein metrics in 4D-case.

Vertex algebra of extended operators in 4d N=2 superconformal field theories. Part I

We construct a class of extended operators in the cohomology of a pair of twisted Schur supercharges of 4d mathcal{N} =2 SCFTs. The extended operators are constructed from the local operators in this cohomology — the Schur operators — by a version of topological descent. They are line, surface, and domain wall world volume integrals of certain super descendants of Schur operators. Their world volumes extend in directions transverse to a spatial plane in Minkowski space-time. As operators in the cohomology of these twisted Schur supercharges, their correlators are (locally) meromorphic functions only of the positions where they intersect this plane. This implies the extended operators enlarge the vertex operator algebra of the Schur operators. We illustrate this enlarged vertex algebra by computing some extended-operator product expansions within a subalgebra of it for the free hypermultiplet SCFT.

SUSY Localization for Coulomb Branch Operators in Omega-Deformed 3d $$\mathcal {N}$$ = 4 Gauge Theories

We perform SUSY localization for Coulomb branch operators of 3d $$\mathcal {N}=4$$ gauge theories in $$\mathbb {R}^3$$ with $$\Omega $$ -deformation. Our study provides a path integral foundation to the so-called abelianization procedure that has been used to study the Coulomb branch. For the dressed monopole operators whose expectation values do not involve non-perturbative corrections, our computations reproduce the results of abelianization. For the expectation values of other operators and the correlation functions of multiple operators in U(N) gauge theories, we compute the non-perturbative corrections due to monopole bubbling using matrix models obtained by string theory (brane) construction. We relate the results of localization to algebraic structures discussed in the mathematical literature, and also uncover a similar relation for line operators in 4d $$\mathcal {N}=2$$ gauge theories. For U(N) (quiver) gauge theories in 3d we demonstrate a direct correspondence between wall-crossing in matrix models and the ordering of operators.

Celestial holography meets twisted holography: 4d amplitudes from chiral correlators

We propose a new program for computing a certain integrand of scattering amplitudes of four-dimensional gauge theories which we call the form factor integrand, starting from 6d holomorphic theories on twistor space. We show that the form factor integrands can be expressed as sums of products of 1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic symmetries uncovered recently in the celestial holography program, and 2.) OPE coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the chiral algebras are in one-to-one correspondence with local operators in 4d. We use this bijection to recover the Parke-Taylor formula, the CSW formula, and certain one-loop scattering amplitudes. Along the way, we explain and derive various aspects of celestial holography, incorporating techniques from the twisted holography program such as Koszul duality. This perspective allows us to easily and efficiently recover the infinite-dimensional chiral algebras of asymptotic symmetries recently extracted from scattering amplitudes of massless gluons and gravitons in the celestial basis. We also compute some simple one-loop corrections to the chiral algebras and derive the three-dimensional bulk theories for which these 2d algebras furnish an algebra of boundary local operators.

Growth of the 116 -BPS index in 4d N=4 supersymmetric Yang-Mills theory

We study the microcanonical superconformal index of $\frac{1}{16}$-supersymmetric operators in 4d $\mathcal{N}=4$ $U(N)$ super Yang-Mills (SYM) theory. We show, numerically for $N\ensuremath{\le}10$, that the large-charge asymptotics are consistent with the entropy of supersymmetric black holes in the dual anti de Sitter space. We then prove, using representation theory, that the index agrees precisely with the multigraviton index, when the charge is less than $2(N+1)$, and begins to deviate for larger values of charge. Thus the $U(N)$ SYM index interpolates between multigraviton values at small charges and black hole growth at large charges.

Observations on BPS observables in 6D

We study possible geometries and R-symmetry breaking patterns that lead to globally BPS surface operators in the six dimensional theory. We find four main classes of solutions in different subspaces of and a multitude of subclasses and specific examples. We prove that these constructions lead to supersymmetry preserving observables and count the number of preserved supercharges. We discuss the underlying geometry, calculate their anomalies and present analogous structures for the holographic dual M2-branes in AdS 7 × S 4. We also comment on the dimensional reduction of these observables to line and surface operators in 4D and 5D theories. This rich spectrum of operators is proposed as the simplest and most natural observables of this mysterious theory.

Harmonic analysis and mean field theory

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and “seed” operators in 4d, and MFT for currents and stress-tensors in 3d.

Chiral rings for surface operators in 4d and 5d SQCD

Chiral rings of two-dimensional (2,2) theories coupled to 4d mathcal{N} = 2 theories with matter hypermultiplets are studied. Specifically, the vacua of the twisted superpotential of the 2d theories with vanishing sum of matter charges are computed by considering the resolvent of the bulk theory. The solutions to the chiral ring equations are also obtained from the instanton partition function via Higgsing for simple surface operators and via the orbifold description for full surface operators. These 2d/4d coupled theories are lifted to 3d/5d theories and vacua are found similarly in two different methods: by solving the 3d chiral ring equations taking into account the effect of 5d resolvent and by computing the 5d instanton partition function in the presence of a surface operator. We also check the Seiberg-like duality for both 2d/4d and 3d/5d coupled systems with a specific Chern-Simons coefficient for the latter.

Geometry of Higgs-branch superconformal primary bundles

It is known that the two- and three-point functions of Higgs-branch superconformal primaries in 4d N=2 superconformal field theories obey non-renormalization theorems on N=2 superconformal manifolds. In this paper we prove a stronger statement, that the bundles of Higgs-branch superconformal primaries over N=2 superconformal manifolds are endowed with a flat connection, or equivalently that Higgs-branch superconformal primaries have vanishing Berry phases under N=2 exactly marginal deformations. This statement fits well with the proposed correspondence between the rigid structures of 2d chiral algebras and the sector of Schur operators in 4d N=2 theories. We also discuss the general interplay between non-renormalization theorems and the curvature of bundles of protected operators and provide a new simpler proof of the vanishing curvature of 1/2-BPS operators in 4d N=4 SYM theory that does not require the use of the 4d tt* equations.

Connecting localization and wall-crossing via D-branes

We demonstrate explicitly that the vacuum expectation values (vevs) of BPS line operators in 4d N=2 super Yang–Mills theory compactified on a circle, computed by localization techniques, can be expanded in terms of Darboux coordinates as proposed by Gaiotto, Moore, and Neitzke [1]. However, we need to augment the expressions for Darboux coordinates with additional monopole bubbling contributions to obtain a precise match. Using D-brane realization of these singular BPS line operators, we derive and incorporate the monopole bubbling contributions as well as predict the degeneracies of framed BPS states contributing to the line operator vevs in the limit of vanishing simultaneous spatial and R-symmetry rotation fugacity parameter.

On exact correlation functions in SU(N) N = 2 $$ \mathcal{N}=2 $$ superconformal QCD

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.

Effective operators in SUSY, superfield constraints and searches for a UV completion

We discuss the role of a class of higher dimensional operators in 4D N=1 super-symmetric effective theories. The Lagrangian in such theories is an expansion in momenta below the scale of “new physics” (Λ) and contains the effective operators generated by integrating out the “heavy states” above Λ present in the UV complete theory. We go beyond the “traditional” leading order in this momentum expansion (in ∂/Λ). Keeping manifest supersymmetry and using superfield constraints we show that the corresponding higher dimensional (derivative) operators in the sectors of chiral, linear and vector superfields of a Lagrangian can be “unfolded” into second-order operators. The “unfolded” formulation has only polynomial interactions and additional massive superfields, some of which are ghost-like if the effective operators were quadratic in fields. Using this formulation, the UV theory emerges naturally and fixes the (otherwise unknown) coefficient and sign of the initial (higher derivative) operators. Integrating the massive fields of the “unfolded” formulation generates an effective theory with only polynomial effective interactions relevant for phenomenology. We also provide several examples of “unfolding” of theories with higher derivative interactions in the gauge or matter sectors that are actually ghost-free. We then illustrate how our method can be applied even when including all orders in the momentum expansion, by using an infinite set of superfield constraints and an iterative procedure, with similar results.

Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d \({{\mathcal N}=4}\) SYM and instantonic solutions in 2D Yang-Mills theory (YM2) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM2, belonging to representations of the dual group L G. This conjecture has been tested in the case G = U(N) = L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and L G = SU(N)/Z N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.

A& B model approaches to surface operators and Toda thoeries

It has recently been argued by Alday et al that the inclusion of surface operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d N=2 gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.