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Nadav Drukker

Publications and source records attributed to Nadav Drukker.

At least 37 records · Page 2Linked to original sources
arXiv2020

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 $\mathcal{N}=(2,0)$ theory. We find four main classes of solutions in different subspaces of $\mathbb{R}^6$ and a multitude of subclasses and specific examples. We prove that these constructions lead to supersymmety 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 $AdS_7\times S^4$. 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 are proposed as the simplest and most natural observables of this mysterious theory.

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arXiv2020

Surface operators in the 6d $\mathcal{N} = (2,0)$ theory

The 6d $\mathcal{N}=(2,0)$ theory has natural surface operator observables, which are akin in many ways to Wilson loops in gauge theories. We propose a definition of a "locally BPS" surface operator and study its conformal anomalies, the analog of the conformal dimension of local operators. We study the abelian theory and the holographic dual of the large $N$ theory refining previously used techniques. Introducing non-constant couplings to the scalar fields allows for an extra anomaly coefficient, which we find in both cases to be related to one of the geometrical anomaly coefficients, suggesting a general relation due to supersymmetry. We also comment on surfaces with conical singularities.

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arXiv2020

Defect CFT techniques in the 6d $\mathcal{N} = (2,0)$ theory

Surface operators are among the most important observables of the 6d $\mathcal{N} = (2,0)$ theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the explicit supersymmetry tranformations of the stress tensor multiplet and the classification of unitary representations of the superconformal algebra preserved by the defect.

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arXiv2020

BPS Wilson loops and quiver varieties

Three dimensional supersymmetric field theories have large moduli spaces of circular Wilson loops preserving a fixed set of supercharges. We simplify previous constructions of such Wilson loops and amend and clarify their classification. For a generic quiver gauge theory we identify the moduli space as a quotient of $C^m$ for some $m$ by an appropriate symmetry group. These spaces are quiver varieties associated to a cover of the original quiver or a subquiver thereof. This moduli space is generically singular and at the singularities there are large degeneracies of operators which seem different, but whose expectation values and correlation functions with all other gauge invariant operators are identical. The formulation presented here, where the Wilson loops are on $S^3$ or squashed $S^3_b$ also allows to directly implement a localization procedure on these observables, which previously required an indirect cohomological equivalence argument.

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arXiv2019

Roadmap on Wilson loops in 3d Chern-Simons-matter theories

This is a compact review of recent results on supersymmetric Wilson loops in ABJ(M) and related theories. It aims to be a quick introduction to the state of the art in the field and a discussion of open problems. It is divided into short chapters devoted to different questions and techniques. Some new results, perspectives and speculations are also presented. We hope this might serve as a baseline for further studies of this topic.

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arXiv2019

Deformations of the circular Wilson loop and spectral (in)dependence

In this paper we study the expectation value of deformations of the circular Wilson loop in ${\cal N}=4$ super Yang-Mills theory. The leading order deformation, known as the Bremsstrahlung function, can be obtained exactly from supersymmetric localization, so our focus is on deformations at higher orders. We find simple expressions for the expectation values for generic deformations at the quartic order at one-loop at weak coupling and at leading order at strong coupling. We also present a very simple algorithm (not requiring integration) to evaluate the two-loop result. We find that an exact symmetry of the strong coupling sigma-model, known as the spectral-parameter independence, is an approximate symmetry at weak coupling, modifying the expectation value starting only at the sextic order in the deformation. Furthermore, we find very simple patterns for how the spectral parameter can appear in the weak coupling calculation, suggesting all-order structures.

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arXiv2017

The energy-momentum multiplet of supersymmetric defect field theories

Defects in field theories break translation invariance, resulting in the non-conservation of the energy-momentum tensor in the directions normal to the defect. This violation is known as the displacement operator. We study 4d ${\cal N}=1$ theories with 3d defects preserving 3d ${\cal N}=1$ supersymmetry by analyzing the embedding of the 3d superspace in the 4d superspace. We use this to construct the energy-momentum multiplet of such defect field theories, which we call the defect multiplet and show how it incorporates the displacement operator. We also derive the defect multiplet by using a superspace Noether procedure.

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arXiv2017

The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines

We study operator insertions into the $1/2$ BPS Wilson loop in ${\cal N}=4$ SYM theory and determine their two-point coefficients, anomalous dimensions and structure constants. The calculation is done for the first few lowest dimension insertions and relies on known results for the expectation value of a smooth Wilson loop. In addition to the particular coefficients that we calculate, our study elucidates the connection between deformations of the line and operator insertions and between the vacuum expectation value of the line and the CFT data of the insertions.

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arXiv2017

More Large $N$ limits of 3d gauge theories

In this paper we study the large $N$ solution to matrix models describing the partition functions of 3d supersymmetric gauge theories on $S^3$. The model we focus on has a single $U(N)$ gauge group and fundamental fields, whose number scales with $N$, also known as the Veneziano limit. The novel point in our model is our choice of masses for the fundamental fields. Instead of vanishing or fixed masses, we consider a linear distribution. We show that the model can still be solved by standard large $N$ techniques and explore the different phases of the model. We also commend about other natural mass distributions.

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arXiv2016

The $\mathcal{N}=2$ Schur index from free fermions

We study the Schur index of 4-dimensional $\mathcal{N}=2$ circular quiver theories. We show that the index can be expressed as a weighted sum over partition functions describing systems of free Fermions living on a circle. For circular $SU(N)$ quivers of arbitrary length we evaluate the large $N$ limit of the index, up to exponentially suppressed corrections. For the single node theory ($\mathcal{N}=4$ SYM) and the two node quiver we are able to go beyond the large $N$ limit, and obtain the complete, all orders large $N$ expansion of the index, as well as explicit finite $N$ results in terms of elliptic functions.

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arXiv2015

3d mirror symmetry as a canonical transformation

We generalize the free Fermi-gas formulation of certain 3d ${\cal N}=3$ supersymmetric Chern-Simons-matter theories by allowing Fayet-Iliopoulos couplings as well as mass terms for bifundamental matter fields. The resulting partition functions are given by simple modifications of the argument of the Airy function found previously. With these extra parameters it is easy to see that mirror-symmetry corresponds to linear canonical transformations on the phase space (or operator algebra) of the 1-dimensional fermions.

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arXiv2015

The ${\cal N}=4$ Schur index with Polyakov loops

Recently the Schur index of ${\cal N}=4$ SYM was evaluated in closed form to all orders including exponential corrections in the large $N$ expansion and for fixed finite $N$. This was achieved by identifying the matrix model which calculates the index with the partition function of a system of free fermions on a circle. The index can be enriched by the inclusion of loop operators and the case of Wilson loops is particularly easy, as it amounts to inserting extra characters into the matrix model. The Fermi-gas approach is applied here to this problem, the formalism is explored and explicit results at large $N$ are found for the fundamental as well as a few other symmetric and antisymmetric representations.

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arXiv2015

The exact Schur index of $\mathcal{N}=4$ SYM

The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least $\mathcal{N}=2$ supersymmetry in four dimensions is a particular refinement of the index, dependent on one parameter $q$ serving as the fugacity for a particular set of charges which commute with the hamiltonian and some supersymmetry generators. This index has a known expression for all Lagrangian and some non-Lagrangian theories as a finite dimensional integral or a complicated infinite sum. In the case of $\mathcal{N}=4$ SYM with gauge group $U(N)$ we rewrite this as the partition function of a gas of $N$ non interacting and translationally invariant fermions on a circle. This allows us to perform the integrals and write down explicit expressions for fixed $N$ as well as the exact all orders large $N$ expansion.

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arXiv2015

A profusion of $1/2$ BPS Wilson loops in $\mathcal{N}=4$ Chern-Simons-matter theories

We initiate the study of $1/2$ BPS Wilson loops in $\mathcal{N}=4$ Chern-Simons-matter theories in three dimensions. We consider a circular or linear quiver with Chern-Simons levels $k$, $-k$ and $0$, and focus on loops preserving one of the two $SU(2)$ subgroups of the $R$-symmetry. In the cases with no vanishing Chern-Simons levels, we find a pair of Wilson loops for each pair of adjacent nodes on the quiver connected by a hypermultiplet (nodes connected by twisted hypermultiplets have Wilson loops preserving another set of supercharges). We expect this classical pairwise degeneracy to be lifted by quantum corrections. In the case with nodes with vanishing Chern-Simons terms connected by twisted hypermultiplets, we find that the usual $1/4$ BPS Wilson loops are automatically enlarged to $1/2$ BPS, as happens also in 3-dimensional Yang-Mills theory. When the nodes with vanishing Chern-Simons levels are connected by untwisted hypermultiplets, we do not find any Wilson loops coupling to those nodes which are classically invariant. Rather, we find several loops whose supersymmetry variation, while non zero, vanishes in any correlation function, so is weakly zero. We expect only one linear combination of those Wilson loops to remain BPS when quantum corrections are included. We analyze the M-theory duals of those Wilson loops and comment on their degeneracy. We also show that these Wilson loops are cohomologically equivalent to certain $1/4$ BPS Wilson loops whose expectation value can be evaluated by the appropriate localized matrix model.

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arXiv2015

Partition functions of 3d $\hat D$-quivers and their mirror duals from 1d free fermions

We study the matrix models calculating the sphere partition functions of 3d gauge theories with $\mathcal{N}=4$ supersymmetry and a quiver structure of a $\hat D$ Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace ($\hat A $) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulations of such mirror pairs are related by a linear symplectic transformation. We then study the large N expansion of the partition function, which as in the case of the $\hat A$-quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of $\hat D$-quiver theories.

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arXiv2015

On Classical Solutions of 4d Supersymmetric Higher Spin Theory

We present a simple construction of solutions to the supersymmetric higher spin theory based on solutions to bosonic theories. We illustrate this for the case of the Didenko-Vasiliev solution in arXiv:0906.3898, for which we have found a striking simplification where the higher-spin connection takes the vacuum value. Studying these solutions further, we check under which conditions they preserve some supersymmetry in the bulk, and when they are compatible with the boundary conditions conjectured to be dual to certain 3d SUSY Chern-Simons-matter theories. We perform the analysis for a variety of theories with $\mathcal{N}$ = 2, $\mathcal{N}$ = 3, $\mathcal{N}$ = 4 and $\mathcal{N}$ = 6 and find a rich spectrum of $1/4$, $1/3$ and $1/2$-BPS solutions.

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arXiv2014

From algebraic curve to minimal surface and back

We derive the Lax operator for a very large family of classical minimal surface solutions in $AdS_3$ describing Wilson loops in $\mathcal{N}=4$ SYM theory. These solutions, constructed by Ishizeki, Kruczenski and Ziama, are associated with a hyperellictic surface of odd genus. We verify that the algebraic curve derived from the Lax operator is indeed none-other than this hyperelliptic surface.

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