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Register a new userDefect CFT techniques in the 6d $mathcal{N} = (2,0)$ theory
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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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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.
Surface operators in the 6d (2,0) theory at large $N$ have a holographic description in terms of M2 branes probing the AdS$_7 times S^4$ M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an $SO(2,2)$ subgroup of 2d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS$_3$ induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the defect CFT data associated to the surface operator. The spectrum of transverse fluctuations of the M2 brane is found to be in one-to-one correspondence with a protected multiplet of operator insertions on the surface, which includes the displacement operator. We compute the one-loop determinants of fluctuations of the M2 brane, and extract the conformal anomaly coefficient of the spherical surface to order $N^0$. We also briefly discuss the RG flow from the non-supersymmetric to the 1/2-BPS defect operator, and its consistency with a $b$-theorem for the defect CFT. Starting with the M2 brane action, we then use AdS$_3$ Witten diagrams to compute the 4-point functions of the elementary bosonic insertions on the surface operator, and extract some of the defect CFT data from the OPE. The 4-point function is shown to satisfy superconformal Ward identities, and we discuss a related subsector of twisted scalar insertions, whose correlation functions are constrained by the residual superconformal symmetry.
Compactifying type $A_{N-1}$ 6d ${cal N}{=}(2,0)$ supersymmetric CFT on a product manifold $M^4timesSigma^2=M^3timestilde{S}^1times S^1times{cal I}$ either over $S^1$ or over $tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4times{cal I}$ or on $M^3timesSigma^2$, respectively. Choosing the radii of $S^1$ and $tilde{S}^1$ inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants $e^2$ and $tilde{e}^2$ are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU($N$) Yang-Mills theories on $M^4times{cal I}$ and on $M^3timesSigma^2$, where $M^4supset M^3=mathbb R_ttimes T_p^2$ with time $t$ and a punctured 2-torus, and ${cal I}subsetSigma^2$ is an interval. In the first case, shrinking ${cal I}$ to a point reduces to Yang-Mills theory or to the Skyrme model on $M^4$, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on $M^3$ and employing the adiabatic method, we derive in the infrared limit a non-linear SU($N$) sigma model with a baby-Skyrme-type term on $Sigma^2$, which can be reduced further to $A_{N-1}$ Toda theory.
The dimensional-deconstruction prescription of Arkani-Hamed, Cohen, Kaplan, Karch and Motl provides a mechanism for recovering the $A$-type (2,0) theories on $T^2$, starting from a four-dimensional $mathcal N=2$ circular-quiver theory. We put this conjecture to the test using two exact-counting arguments: In the decompactification limit, we compare the Higgs-branch Hilbert series of the 4D $mathcal N=2$ quiver to the half-BPS limit of the (2,0) superconformal index. We also compare the full partition function for the 4D quiver on $S^4$ to the (2,0) partition function on $S^4 times T^2$. In both cases we find exact agreement. The partition function calculation sets up a dictionary between exact results in 4D and 6D.
We study type-B conformal anomalies associated with $frac{1}{2}$-BPS Coulomb-branch operators in 4D $mathcal N=2$ superconformal field theories. When the vacuum preserves the conformal symmetry these anomalies coincide with the two-point function coefficients in the Coulomb-branch chiral ring. They are non-trivial functions of exactly-marginal couplings that can be determined from the $S^4$ partition function. In this paper, we examine the fate of these anomalies in vacua of the Higgs-branch moduli space, where conformal symmetry is spontaneously broken. We argue non-perturbatively that these anomalies are covariantly constant on conformal manifolds. In some cases, this can be used to show that they match in the broken and unbroken phases. Thus, we uncover a new class of data on the Higgs branch of 4D $mathcal N=2$ conformal field theories that are exactly computable. An interesting application of this matching occurs in $mathcal N=2$ circular quivers that deconstruct the 6D (2,0) theory on a torus. In that context, we argue that 4D supersymmetric localisation can be used to calculate non-trivial data involving $frac{1}{2}$-BPS operators of the 6D theory as exact functions of the complex structure of the torus.
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