We studied the quantum dynamics of six dimensional $\mathcal{N}=(2, 0)$ superconformal field theory (the QNG theory). We developed the spinor technique for six-dimensional quantum field theories. By combining this technique with the canonical quantization procedure, we can overcome the subtlety of the chiral nature of $\mathcal{N}=(2, 0)$ free tensor multiplet and work out its quantum mechanical theory. We then studied the $\mathbf{T}^2$ compactification of the QNG theory and argued that the resulting four dimensional quantum field theory is indeed the $\mathcal{N}=4$ super-Yang-Mills theory with the $SL(2, \mathbf{Z})$ duality coming from the mapping class symmetries of $\mathbf{T}^2$. We also investigated the BPS self-dual string excitations by proposing a CFT description to their world sheet theory. At last, we constructed the super-twistor space $\hat{\mathbb{PT}}$ that corresponded to the superconformal group $U^*Sp(4|2, \mathbf{H})\subset OSp(8|4, \mathbb{C})$, encoded the full free tensor multiplet into it and made some speculations on the possible super-twistor formulation of the QNG theory.
The local supertwistor formalism, which involves a superconformal connection acting on the bundle of such objects over superspace, is used to investigate superconformal geometry in six dimensions. The geometry corresponding to (1, 0) and (2, 0) off-shell conformal supergravity multiplets, as well the associated finite super-Weyl transformations, are derived.
Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group SU(2m,2n/2N) in the field of quaternions H. The specific construction contains naturally the supertwistor one of the previous work by Litov and Pervushin [1] and it is shown that in the case of extended supersymmetry such an approach leads to the separation of a class of superspaces and and its groups of motion. We briefly discuss this particular extension to the domain of quaternionic superspaces as nonli…
In this paper we give an explicit expression for a star product on the super Minkowski space written in the supertwistor formalism. The big cell of the super Grassmannian Gr(2|0, 4|1) is identified with the chiral, super Minkowki space. The super Grassmannian is an homogeneous space under the action of the complexification SL(4|1) of SU(2,2|1), the superconformal group in dimension 4, signature (1,3) and supersymmetry N=1. The quantization is done by substituting the groups and homogeneous spaces by their quantum deformed counterparts. The calculations are done in Manin's formalism. When we re…
On compactified harmonic/projective superspace, 5D superconformal theories, and all that
在超扭量方法中，我们分析了4D N = 2紧致调和/射影超空间的超共形结构。在5D超共形对称的情况下，我们推导出超共形Killing矢量和相关的积木出现在初级超场的变换律。各种离壳超共形多重态在5D调和超空间和射影超空间中都有出现，包括所谓的热带（向量）多重态和极（超）多重态。超共形作用的家庭被描述在5D调和和射影超空间设置。我们还提出了规范中心荷的5D超共形理论的例子。
Superconformal field theory in three dimensions: Correlation functions of conserved currents
对于三维时空中N-扩展超共形场论，当N= 1，2，3时，我们计算了超流和味流多重态的两点和三点关联函数.我们表明，超对称性施加额外的限制的守恒电流的守恒因子相比，非超对称的情况下，研究奥斯本和Petkou在hep-th/9307010。结果表明，超流的三点函数由一个与守恒方程和所有对称性相一致的函数形式确定。类似地，在N=1和N=3的情况下，风味电流多重峰的三点函数也由单个函数形式确定。N=2的情况下的具体特征是，两个独立的结构是允许的味电流多重态的三点函数，但只有其中之一有助于在这些多重态中包含的守恒电流的三点函数。由于超引力和超杨-米尔斯-沃德恒等式被期望将所考虑的两点和三点函数的系数联系起来，因此三维超共形场论得到的结果与二维共形场论中的结果类似。此外，我们提出了一个新的超扭量构造紧Minkowski超空间。它适合于在三维闵可夫斯基空间以外的时空中发展超共形场论，如S^1 x S^2及其泛覆盖空间R x S^2。
We construct 7-dimensional quantum field theories encoding the anomalies of conformal field theories with (2,0) supersymmetry in six dimensions. We explain how the conformal blocks of the (2,0) theories arise in this context. A result of independent interest is a detailed specification of the data required to define a (2,0) theory with topologically non-trivial spacetime and R-symmetry bundle.
Quantum gravity in AdS$_7 \times$S$^4$ is dual to a 6d superconformal field theory, known as the 6d $(2,0)$ theory, which is very challenging to describe because it lacks a conventional Lagrangian description. On the other hand, certain null reductions of the 6d $(2,0)$ theory give rise to 5d Lagrangian theories with $SU(1,3)$ spacetime symmetry, $SO(5)$ R-symmetry, and 24 supercharges. This appears to be closely related to the superconformal group of a 3d superconformal Chern-Simons theory known as the ABJM theory, which is believed to be integrable in the planar limit, if one exchanges the r…
We present a conjectural description of the space of local operators on a stack of finitely many fivebranes in $M$ theory at the level of the holomorphic twist. Our approach is through the lens of twisted holography and utilizes a description of the minimal twist of eleven-dimensional supergravity. We find that the spaces of local operators are modules for the exceptional linearly compact super Lie algebra $E(3|6)$. From the conjectural description of local operators we deduce closed formulas for the superconformal index of six-dimensional $\mathcal{N}=(2,0)$ theories of type $A_{N-1}$.
Superconformal geometries in spacetime dimensions $D=3,4,{5}$ and $6$ are discussed in terms of local supertwistor bundles over standard superspace. These natually admit superconformal connections as matrix-valued one-forms. In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and $S$-supersymmetry transformations become subsumed into super-Weyl transformations. The number of component fields can be reduced to those of the minimal off-shell confo…
Given a conserved and traceless energy-momentum tensor and a conformal Killing vector, one obtains a conserved current. We generalise this construction to superconformal theories in three, four, five and six dimensions with various amounts of supersymmetry by working in the appropriate superspaces.
Within the framework of six-dimensional ${\cal N}=(1,0)$ conformal supergravity, we introduce new off-shell multiplets ${\cal O}{}^{*}(n)$, where $n=3,4,\dots,$ and use them to construct higher-rank extensions of the linear multiplet action. The ${\cal O}{}^{*}(n)$ multiplets may be viewed as being dual to well-known superconformal ${\cal O}(n)$ multiplets. We provide prepotential formulations for the ${\cal O}(n)$ and ${\cal O}{}^{*}(n)$ multiplets coupled to conformal supergravity. For every ${\cal O}{}^{*}(n)$ multiplet, we construct a higher derivative invariant which is superconformal on …
A closed formula of the universal part of supersymmetric Rényi entropy $S_q$ for six-dimensional $(1,0)$ superconformal theories is proposed. Within our arguments, $S_q$ across a spherical entangling surface is a cubic polynomial of $ν=1/q$, with $4$ coefficients expressed as linear combinations of the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies. As an application, we establish linear relations between the $c$-type Weyl anomalies and the 't Hooft anomaly coefficients. We make a conjecture relating the supersymmetric Rényi entropy to an equivariant integral of…
We bootstrap ${\cal N}=(1,0)$ superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming $E_8$ flavor group, we present universal bounds on the central charge $C_T$ and the flavor central charge $C_J$. Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on $C_J$, and numerically determine the spectrum of long multiplets in the rank-one E-string theory. We comment on the possibility of solving the higher-rank E-string theories by bootstrap and thereby probing M-theory o…
In the framework of six-dimensional conformal field theories with ${\cal N}=(1,0)$ supersymmetry we develop the map between the holographic description, the field theoretical description and the associated Hanany-Witten set-ups. General expressions that calculate various observables are presented. The study of string solitons singles out a special background of Massive IIA on which we show (by explicitly finding a Lax pair) that the Neveu-Schwarz part of the string sigma model is classically integrable. We study the particular dual conformal field theory and compute some of its observables.
We explore the connection between the holographic Weyl anomaly and the superconformal index in six-dimensional $(1,0)$ theories. Using earlier results from holographic computations of the $\mathcal O(1)$ contributions $δa$ and $δ(c-a)$ to the corresponding six-dimensional Weyl anomaly coefficients, we derive a pair of differential operators that extracts these values from the large-$N$ single-trace index. In doing so, we also highlight the structure of these corrections in terms of group theory invariants of the superconformal representations.
We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a f…
We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.
I summarize and discuss some recent results on formulating actions of six-dimensional superconformal field theories using the language of higher gauge theory. The latter guarantees mathematical consistency of our constructions and we review crucial aspects of this framework, such as $L_\infty$-algebras and corresponding kinematical data given by higher connections. We then show that there is a mathematically consistent non-Abelian extension of the self-dual string equation which satisfies many physical expectations. Our construction favors a particular higher gauge group leading us to higher p…
We describe three analytic classes of infinitely many AdS_d supersymmetric solutions of massive IIA supergravity, for d = 7, 5, 4. The three classes are related by simple universal maps. For example, the AdS_7 x M_3 solutions (where M_3 is topologically S^3) are mapped to AdS_5 x Sigma_2 x M'_3, where Sigma_2 is a Riemann surface of genus g >= 2 and the metric on M'_3 is obtained by distorting M_3 in a certain way. The solutions can have localized D6 or O6 sources, as well as an arbitrary number of D8-branes. The AdS_7 case (previously known only numerically) is conjecturally dual to an NS5-D6…
We calculate the superconformal indices of a class of six-dimensional ${\cal N}=(1,0)$ superconformal field theories realized on M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularity by using the method developed in previous works of the authors and collaborators. We use the AdS/CFT correspondence, and finite $N$ corrections are included as the contribution of M2-branes wrapped on two-cycles in $S^4/\mathbb{Z}_k$. We confirm that the indices are consistent with the expected flavor symmetries.
We consider conformal and 't Hooft anomalies in six-dimensional ${\cal N}=(1,0)$ superconformal field theories, focusing on those conformal anomalies that determine the two- and three-point functions of conserved flavor and $SU(2)_R$ currents, as well as stress tensors. By analyzing these correlators in superspace, we explain why the number of independent conformal anomalies is reduced in supersymmetric theories. For instance, non-supersymmetric CFTs in six dimensions have three independent conformal $c$-anomalies, which determine the stress-tensor two- and three-point functions, but in superc…
