Superconformal Chern-Simons theories and the squashed seven sphere

We study implications of $$ \mathcal{N}=4 $$ superconformal symmetry in three dimensions, thus extending our earlier results in [1] devoted to the $$ \mathcal{N}\le 3 $$ cases. We show that the three-point function of the supercurrent in $$ \mathcal{N}=4 $$ superconformal field theories contains two linearly independent forms. However, only one of these structures contributes to the three-point function of the energy-momentum tensor and the other one is present in those $$ \mathcal{N}=4 $$ superconformal theories which are not invariant under the mirror map. We point out that general $$ \mathcal{N}=4 $$ superconformal field theories admit two inequivalent flavour current multiplets and show that the three-point function of each of them is determined by one tensor structure. As an example, we compute the two- and three-point functions of the conserved currents in $$ \mathcal{N}=4 $$ superconformal models of free hypermultiplets. We also derive the universal relations between the coefficients appearing in the two- and threepoint correlators of the supercurrent and flavour current multiplets in all superconformal theories with $$ \mathcal{N}\le 4 $$ supersymmetry. Our derivation is based on the use of Ward identities in conjunction with superspace reduction techniques.

We study a class of classical solutions of three-dimensional N=6 superconformal Chern-Simons theory coupled with U(N) \times U(N) bi-fundamental matter fields. Especially, time evolutions of fuzzy spheres are discussed for both massless and massive cases. For the massive case, there are a variety of solutions having different behaviors according to the value of the mass. In addition to the time-dependent solutions, we analyze non-BPS static solutions which represent parallel M5/M5 or M5/anti-M5-branes suspended by multiple M2-branes. These solutions are similar to the fundamental strings connecting two parallel (anti) Dp-branes in perturbative string theory. A moving M5-brane and domain wall solutions with constant velocity that are obtained by the Lorentz boost of the known BPS solutions are briefly addressed.

We review recent progress in formulating the worldvolume theory of M2-branes using the Nambu bracket. Although it is generally agreed that this formulation should be replaced by another using the superconformal Chern-Simons theory, we try to pursue a possible complementary role played by the Nambu bracket. Since the partition function of the superconformal Chern-Simons theory implies a hidden super gauge group, we study the supersymmetric generalization of the Nambu bracket. We explicitly construct the superunitary Nambu algebra. kiyoshig@sci.osaka-cu.ac.jp moriyama@sci.osaka-cu.ac.jp yanok@sci.osaka-cu.ac.jp

The spectrum of planar mathcal{N} = 6 superconformal Chern-Simons theory, dual to type IIA superstring theory on AdS4 × CP3, is accessible at finite coupling using integrability. Starting from the results of [arXiv:1403.1859], we study in depth the basic integrability structure underlying the spectral problem, the Quantum Spectral Curve. The new results presented in this paper open the way to the quantitative study of the spectrum for arbitrary operators at finite coupling. Besides, we show that the Quantum Spectral Curve is embedded into a novel kind of Q-system, which reflects the OSp(4|6) symmetry of the theory and leads to exact Bethe Ansatz equations. The discovery of this algebraic structure, more intricate than the one appearing in the AdS5/CFT4 case, could be a first step towards the extension of the method to AdS3/CFT2.

In the so-called (2, 2) theory, which is the U(N)4 circular quiver superconformal Chern-Simons theory with levels (k, 0, −k, 0), it was known that the instanton effects are described by the free energy of topological strings whose Gopakumar-Vafa invariants coincide with those of the local D5 del Pezzo geometry. By considering two types of one-parameter rank deformations U(N)×U(N + M)×U(N + 2M)×U(N + M) and U(N + M)×U(N)×U(N + M)×U(N), we classify the known diagonal BPS indices by degrees. Together with other two types of one-parameter deformations, we further propose the topological string expression when both of the above two deformations are turned on.

Anomalous dimensions at four loops in [formula omitted] superconformal Chern–Simons theories

Chern-Simons theory with certain gauge groups is known to be equivalent to a first-order formulation of three-dimensional Einstein gravity with a cosmological constant, where both are purely topological. Here, we extend this correspondence to theories with dynamical degrees of freedom. As an example, we show that three-dimensional Yang-Mills theory with gauge group SL(2,R) is equivalent to the first-order formulation of three-dimensional Einstein gravity with no cosmological constant coupled to a background stress-energy tensor density (which breaks the diffeomorphism symmetry). The local degree of freedom of three-dimensional Yang-Mills theory corresponds to degenerate “gravitational waves” in which the metric is degenerate and the spin connection is no longer completely determined by the metric. Turning on a cosmological constant produces the third-way (for Λ<0) or the imaginary third-way (for Λ>0) gauge theories with a background stress-energy tensor density. Published by the American Physical Society 2025

We use group theoretic methods to calculate the spectrum of short multiplets around the extremum of = 8 gauged supergravity potential which possesses = 2 supersymmetry and SU(3) global symmetry. Upon uplifting to M-theory, it describes a warped product of AdS4 and a certain squashed and stretched 7-sphere. We find quantum numbers in agreement with those of the gauge invariant operators in the = 2 superconformal Chern-Simons theory recently proposed to be the dual of this M-theory background. This theory is obtained from the U(N) × U(N) theory through deforming the superpotential by a term quadratic in one of the superfields. To construct this model explicitly, one needs to employ monopole operators whose complete understanding is still lacking. However, for the U(2) × U(2) gauge theory we make a proposal for the form of the monopole operators which has a number of desired properties. In particular, this proposal implies enhanced symmetry of the U(2) × U(2) ABJM theory for k = 1,2; it makes its similarity to and subtle difference from the BLG theory quite explicit.

We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product K3 $\times$ K3, with one of the K3's chosen from these families of elliptic K3 surfaces without a section, realises an elliptic Calabi-Yau 4-fold without a section. We then compactify F-theory on such K3 $\times$ K3's. We determine the gauge groups and matter fields which arise on 7-branes for these models of F-theory compactifications without a section. Since each K3 $\times$ K3 constructed does not have a section, gauge groups arising on 7-branes for F-theory models on constructed K3 $\times$ K3's do not have $U(1)$-part. Interestingly, exceptional gauge group $E_6$ appears for some cases.

This chapter presents various aspects of Chern-Simons theory. In a groundbreaking paper, Witten (1989) showed that Chern-Simons gauge theory, which is a quantum field theory in three dimensions, provides a physical description of a wide class of invariants of three-manifolds and of knots and links in three-manifolds. The partition function and correlation functions of Wilson loops in Chern-Simons theory can be computed in a variety of ways. In order to define the partition function of Chern-Simons theory at the quantum level, one has to specify a framing of the three-manifold. It turns out that the evaluation of correlation functions also involves a choice of framing of the knots. A good starting point for understanding framing is to take Chern-Simons theory with gauge group U. The relation between Chern-Simons theory and string theory involves the vacuum expectation values of Wilson loop operators for arbitrary irreducible representations of U(N). This means that N has to be bigger than the number of boxes of any representation under consideration.

We introduce the general N = 1 gauge theory superconformallycoupled to supergravity. The theory has local SU(2,2|1) symmetry andno dimensional parameters. The superconformal origin of theFayet-Iliopoulos (FI) terms is clarified. The phase of this theory withspontaneously broken conformal symmetry gives various formulations ofN = 1 supergravity interacting with matter, depending on the choice ofthe R-symmetry fixing.We have found that the locally superconformal theory is usefulfor describing the physics of the early universe with a conformally flatFRW metric. Few applications of superconformal theory to cosmologyinclude the study of (a) particle production after inflation,particularly the nonconformal helicity-±½ states ofgravitino,(b) the super-Higgs effect in cosmology and the derivation ofthe equations for the gravitino interacting with any number of chiral and vector multipletsin the gravitational background with varying scalar fields, (c)the weak-coupling limit of supergravity MP→∞ andgravitino-goldstino equivalence. This explains why gravitino productionin the early universe is not suppressed in the limit of weakgravitational coupling.We discuss the possible existence of anunbroken phase of the superconformal theories, interpreted as a strong-coupling limitof supergravity MP→0.

N = 4 superconformal coset theories

This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two-entry Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the role of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity, and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity. 3-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger–Lambert–Gustavsson model. As a result, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (it turns out that Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the Lie or n-Lie algebra bracket is relaxed, one is led to a more general type of n-algebras, the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras. The standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose generalized Jacobi identity reproduces the pattern of the generalized Lie algebras, and the Nambu–Poisson structures, which satisfy the Filippov identity and determine Filippov algebras. Finally, the recent work of Bagger–Lambert and Gustavsson on superconformal Chern–Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A4 model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization.

We study the charged chiral matter spectrum of four-dimensional F-theory compactifications on elliptically fibered Calabi-Yau fourfolds by using the dual M-theory description. A chiral spectrum can be induced by M-theory four-form flux on the fully resolved Calabi-Yau fourfold. In M-theory this flux yields three-dimensional Chern-Simons couplings in the Coulomb branch of the gauge theory. In the F-theory compactification on an additional circle these couplings are only generated by one-loop corrections with charged fermions running in the loop. This identification allows us to infer the net number of chiral matter fields of the four-dimensional effective theory. The chirality formulas can be evaluated by using the intersection numbers and the cones of effective curves of the resolved fourfolds. We argue that a study of the effective curves also allows to follow the resolution process at each co-dimension. To write simple chirality formulas we suggest to use the effective curves involved in the resolution process to determine the matter surfaces and to connect with the group theory at co-dimension two in the base. We exemplify our methods on examples with SU(5) and SU(5)xU(1) gauge group.

We discuss a generalization of Chern-Simons theory in three dimensions based on Leibniz (or Loday) algebras, which are generalizations of Lie algebras. Special cases of such theories appear in gauged supergravity, where the Leibniz algebra is defined in terms of the global (Lie) symmetry algebra of the ungauged limit and an embedding tensor. We show that the Leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a Lie algebra that describes the enhanced symmetry of an `ungauged phase' of the theory. Moreover, we show that a `topological phase' of ${\rm E}_{8(8)}$ exceptional field theory can be interpreted as a Chern-Simons theory for an algebra unifying the three-dimensional Poincar\'e algebra and the Leibniz algebra of ${\rm E}_{8(8)}$ generalized diffeomorphisms.