Superconformal Chern-Simons Theory Research Articles

Yangian Symmetry in Five Dimensions.

Quantum gravity in AdS_{7}×S^{4} is dual to a six-dimensional (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 Aharony-Bergman-Jafferis-Maldacena theory, which is believed to be integrable in the planar limit, if one exchanges the role of conformal and R symmetry. In this Letter, we construct a representation of the 5D superconformal group using 6D supertwistors and show that it admits an infinite dimensional extension known as Yangian symmetry, which opens up the possibility that these 5D theories are exactly solvable in the planar limit.

3d dualities with decoupled sectors and brane transitions

We find exact relations among the sphere partition functions of three-dimensional mathcal{N} = 4 superconformal Chern-Simons theories with circular quiver diagrams. These relations suggest new dualities in gauge theories which are the products of circular quiver gauge theories and decoupled linear quiver gauge theories. The dualities can be interpreted as simple brane transitions in the Hanany-Witten brane construction in type IIB string theory. Interestingly, the brane transitions cannot be generated by the Hanany-Witten transition or the SL (2, ℤ) transformation. Our results can be regarded as generalization and clarification of dualities implied from Weyl group symmetries of quantum curves. To obtain the exact results, we employ the supersymmetric localization technique and the Fermi gas approach.

Fermi gas approach to general rank theories and quantum curves

It is known that matrix models computing the partition functions of three-dimensional mathcal{N} = 4 superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.

Quantum mirror map for del Pezzo geometries

Mirror maps play an important role in studying supersymmetric gauge theories. In these theories the dynamics is often encoded in an algebraic curve where two sets of periods enjoy the symplectic structure. The A-periods contribute to redefinitions of chemical potentials known as mirror maps. Using the quantization of the D5 del Pezzo geometry, which enjoys the symmetry of the D5 Weyl group, we are able to identify clearly the group-theoretical structure and the multi-covering structure for the mirror map. With the structures, we can apply the mirror map to superconformal Chern–Simons theories describing the worldvolume of multiple M2-branes on various backgrounds, where we find that the redefinition of the chemical potential is obtained directly from the mirror map. Besides, we have interesting observations for the mirror map: the representations appearing in the quantum mirror map are the same as those appearing in the BPS indices except for the trivial case of degree 1 and the coefficients are all integers.

Jacobi-Trudi Identity in Super Chern-Simons Matrix Model

It was proved by Macdonald that the Giambelli identity holds if we define the Schur functions using the Jacobi-Trudi identity. Previously for the super Chern-Simons matrix model (the spherical one-point function of the superconformal Chern-Simons theory describing the worldvolume of the M2-branes) the Giambelli identity was proved from a shifted version of it. With the same shifted Giambelli identity we can further prove the Jacobi-Trudi identity, which strongly suggests an integrable structure for this matrix model.

ABJ quadrality

We study physical consequences of adding orientifolds to the ABJ triality, which is among 3d mathcal{N}=6 superconformal Chern-Simons theory known as ABJ theory, type IIA string in AdS4 × ℂℙ3 and mathcal{N}=6 supersymmetric (SUSY) Vasiliev higher spin theory in AdS4. After adding the orientifolds, it is known that the gauge group of the ABJ theory becomes O(N1) × USp(2N2) while the background of the string theory is replaced by AdS4 × ℂℙ3/Z2, and the supersymmetries in the both theories reduce to mathcal{N}=5 . We propose that adding the orientifolds to the mathcal{N}=6 Vasiliev theory leads to mathcal{N}=5 SUSY Vasiliev theory. It turns out that the mathcal{N}=5 case is more involved because there are two formulations of the mathcal{N}=5 Vasiliev theory with either O or USp internal symmetry. We show that the two mathcal{N}=5 Vasiliev theories can be understood as certain projections of the mathcal{N}=6 Vasiliev theory, which we identify with the orientifold projections in the Vasiliev theory. We conjecture that the O(N1) × USp(2N2) ABJ theory has the two vector model like limits: N2 ≫ N1 and N1 ≫ N2 which correspond to the semi-classical mathcal{N}=5 Vasiliev theories with O(N1) and USp(2N2) internal symmetries respectively. These correspondences together with the standard AdS/CFT correspondence comprise the ABJ quadrality among the mathcal{N}=5 ABJ theory, string/M-theory and two mathcal{N}=5 Vasliev theories. We provide a precise holographic dictionary for the correspondences by comparing correlation functions of stress tensor and flavor currents. Our conjecture is supported by various evidence such as agreements of the spectra, one-loop free energies and SUSY enhancement on the both sides. We also predict the leading free energy of the mathcal{N}=5 Vasiliev theory from the CFT side. As a byproduct, we give a derivation of the relation between the parity violating phase in the mathcal{N}=6 Vasiliev theory and the parameters in the mathcal{N}=6 ABJ theory, which was conjectured in [1].

Superconformal Chern-Simons theories from del Pezzo geometries

We present an explicit expression for the grand potential of the U(N )3 superconformal Chern-Simons theory with the Chern-Simons levels being (k, 0, −k). From the viewpoint of the Newton polygon, it is expected that the grand potential is given by the free energy of the topological string theory on the local D5 del Pezzo geometry, though the explicit identification was a puzzle for years. We show how the expectation is realized explicitly. As a bonus, we can also study the {mathbb{Z}}_2 orbifold of this theory and find the grand potential is now given in terms of the local E7 del Pezzo geometry.

The full Quantum Spectral Curve for AdS4/CFT3

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.

Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

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.

A requiem forAdS4×CP3fermionic self-T duality

Strong evidence for dual superconformal symmetry in $\mathcal{N} = 6$ superconformal Chern-Simons theory has fueled expectations that the AdS/CFT dual geometry $AdS_4 \times \mathbb{C} P^3$ is self-dual under T-duality. We revisit the problem to identify commuting bosonic and fermionic isometries in a systematic fashion and show that fermionic T-duality, a symmetry originally proposed by Berkovits & Maldacena, inevitably leads to a singularity in the dilaton transformation. We show that TsT deformations commute with fermionic T-duality and comment on T-duality in the corresponding sigma model. Our results rule out self-duality based on fermionic T-duality for $AdS_4 \times \mathbb{C} P^3$ or its TsT deformations, but leave the door open for new possibilities.

Prospects of the Nambu bracket

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

Exact relations between M2-brane theories with and without orientifolds

We study partition functions of low-energy effective theories of M2-branes, whose type IIB brane constructions include orientifolds. We mainly focus on circular quiver superconformal Chern-Simons theory on $S^3$, whose gauge group is $O(2N+1)\times USp(2N)\times \cdots \times O(2N+1)\times USp(2N)$. This theory is the natural generalization of the $\mathcal{N}=5$ ABJM theory with the gauge group $O(2N+1)_{2k} \times USp(2N)_{-k}$. We find that the partition function of this type of theory has a simple relation to the one of the M2-brane theory without the orientifolds, whose gauge group is $U(N)\times \cdots \times U(N)$. By using this relation, we determine an exact form of the grand partition function of the $O(2N+1)_{2} \times USp(2N)_{-1}$ ABJM theory, where its supersymmetry is expected to be enhanced to $\mathcal{N}=6$. As another interesting application, we discuss that our result gives a natural physical interpretation of a relation between the grand partition functions of the $U(N+1)_4 \times U(N)_{-4}$ ABJ theory and $U(N)_2 \times U(N)_{-2}$ ABJM theory, recently conjectured by Grassi-Hatsuda-Mari\~no. We also argue that partition functions of $\hat{A}_3$ quiver theories have representations in terms of an ideal Fermi gas systems associated with $\hat{D}$-type quiver theories and this leads an interesting relation between certain $U(N)$ and $USp(2N)$ supersymmetric gauge theories.

Orientifold ABJM matrix model: chiral projections and worldsheet instantons

We study the partition function of the orientifold ABJM theory, which is a superconformal Chern-Simons theory associated with the orthosymplectic supergroup. We find that the partition function associated with any orthosymplectic supergroup can be realized as the partition function of a Fermi gas system whose density matrix is identical to that associated with the corresponding unitary supergroup with a projection to the even or odd chirality. Furthermore we propose an identity which gives directly all of the Gopakumar-Vafa invariants for the worldsheet instanton effects in the chirally projected theories.

Instanton effects in ABJM theory with general R-charge assignments

We study the large N expansion of the partition function of the quiver superconformal Chern-Simons theories deformed by two continuous parameters which correspond to the general R-charge assignment to the matter fields. Though the deformation breaks the conformal symmetry, we find that the partition function shares various structures with the superconformal cases, such as the Airy function expression of the perturbative expansion in 1/N with the overall constant A(k) related to the constant map in the ABJM case through a simple rescaling of k. We also identify five kinds of the non-perturbative effects in 1/N which correspond to the membrane instantons. The instanton exponents and the singular structure of the coefficients depend on the continuous deformation parameters, in contrast to the superconformal case where all the parameters are integers associated with the orbifold action on the moduli space. This implies that the singularity of the instanton effects would be observable also in the gravity side.

Superconformal Chern-Simons-matter theories inN=4superspace

In three dimensions, every known N=4 supermultiplet has an off-shell completion. However, there is no off-shell N=4 formulation for the known extended superconformal Chern-Simons (CS) theories with eight and more supercharges. To achieve a better understanding of this issue, we provide N=4 superfield realisations for the equations of motion which correspond to various N=4 and N=6 superconformal CS theories, including the Gaiotto-Witten theory and the ABJM theory. These superfield realisations demonstrate that the superconformal CS theories with N>3 (except for the Gaiotto-Witten theory) require a reducible long N=4 vector multiplet, from which the standard left and right N=4 vector multiplets are obtained by constraining the field strength to be either self-dual or anti self-dual. Such a long multiplet naturally originates upon reduction of any off-shell N>4 vector multiplet to N=4 superspace. For the long N=4 vector multiplet we develop a prepotential formulation. It makes use of two prepotentials being subject to the constraint which defines the so-called hybrid projective multiplets introduced in the framework of N=4 supergravity-matter systems in arXiv:1101.4013. We also couple N=4 superconformal CS theories to N=4 conformal supergravity.

Ward identities and gauge flow for M-theory inN=3superspace

We derive the BRST symmetry, Slavnov-Taylor identities and Nielsen identities for the ABJM theories in ${\cal N}{=}3$ harmonic superspace. Further, the gauge dependence of one-particle irreducible amplitudes in such superconformal Chern-Simons theory is shown to be generated by a canonical flow with respect to the extended Slavnov-Taylor identity, induced by the extended BRST transformations (including the BRST transformations of the gauge parameters).

Superconformal Chern-Simons partition functions of affine D-type quiver from Fermi gas

We consider the partition function of the superconformal Chern-Simons theories with the quiver diagram being the affine D-type Dynkin diagram. Rewriting the partition function into that of a Fermi gas system, we show that the perturbative expansions in 1/N are summed up to an Airy function, as in the ABJM theory or more generally the theories of the affine A-type quiver. As a corollary, this provides a proof for the previous proposal in the large N limit. For special values of the Chern-Simons levels, we further identify three species of the membrane instantons and also conjecture an exact expression of the overall constant, which corresponds to the constant map in the topological string theory.

Exact instanton expansion of superconformal Chern-Simons theories from topological strings

It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the Calabi-Yau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the same expression from the refined topological string theory as the ABJM matrix model with different topological invariants while that of the other is more general. We also observe an unexpected relation between these two theories.

Heavy operators in superconformal Chern-Simons theory

We study the anomalous dimensions for scalar operators in ABJM theory in the SU(2) sector. The operators we consider have a classical dimension that grows as N in the large N limit. Consequently, the large N limit is not captured by summing planar diagrams - non-planar contributions have to be included. We find that the mixing matrix at two-loop order is diagonalized using a double coset ansatz, reducing it to the Hamiltonian of a set of decoupled oscillators. The spectrum of anomalous dimensions, when interpreted in the dual gravity theory, shows that the energy of the fluctuations of the corresponding giant graviton is dependent on the size of the giant. The first subleading corrections to the large N limit are also considered. These subleading corrections to the dilatation operator do not commute with the leading terms, indicating that integrability may not survive beyond the large N limit.

Partition functions of superconformal Chern-Simons theories from Fermi gas approach

We study the partition function of three-dimensional ${\mathcal N}=4$ superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes. In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as $Z(N)=e^{A}C^{-1/3}\mathrm{Ai}[C^{-1/3}(N-B)]$ with coefficients $C$, $B$ and $A$ and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves. We find that many of the interesting properties in the ABJM theory are extended to the general superconformal Chern-Simons theories. Especially, we find an explicit expression of $B$ for general ${\mathcal N}=4$ theories, a conjectural form of $A$ for a special class of theories, and cancellation in the non-perturbative coefficients for the simplest theory next to the ABJM theory.