We propose a new formalism to study the ABJM matrix model. Contrary to expressing the fractional brane background with the Wilson loops in the open string formalism, we formulate the Wilson loop expectation value from the viewpoint of the closed string background. With this new formalism, we can prove some duality relations in the matrix model.

We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional N = 2 superconformal field theories. These (semi-)short multiplets are called "Schur multiplets" and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these threepoint functions, we read off the corresponding OPE selection rules for the Schur multiplets. Our results particularly imply that there are non-trivial selection rules on the quantum numbers of Schur operators in these multiplets. We also give a conjecture on the selection rules for general Schur multiplets.

We construct the chiral algebra associated with the A1-type class $$ \mathcal{S} $$ S theory for the genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of SU(2) which is not associated with any conserved one-form current in four dimensions. We find that this novel SU(2) strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.

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.Recently, a proposal has been put forward to formulate the worldvolume theory of multiple M2-branes using the Nambu bracket. This theory is called BLG theory after its founders, Bagger, Lambert, and Gustavsson [2,3]. See Ref. [4] for an extensive review of the BLG theory. It is surprising to see Nambu's idea revived after three decades at the frontier of string theory.The BLG theory is a 2+1D theory whose Lagrangian is schematically3) 1 See Refs. [8-10] for earlier studies on the fundamental identity and these requirements. 2 Moreover, all of the perturbative [16, 17] and nonperturbative [16, 18-22] corrections are determined. See Ref. [23] for a review.

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