Membrane Quantum Mechanics

We consider the multiple M2-branes wrapped on a compact Riemann surface and study the arising quantum mechanics by taking the limit where the size of the Riemann surface goes to zero. The IR quantum mechanical models resulting from the BLG-model and the ABJM-model compactified on a torus are N = 16 and N = 12 superconformal gauged quantum mechanics. After integrating out the auxiliary gauge fields we find OSp(16|2) and SU(1,1|6) quantum mechanics from the reduced systems. The curved Riemann surface is taken as a holomorphic curve in a Calabi-Yau space to preserve supersymmetry and we present a prescription of the topological twisting. We find the N = 8 superconformal gauged quantum mechanics that may describe the motion of two wrapped M2-branes in a K3 surface.Comment: 54 pages, v2: errors corrected and notations improve

(0,4) brane box models

Two-dimensional $\mathcal{N}=(0,4)$ supersymmetric quiver gauge theories are realized as D3-brane box configurations (two dimensional intervals) which are bounded by NS5-branes and intersect with D5-branes. The periodic brane configuration is mapped to D1-D5-D5$'$ brane system at orbifold singularity via T-duality. The matter content and interactions are encoded by the $\mathcal{N}=(0,4)$ quiver diagrams which are determined by the brane configurations. The Abelian gauge anomaly cancellation indicates the presence of Fermi multiplets at the NS-NS$'$ junction. We also discuss the brane construction of $\mathcal{N}=(0,4)$ supersymmetric boundary conditions in 3d $\mathcal{N}=4$ gauge theories involving two-dimensional boundary degrees of freedom that cancel gauge anomaly.Comment: 42 pages, 23 figure

Superconformal Quantum Mechanics from M2-branes

We discuss the superconformal quantum mechanics arising from the M2-branes. We begin with a comprehensive review on the superconformal quantum mechanics and emphasize that conformal symmetry and supersymmetry in quantum mechanics contain a number of exotic and enlightening properties which do not occur in higher dimensional field theories. We see that superfield and superspace formalism is available for $\mathcal{N}\le 8$ superconformal mechanical models. We then discuss the M2-branes with a focus on the world-volume descriptions of the multiple M2-branes which are superconformal three-dimensional Chern-Simons matter theories. Finally we argue that the two topics are connected in M-theoretical construction by considering the multiple M2-branes wrapped around a compact Riemann surface and study the emerging IR quantum mechanics. We establish that the resulting quantum mechanics realizes a set of novel $\mathcal{N}\ge 8$ superconformal quantum mechanical models which have not been reached so far. Also we discuss possible applications of the superconformal quantum mechanics to mathematical physics.Comment: PhD Thesis, 294 page

Supersymmetric boundary conditions in three-dimensional N=2 theories

We study supersymmetric boundary conditions in three-dimensional N=2 Landau-Ginzburg models and Abelian gauge theories. In the Landau-Ginzburg model the boundary conditions that preserve (1,1) supersymmetry (A-type) and (2,0) supersymmetry (B-type) on the boundary are classified in terms of subspaces of the target space (“brane”). An A-type brane is a Lagrangian submanifold on which the imaginary part of the superpotential is constant, while a B-type brane is a holomorphic submanifold on which the superpotential is constant. We also consider the N=2 Maxwell theory with boundary and the Abelian duality. Finally we make some comments on N=2 SQED with boundary condition and the mirror symmetry

Exact N=2∗\mathcal{N}=2^{*} Schur line defect correlators

We study the Schur line defect correlation functions in $\mathcal{N}=4$ and $\mathcal{N}=2^*$ $U(N)$ super Yang-Mills (SYM) theory. We find exact closed-form formulae of the correlation functions of the Wilson line operators in the fundamental, antisymmetric and symmetric representations via the Fermi-gas method in the canonical and grand canonical ensembles. All the Schur line defect correlators are shown to be expressible in terms of multiple series that generalizes the Kronecker theta function. From the large $N$ correlators we obtain generating functions for the spectra of the D5-brane giant and the D3-brane dual giant and find a correspondence between the fluctuation modes and the plane partition diamonds.Comment: 77 pages, 3 figure

N=2∗\mathcal{N}=2^{*} Schur indices

We find closed-form expressions for the Schur indices of 4d $\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.Comment: 61 page

3d exceptional gauge theories and boundary confinement

We find boundary confining dualities of 3d N = 2 supersymmetric gauge theories with exceptional gauge groups. The half-indices which enumerate the boundary BPS local operators in the presence of Neumann and Dirichlet boundary conditions for gauge fields are identified with the Askey-Wilson type q-beta integrals and Macdonald type sums respectively. New conjectural identities of E6 and E7 integrals and sums are derived from the boundary confining dualities. We also consider theories with a vector multiplet and adjoint chiral, which correspond to an N = 4 vector multiplet, with appropriate boundary conditions. We argue for the boundary confinement of the N = 4 vector multiplet and comment on such theories also with classical gauge groups