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A<sc>bstract</sc> We study 1 + 1-dimensional SU(N) gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumferenceL. Using periodic boundary conditions, we derive the effective action for the quantum mechanics of the holonomy and the fermion zero modes in perturbation theory up to order (gL)³. When the adjoint fermion mass-squared is tuned tog²N/(2π), the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into theℤ_Ncenter symmetry sectors (universes) labeled byp= 0, . . . ,N– 1 and show that in one of the sectors the supersymmetry is unbroken, while in the others it is broken spontaneously. These results give us new insights into the (1, 1) supersymmetry of adjoint QCD₂, which has previously been established using light-cone quantization. When the adjoint mass is set to zero, our effective Hamiltonian does not depend on the fermions at all, so that there are 2^N−1degenerate sectors of the Hilbert space. This construction appears to provide an explicit realization of the extended symmetry of the massless model, where there are 2^2N−2operators that commute with the Hamiltonian. We also generalize our results to other gauge groupsG, for which supersymmetry is found at the adjoint mass-squaredg²h^∨/(2π), whereh^∨is the dual Coxeter number ofG. 

A<sc>bstract</sc> We analyze correlation functions of SU(k) × SU(2)_Fflavor currents in a family of three-dimensional$$ \mathcal{N} $$ $N$= 4 superconformal field theories, combining analytic bootstrap methods with input from supersymmetric localization. Via holographic duality, we extract gluon and graviton scattering amplitudes of M-theory on AdS₄×S⁷/ℤ_kwhich contains a ℂ²/ℤ_korbifold singularity. From these results, we derive aspects of the effective description of M-theory on the orbifold singularity beyond its leading low energy limit. We also determine a threshold correction to the holographic correlator from the combined contribution of two-loop gluon and tree-level bulk graviton exchange. 

A<sc>bstract</sc> We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension unprotected operator in$$ \mathcal{N} $$ $N$= 4 SU(N) super-Yang-Mills theory for a wide range ofNand Yang-Mills couplingsg_YM. We find that our bounds are approximately saturated by weak coupling results at smallg_YM. Furthermore, at largeNour bounds interpolate between integrability results for the Konishi operator at smallg_YMand strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at largeg_YM. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling. 

A<sc>bstract</sc> We introduce a Hamiltonian lattice model for the (1 + 1)-dimensional SU(N_c) gauge theory coupled to one adjoint Majorana fermion of massm. The discretization of the continuum theory uses staggered Majorana fermions. We analyze the symmetries of the lattice model and find lattice analogs of the anomalies of the corresponding continuum theory. An important role is played by the lattice translation by one lattice site, which in the continuum limit involves a discrete axial transformation. On a lattice with periodic boundary conditions, the Hilbert space breaks up into sectors labeled by theN_c-alityp= 0, …N_c− 1. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, form= 0 and evenN_c, the sectorspandp′ are degenerate if |p−p′| =N_c/2. In theN_c= 2 case, we explicitly construct the action of the Hamiltonian on a basis of gauge-invariant states, and we perform both a strong coupling expansion and exact diagonalization for lattices of up to 12 lattice sites. Upon extrapolation of these results, we find good agreement with the spectrum computed previously using discretized light-cone quantization. One of our new results is the first numerical calculation of the fermion bilinear condensate. 

A bstract The mass spectrum of 1 + 1-dimensional SU( N ) gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large N limit using Light-Cone Quantization. Here we extend this approach to theories with small values of N , exhibiting explicit results for N = 2 , 3, and 4. In the context of Discretized Light-Cone Quantization, we develop a procedure based on the Cayley-Hamilton theorem for determining which states of the large N theory become null at finite N . For the low-lying bound states, we find that the squared masses divided by g 2 N , where g is the gauge coupling, have very weak dependence on N . The coefficients of the 1 /N 2 corrections to their large N values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a Kac-Moody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to g 2 N/π , we find evidence that the spectrum exhibits boson-fermion degeneracies, in agreement with the supersymmetry of the model at any value of N . 

A bstract Three-dimensional $$ \mathcal{N} $$ N = 4 superconformal field theories contain 1d topological sectors consisting of twisted linear combinations of half-BPS local operators that can be inserted anywhere along a line. After a conformal mapping to a round three-sphere, the 1d sectors are now defined on a great circle of S 3 . We show that the 1d topological sectors are preserved under the squashing of the sphere. For gauge theories with matter hypermultiplets, we use supersymmetric localization to derive an explicit description of the topological sector associated with the Higgs branch. Furthermore, we find that the dependence of the 1d correlation functions on the squashing parameter b can be removed after appropriate rescalings. One can introduce real mass and Fayet-Iliopolous parameters that, after appropriate rescalings, modify the 1d theory on the squashed sphere precisely as they do on the round sphere. In addition, we also show that when a generic 3d $$ \mathcal{N} $$ N = 4 theory is deformed by real mass parameters, this deformation translates into a universal deformation of the corresponding 1d theory. 

A bstract We study the mass-deformed sphere free energy of three-dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals. Building on previous observations, we conjecture a proportionality relation between the sphere free energy on the boundary and the prepotential of the four-dimensional $$ \mathcal{N} $$ N = 2 supergravity theory in the bulk. We verify this formula by explicit computation in several examples of supergravity theories with vector multiplets and hypermultiplets.