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A<sc>bstract</sc> In our earlier work [1], we introduced a lattice Hamiltonian for Adjoint QCD₂using staggered Majorana fermions. We found the gauge invariant space of states explicitly for the gauge group SU(2) and used them for numerical calculations of observables, such as the spectrum and the expectation value of the fermion bilinear. In this paper, we carry out a more in-depth study of our lattice model, extending it to any compact and simply-connected gauge groupG. We show how to find the gauge invariant space of states and use it to study various observables. We also use the lattice model to calculate the mixed ’t Hooft anomalies of Adjoint QCD₂for arbitraryG. We show that the matrix elements of the lattice Hamiltonian can be expressed in terms of the Wigner 6j-symbols ofG. ForG= SU(3), we perform exact diagonalization for lattices of up to six sites and study the low-lying spectrum, the fermion bilinear condensate, and the string tension. We also show how to write the lattice strong coupling expansion for ground state energies and operator expectation values in terms of the Wigner 6j-symbols. For SU(3) we carry this out explicitly and find good agreement with the exact diagonalizations, and for SU(4) we give expansions that can be compared with future numerical studies. 

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 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 Two-dimensional SU( N ) gauge theory coupled to a Majorana fermion in the adjoint representation is a nice toy model for higher-dimensional gauge dynamics. It possesses a multitude of “gluinoball” bound states whose spectrum has been studied using numerical diagonalizations of the light-cone Hamiltonian. We extend this model by coupling it to N f flavors of fundamental Dirac fermions (quarks). The extended model also contains meson-like bound states, both bosonic and fermionic, which in the large- N limit decouple from the gluinoballs. We study the large- N meson spectrum using the Discretized Light-Cone Quantization (DLCQ). When all the fermions are massless, we exhibit an exact $$ \mathfrak{osp} $$ osp (1|4) symmetry algebra that leads to an infinite number of degeneracies in the DLCQ approach. More generally, we show that many single-trace states in the theory are threshold bound states that are degenerate with multi-trace states. These exact degeneracies can be explained using the Kac-Moody algebra of the SU( N ) current. We also present strong numerical evidence that additional threshold states appear in the continuum limit. Finally, we make the quarks massive while keeping the adjoint fermion massless. In this case too, we observe some exact degeneracies that show that the spectrum of mesons becomes continuous above a certain threshold. This demonstrates quantitatively that the fundamental string tension vanishes in the massless adjoint QCD 2 without explicit four-fermion operators. 

We investigated the neural representation of locomotion in the nematode C. elegans by recording population calcium activity during movement. We report that population activity more accurately decodes locomotion than any single neuron. Relevant signals are distributed across neurons with diverse tunings to locomotion. Two largely distinct subpopulations are informative for decoding velocity and curvature, and different neurons’ activities contribute features relevant for different aspects of a behavior or different instances of a behavioral motif. To validate our measurements, we labeled neurons AVAL and AVAR and found that their activity exhibited expected transients during backward locomotion. Finally, we compared population activity during movement and immobilization. Immobilization alters the correlation structure of neural activity and its dynamics. Some neurons positively correlated with AVA during movement become negatively correlated during immobilization and vice versa. This work provides needed experimental measurements that inform and constrain ongoing efforts to understand population dynamics underlying locomotion in C. elegans .