Igor R. Klebanov

We reexamine the large N limit of SU ( N ) symmetric quantum mechanics of a Hermitian matrix whose singlet sector is well known to be exactly solvable via free fermions. When the Fermi level approaches a maximum of the potential, there is critical behavior corresponding to string theory in two dimensions. We uncover new phenomena in the adjoint sector by solving the Marchesini-Onofri equation both numerically and analytically using semiclassical approximations: at criticality, the spectrum is governed by Regge trajectories with energy eigenvalues growing according to Δ 2 ∼ n / α ′ . In the dual 2D string theory, we interpret these states as oscillatory excitations of a “short” folded open string. Up to subleading corrections, this Regge behavior is essentially universal and is insensitive to the particular potential we choose to approach criticality. Slightly away from criticality, the highly excited states transition into “long strings” that extend far into the Liouville direction.

We discuss dimensional continuation of the massless scalar field theory with the i ϕ 5 interaction term. It preserves the so-called P T symmetry, which acts by ϕ → − ϕ accompanied by i → − i . Below its upper critical dimension 10 / 3 , this theory has interacting infrared fixed points. We argue that the fixed point in d = 2 describes the nonunitary minimal conformal model M ( 2 , 7 ) . We identify the operators ϕ and ϕ 2 with the Virasoro primaries ϕ 1 , 2 and ϕ 1 , 3 , respectively, and i ϕ 3 with a quasiprimary operator, which is a Virasoro descendant of ϕ 1 , 3 . Our identifications appear to be consistent with the operator product expansions and with considerations based on integrability. Using constrained Padé extrapolations, we provide estimates of the critical exponents in d = 3 . We also comment on possible lattice descriptions of M ( 2 , 7 ) and discuss RG flows to and from this conformal field theory (CFT). Finally, we conjecture that the minimal models M ( 2 , 2 n + 1 ) are described by the massless scalar field theories with the i ϕ 2 n − 1 interaction terms.

The Yang-Lee universality class arises when an imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the d < 6 Ising model. In d = 2 , this nonunitary conformal field theory (CFT) is exactly solvable via the M ( 2 , 5 ) minimal model. As found long ago by von Gehlen using exact diagonalization, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the PT symmetry. In this paper, we explore the analogous quantum critical behavior in higher-dimensional non-Hermitian Hamiltonians on regularized spheres S d − 1 . For d = 3 , we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron, and dodecahedron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Padé extrapolations of the 6 − ε expansions in Fisher’s i ϕ 3 Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the M ( 2 , 5 ) minimal model. For d = 4 , we obtain new results by replacing S 3 with the self-dual polytope called the 24-cell.

A bstract: The dimensional continuation approach to calculating the free energy of d -dimensional Euclidean CFT on the round sphere S ^d has been used to develop its 4 – ϵ expansion for a number of well-known non-supersymmetric theories, such as the O ( N ) model. The resulting estimate of the sphere free energy F in the 3D Ising model has turned out to be in good agreement with the numerical value obtained using fuzzy sphere regularization. In this paper, we develop the 6 – ϵ expansions for CFTs on S ^d described by scalar field theory with cubic interactions and use their resummations to estimate the values of F . In particular, we study the theories with purely imaginary coupling constants, which describe non-unitary universality classes arising when certain conformal minimal models are continued above two dimensions. The Yang-Lee model M (2, 5) is described by a field theory with one scalar field, while the D -series M (3, 8) model is described by two scalar fields. We also study the OSp(1|2) symmetric cubic theory of one commuting and two anti-commuting scalar fields, which appears to describe the critical behavior of random spanning forests. In the course of our work, we revisit the calculation of beta functions of marginal operators containing the curvature. We also use another method for approximating F , which relies on perturbation theory around the bilocal action near the long-range/short-range crossover. The numerical values it gives for F tend to be in good agreement with other available methods.

In our earlier work [1], we introduced a lattice Hamiltonian for Adjoint QCD₂ using 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 group G. 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 QCD₂ for arbitrary G. We show that the matrix elements of the lattice Hamiltonian can be expressed in terms of the Wigner 6j-symbols of G. For G = 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.

We study 1 + 1-dimensional SU(N) gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumference L. 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 to g ² N/(2π), the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into the ℤ _N center symmetry sectors (universes) labeled by p = 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−1 degenerate 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−2 operators that commute with the Hamiltonian. We also generalize our results to other gauge groups G, for which supersymmetry is found at the adjoint mass-squared g ² h ^∨/(2π), where h ^∨ is the dual Coxeter number of G.

A bstract: We introduce a Hamiltonian lattice model for the (1 + 1)-dimensional SU( N _c ) gauge theory coupled to one adjoint Majorana fermion of mass m . 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 the N _c -ality p = 0, … N _c − 1. Our symmetry analysis implies various exact degeneracies in the spectrum of the lattice model. In particular, it shows that, for m = 0 and even N _c , the sectors p and p ′ are degenerate if | p − p ′| = N _c /2. In the N _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.