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

United States

Ross Dempsey

Silviu Pufu

Benjamin Søgaard

Igor Klebanov

More about the lattice Hamiltonian for Adjoint QCD

In this work, we develop a tool to study string perturbation theory of the Klebanov-Strassler solution in the large radius approximation based on open-closed superstring field theory. Combining the large radius expansion and a double scaling limit, we find a perturbative background solution of open-closed superstring field theory that corresponds to the Klebanov-Strassler solution. To illustrate the utilities of this approach, we break the supersymmetry of the background by placing a stack of anti-D3-branes at the tip of the throat without invoking S-duality. We highlight how our solutions qualitatively differ from the study of Kachru-Pearson-Verlinde, which studied S-dualized anti-D3-brane in the KS background. We then find a perturbative open string background solution to the third order in the large radius approximation. The perturbative background solution to the open string field theory we found is expected to be dual to a D5-brane probing the KS solution.

Manki Kim

String perturbation theory of Klebanov-Strassler throat

We study the dynamics of a first-order phase transition in a strongly coupled gauge theory at non-zero temperature and chemical potential, computing nucleation rates and wall speeds from first principles. The gauge theory is the four-dimensional superconformal SU($\textit{N}$) × SU($\textit{N}$) Klebanov-Witten theory, which at low temperatures displays an instability to forming scalar condensates that higgses the theory. The computation is made possible by utilizing the gravity dual, type IIB string theory on asymptotically AdS × $\textit{T}$ spacetimes. The instability is detected through the nucleation and subsequent localization of D-branes in the bulk, which in the probe limit is amenable to calculations. The nucleation rates suggest a preference for greater asymmetry between the two gauge groups as the temperature is lowered beyond its critical value. The wall speed stays below the speed of sound of the conformal plasma across all parameter values and exhibits nonlinear dependence on the energy difference between the two phases.

Finland

Oscar Henriksson

Niko Jokela

Julia Junttila

Dynamics of a Higgs phase transition in the Klebanov-Witten theory

It has been proposed that the Ginzburg-Landau description of the non-unitary conformal minimal model (3, 8) is provided by the Euclidean theory of two real scalar fields with third-order interactions that have imaginary coefficients. The same lagrangian describes the non-unitary model (3, 10), which is a product of two Yang-Lee theories (2, 5), and the Renormalization Group flow from it to (3, 8). This proposal has recently passed an important consistency check, due to Y. Nakayama and T. Tanaka, based on the anomaly matching for non-invertible topological lines. In this paper, we elaborate the earlier proposal and argue that the two-field theory describes the series modular invariants of both (3, 8) and (3, 10). We further propose the Ginzburg-Landau descriptions of the entire class of series minimal models ( , 3 – 1) and ( , 3 + 1), with odd integer . They involve $$ \mathcal{PT} $$ symmetric theories of two scalar fields with interactions of order multiplied by imaginary coupling constants.

Andrei Katsevich

Zimo Sun

Ginzburg-Landau description of a class of non-unitary minimal models

We study 1 + 1-dimensional SU($\textit{N}$) gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumference $\textit{L}$. 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 ($\textit{gL}$) . When the adjoint fermion mass-squared is tuned to $\textit{g}$ $\textit{N}$/(2$\textit{π}$), the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into the $\textit{ℤ}$ center symmetry sectors (universes) labeled by $\textit{p}$ = 0, . . . , $\textit{N}$ – 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 QCD , 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 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 operators that commute with the Hamiltonian. We also generalize our results to other gauge groups $\textit{G}$, for which supersymmetry is found at the adjoint mass-squared $\textit{g}$ $\textit{h}$ /(2$\textit{π}$), where $\textit{h}$ is the dual Coxeter number of $\textit{G}$.

Small circle expansion for adjoint QCD

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.

Lattice Hamiltonian for adjoint QCD2

We explore a new approach to boundaries and interfaces in the O(N) model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk dimension is 4 − ϵ, and they explicitly break the O(N) symmetry of the bulk theory down to O(N − 1). We show that the one-loop beta functions of the cubic couplings are affected by the quartic bulk interactions. For the interfaces, we find real fixed points up to the critical value N crit ≈ 7, while for N &gt; 4 there are IR stable fixed points with purely imaginary values of the cubic couplings. For the boundaries, there are real fixed points for all N, but we don’t find any purely imaginary fixed points. We also consider the theories of M pairs of symplectic fermions and one real scalar, which have quartic OSp(1|2M) invariant interactions in the bulk. We then add the Sp(2M) invariant localized cubic interactions. The beta functions for these theories are related to those in the O(N) model via the replacement of N by 1 − 2M. In the special case M = 1, there are boundary or interface fixed points that preserve the OSp(1|2) symmetry, as well as other fixed points that break it.

Sweden

Sabine Harribey

Boundaries and interfaces with localized cubic interactions in the O(N) model

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.

Loki Lin

Adjoint Majorana QCD2 at finite N

A pair of the 2D non-unitary minimal models M(2, 5) is known to be equivalent to a variant of the M(3, 10) minimal model. We discuss the RG flow from this model to another non-unitary minimal model, M(3, 8). This provides new evidence for its previously proposed Ginzburg-Landau description, which is a ℤ2 symmetric theory of two scalar fields with cubic interactions. We also point out that M(3, 8) is equivalent to the (2, 8) superconformal minimal model with the diagonal modular invariant. Using the 5-loop results for theories of scalar fields with cubic interactions, we exhibit the 6 − ϵ expansions of the dimensions of various operators. Their extrapolations are in quite good agreement with the exact results in 2D. We also use them to approximate the scaling dimensions in d = 3, 4, 5 for the theories in the M(3, 8) universality class.

Vladimir Narovlansky

Grigory Tarnopolsky

Ginzburg-Landau description and emergent supersymmetry of the (3, 8) minimal model

In this paper we consider the Klebanov-Tseytlin background and its non-Abelian <math><mi>T</mi></math>-dual geometry along a suitably chosen <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> subgroup of isometries. We analyze the Penrose limits along various null geodesics of both geometries. We observe that the Klebanov-Tseytlin geometry does not admit any pp-wave solutions. However, the <math><mi>T</mi></math>-dual background gives rise to a pp-wave solution upon taking the Penrose limit along some appropriate null geodesic. We comment on the possible gauge theory dual for our pp-wave background.