We study higher-derivative corrections to the graviton scattering amplitude in M-theory, via the stress tensor correlator of 3d $$ \mathcal{N} $$ = 8 U($\textit{N}$) × U($\textit{N}$) ABJM theory (dual to graviton scattering in M-theory on AdS × $\textit{S}$ /$\textit{ℤ}$ ). We use the conformal bootstrap combined with an integral constraint derived from supersymmetric localization in order to constrain semishort OPE coefficients appearing in the stress tensor correlator. We obtain islands that are significantly more precise than those in previous studies that did not use the integral constraint. Using these islands, we can estimate the powers and coefficients in a large central charge expansion. This allows us to accurately read off the N LO contribution, from the protected $\textit{D}$ $\textit{R}$ correction, and also estimate the N LO contribution, from the unprotected $\textit{D}$ $\textit{R}$ correction.

United Kingdom

Shai Chester

United States

Ross Dempsey

Silviu Pufu

Higher-derivative corrections in M-theory from precision numerical bootstrap

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.

Benjamin Søgaard

Igor Klebanov

More about the lattice Hamiltonian for Adjoint QCD

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

Level repulsion in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills via integrability, holography, and the bootstrap

We examine the phase structure of the two-flavor Schwinger model as a function of the <math><mi>θ</mi></math> angle and the two masses, <math><msub><mi>m</mi><mn>1</mn></msub></math> and <math><msub><mi>m</mi><mn>2</mn></msub></math>. In particular, we find interesting effects at <math><mi>θ</mi><mo>=</mo><mi>π</mi></math>: along the SU(2)-invariant line <math><msub><mi>m</mi><mn>1</mn></msub><mo>=</mo><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mi>m</mi></math>, in the regime where <math><mi>m</mi></math> is much smaller than the charge <math><mi>g</mi></math>, the theory undergoes logarithmic renormalization group flow of the Berezinskii-Kosterlitz-Thouless type. As a result, dimensional transmutation takes place, leading to a nonperturbatively small mass gap <math><mrow><mo>∼</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>A</mi><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup></mrow></math>. The SU(2)-invariant line lies within a region of the phase diagram where the charge conjugation symmetry is spontaneously broken and whose boundaries we determine numerically. Our numerical results are obtained using the Hamiltonian lattice gauge formulation that includes the mass shift <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>lat</mi></mrow></msub><mo>=</mo><mi>m</mi><mo>−</mo><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>a</mi><mo>/</mo><mn>4</mn></mrow></math> dictated by the discrete chiral symmetry.

Igor R. Klebanov

Silviu S. Pufu

Benjamin T. Søgaard

Bernardo Zan

Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature

We show that the classical double copy relationship for Kerr-Schild spacetimes can be dimensionally reduced to give a natural notion of the double copy for Kaluza-Klein theory with gravity coupled to a gauge field and a dilaton. Under dimensional reduction the Kerr-Schild (KS) ansatz becomes the stringy Kerr-Schild (sKS) ansatz introduced by Wu. This ansatz captures many Kaluza-Klein black hole solutions, including single-charge black holes arising in both gauged and ungauged supergravity theories. We identify the single copy gauge field and scalar field of an arbitrary sKS solution. We show that the boost-reduction procedure for generating charged black hole solutions can be generalized to any stationary KS seed, and used to combine the metric with the zeroth and single copies of that seed into a single sKS solution. Furthermore, we comment on multi-charge solutions that can be written in a multi-sKS form, proposing a double copy interpretation involving multiple single copy sectors.

Peter Weck

Compactifying the Kerr-Schild double copy

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

We combine supersymmetric localization results with numerical bootstrap techniques to compute upper bounds on the low-lying CFT data of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills theory as a function of the complexified gauge coupling τ. In particular, from the stress tensor multiplet four-point function, we extract the scaling dimension of the lowest-lying unprotected scalar operator and its OPE coefficient. While our method can be applied in principle to any gauge group G, we focus on G = SU(2) and SU(3) for simplicity. At weak coupling, the upper bounds we find are very close to the corresponding four-loop results. We also give preliminary evidence that these upper bounds become small islands under reasonable assumptions.

Israel

Bootstrapping   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills on the conformal manifold

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   <math> <mi>osp</mi> </math>  $$ \mathfrak{osp} $$ (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 QCD2 without explicit four-fermion operators.