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A bstract If the coupling constants in QFT are promoted to functions of space-time, the dependence of the path integral on these couplings is highly constrained by conformal symmetry. We begin the present note by showing that this idea leads to a new proof of Zamolodchikov’s theorem. We then review how this simple observation also leads to a derivation of the a -theorem. We exemplify the general procedure in some interacting theories in four space-time dimensions. We concentrate on Banks-Zaks and weakly relevant flows, which can be controlled by ordinary and conformal perturbation theories, respectively. We compute explicitly the dependence of the path integral on the coupling constants and extract the change in the a -anomaly (this agrees with more conventional computations of the same quantity). We also discuss some general properties of the sum rule found in [ 1 ] and study it in several examples.

A bstract We discuss some general aspects of renormalization group flows in four dimensions. Every such flow can be reinterpreted in terms of a spontaneously broken conformal symmetry. We analyze in detail the consequences of trace anomalies for the effective action of the Nambu-Goldstone boson of broken conformal symmetry. While the c -anomaly is algebraically trivial, the a -anomaly is “non-Abelian”, and leads to a positive-definite universal contribution to the S -matrix element of 2 → 2 dilaton scattering. Unitarity of the S -matrix results in a monotonically decreasing function that interpolates between the Euler anomalies in the ultraviolet and the infrared, thereby establishing the a -theorem.

A bstract We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group SU( N ). The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits ∼ 2 2 N non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all N these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, m ≪ g YM , the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the k -string tension for N ≤ 5. Our results suggest that the k -string tension, T k , is T k ∼ | m | sin( πk/N ) for all N . We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for SU( N ≤ 5) gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of N . The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

A bstract We consider the dynamics of 2+1 dimensional SU( N ) gauge theory with Chern-Simons level k and N f fundamental fermions. By requiring consistency with previously suggested dualities for N f ≤ 2 k as well as the dynamics at k = 0 we propose that the theory with N f > 2 k breaks the U( N f ) global symmetry spontaneously to U( N f / 2 + k ) × U( N f / 2 − k ). In contrast to the 3+1 dimensional case, the symmetry breaking takes place in a range of quark masses and not just at one point. The target space never becomes parametrically large and the Nambu-Goldstone bosons are therefore not visible semi-classically. Such symmetry breaking is argued to take place in some intermediate range of the number of flavors, 2 k < N f < N ∗ ( N, k ), with the upper limit N ∗ obeying various constraints. The Lagrangian for the Nambu-Goldstone bosons has to be supplemented by nontrivial Wess-Zumino terms that are necessary for the consistency of the picture, even at k = 0. Furthermore, we suggest two scalar dual theories in this range of N f . A similar picture is developed for SO( N ) and Sp( N ) gauge theories. It sheds new light on monopole condensation and confinement in the SO( N ) & Spin( N ) theories.

A bstract In recent years it has been shown that strongly coupled systems become analytically tractable in the regime of large quantum numbers, such as large spin or large charge. The effective theories that emerge in these two limits are Regge theory and superfluid theory, respectively. Here we make a proposal for a new phase, the “giant vortex,” describing an intermediate regime with large spin and charge. The new phase connects superfluid theory with the large-spin expansion. The giant vortex admits a semi-classical effective theory description with peculiar chiral excitations (moving at the speed of light) and a Fock space of states that is reminiscent of the multi-twist operators in Regge theory, including the leading and daughter Regge trajectories. A similar giant vortex phase appears for Bose-Einstein condensates in a rotating trap, and our results should be applicable in that context as well. We show that the transition from the giant vortex to the Regge regime is accompanied by the scaling dimension turning from being larger than to being smaller than the mean field theory value, i.e. gravity switches from being the weakest force at small AdS distance to being the strongest force at large AdS distance.

A bstract SU( N ) gauge theory is time reversal invariant at θ = 0 and θ = π . We show that at θ = π there is a discrete ’t Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π , several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O (2)-symmetric fixed points. It may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies.

A bstract We study the correlation functions of Coulomb branch operators of four-dimensional N = 2 Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. “Extremal” correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a “dual” description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a ’t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the ’t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong ’t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.

A bstract We study SU( N ) Quantum Chromodynamics (QCD) in 3+1 dimensions with N f degenerate fundamental quarks with mass m and a θ -parameter. For generic m and θ the theory has a single gapped vacuum. However, as θ is varied through θ = π for large m there is a first order transition. For N f = 1 the first order transition line ends at a point with a massless η ′ particle (for all N ) and for N f > 1 the first order transition ends at m = 0, where, depending on the value of N f , the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the 4 d bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different 3 d phases. These turn out to be the phases of the recently studied 3 d Chern-Simons matter theories, thus relating the dynamics of QCD 4 and QCD 3 , and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless 4 d theory has an SU( N f ) sigma model, the domain wall theory at low (nonzero) mass supports a 3 d massless ℂ ℙ N f − 1 nonlinear σ -model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.

A bstract We consider the critical O ( N ) model in the presence of an external magnetic field localized in space. This setup can potentially be realized in quantum simulators and in some liquid mixtures. The external field can be understood as a relevant perturbation of the trivial line defect, and thus triggers a defect Renormalization Group (RG) flow. In agreement with the g -theorem, the external localized field leads at long distances to a stable nontrivial defect CFT (DCFT) with g < 1. We obtain several predictions for the corresponding DCFT data in the epsilon expansion and in the large N limit. The analysis of the large N limit involves a new saddle point and, remarkably, the study of fluctuations around it is enabled by recent progress in AdS loop diagrams. Our results are compatible with results from Monte Carlo simulations and we make several predictions that can be tested in the future.

A bstract We consider supersymmetric theories on a space with compact space-like slices. One can count BPS representations weighted by (−1) F , or, equivalently, study supersymmetric partition functions by compactifying the time direction. A special case of this general construction corresponds to the counting of short representations of the superconformal group. We show that in four-dimensional N = 1 theories the “high temperature” asymptotics of such counting problems is fixed by the anomalies of the theory. Notably, the combination a − c of the trace anomalies plays a crucial role. We also propose similar formulae for six-dimensional (1 , 0) theories.