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We construct field theories in $2+1$ dimensions with multiple conformal symmetries acting on only one of the spatial directions. These can be considered a conformal extension to subsystem scale invariances, borrowing the language often used for fractons.
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The loss of criticality in the form of weak first-order transitions or the end of the conformal window in gauge theories can be described as the merging of two fixed points that move to complex values of the couplings. When the complex fixed points are close to the real axis, the system typically exhibits walking behavior with Miransky (or Berezinsky-Kosterlitz-Thouless) scaling. We present a novel realization of these phenomena at strong coupling by means of the gauge/gravity duality, and give evidence for the conjectured existence of complex conformal field theories at the fixed points.
It is widely expected that at sufficiently high temperatures order is always lost, e.g. magnets loose their ferromagnetic properties. We pose the question of whether this is always the case in the context of quantum field theory in $d$ space dimensions. More concretely, one can ask whether there exist critical points (CFTs) which break some global symmetry at arbitrary finite temperature. The most familiar CFTs do not exhibit symmetry breaking at finite temperature, and moreover, in the context of the AdS/CFT correspondence, critical points at finite temperature are described by an uncharged black brane which obeys a no-hair theorem. Yet, we show that there exist CFTs which have some of their internal symmetries broken at arbitrary finite temperature. Our main example is a vector model which we study both in the epsilon expansion and arbitrary rank as well as the large rank limit (and arbitrary dimension). The large rank limit of the vector model displays a conformal manifold, a moduli space of vacua, and a deformed moduli space of vacua at finite temperature. The appropriate Nambu-Goldstone bosons including the dilaton-like particle are identified. Using these tools we establish symmetry breaking at finite temperature for finite small $epsilon$. We also prove that a large class of other fixed points, which describe some of the most common quantum magnets, indeed behave as expected and do not break any global symmetry at finite temperature. We discuss some of the consequences of finite temperature symmetry breaking for the spectrum of local operators. Finally, we propose a class of fixed points which appear to be possible candidates for finite temperature symmetry breaking in $d=2$.
Recently, Bern et al observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry. It corresponds to a projection of the latter along a certain lightlike direction. Previous studies were performed at the level of the loop integrand, and a Ward identity for the integral was formulated. We investigate the implications of the symmetry at the level of the integrated quantities. In particular, we focus on the phenomenologically important case of five-particle scattering. The symmetry simplifies the four-variable problem to a three-variable one. In the context of the recently proposed space of pentagon functions, the symmetry is much stronger. We find that it drastically reduces the allowed function space, leading to a well-known space of three-variable functions. Furthermore, we show how to use the symmetry in the presence of infrared divergences, where one obtains an anomalous Ward identity. We verify that the Ward identity is satisfied by the leading and subleading poles of several nontrivial five-particle integrals. Finally, we present examples of integrals that possess both ordinary and dual conformal symmetry.
We present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, $P$, and conformal Casimir, $mathcal{C}$. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with $mathcal{C} leq mathcal{C}_{max}$, one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free $O(N)$ CFT deformed by a mass term and a non-perturbative quartic interaction at large-$N$. In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.
Using the recently developed fractional Virasoro algebra cite{la_nave_fractional_2019}, we construct a class of nonlocal CFTs with OPEs of the form $T_k(z)Phi(w) sim frac{ h_gamma Phi}{(z-w)^{1+gamma}}+frac{partial_w^gamma Phi}{z-w},$ and $T_k(z)T_k(w) sim frac{ c_kZ_gamma}{(z-w)^{3gamma+1}}+frac{(1+gamma ) T_k(w)}{(z-w)^{1+gamma}}+frac{partial^gamma_w T_k}{z-w}$ which naturally results in a central charge, $c_k$, that is state-dependent, with $k$ indexing a particular grading. Our work indicates that only those theories which are nonlocal have state-dependent central charges, regardless of the pseudo-differential operator content of their action. All others, including certain fractional Laplacian theories, can be mapped onto an equivalent local one using a suitable covering/field redefinition. In addition, we discuss various perturbative implications of deformations of fractional CFTs that realize a fractional Virasoro algebra through the lense of a degree/state-dependent refinement of the 2 dimensional C-theorem.
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