Integrability of conformal fishnet theory

Abstract

Abstract:We study integrability of fishnet-type Feynman graphs arising in planar fourdimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed N = 4 SYM theory. We show that the transfer matrix "building" the fishnet graphs emerges from the R−matrix of non-compact conformal SU(2, 2) Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of N = 4 SYM. Using QSC and spin chain methods, we construct Baxter equation for Q−functions of the conformal spin chain needed for computation of the anomalous dimensions of operators of the type tr(φ J 1 ) where φ 1 is one of the two scalars of the theory. For J = 3 we derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions with J = 3 to practically unlimited precision at any coupling. These equations also describe an infinite tower of local conformal 1 Unité Mixte de Recherche 3681 du CNRS.Open Access, c The Authors. Article funded by SCOAP 3 .https://doi.org/10.1007/JHEP01 (2018)095 JHEP01 (2018)095 operators all carrying the same charge J = 3. The method should be applicable for any J and, in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable system of noncompact SU(2, 2) spins. This bears similarities with the classical strings arising in the strongly coupled limit of N = 4 SYM.

The holographic dual of strongly γ-deformed $$ \mathcal{N} $$ = 4 SYM theory: derivation, generalization, integrability and discrete reparametrization symmetry

“…This section generalizes the results of [8], where the non-perturbative spectrum was obtained for the operators J = 3, M = 0 10 , and [7], where the general J, M = 0 case was considered. An important result of [7,8], which we are going to use here is the quantization condition, which is needed in addition to the Baxter TQ-relations. We will propose the generalization of this quantization condition and also present the most general form of the TQ-relations valid for any values of the charges J 1 , J 2 such that J 1 > |J 2 |.…”

“…1 Furthermore, extending the integrability construction to all operators allows us, in particular, to solve the spectrum with magnons included, which is a new result by itself. The solution takes the form of a Baxter TQ-relations, subjected to the quantization conditions of [7] and generalizing results of [8]. It allows us to compute the conformal dimension of almost all single trace operators 2 .…”

“…This property implies the quantum integrability of our model, as we identified the Hamiltonian of the system within a large family of mutually commuting operators. 8 In the current formulation the ordering of the magnons is quite important. Different distributions of magnons within the chain will produce different Hamiltonian and Toperators.…”

“…is grateful to V.Kazakov and Z.Tsuboi for discussing the possibility of inclusion of inhomogeneities in relation to the magnons. 8 Of course one should verify that there are sufficiently many independent integrals of motion. This is usually rather hard to verify rigorously, but we strongly believe this is the case.…”

A review of the AdS/CFT Quantum Spectral Curve

“…This is a remarkable 'baby' version of SYM which is non-unitary but still conformal (up to some subtleties) and integrable, the integrability being visible at the level of Feynman graphs. Implementing the limit in the QSC led to a plethora of results for this model, allowing one to make use of the powerful methods developed for SYM [82] (see also the review [27]). A very similar limit in the QSC for the quark-antiquark potential was studied earlier in [76].…”