Non-conformality of ${{\gamma }_{i}}$-deformed 𝒩 = 4 SYM theory

Abstract

Abstract. We show that the γ i -deformation, which was proposed as candidate gauge theory for a non-supersymmetric three-parameter deformation of the AdS/CFT correspondence, is not conformally invariant due to a running double-trace couplingnot even in the 't Hooft limit. Moreover, this non-conformality cannot be cured when we extend the theory by adding at tree-level arbitrary multi-trace couplings that obey certain minimal consistency requirements. Our findings suggest a possible connection between this breakdown of conformal invariance and a puzzling divergence recently encountered in the integrability-based descriptions of two-loop finite-size corrections for the single-trace operator of two identical chiral fields. We propose a test to clarify this.

Integrability of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring and its deformations

“…In particular, it is interesting to note that for γ-deformed SYM the operator Tr(Z J ) is insensitive to the non-conformality of the theory for J > 2, with anomalous dimensions precisely matching the energies of eqn. (4.67) above, while for J = 2 the result is scheme dependent [157].…”

“…Just as we discussed for orbifolds above, without supersymmetry various subtleties can enter the game, and it appears they do; conformal symmetry of the generically γ-deformed nonsupersymmetric proposed dual field theories [153,156] is broken at the quantum level [157], despite previous results [158]. From the point of view of AdS/CFT this can have various interpretations briefly discussed in [157].…”

“…From the point of view of AdS/CFT this can have various interpretations briefly discussed in [157]. One interesting possibility closely related to the situation for non-supersymmetric orbifolds is that the resulting non-supersymmetric string background is unstable due to the presence of tachyons (which are at least manifestly present in string theory on γ-deformed flat space [159]).…”

“…As in the orbifold case, here we may similarly hope to get some concrete insight with an exact description of the string spectrum on these backgrounds. Also, the effects of this non-conformality appears to be limited to particular subsets of states in the theory [157,157], outside of which we can still talk sensibly of scaling dimensions. Indeed, outside of these subsets, explicit results [160,161] obtained in the string theory picture we are describing here can again be successfully matched with perturbative gauge theory [157].…”

RG stability of integrable fishnet models

“…The aforementioned construction has, however, an important weakness: generic γ-deformations [11,12], and in particular the fishnet model for the N = 4 γ-deformation [13], are unstable under RG flow, and therefore need to be corrected by adding double trace operators to the Lagrangian. Such terms are not unprecedented in deformations of N = 4 or ABJM: similar double trace corrections arise in the discrete deformations implemented by orbifolding the compact space in the holographic dual [14][15][16][17].…”