Generalizing the holographic fishchain

Abstract

We attempt to generalize the integrable Gromov–Sever models, the so-called fishchain models, which are dual to biscalar fishnets. We show that they can be derived in any dimension, at least for some integer deformation parameter of the fishnet lattice. In particular, we focus on the study of fishchain models in AdS$_7$ that are dual to the six-dimensional fishnet models.

Appendix: Spectrum of nonisotropic lattice for a biscalar fishnet theory

The star–triangle identity [25] is given by

$$\int\,d^dx_0x_{01}^ax_{02}^bx_{03}^c =\pi^{d/2}\frac{\Gamma(a/2+d/2)\Gamma(b/2+d/2)\Gamma(c/2+d/2)} {\Gamma(-a/2)\Gamma(-b/2)\Gamma(-c/2)} x^{-a-d}_{12}x^{-b-d}_{23}x^{-c-d}_{13}, $$

(A.1)

and as $x_3\to\infty$, we simplify it to

$$\int\,d^dx_0x_{01}^ax_{02}^b =\pi^{d/2}\frac{\Gamma(a/2+d/2)\Gamma(b/2+d/2)\Gamma(-a/2-b/2-d/2)} {\Gamma(-a/2)\Gamma(-b/2)\Gamma(a/2+b/2+d)}x_{12}^{a+b+d}. $$

(A.2)

Then we can use the identity in this form to obtain the spectrum of the wave function for $J=2$. Conformal symmetry fixes the form of the CFT wave function as

$$\Psi_{S,\Delta,x_0}(x_1,x_2) =\frac{x_{12}^{\Delta-S-d+2\omega}}{x_{01}^{(\Delta-S)/2}x_{02}^{(\Delta-S)/2}} \biggl(2\frac{(nx_{02})}{x_{02}^2}-2\frac{(nx_{01})}{x_{01}^2}\biggr)^S. $$

(A.3)

This wave function plays the role of an eigenvector of the graph-building operator. Therefore, the eigenvalues of the operator can be defined as

$$\mathcal H \mathbin{\stackrel{\scriptscriptstyle{\circ}}{{}_{\vphantom{.}}}} \Psi_{S,\Delta,x_0}=h_{\Delta,S}^{-1}\Psi_{S,\Delta,x_0}, $$

(A.4)

and the graph-building operator is given in the generalized form

$$\mathcal H=\int d^dx_1\,d^dx_2\,\frac{1}{x_{12}^{4\omega}x_{13}^{d-2\omega}x_{24}^{d-2\omega}}. $$

(A.5)

Using inversion with respect to $x_0$, the star–triangle relation, and integration by parts, as it was done in [8], [11], a spectrum for the nonisotropic lattice integral can be obtained as

$$h_{\Delta,S}=\frac{\Gamma(\omega)^2}{\Gamma(d/2-\omega)^2} \frac{\Gamma(d/4+S/2-\Delta/2)\Gamma(d/4-2\omega+S/2+\Delta/2)} {\Gamma(d/4+S/2+\Delta/2)\Gamma(d/4+2\omega+S/2-\Delta/2)}. $$

(A.6)

In the $\omega=d/4$ limit, the result in [11] can be reproduced.