viewed as a partition function Z
L,T
for a 2d vertex model [13], with the bulk spacetime
points acting as classical “spins”, the propagators as nearest neighbour couplings and with
the graph’s external lines setting up the boundary conditions. Different observables of
the planar fishnet theory correspond to different boundary conditions and all the graphs
obeying the same boundary conditions are summed over.
An important observation concerning the large order behaviour of the fishnet diagrams
was made by Zamolodchikov [13], see also [28] for a recent discussion, who computed, using
integrable vertex model techniques, the free energy per site in the thermodynamical limit
log Z
L,T
∼ −LT log g
2
cr
, (1.2)
for graphs subject to periodic boundary conditions, and found that
g
cr
=
Γ(
3
4
)
√
πΓ(
5
4
)
' 0.76 . . . . (1.3)
This constant determines a critical coupling for thermodynamically large observables in
the fishnet theory and one might expect, in analogy with matrix models [29–32], that a
“dual” continuum description is taking over at that point.
2
In this paper we examine the thermodynamical limit of fishnet graphs using integrable
methods borrowed from the N = 4 SYM theory and argue that the continuum description
is given by the 2d (bosonic) AdS
5
sigma model.
Our discussion will center around the scaling dimension ∆ of the BMN operator tr φ
L
1
which maps to the ground-state energy of a ferromagnetic non-compact spin chain [24].
Integrability will allow us to study the thermodynamical limit of this energy for generic
coupling g by means of a linear integral equation. It will confirm the existence of a non-
trivial thermodynamical scaling ∆ ∼ Lf(g), for sufficiently “strong” coupling, and the
emergence of a critical behaviour ∼
√
g
cr
− g close to the critical point, in line with the
results of [15, 24] for L = 2, 3.
3
A sketch of the thermodynamical behaviour of the scaling
dimension is shown in figure 2.
Dualizing our equation, by means of a particle-hole transformation, will reveal the
nature of the critical point and suggest the interpretation of the BMN operator as describing
the “tachyon” ground-state of the AdS sigma model,
tr φ
L
1
↔ V
∆
∼ e
−i∆t
, (1.4)
labelled by the global time energy ∆ of the BMN operator and implicitly by the size L of
the worldsheet. Though we will actually never cross the line where the AdS mass squared
turns negative, we will stick to the name of tachyon for the dual object.
4
The correspondence (1.4) is best summarized by the formula
log g
2L
= log g
2L
cr
+ E
2d
(∆, L) , (1.5)
2
Note that the critical coupling does not refer to a point at which the 4d theory is becoming critical;
the fishnet theory is conformal for any g
2
. It is a point at which the planar diagrams become dense.
3
The location of the branch point g
cr
(L) is function of the length L; in particular [15], g
cr
(L = 2) = 0.
4
The tachyonic domain maps to g > g
cr
(L), where the scaling dimension has an imaginary part, ∆ =
2 + iν; see [15, 19, 24] for discussions.
– 3 –
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