disordered interaction. In large N diagrammatics, the dominance of “melonic” diagram
make the model solvable at strong coupling limit [2–5, 9–12]. Also, this model features
emergent reparametrization symmetry in the strict strong coupling limit after disorder
average [3, 4, 9–12]. This reparametrization symmetry is broken spontaneously and ex-
plicitly at strong but finite coupling limit, which leads to Schwarzian effective action for
Pseudo-Nambu-Goldstone modes [3, 4, 9–12]. Due to this mode, the SYK model is maxi-
mally chaotic, and the Lyapunov exponent of out-of-time-ordered correlator saturates chaos
bound [4, 11]. The same feature has been found in unitary quantum mechanical model
of fermi tensors without disorder [13–21]. In tensor models, the “melonic” diagrams also
dominate in large N, which leads to maximal chaos like the SYK model [13–15, 19]. This
maximal chaos [22–24] indicates that both quantum mechanical models could be dual to
gravity theory near horizon limit of extremal black hole, and the dual models have been
proposed to be dilaton gravity [25, 26], Liouville theories [27] and 3D gravity [28]. Be-
cause of these attractive features, the generalizations of the SYK and the tensor models
have been studied in various context (e.g., random matrix behavior [29–34], flavor [35, 36],
lattice generalization in higher dimensions [37–45], Schwarzian effective action [46–49] and
supersymmetry [50, 51], massive field instead of random coupling [52, 53], higher point
function [54] and 1/N corrections [55, 56].)
Most generalizations of the SYK model share the same feature: bi-local in time space.
This bi-local structure is naturally appears in SYK model because the SYK model is es-
sentially a large N vector model. One of the systematic analysis of such large N models
was introduced as collective field theory in [57], which captures invariant physical degrees
of freedom and provides the effective action thereof. The collective field theory has suc-
cessfully analyzed the large N models in the context of AdS/CFT [58–67]. Especially, a
bi-local collective field theory for three-dimensional U(N)/O(N) vector model gave rich
understanding of higher spin AdS
4
/CFT
3
correspondence [68–78]. However, in collective
field theory, the bi-local structure is not restricted to space-time. In general, one can con-
struct bi-local space of other abstract space in addition to spacetime. For example, in the
bi-local thermofield CFT [77], the bi-local field is given by Ψ(x
1
, a; x
2
, b) where x
1
and x
2
corresponds to spacetime as usual. a, b(= 1, 2) represents labels of two copies of system
in thermofield CFT, which corresponds to CFT lives on the left and right boundary of
eternal black hole. Furthermore, we have also constructed bi-local field Ψ(τ
1
, a; τ
2
, b) from
the time (τ
1
, τ
2
) and replica space (a, b = 1, 2, ··· , n) in the SYK model [10].
In this paper, we will develop the bi-local collective superfield theory
1
for one-
dimensional vector model by constructing bi-local superspace, especially will focus on su-
persymmetric SYK model introduced by [50]. This bi-local collective superfield theory
enable us to analyze the effective action of SUSY SYK model in large N systematically.
Furthermore, in the bi-local collective theory, the matrix structure in the bi-local space
naturally appears so that the bi-local collective theory can be seen as a matrix theory in
the bi-local space. Hence, one can analyze the SUSY SYK model in the supermatrix for-
mulation. This supermatrix formulation drastically simplifies analysis. We find that N = 1
1
Note that the collective theory for large N supermatrix model was already studied in [79–81].
– 2 –
