SYK model in the double-scaling limit [9], by evaluating the Schwarzian path integral ex-
actly [10], and by reducing the problem to Liouville quantum mechanics [11, 12]. The last
method is the most powerful one as it can also be used for calculation of matrix elements.
Two different reductions of the Schwarzian theory to a 2D CFT with a large central charge
were proposed in [13]. Our approach will be similar to that of [11, 12], but we consider a
more general problem, one that has two parameters but fewer infinities to worry about. As
a consequence, the wavefunction, including the overall factor, is defined unambiguously.
The Schwarzian action also arises from two-dimensional Jackiw-Teitelboim theory,
which involves the metric tensor g and a dilaton field Φ [14–16]. The Euclidean action is
I
JT
[g, Φ] = −
1
4π
ˆ
D
Φ(R + 2)
√
g d
2
x −
1
2π
ˆ
∂D
ΦK d`, (1.5)
where D is a disk, d` is the boundary length element, and K is the extrinsic curvature.
The boundary term is such that the variation of the action depends only on δg and δΦ
but not their derivatives; this is necessary to define boundary conditions. The condition
Φ|
∂D
= Φ
∗
(for some constant Φ
∗
) is imposed and the total boundary length L is fixed.
The bulk term in (1.5) gives the constraint R = −2 but vanishes on-shell. Thus one
can isometrically embed (or more generally, immerse) D in the Poincare disk so that the
action becomes −
Φ
∗
2π
´
∂D
K d`. It is convenient to also add a trivial term proportional to L:
I
g
= I
JT
+ γL = −γ
ˆ
∂D
(K − 1) d`, γ =
Φ
∗
2π
. (1.6)
Now, consider polar coordinates r, ϕ on the Poincare disk as functions on the curve ∂D,
which is parametrized by the proper length `. If L 1, it is reasonable to assume that
r(`) is close to 1 and that the curve is roughly parallel to the unit circle. Then
K − 1 ≈ Sch(e
iϕ(`)
, `). (1.7)
(For the reader’s convenience, this equation is derived in the beginning of the next section.)
We conclude that action (1.6) is approximately equal to the Schwarzian action.
Not making any approximations, one can still simplify action (1.6). By the Gauss-
Bonnet theorem,
´
∂D
K d` equals 2π plus the area enclosed by the curve. Then we ar-
rive at the following geometric action and global constraint for a closed curve X in the
Poincare disk:
I
g
[X] = −γ
area[X] −L + 2π
, length[X] = L. (1.8)
We assume that γ > 0 and take area[X] to be positive if X goes counterclockwise. As has
just been explained, this model is classically equivalent to Jackiw-Teitelboim theory. How-
ever, the functional integrals appear to be different. Indeed, each of the integrals should
include all curves (even self-intersecting ones) for which the corresponding action makes
sense. The area is defined for all closed curves, whereas in the dilaton problem, a curve
should bound an immersed disk. On the other hand, both models are quantum mechani-
cally equivalent to the Schwarzian model if γ and L are large. The rough argument is that
under this assumption, typical curves have K ≈ 1 and do not wiggle too much, so that one
can use equation (1.7). We will refer to the condition γ, L 1 as the Schwarzian limit.
– 3 –
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