JHEP03(2019)044
Figure 2. Using conservation we can push the symplectic form from half of the Euclidean boundary
to an arbitrary bulk slice Σ anchored to the boundary at t
E
= 0.
This leads to the main observation in [21], namely that for holographic field theories,
the standard dictionary relates (2.11) with the bulk symplectic flux along half of the Eu-
clidean boundary. The bulk symplectic flux is defined via Wald’s procedure [31, 32], i.e.
by first defining the presymplectic form θ from the boundary term in the variation of the
bulk Lagrangian
δL
bulk
d
d+1
x = −E
φ
δφd
d+1
x + dθ(φ, δφ), (2.13)
where E
φ
is are the bulk equations of motion, and then defining the d-form
ω
bulk
(φ, δ
1
φ, δ
2
φ) = δ
1
θ(φ, δ
2
φ) − δ
2
θ(φ, δ
1
φ), (2.14)
whose integral on a codimension one surface gives the symplectic flux. Since this symplectic
flux is conserved on shell, we can push the surface into the Euclidean bulk. See figure 2 for
an illustration of this. Note that for the bulk symplectic form to be equal to the boundary
one, we only need for the union of the half-boundary and the bulk Cauchy slice to be a
closed manifold, but for the rest of the paper we will focus on the case where the topology
of the bulk Cauchy slice and the half boundary are the same.
3
During this procedure, one needs to solve for the bulk geometry that has boundary
conditions λ for the bulk fields. On slices that continue nicely to Lorentzian signature, this
gives a prescription for associating bulk initial data to the state |λi, as originally proposed
in [35] and further studied in [36]. This Lorentzian initial data is real precisely if λ
+
is the
conjugate of λ
−
. This continuation gets rid of the i in (2.11) and we obtain an equality
between the Berry curvature on the space of half sided sources and the bulk symplectic
form for the corresponding Lorentzian initial data on a Cauchy slice Σ
Ω(δλ
1
, δλ
2
) =
Z
Σ
ω
Lor
(φ, δφ
1
, δφ
2
).
(2.15)
Similarly, the complex structure (2.10) provides a quantum polarization of the bulk phase
space, which corresponds to separation of on-shell field variations into positive and neg-
ative energy modes, whenever a time like Killing vector is available. The K¨ahler metric
3
Some examples of when this is not the case are the TFD state below the Hawking-Page transition or
the AdS soliton [33, 34].
– 7 –
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