Free-kick condition for entanglement entropy
in higher curvature gravity
Seyed Morteza Hosseini
University of Milano-Bicocca
August 13, 2015
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
1 / 26
Outline
1. Entanglement entropy in CFT’s
2. Holographic entanglement entropy for higher curvature gravity
3. Geodesics in the OTT geometry
4. Entanglement entropy of the BTZ black hole in NMG
5. Entanglement entropy of the OTT black hole
Based on
S. M. Hosseini and A. Veliz-Osorio, “Free-kick condition for entanglement entropy
in higher curvature gravity,” arXiv:1505.00826 [hep-th].
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
2 / 26
Entanglement entropy in CFT’s
Entanglement entropy −→ Fundamental quantum property
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
3 / 26
Entanglement entropy in CFT’s
Entanglement entropy −→ Fundamental quantum property
Imagine that the system of interest is in a pure state |Ψi. We can find
the reduced density matrix obtained by tracing out the degrees of
freedom in Ā
ρA = TrHĀ |ΨihΨ| .
Von Neumann entropy of ρA
SEE (A) = − Tr(ρA log ρA ) .
We refer to this quantity as the entanglement entropy of A.
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
3 / 26
Entanglement entropy in CFT’s
In a (1+1)-dimensional CFT the EE of an interval of length ` can be
computed analytically!
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
4 / 26
Entanglement entropy in CFT’s
In a (1+1)-dimensional CFT the EE of an interval of length ` can be
computed analytically!
Zero temperature:
(n)
SA
1
=
log Tr [ρnA ] ,
1−n
Tr [ρnA ]
− 6c (n−1/n)
`
∝
.
The EE can be extracted from the above expression by taking the
n → 1 limit,
c
`
SEE (A) = log
,
3
where c is the central charge of the CFT and an ultraviolet cut-off.
(Holzhey et al. ’94, Calabrese, Cardy ’04)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
4 / 26
Entanglement entropy in CFT’s
Finite temperature T = β −1 :
SEE (A) =
( πc `
,
c
β
π`
' c 3 β`
log
sinh
log
,
3
π
β
3
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
` β classical extensive
` β T = 0, non-extensive
University of Milano-Bicocca
5 / 26
Entanglement entropy in CFT’s
Finite temperature T = β −1 :
SEE (A) =
( πc `
,
c
β
π`
' c 3 β`
log
sinh
log
,
3
π
β
3
` β classical extensive
` β T = 0, non-extensive
Finite size:
SEE (A) =
c
ξ
π`
log
sinh
.
3
π
ξ
Symmetric ` → ξ − `.
Maximal for ` = ξ/2.
(Calabrese, Cardy ’04)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
5 / 26
Holographic EE for higher curvature gravity
Holographic EE
(Ryu, Takayanagi ’06)
1
min Area (Σ) ,
4G ∂A=∂Σ
Σ is a co-dimension two hypersurface which we demand to be
anchored at ∂A.
SEE (A) =
A
AAdS
Σ
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
6 / 26
Holographic EE for higher curvature gravity
Holographic EE
(Ryu, Takayanagi ’06)
1
min Area (Σ) ,
4G ∂A=∂Σ
Σ is a co-dimension two hypersurface which we demand to be
anchored at ∂A.
SEE (A) =
A
AAdS
Σ
In the presence of higher curvature corrections,
entropy computations must be modified!
(Wald ’93)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
6 / 26
Holographic EE for higher curvature gravity
Let’s consider a general four-derivative gravity action
S=
1
16πG
Z
√ dd+1 x −g R − 2Λ + c1 R2 + c2 Rµν Rµν + c3 Rµνρσ Rµνρσ .
(1)
The correct value of the entanglement entropy is now determined by
extremizing the functional
SEE =
1
4G
Z
√
1
dd−1 y h 1 + 2c1 R + c2 R|| − K2 + 2c3 R|| || − Tr (K)2 .
2
Σ
(Camps ’13, Fursaev et al. ’13, Dong ’13)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
7 / 26
Holographic EE for higher curvature gravity
1. Find a basis for the vector space normal to the surface Σ
gµν nµ(α) nν(β) = ηαβ .
2. Induced metric
hµν = gµν − η αβ n(α)
µ
n(β)
ν
.
3. Relevant contributions from the ambient Riemann curvature
µ
ν
ρ
σ
n(δ)
n(β)
n(γ) Rµνρσ ,
R|| || = η αδ η βγ n(α)
µ
ν
R|| = η αβ n(α)
n(β) Rµν .
4. Extrinsic curvature
K(α) µν = hµλ hνρ ∇ρ n(α) λ ,
K2 ≡ η αβ (K(α) )µµ (K(β) )νν ,
2
Tr (K) ≡ η αβ (K(α) )µν (K(β) )νµ .
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
8 / 26
Holographic EE for higher curvature gravity
In the following we shall be concerned with solutions to the NMG,
Z
√
1
1
d3 x −g R − 2Λ + 2 K ,
S=
16πG
m
where
3
K = Rµν Rµν − R2 .
8
(Bergshoeff et al. ’09)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
9 / 26
Holographic EE for higher curvature gravity
In the following we shall be concerned with solutions to the NMG,
Z
√
1
1
d3 x −g R − 2Λ + 2 K ,
S=
16πG
m
where
3
K = Rµν Rµν − R2 .
8
(Bergshoeff et al. ’09)
In the notation of Eq. (1) this theory corresponds to the coefficients
1
3
,
c2 = 2 ,
c3 = 0 .
2
8m
m
Therefore, the entanglement entropy functional reduces to
c1 = −
SEE =
1
4G
√
1
1
3
dτ h 1 + 2 R|| − K2 − R
.
m
2
4
Σ
Z
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
(2)
University of Milano-Bicocca
9 / 26
Geodesics in the OTT geometry
For m2 L2 = 1/2, L being the AdS radius, NMG admits the following
black hole solution, (Oliva et al. ’09)
ds2 = −f (r)dt2 + f (r)−1 dr2 + r2 dφ2 ,
with
r2
.
L2
In the b → 0 limit the solution is reduced to the BTZ black hole. In
the case of b > 0 there is a single event horizon located at
p
1
r+ =
−bL2 + b2 L4 + 4µL2 .
2
f (r) = −µ + br +
We can associate a Hawking temperature and entropy to it
T =
1 p 2 2
b L + 4µ ,
4πL
S=
πL p 2 2
b L + 4µ .
2G
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
10 / 26
Geodesics in the OTT geometry
Geodesics are found by extremizing the functional
Z
√
I(A) =
dd−1 y h ,
ΣA
where A is a spacelike interval at asymptotic infinity and ΣA has its
end-points fixed at ∂A.
We could either parametrize these curves by the angular or by the
radial AdS coordinates,
I
boundary parametrization
I
bulk parametrization.
Choosing the bulk parametrization, we find that the induced metric is
given by
h = r2 φ0 (r)2 − f (r)−1 ,
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
11 / 26
Geodesics in the OTT geometry
Setting L = 1, we find
1
,
φ0 (r) = p
r (r − r+ )(r + r+ + b)(ar2 − 1)
Therefore, the profile of a geodesic reaching down to a radius r∗ ≥ r+
is given by
Z r
φ(r) =
dr̃ φ0 (r̃) .
r∗
The integration constant a can be determined by imposing the
boundary condition
dr = 0,
dφ r∗
which implies that a = r∗−2 .
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
12 / 26
Geodesics in the OTT geometry
Performing this integration explicitly, we find
φ(r) =
2 r∗ F (z|η) + (r+ − r∗ ) Π(n; z|η)
p
,
r+
(r+ + r∗ )(b + r+ + r∗ )
(3)
where F (z|η) and Π(n; z|η) are incomplete elliptic integrals of the first
and the third kind, respectively and
s
!
(r − r∗ )(r+ + r∗ )
2r+
n=
,
z = arcsin
,
r+ + r∗
2r∗ (r − r+ )
η=
2r∗ (b + 2r+ )
.
(r+ + r∗ )(b + r+ + r∗ )
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
13 / 26
Geodesics in the OTT geometry

ϕ(r* )
OTT geodesics
2 r∗ F (z|η) + (r+ − r∗ ) Π(n; z|η)
p
,
r+
(r+ + r∗ )(b + r+ + r∗ )
.
φ̃geodesic (r∗ ) = φ(r, r∗ )
φ(r, r∗ ) =
r+
r→∞
r*
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
14 / 26
Geodesics in the OTT geometry

ϕ(r* )
BTZ geodesics (b = 0)
s
2 
r∗2 − r+rr∗
1
.
φ(r, r∗ ) =
arccosh 
2
r+
r∗2 − r+
φ̃(r∗ ) = φ(r, r∗ )
.
r+
r→∞
r*
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
15 / 26
Geodesics in the OTT geometry

ϕ(r* )
BTZ geodesics (b = 0)
s
2 
r∗2 − r+rr∗
1
.
φ(r, r∗ ) =
arccosh 
2
r+
r∗2 − r+
φ̃(r∗ ) = φ(r, r∗ )
.
r+
r→∞
r*
Depth ←→ entangling size
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
15 / 26
Geodesics in the OTT geometry

ϕ(r* )
BTZ geodesics (b = 0)
s
2 
r∗2 − r+rr∗
1
.
φ(r, r∗ ) =
arccosh 
2
r+
r∗2 − r+
φ̃(r∗ ) = φ(r, r∗ )
.
r+
r→∞
r*
Depth ←→ entangling size
Boundary parametrization
− 12
cosh2 (r+ φ)
,
r(φ) = r+ 1 −
cosh2 (r+ φ0 )


1
r
∗
.
φ0 =
arccosh  q
r+
2
2
r −r
∗
+
Observe that φ̃ = φ0 .
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
15 / 26
Entanglement entropy of the BTZ black hole in NMG
SEE =
1
4G
√
1
3
1
dτ h 1 + 2 R|| − K2 − R
.
m
2
4
Σ
Z
The induced metric h and R|| are given by
h=
r2
L2
1
+ r2 φ0 (r)2 ,
−µ
R|| = −
4
,
L2
while the contraction of the extrinsic curvature reads
i2
2
2L2 r r2 − µL2 φ00 (r) + 2r2 r2 − µL2 φ0 (r)3 + 6L2 r2 − 4µL4 φ0 (r)
K2 (r) =
.
3
4L2 r2 r2 − µL2 φ0 (r)2 + L2
h
Very complicated equations of motion!
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
16 / 26
Entanglement entropy of the BTZ black hole in NMG
The higher-derivative terms contribution is topological and can be
written as a total derivative. (Erdmenger et al. ’14)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
17 / 26
Entanglement entropy of the BTZ black hole in NMG
The higher-derivative terms contribution is topological and can be
written as a total derivative. (Erdmenger et al. ’14)
Thus, the solutions of the geodesic equation solve also these more
convoluted equations of motion!
Z
rdr
1 r
q
SEE =
,
G r∗
2 (r 2 − r 2 )
r2 − r+
∗
q
r
p
1
2 +
r2 − r+
r2 − r∗2 ,
= log
G
r∗
where we introduced an ultraviolet cut-off r 1.
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
17 / 26
Entanglement entropy of the BTZ black hole in NMG
Boundary parametrization
r∗2 =
2
r+
1 − cosh
=⇒
−2
r+ φ̃
SEE
,
β=
2π
,
r+
=
1
,
r
` = 2φ̃ ,
1
β
π`
= log
sinh
,
G
π
β
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
18 / 26
Entanglement entropy of the BTZ black hole in NMG
Boundary parametrization
r∗2 =
2
r+
1 − cosh
=⇒
−2
r+ φ̃
SEE
,
β=
2π
,
r+
=
1
β
π`
= log
sinh
,
G
π
β
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
1
,
r
` = 2φ̃ ,
!
c=
3
.
G
University of Milano-Bicocca
18 / 26
Entanglement entropy of the BTZ black hole in NMG
Boundary parametrization
r∗2 =
2
r+
1 − cosh
=⇒
−2
r+ φ̃
SEE
,
β=
2π
,
r+
=
1
β
π`
= log
sinh
,
G
π
β
Central charge for NMG:
c=
3
2G
1+
1
2m2
1
,
r
` = 2φ̃ ,
!
c=
3
.
G
.
(Giribet et al. ’09, Bergshoeff et al. ’09)
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
18 / 26
Entanglement entropy of the BTZ black hole in NMG
Boundary parametrization
r∗2 =
2
r+
1 − cosh
=⇒
−2
r+ φ̃
SEE
,
β=
2π
,
r+
=
1
,
r
1
β
π`
= log
sinh
,
G
π
β
Central charge for NMG:
c=
3
2G
1+
1
2m2
` = 2φ̃ ,
!
c=
3
.
G
.
(Giribet et al. ’09, Bergshoeff et al. ’09)
Holographic EE for BTZ black hole in NMG
SEE
1
=
2G
1
1+
2m2
β
log
sinh
π
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
π`
β
.
University of Milano-Bicocca
18 / 26
Entanglement entropy of the OTT black hole
The induced metric h and R|| are given by
h=
1
r b+
R|| = −
r
L2
+ r2 φ0 (r)2 ,
−µ
bL2
b
4
− −
,
r
L2
2r r2 φ0 (r)2 (bL2 r − µL2 + r2 ) + L2
while the extrinsic curvature contraction reads
K2 (r) =
1
4L2
2 0
r2 φ0 (r)2
3
+ 2r φ (r)
Equations of motion:
(bL2 r
h
−
2
µL2
2
bL r − µL + r
d
dr
δL
δφ0 0
+
r2 )
2 2
−
+
3
L2
2L2 rφ00 (r) bL2 r − µL2 + r2
+ φ0 (r) 5bL4 r − 4µL4 + 6L2 r2
δL
δφ0
i2
.
= k,
where k is an integration constant.
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
19 / 26
Entanglement entropy of the OTT black hole
Small b expansion
φ(r) = φ(0) (r) + b φ(1) (r) + O b2 ,
k = k (0) + b k (1) + O b2 .
Obviously, the zeroth order contribution reduces to the NMG-BTZ
black hole and
k (0) = 2r∗ .
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
20 / 26
Entanglement entropy of the OTT black hole
Small b expansion
φ(r) = φ(0) (r) + b φ(1) (r) + O b2 ,
k = k (0) + b k (1) + O b2 .
Obviously, the zeroth order contribution reduces to the NMG-BTZ
black hole and
k (0) = 2r∗ .
Setting L = 1, we find
φ(1) (r) = − p
1
r2 − r∗2
s
"
−
2 (r + r ) − r
2
r − r+ k(1) r+
∗ r∗ + rr+
+
+ c1
r + r+
2r2 r2 − r2
+
+ c2 log
q
2 +
r2 − r+
q
+
r2 − r∗2
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
∗
#
−
1 (0)
φ (r) + c3 .
2r+
(4)
University of Milano-Bicocca
20 / 26
Entanglement entropy of the OTT black hole
The pressing question is: How are we going to fix these constants?
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
21 / 26
Entanglement entropy of the OTT black hole
The pressing question is: How are we going to fix these constants?
Three of them are rather easy to fix by setting
r(φ)
= r∗ ,
r0 (φ)
= 0,
r000 (φ)
= 0.
φ=0
φ=0
φ=0
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
21 / 26
Entanglement entropy of the OTT black hole
The pressing question is: How are we going to fix these constants?
Three of them are rather easy to fix by setting
r(φ)
= r∗ ,
r0 (φ)
= 0,
r000 (φ)
= 0.
φ=0
φ=0
φ=0
What about the initial acceleration or
the concavity of the entangling curve,
r00 (φ)
≡ λ ??
φ=0
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
21 / 26
Entanglement entropy of the OTT black hole
The pressing question is: How are we going to fix these constants?
Three of them are rather easy to fix by setting
r(φ)
= r∗ ,
r0 (φ)
= 0,
r000 (φ)
= 0.
φ=0
φ=0
φ=0
What about the initial acceleration or
the concavity of the entangling curve,
r00 (φ)
≡ λ ??
φ=0
Free-kick condition
φ̃(r∗ ) = φ̃geodesic (r∗ ) .
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
21 / 26
Entanglement entropy of the OTT black hole
Back in the boundary parametrization, the boundary conditions read
1 3φ00 (r)2 − φ000 (r)φ0 (r) φ(r)
= 0,
= 0,
= 0.
φ0 (r) r=r∗
φ0 (r)5
r=r∗
r=r∗
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
22 / 26
Entanglement entropy of the OTT black hole
These are are satisfied by setting
2
r∗2 − k (1) r+
q
,
c2 = c3 = 0 ,
2
2
2 r+
r∗2 − r+
√
2
3 1 3 5
2r∗ (r+ − r∗ )
3r∗
(1)
F
−
;
,
;
;
sin(w),
p
sin(w)
k =−
1
3/2
r+
2 2 2 2
3r+ (r+ + r∗ )
h
i
2
2r∗ (r+ − r∗ ) r∗ F (w|p) + (r+ + r∗ ) E(w|p)
,
+
2 (r + r )
r+
+
∗
c1 =
where E is the incomplete elliptic integral of the second kind, F1 is
the Appell hypergeometric function, and
r
4r+ r∗
r+ + r∗
w = arcsin
,
p=
2 .
2r∗
(r+ + r∗ )
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
23 / 26
Entanglement entropy of the OTT black hole
Additionally, we find that to linear order in b the acceleration is given
by
r∗ (1)
φ00 (r) 2
r∗ k − 2r+ + r∗ b .
r00 (φ)
= − 0 3 = r∗ r∗2 − r+
+
φ (r) r=r∗
2
φ=0
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
24 / 26
Entanglement entropy of the OTT black hole
Additionally, we find that to linear order in b the acceleration is given
by
r∗ (1)
φ00 (r) 2
r∗ k − 2r+ + r∗ b .
r00 (φ)
= − 0 3 = r∗ r∗2 − r+
+
φ (r) r=r∗
2
φ=0
Finally, we find the entanglement entropy of the OTT black hole to
linear order in b
 q
2 2

4
3 2
4
(1) 3
(1)
2
2
1  b r − r+ r r+ r+ − k r∗ + r r∗ − k r+ r∗ + r+ r∗ − r+ r∗
SEE (r+ , r∗ ) =
p
2 
2
2Gr+

r (r + r+ ) r2 − r∗2 r∗2 − r+
r
q

q
2 − r2

br∗ k(1) r+
∗
2
2
2
2
2
q
+p
+ 2r+ log
r − r+ + r − r∗
+ O(b2 ) .

2
2
2
2

r − r∗ r∗ − r+
r∗
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
24 / 26
Entanglement entropy of the OTT black hole
Bulk variables (r+ , r∗ ) ←→ Boundary variables (β, `).
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
25 / 26
Entanglement entropy of the OTT black hole
Bulk variables (r+ , r∗ ) ←→ Boundary variables (β, `).
We test whether this expression matches the CFT expectation
c
β
π`
log
sinh
,
3
π
β
with
4π
3
,
β=
,
G
b + 2r+
4 r∗ F (z̃|η) + (r+ − r∗ ) Π(n; z̃|η)
p
`=
,
r+
(r+ + r∗ )(b + r+ + r∗ )
c=
z̃ = lim z ,
r→∞
corresponds to the sought after SEE (β, `).
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
25 / 26
Entanglement entropy of the OTT black hole
Bulk variables (r+ , r∗ ) ←→ Boundary variables (β, `).
We test whether this expression matches the CFT expectation
c
β
π`
log
sinh
,
3
π
β
with
4π
3
,
β=
,
G
b + 2r+
4 r∗ F (z̃|η) + (r+ − r∗ ) Π(n; z̃|η)
p
`=
,
r+
(r+ + r∗ )(b + r+ + r∗ )
c=
z̃ = lim z ,
r→∞
corresponds to the sought after SEE (β, `).
Holographic EE matches exactly SEE (β, `) up to O(b2 )!!
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
25 / 26
Thank you for your attention!
Seyed Morteza Hosseini
Free-kick condition for entanglement entropy in higher curvature gravity
University of Milano-Bicocca
26 / 26