gong show eduardo teste - strings 2017 · entanglement entropy of a sphere s cft(r)=µ d2r d2 + ......
TRANSCRIPT
The “a” theorem and the Markov property of the vacuum
Eduardo Testé
with Horacio Casini and Gonzalo TorrobaInstituto Balseiro, Centro Atómico Bariloche, S.C. de Bariloche, Argentina
based on
entanglement entropy of a sphere
SCFT(r) = µd�2rd�2 + ...
(�1)
d/2�14 a log(r/✏)
(�1)(d�1)/2 F{ordered under RG flows?
the entropic “c” and “F” theorems (Casini, Huerta)
1.
entanglement entropy of a sphere
SCFT(r) = µd�2rd�2 + ...
(�1)
d/2�14 a log(r/✏)
(�1)(d�1)/2 F{ordered under RG flows?
the entropic “c” and “F” theorems (Casini, Huerta)
1.
2. The Strong Subadditivity (SSA) inequality
S(A) + S(B) > S(A ^B) + S(A _B)
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^B) + S(A _B) ?
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^B) + S(A _B) ?
A ^BAB
null plane
in general
(Markov state)
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^B) + S(A _B) ?
A ^BAB
null plane
in general
null cone
for a CFT
BA ^B
A
(Markov state)
S(A) + S(B) > S(A ^B) + S(A _B)vacuum of the
RG running QFT
BA ^B
A
S(A) + S(B) = S(A ^B) + S(A _B) vacuum of the UV CFT
(Markovian)
Application: entropic proof of the “a” theorem
Application: entropic proof of the “a” theorem
S(A) + S(B) > S(A ^B) + S(A _B)
BA ^B
A
S(A) + S(B) = S(A ^B) + S(A _B)
�S(A) +�S(B) > �S(A ^B) +�S(A _B)
vacuum of the RG running
QFT
vacuum of the UV CFT
(Markovian)
take symmetric form of this
Application: entropic proof of the “a” theorem
�S(A) +�S(B) > �S(A ^B) +�S(A _B)
r�S00(r)� (d� 3)�S0(r) 6 0
= �S̃ ! �S =
SRG QFT
SUV CFT
SRG QFT
SUV CFT
No angle contribution problem
The differences in local curvatures are UV and cancels
as in the F theorem
Application: entropic proof of the “a” theorem
SCFT(r) = µ2r2 � 4 a log(r/✏)
aUV > aIR
r�S00(r)� (d� 3)�S0(r) 6 0
d = 4
Application: entropic proof of the “a” theorem
SCFT(r) = µ2r2 � 4 a log(r/✏)
aUV > aIR
r�S00(r)� (d� 3)�S0(r) 6 0
d = 2 d = 3
cUV > cIR FUV > FIR
d = 4
Application: entropic proof of the “a” theorem
SCFT(r) = µ2r2 � 4 a log(r/✏)
aUV > aIR
r�S00(r)� (d� 3)�S0(r) 6 0
d = 2 d = 3
cUV > cIR FUV > FIR
d = 4
unified picture of RG irreversibility
Application: entropic proof of the “a” theorem
SCFT(r) = µ2r2 � 4 a log(r/✏)
aUV > aIR
r�S00(r)� (d� 3)�S0(r) 6 0
d = 2 d = 3
cUV > cIR FUV > FIR
d = 4
unified picture of RG irreversibilityThank you
(I will be outside with a poster)