Bremsstrahlung function for ABJM theorybased on work in progress with L. Griguolo, M. Preti and D. Seminara
Lorenzo Bianchi
Universität Hamburg
March 3rd , 2017.YRISW, Dublin
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 1 / 13
Introduction and goals
Exact results in interacting QFTs are notoriously hard to achieve.
We know few examples of superconformal field theories where integrability shows up(in this talk we look at 4d N = 4 SYM and 3d ABJM theory)For a restricted class (BPS) of observables, exact results may be achieved bysupersymmetric localization.
One of these observables is the circular Wilson loop.
In N = 4 SYM a beautiful formula relates the circular Wilson loop to the energyemitted by a moving heavy particle, i.e. the Bremsstrahlung function [Correa, Henn,Maldacena, Sever, 2012]
B =1
2⇡2�@� log hW�i
Goal of this talk
Prove a similar formula for ABJM theory.
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 2 / 13
Introduction and goals
Exact results in interacting QFTs are notoriously hard to achieve.
We know few examples of superconformal field theories where integrability shows up(in this talk we look at 4d N = 4 SYM and 3d ABJM theory)For a restricted class (BPS) of observables, exact results may be achieved bysupersymmetric localization.
One of these observables is the circular Wilson loop.
In N = 4 SYM a beautiful formula relates the circular Wilson loop to the energyemitted by a moving heavy particle, i.e. the Bremsstrahlung function [Correa, Henn,Maldacena, Sever, 2012]
B =1
2⇡2�@� log hW�i
Goal of this talk
Prove a similar formula for ABJM theory.
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 2 / 13
Conformal defects
A defect breaks translation invariance
@µTµa(x⌫) = �⌃(x) D
a(x i ),
Da(x i ) is the displacement operator
It implements small modifications of the defect
� hX iW = �Z
dd�2x �xa(x i ) hDa(x i )X iW
Its two-point function is fixed by symmetry
hDa(x i )Db(0)iW = CD�ab
|x i |2(d�1) .
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 3 / 13
Conformal defects
A defect breaks translation invariance
@µTµa(x⌫) = �⌃(x) D
a(x i ),
Da(x i ) is the displacement operator
It implements small modifications of the defect
� hX iW = �Z
dd�2x �xa(x i ) hDa(x i )X iW
Its two-point function is fixed by symmetry
hDa(x i )Db(0)iW = CD�ab
|x i |2(d�1) .
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 3 / 13
Conformal defects
The defect breaks translation invariance
@µTµa(x⌫) = �⌃(x) D
a(x i ),
Local operators acquire a non-vanishing one-pointfunction.
The kinematics is fixed by conformal invariance
hOiW ⌘hW OihW i =
COr�
For the stress tensor
hTijiW = �h2⇡
�ijrd
hTabiW =h2⇡
(d�1) �ab�d nanbrd
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 4 / 13
Emitted energy and CD
Energy emitted by a moving electron (Liénard formula)
�E = � 2e2
3m2dpµ
d⌧dpµd⌧
=2e2
3
Zdt
|v̇|2 � |v ⇥ v̇|2
(1� |v|2)3
For a heavy probe moving in a conformal field theory (Wilsonline) we can use conformal defect techniques. Consider a smalldeformation �x = 2 ✏ cos!t, the absorption probability is
pabs = T |✏2|Z
dte i!t hD(t)D(0)iW =⇡3|✏|2T!3CD
Energy emitted by a heavy probe in a CFT
�E =⇡6CD
Zdt
|v̇|2 � |v ⇥ v̇|2
(1� |v|2)3
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 5 / 13
Emitted energy and CD
Energy emitted by a moving electron (Liénard formula)
�E = � 2e2
3m2dpµ
d⌧dpµd⌧
=2e2
3
Zdt
|v̇|2 � |v ⇥ v̇|2
(1� |v|2)3
For a heavy probe moving in a conformal field theory (Wilsonline) we can use conformal defect techniques. Consider a smalldeformation �x = 2 ✏ cos!t, the absorption probability is
pabs = T |✏2|Z
dte i!t hD(t)D(0)iW =⇡3|✏|2T!3CD
Energy emitted by a heavy probe in a CFT
�E =⇡6CD
Zdt
|v̇|2 � |v ⇥ v̇|2
(1� |v|2)3
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 5 / 13
Cusped Wilson lines
hWcuspi = e��cusp(�) logL✏
In the limit of small angle the expectation value is again controlled by the displacementoperator.
�cusp(�) ⇠ �B�2 = �12�2
Zd⌧ hD(⌧)D(0)i = �CD
12�2
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 6 / 13
In summary
�cusp(�) ⇠ �B �2 �E ⇠ 2⇡ BZ
dtv̇ 2 hDa(x i )Db(0)iW =12B �ab
|x i |2(d�1)
Generalized cusp in N = 4 SYM
W = TrPe iHA·x+
H|dx|nA�A A = 1, ..., 6
✓ = nAn0A
�cusp(�, ✓) ⇠ B(✓2 � �2) + O((�2 � ✓2)2)
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 7 / 13
In summary
�cusp(�) ⇠ �B �2 �E ⇠ 2⇡ BZ
dtv̇ 2 hDa(x i )Db(0)iW =12B �ab
|x i |2(d�1)
Generalized cusp in N = 4 SYM
W = TrPe iHA·x+
H|dx|nA�A A = 1, ..., 6
✓ = nAn0A
�cusp(�, ✓) ⇠ B(✓2 � �2) + O((�2 � ✓2)2)
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 7 / 13
ABJ(M) theory
Three-dimensional N = 6 super Chern-Simons theory with matter.Gauge group U(N)⇥ U(M), but here M = N.
R-symmetry group SU(4) ⇠ SO(6). CI (C̄ I ) in (anti-)fundamental.String theory dual in AdS4 ⇥ CP3.Integrable structure at large N, with non-trivial interpolating function h(�).
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 8 / 13
Wilson loops in ABJM theory
W = 12N
Tr
P exp
✓�i
Zd⌧L(⌧)
◆�
In this case L(⌧) is a supermatrix in U(N|N)
L =✓Aµẋµ � iMJ ICI C̄ J �i⌘I ̄I
�i I ⌘̄I µẋµ � iMJ I C̄ JCI
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 9 / 13
Wilson loops in ABJM theory
W = 12N
Tr
P exp
✓�i
Zd⌧L(⌧)
◆�
In this case L(⌧) is a supermatrix in U(N|N)
L =✓Aµẋµ � iMJ ICI C̄ J �i⌘I ̄I
�i I ⌘̄I µẋµ � iMJ I C̄ JCI
◆
Straight line - 12 BPS configuration
MI J =
0
BB@
�1 0 0 00 1 0 00 0 1 00 0 0 1
1
CCA ,
⌘↵I =
0
BB@
1000
1
CCA
I
�1 1
�↵, ⌘̄I↵ = i
�1 0 0 0
�I✓11
◆
↵
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 9 / 13
Wilson loops in ABJM theory
W = 12N
Tr
P exp
✓�i
Zd⌧L(⌧)
◆�
In this case L(⌧) is a supermatrix in U(N|N)
L =✓Aµẋµ � iMJ ICI C̄ J �i⌘I ̄I
�i I ⌘̄I µẋµ � iMJ I C̄ JCI
◆
Generalized cusp
MI J =
0
BB@
� cos ✓ sin ✓ 0 0sin ✓ cos ✓ 0 00 0 1 00 0 0 1
1
CCA ,
⌘↵I =
0
BB@
cos ✓2� sin ✓2
00
1
CCA
I
�1 1
�↵, ⌘̄I↵ = i
�cos ✓2 � sin
✓2 0 0
�I✓11
◆
↵
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 9 / 13
Bremsstrahlung function for ABJM theory
�cusp(�, ✓) ⇠ B(✓2 � �2) + O((�2 � ✓2)2)L(⌧)= L(0)(⌧) + ✓L(1)(⌧) + O(✓2)
�cusp(�, ✓)⇠ @2@✓2 log hWi���✓=0
= � R 10 ds1 R s10 ds2 hL(1)(s1)L(1)(s2)iW0
Write B in terms of defect two-point functions of local operators
Scalar operator
hO(⌧1)Ō(⌧2)iWline =cs
|⌧1 � ⌧2|2�O
for us O(⌧) = CC̄ and �O = 1
Fermionic operator
h ̄↵(⌧1) �(⌧2)iWline =icf (x1(⌧1)� x2(⌧2))µ�µ
|⌧1 � ⌧2|2� +1
for us � = 1
B(�) =1N
✓cs(�)�
14cf (�)
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 10 / 13
Bremsstrahlung function for ABJM theory
�cusp(�, ✓) ⇠ B(✓2 � �2) + O((�2 � ✓2)2)L(⌧)= L(0)(⌧) + ✓L(1)(⌧) + O(✓2)
�cusp(�, ✓)⇠ @2@✓2 log hWi���✓=0
= � R 10 ds1 R s10 ds2 hL(1)(s1)L(1)(s2)iW0Write B in terms of defect two-point functions of local operators
Scalar operator
hO(⌧1)Ō(⌧2)iWline =cs
|⌧1 � ⌧2|2�O
for us O(⌧) = CC̄ and �O = 1
Fermionic operator
h ̄↵(⌧1) �(⌧2)iWline =icf (x1(⌧1)� x2(⌧2))µ�µ
|⌧1 � ⌧2|2� +1
for us � = 1
B(�) =1N
✓cs(�)�
14cf (�)
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 10 / 13
Bremsstrahlung function for ABJM theory
�cusp(�, ✓) ⇠ B(✓2 � �2) + O((�2 � ✓2)2)L(⌧)= L(0)(⌧) + ✓L(1)(⌧) + O(✓2)
�cusp(�, ✓)⇠ @2@✓2 log hWi���✓=0
= � R 10 ds1 R s10 ds2 hL(1)(s1)L(1)(s2)iW0Write B in terms of defect two-point functions of local operators
Scalar operator
hO(⌧1)Ō(⌧2)iWline =cs
|⌧1 � ⌧2|2�O
for us O(⌧) = CC̄ and �O = 1
Fermionic operator
h ̄↵(⌧1) �(⌧2)iWline =icf (x1(⌧1)� x2(⌧2))µ�µ
|⌧1 � ⌧2|2� +1
for us � = 1
B(�) =1N
✓cs(�)�
14cf (�)
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 10 / 13
Circular Wilson loops for ABJM
W� = TrP exp
✓�i
Zd⌧L(⌧)
◆�
L(⌧) is again a supermatrix in U(N|N)
L =✓Aµẋµ � iMJ ICI C̄ J �i⌘I ̄I
�i I ⌘̄I µẋµ � iMJ I C̄ JCI
◆
Circular WL - 12 BPS configuration
MI J =
0B@�1 0 0 00 1 0 00 0 1 00 0 0 1
1CA ,
⌘↵I = ei⌧2
0B@1000
1CAI
�1 �ie�i⌧� , ⌘̄I↵ = ie �i⌧2 �1 0 0 0�I ✓ 1iei⌧
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 11 / 13
Circular Wilson loops for ABJM
W� = TrP exp
✓�i
Zd⌧L(⌧)
◆�
L(⌧) is again a supermatrix in U(N|N)
L =✓Aµẋµ � iMJ ICI C̄ J �i⌘I ̄I
�i I ⌘̄I µẋµ � iMJ I C̄ JCI
◆
“Latitude” WL - 14 BPS configuration (⌫ = cos ✓ sin 2↵)
MI J =
0BB@�⌫ e�i⌧p1 � ⌫2 0 0
ei⌧p1 � ⌫2 ⌫ 0 00 0 1 00 0 0 1
1CCA ,
⌘↵I =ei⌫⌧2p2
0BB@p1 + ⌫
�p1 � ⌫ei⌧00
1CCAI
�1 �ie�i⌧� , ⌘̄I↵ = i e �i⌫⌧2p
2
�p1 + ⌫ �p1 � ⌫e�i⌧ 0 0�I ✓ 1
iei⌧
◆
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 11 / 13
Final result and comments
14⇡2
@@⌫
log hW�i����⌫=1
=1N
✓cs(�)�
14cf (�)
◆= B(�)
Comments
This formula was conjectured based on a two-loop computation [M. Bianchi, Griguolo,Leoni, Penati, Seminara, 2014]
The 14 BPS Wilson line can be computed by localization, although exploiting somenon-trivial cohomological equivalence.
Comparison with the result from integrability would allow to finally determine theinterpolating function h(�) (the full QSC for this system has been derived in[Bombardelli, Cavaglià, Fioravanti, Gromov, Tateo, 2016/2017]), testing the conjecture of[Gromov, Sizov, 2014].
Our formula for the Bremsstrahlung function in terms of cs and cf simplifies theperturbative computation of B(�).
The Bremsstrahlung function in ABJM has been related to the one-point function ofthe stress tensor B = 2h [Maldacena, Lewkowycz 2013]. This relation is much moremysterious than the others and particularly interesting from a dCFT point of view.
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 12 / 13
THANK YOU
Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/2017 13 / 13