bremsstrahlung function for abjm theorytristan/yrisw_2017/talks/... · 2017. 3. 7. ·...

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.


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) .


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


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


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


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)


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(�).


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 Âµẋµ � iMJ I C̄ JCI

◆



W = 12N

Tr

P exp

✓�i

Zd⌧L(⌧)

◆�




◆

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

◆

↵



W = 12N

Tr

P exp

✓�i

Zd⌧L(⌧)

◆�




◆

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

◆

↵


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 (�)

◆


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 (�)

◆


Circular Wilson loops for ABJM

W� = TrP exp

✓�i

Zd⌧L(⌧)

◆�

L(⌧) is again a supermatrix in U(N|N)



◆

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⌧

◆


Circular Wilson loops for ABJM

W� = TrP exp

✓�i

Zd⌧L(⌧)

◆�

L(⌧) is again a supermatrix in U(N|N)



◆

“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⌧

◆


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.


THANK YOU


bremsstrahlung function for abjm theorytristan/yrisw_2017/talks/... · 2017. 3. 7. ·...

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