Bremsstrahlung function for ABJM theory based on work in progress with L. Griguolo, M. Preti and D. Seminara

Transcription

1 Bremsstrahlung function for ABJM theory based on work in progress with L. Griguolo, M. Preti and D. Seminara Lorenzo Bianchi Universität Hamburg March 3 rd,2017. YRISW, Dublin Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

2 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 =4SYMand3dABJMtheory) For a restricted class (BPS) of observables, exact results may be achieved by supersymmetric localization. One of these observables is the circular Wilson loop. In N = 4 SYM a beautiful formula relates the circular Wilson loop to the energy emitted by a moving heavy particle, i.e. the Bremsstrahlung function [Correa, Henn, Maldacena, Sever, 2012] B = 1 2 log hw i Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

3 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 =4SYMand3dABJMtheory) For a restricted class (BPS) of observables, exact results may be achieved by supersymmetric localization. One of these observables is the circular Wilson loop. In N = 4 SYM a beautiful formula relates the circular Wilson loop to the energy emitted by a moving heavy particle, i.e. the Bremsstrahlung function [Correa, Henn, Maldacena, Sever, 2012] B = 1 2 log hw i Goal of this talk Prove a similar formula for ABJM theory. Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

4 Conformal defects A defect breaks translation µt µa (x )= (x) D a (x i ), Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

5 Conformal defects A defect breaks translation µt µa (x )= (x) D a (x i ), D a (x i )isthedisplacement operator It implements small modifications of the defect Z hx i W = d d 2 x x a (x i ) hd a (x i )X i W Its two-point function is fixed by symmetry hd a (x i ) D b (0)i W = C D ab. x i 2(d 1) Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

6 Conformal defects The defect breaks translation µt µa (x )= (x) D a (x i ), Local operators acquire a non-vanishing one-point function. The kinematics is fixed by conformal invariance For the stress tensor hoi W hwoi hw i = C O r ht ij i W = h ij 2 r d ht ab i W = h (d 1) ab dn an b 2 r d Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

7 Emitted energy and C D Energy emitted by a moving electron (Liénard formula) E = 2e2 dp µ Z dp µ 3m 2 d d = 2e2 dt v 2 v v 2 3 (1 v 2 ) 3 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

8 Emitted energy and C D Energy emitted by a moving electron (Liénard formula) E = 2e2 dp µ Z dp µ 3m 2 d d = 2e2 dt v 2 v v 2 3 (1 v 2 ) 3 For a heavy probe moving in a conformal field theory (Wilson line) we can use conformal defect techniques. Consider a small deformation x =2 cos!t, theabsorptionprobabilityis Z p abs = T 2 dte i!t hd(t)d(0)i W = 3 2 T! 3 C D Energy emitted by a heavy probe in a CFT E = Z 6 C D dt v 2 v v 2 (1 v 2 ) 3 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

9 Cusped Wilson lines hw cuspi = e cusp( )log L In the limit of small angle the expectation value is again controlled by the displacement operator. cusp( ) B 2 = 1 Z 2 d hd( )D(0)i = C D Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

10 In summary Z cusp( ) B 2 E 2 B dt v 2 hd a (x i ) D b (0)i W = ab 12 B x i 2(d 1) Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

11 In summary Z cusp( ) B 2 E 2 B dt v 2 hd a (x i ) D b (0)i W = ab 12 B x i 2(d 1) Generalized cusp in N =4SYM W =TrPe i H A x+ H dx n A A A =1,...,6 cusp(, ) B( 2 2 )+O(( 2 2 ) 2 ) = n A n 0 A Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

12 ABJ(M) theory Three-dimensional N =6superChern-Simonstheorywithmatter. Gauge group U(N) U(M), but here M = N. R-symmetry group SU(4) SO(6). C I ( C I ) in (anti-)fundamental. String theory dual in AdS 4 CP 3. Integrable structure at large N, with non-trivial interpolating function h( ). Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

13 Wilson loops in ABJM theory W = 1 apple 2N Tr P exp Z i d L( ) In this case L( ) isasupermatrixinu(n N) Aµẋ µ im I J J C I C i I I L = i I I Â µẋ µ I im J C J C I Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

14 Wilson loops in ABJM theory W = 1 apple 2N Tr P exp Z i d L( ) In this case L( ) isasupermatrixinu(n N) Aµẋ µ im I J J C I C i I I L = i I I Â µẋ µ I im J C J C I Straight line - 1 BPS configuration I = B0 0A 0 I M I J = B A, , I = i I 1 1 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

15 Wilson loops in ABJM theory W = 1 apple 2N Tr P exp Z i d L( ) In this case L( ) isasupermatrixinu(n N) Aµẋ µ im I J J C I C i I I L = i I I Â µẋ µ I im J C J C I Generalized cusp 0 1 cos 2 I = B sin 2 0 A 0 I M I J = 0 1 cos sin 0 0 sin cos 0 0 C A, , I = i cos 2 sin I 1 1 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

16 Bremsstrahlung function for ABJM theory cusp(, ) B( 2 2 )+O(( 2 2 ) 2 ) L( )= L (0) ( ) + L (1) ( ) +O( 2 ) 2 log hwi = R 1 R s1 ds = ds 2 hl (1) (s 1)L (1) (s 2)i W0 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

17 Bremsstrahlung function for ABJM theory cusp(, ) B( 2 2 )+O(( 2 2 ) 2 ) L( )= L (0) ( ) + L (1) ( ) +O( 2 ) 2 log hwi = R 1 R s1 ds = ds 2 hl (1) (s 1)L (1) (s 2)i W0 Write B in terms of defect two-point functions of local operators Scalar operator Fermionic operator c s ho( 1)Ō( 2)i W line = O for us O( ) =C C and O =1 h ( 1) ( 2)i Wline = ic f (x 1( 1) x 2( 2)) µ µ for us =1 Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

18 Bremsstrahlung function for ABJM theory cusp(, ) B( 2 2 )+O(( 2 2 ) 2 ) L( )= L (0) ( ) + L (1) ( ) +O( 2 ) 2 log hwi = R 1 R s1 ds = ds 2 hl (1) (s 1)L (1) (s 2)i W0 Write B in terms of defect two-point functions of local operators Scalar operator Fermionic operator c s ho( 1)Ō( 2)i W line = O for us O( ) =C C and O =1 h ( 1) ( 2)i Wline = ic f (x 1( 1) x 2( 2)) µ µ for us =1 B( )= 1 N c s( ) 1 4 c f ( ) Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

19 Circular Wilson loops for ABJM W apple =Tr P exp Z i d L( ) L( ) isagainasupermatrixinu(n N) Aµẋ µ im I J J C I C i I I L = i I I Â µẋ µ I im J C J C I Circular WL BPS configuration I = e i A 0 I M J B C I A, ie i, I = ie i I 1 ie i Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

20 Circular Wilson loops for ABJM W apple =Tr P exp Z i d L( ) L( ) isagainasupermatrixinu(n N) Aµẋ µ im I J J C I C i I I L = i I I Â µẋ µ I im J C J C I Latitude WL BPS configuration ( =cos sin 2 ) 0 e i p M J I = Be i p A, p 1 I = e i 2 p 1+ p B 1 e i C 0 A 1 ie i, I = i e i 2 p p p 1+ 1 e i 2 0 I 0 0 I 1 ie i Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

21 Final result and comments 1 log hw i = 1 c s( ) =1 N 1 4 c f ( ) = B( ) Comments This formula was conjectured based on a two-loop computation [M. Bianchi, Griguolo, Leoni, Penati, Seminara, 2014] The 1 BPS Wilson line can be computed by localization, althoughexploitingsome 4 non-trivial cohomological equivalence. Comparison with the result from integrability would allow to finally determine the interpolating 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 c s and c f simplifies the perturbative computation of B( ). The Bremsstrahlung function in ABJM has been related to the one-point function of the stress tensor B =2h [Maldacena, Lewkowycz 2013]. Thisrelationismuchmore mysterious than the others and particularly interesting from a dcft point of view. Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13

22 THANK YOU Lorenzo Bianchi (HH) Bremsstrahlung for ABJM 03/03/ / 13