Exact results for supersymmetric cusps in ABJM theory based on and

Transcription

1 Exact results for supersymmetric cusps in ABJM theory based on and Marco S. Bianchi Centre for Research in String Theory Ascona, Switzerland July 4 th, 2017

2 Collaborators (1) Luca Griguolo (Parma U.) Silvia Penati (Milano-Bicocca U.) Domenico Seminara (Firenze U.)

3 Collaborators (2) Andrea Mauri (Milano-Bicocca U.) Michelangelo Preti (DESY) Matias Leoni (Buenos Aires U.)

4 Outline Introduction Supersymmetric Wilson loops in ABJM theory Supersymmetric cusps in ABJM model Three-loop test Conclusions

5 Introduction

6 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/2 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6

7 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/2 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 coupling to scalars Φ I of N = 4 SYM via SO(6) vector preserves 1/2 of supercharges (locally), i.e. 1/2-BPS

8 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/2 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 coupling to scalars Φ I of N = 4 SYM via SO(6) vector preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces zi z0 aba M N

9 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/2 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 Γ : x 0 = 0 x 1 = τ x 2 = 0 x 4 = 0 τ n 1 = 1 n 2 = 0 n 3 = 0 n 4 = n 5 = n 6 = 0 preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces contours globally preserving susy: 1/2-BPS line

10 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/2 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 Γ : x 0 = 0 x 1 = cos τ x 2 = sin τ x 4 = 0 0 τ 2π n 1 = 1 n 2 = 0 n 3 = 0 n 4 = n 5 = n 6 = 0 preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces contours globally preserving susy: 1/2-BPS line, maximal circle

11 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/4 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 Γ : x 0 = 0 x 1 = sin θ cos τ x 2 = sin θ sin τ x 4 = cos θ 0 τ 2π n 1 = cos θ n 2 = sin θ cos τ n 3 = sin θ sin τ n 4 = n 5 = n 6 = 0 preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces contours globally preserving susy: 1/4-BPS line, maximal circle or latitude (S 3 and SO(6))

12 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/4 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 Γ : x 0 = 0 x 1 = sin θ cos τ x 2 = sin θ sin τ x 4 = cos θ 0 τ 2π n 1 = cos θ n 2 = sin θ cos τ n 3 = sin θ sin τ n 4 = n 5 = n 6 = 0 preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces contours globally preserving susy: 1/4-BPS line, maximal circle or latitude (S 3 and SO(6)) conformal map from sphere to plane anomalous transformations [Drukker, Gross 01]

13 Supersymmetric Wilson loops Let s use our favourite theory as an example: N = 4 SYM one can define locally supersymmetric Wilson loops! W 1/4 = 1 [ i N Tr P exp g dτ Γ ( A µ ẋ µ (τ) + i n I (τ) ẋ Φ I ) ] I = 1,... 6 Γ : x 0 = 0 x 1 = sin θ cos τ x 2 = sin θ sin τ x 4 = cos θ 0 τ 2π n 1 = cos θ n 2 = sin θ cos τ n 3 = sin θ sin τ n 4 = n 5 = n 6 = 0 preserves 1/2 of supercharges (locally), i.e. 1/2-BPS finite expectation value on spacial smooth contours natural object in N = 4: massive quarks via Higgsing strong coupling AdS/CFT dual: [Maldacena; Rey, Yee 98] vev computed via minimal surfaces contours globally preserving susy: 1/4-BPS line, maximal circle or latitude (S 3 and SO(6)) conformal map from sphere to plane anomalous transformations [Drukker, Gross 01] they are amenable of exact results via localization [Erickson, Semenoff, Zarembo 00; Pestun 07]

14 theories on a compact manifold Localization invariance under fermionic symmetry (conserved supercharge) -dim path integral reduces to a finite-dim matrix model (MM) [Pestun 05] can compute expectation values of operators preserving supercharge as MM average often this simplifies in the planar limit (saddle point approx)

15 theories on a compact manifold Localization invariance under fermionic symmetry (conserved supercharge) -dim path integral reduces to a finite-dim matrix model (MM) [Pestun 05] can compute expectation values of operators preserving supercharge as MM average often this simplifies in the planar limit (saddle point approx) Example: can compute the exact vev of supersymmetric Wilson loops in N = 4 SYM from a (gaussian) matrix model (λ = g 2 N) W 1/2 = 1 [Erickson, Semenoff, Zarembo 00; Drukker, Gross 01 ]

16 theories on a compact manifold Localization invariance under fermionic symmetry (conserved supercharge) -dim path integral reduces to a finite-dim matrix model (MM) [Pestun 05] can compute expectation values of operators preserving supercharge as MM average often this simplifies in the planar limit (saddle point approx) Example: can compute the exact vev of supersymmetric Wilson loops in N = 4 SYM from a (gaussian) matrix model (λ = g 2 N) W 1/2 = 1 ( N L1 N 1 λ ) e λ 2 8N = λ I ( ) 1 1( λ) + O 4N N 2 [Erickson, Semenoff, Zarembo 00; Drukker, Gross 01; Pestun 07]

17 theories on a compact manifold Localization invariance under fermionic symmetry (conserved supercharge) -dim path integral reduces to a finite-dim matrix model (MM) [Pestun 05] can compute expectation values of operators preserving supercharge as MM average often this simplifies in the planar limit (saddle point approx) Example: can compute the exact vev of supersymmetric Wilson loops in N = 4 SYM from a (gaussian) matrix model (λ = g 2 N) W 1/4 = W 1/2 λ λ cos 2 θ [Drukker, Giombi, Ricci, Trancanelli 07; Pestun 09]

18 Cusps cusp anomalous dimension even supersymmetric WL develop UV divergences for cusped contours renormalization: cusp anomalous dimension Γ cusp(φ) [Polyakov 80; Korchemsky and Radyushkin 87]

19 Cusps cusp anomalous dimension even supersymmetric WL develop UV divergences for cusped contours renormalization: cusp anomalous dimension Γ cusp(φ) [Polyakov 80; Korchemsky and Radyushkin 87] If the rays are light-like, φ i, additional divergence: Γ cusp(φ) ϕ Γ cusp where φ = i ϕ: controls IR divergence of amplitudes of massless particles [Magnea, Sterman, 90; Bern, Dixon, Smirnov, 05] anomalous dimension of twist-2 operators [Korchemsky 89; Korchemsky and Marchesini 93]

20 Cusps cusp anomalous dimension even supersymmetric WL develop UV divergences for cusped contours renormalization: cusp anomalous dimension Γ cusp(φ) [Polyakov 80; Korchemsky and Radyushkin 87] If the rays are light-like, φ i, additional divergence: Γ cusp(φ) ϕ Γ cusp where φ = i ϕ: controls IR divergence of amplitudes of massless particles [Magnea, Sterman, 90; Bern, Dixon, Smirnov, 05] anomalous dimension of twist-2 operators [Korchemsky 89; Korchemsky and Marchesini 93] in planar N = 4 ( t Hooft coupling λ = g 2 N): sl(2) sector anomalous dimensions (large spin) AdS/CFT description GKP string (folded string rotating in AdS) [Gubser, Klebanov, Polyakov 02] computed up to four loops at weak coupling (λ 4 ) and up to two loops at strong coupling order ( 1 λ ) [Bern, Carrasco, Czakon, Dixon, Johanson, Kosower, Smirnov... Kruczenski; Giombi, Ricci, Roiban, Tirziu, Tseytlin, Vergu...] integrability...

21 Integrability N = 4 SYM integrability spectral problem of planar N = 4 SYM is described by an integrable spin chain [Minahan, Zarembo 02; Beisert, Staudacher 03 + many more]

22 Integrability N = 4 SYM integrability spectral problem of planar N = 4 SYM is described by an integrable spin chain all order results for anomalous dimensions of composite operators mutual test of AdS/CFT (proof?) [Minahan, Zarembo 02; Beisert, Staudacher 03 + many more] [from Beisert et al. review]

23 Integrability N = 4 SYM integrability spectral problem of planar N = 4 SYM is described by an integrable spin chain all order results for anomalous dimensions of composite operators mutual test of AdS/CFT (proof?) for (light-like) cusp anomalous dimension BES equation, valid to all orders in λ! [Beisert, Eden and Staudacher 05] [Minahan, Zarembo 02; Beisert, Staudacher 03 + many more] [from Beisert et al. review]

24 Integrability N = 4 SYM integrability spectral problem of planar N = 4 SYM is described by an integrable spin chain all order results for anomalous dimensions of composite operators mutual test of AdS/CFT (proof?) for (light-like) cusp anomalous dimension BES equation, valid to all orders in λ! [Beisert, Eden and Staudacher 05] [Minahan, Zarembo 02; Beisert, Staudacher 03 + many more] [from Beisert et al. review] For space-like (supersymmetric) cusps?

25 Supersymmetric cusps in N = 4 SYM Two Wilson lines at an angle φ: IR behaviour of scattering of massive colored particles [Korchemsky, Radyushkin 92] θ Regge limit of 4-pt massive amplitudes [Henn, Naculich, Schnitzer, Spradlin 10] quark-antiquark potential on sphere at angle π φ φ

26 Supersymmetric cusps in N = 4 SYM Two Wilson lines at an angle φ: IR behaviour of scattering of massive colored particles [Korchemsky, Radyushkin 92] θ Regge limit of 4-pt massive amplitudes [Henn, Naculich, Schnitzer, Spradlin 10] quark-antiquark potential on sphere at angle π φ φ locally 1/2-BPS cusp constructed with two 1/2-BPS rays at a (geometric) angle φ change of angle in the internal space as well: n 1 n 2 n 1 n 2 = cos θ [Drukker, Gross, Ooguri 99] BPS condition for φ 2 = θ 2 the cusp is supersymmetric (1/4-BPS): Γ cusp(φ = ±θ) = 0 computation at weak and strong coupling [Correa, Henn, Maldacena, Sever 12 Henn, Huber 13 Grozin, Henn, Korchemsky, Marquard 15 Drukker, Forini 09-10]

27 The Bremsstrahlung function Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2π B dt v 2

28 Bremsstrahlung function The Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2e2 3 dt v 2

29 Bremsstrahlung function The Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2π B dt v 2 exact computation by both integrability and localization [Correa, Henn, Maldacena, Sever 12]

30 Bremsstrahlung function The Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2π B dt v 2 exact computation by both integrability and localization 1. localization based [Correa, Henn, Maldacena, Sever 12] exploits BPS condition Γ cusp(φ 2 = θ 2 ) = 0 derivation passes through relation of B with displacement operator on line defect and latitude 1/4-BPS WLs exact expression: B = 1 2π 2 λ λ W1/2 = 1 λ I2 ( λ) 4π 2 I 1 ( + O(N 2 ) λ)

31 Bremsstrahlung function The Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2π B dt v 2 exact computation by both integrability and localization 1. localization based [Correa, Henn, Maldacena, Sever 12] exploits BPS condition Γ cusp(φ 2 = θ 2 ) = 0 derivation passes through relation of B with displacement operator on line defect and latitude 1/4-BPS WLs exact expression: B = 1 2π 2 λ λ W1/2 = 1 λ I2 ( λ) 4π 2 I 1 ( + O(N 2 ) λ) 2. integrability based: spectrum of operators on WL with boundary reflection matrix set of TBA equations, QSC [Correa, Maldacena, Sever; Drukker; Gromov, Sever 12 Gromov, Levkovich-Maslyuk, Sizov 13]

32 Bremsstrahlung function The Bremsstrahlung function We define B as the small angle limit of the cusp anomalous dimension: Γ cusp = B(g, N) φ in a CFT energy emitted by a heavy quark: E = 2π B dt v 2 exact computation by both integrability and localization 1. localization based [Correa, Henn, Maldacena, Sever 12] exploits BPS condition Γ cusp(φ 2 = θ 2 ) = 0 derivation passes through relation of B with displacement operator on line defect and latitude 1/4-BPS WLs exact expression: B = 1 2π 2 λ λ W1/2 = 1 λ I2 ( λ) 4π 2 I 1 ( + O(N 2 ) λ) 2. integrability based: spectrum of operators on WL with boundary reflection matrix set of TBA equations, QSC [Correa, Maldacena, Sever; Drukker; Gromov, Sever 12 Gromov, Levkovich-Maslyuk, Sizov 13] Comment: comparing the two exact results one can determine potential finite renormalization of coupling constants

33 To summarize N = 4 SYM is cool spectral problem is integrable exact susy Wilson loop vev via localization Bremsstrahlung function can be computed exactly by BOTH can this program be extended to other theories? localization integrability Natural candidates ABJM theory in 3 dimensions N = 2 SCFTs in 4 dimensions [Fiol, Gerchkovits, Komargodski 15]

34 Supersymmetric Wilson loops in ABJM theory

35 ABJM theory [Aharony, Bergman, Jafferis, Maldacena 08] d = 3, N = 6 SCFT OSp(4 6) Chern Simons with gauge group U(N) k U(N) k, k Z (A, Â) Bifundamental matter fields (Y I, ψ I ), I = 1, 2, 3, 4, (N, N)

36 ABJM theory [Aharony, Bergman, Jafferis, Maldacena 08] d = 3, N = 6 SCFT OSp(4 6) Chern Simons with gauge group U(N) k U(N) k, k Z (A, Â) Bifundamental matter fields (Y I, ψ I ), I = 1, 2, 3, 4, (N, N) low energy theory on N M2 branes probing a C 4 /Z k AdS 4/CFT 3 correspondence: M th on AdS4 S 7 /Z k N k 5 IIA string-th on AdS 4 CP 3 k N k 5 N = 6 ABJM SCFT in 3d k N Type IIA ST on AdS 4 CP 3 k N k 5 M-theory on AdS 4 S 7 /Z k N k 5 λ = N k

37 ABJM theory [Aharony, Bergman, Jafferis, Maldacena 08] d = 3, N = 6 SCFT OSp(4 6) Chern Simons with gauge group U(N) k U(N) k, k Z (A, Â) Bifundamental matter fields (Y I, ψ I ), I = 1, 2, 3, 4, (N, N) low energy theory on N M2 branes probing a C 4 /Z k AdS 4/CFT 3 correspondence: M th on AdS4 S 7 /Z k N k 5 IIA string-th on AdS 4 CP 3 k N k 5 integrable in the planar limit [Minahan, Zarembo; Gromov and Vieira 08 + many more] N = 6 ABJM SCFT in 3d k N INTEGRABILITY (N ) Type IIA ST on AdS 4 CP 3 k N k 5 M-theory on AdS 4 S 7 /Z k N k 5 λ = N k

38 The h(λ) function The interpolating h-function ABJM (or AdS 4 CP 3 ) integrability features a nontrivial interpolating function of the t Hooft coupling λ = N k [Gaiotto, Giombi, Yin; Grignani, Harmark, Orselli; Nishioka, Takayanagi; Gromov, Vieira 08] magnon dispersion relation for ABJM spin chain E = h 2 2 (λ) sin 2 p 2 scaling function (twist-2 / light-like cusp anomalous dimension) Γ ABJM(λ) = 1 2 Γ N =4(λ YM) λ YM h(λ) 4π [Gromov and Vieira 08] λ YM = g 2 N h(λ) appears in all integrability based predictions, it is needed for comparisons with results obtained by other methods

39 The h(λ) function The interpolating h-function ABJM (or AdS 4 CP 3 ) integrability features a nontrivial interpolating function of the t Hooft coupling λ = N k [Gaiotto, Giombi, Yin; Grignani, Harmark, Orselli; Nishioka, Takayanagi; Gromov, Vieira 08] magnon dispersion relation for ABJM spin chain E = h 2 2 (λ) sin 2 p 2 scaling function (twist-2 / light-like cusp anomalous dimension) Γ ABJM(λ) = 1 2 Γ N =4(λ YM) λ YM h(λ) 4π [Gromov and Vieira 08] λ YM = g 2 N h(λ) appears in all integrability based predictions, it is needed for comparisons with results obtained by other methods conjecture for its value to all orders, inspired by analogy with localization λ = sinh 2πh(λ) 2π ( 1 3F 2 2, 1 2, 1 2 ; 1, 3 ) 2 ; sinh2 2πh(λ) [Gromov, Sizov 14]

40 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1

41 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) 2 24 tests against perturbative results: 12 λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1

42 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1 tests against perturbative results: SU(2) SU(2) sector dispersion relation to 2 loops [Minahan, Ohlsson-Sax, Sieg 09] IR divergence of 2-loop scattering amplitudes/ UV of cusped WL [MSB, Leoni, Mauri, Penati, Santambrogio; Chen, Huang 12; Henn, Plefka, Wiegandt 10]

43 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1 tests against perturbative results: SU(2) SU(2) sector dispersion relation to 4 loops [Leoni, Mauri, Minahan, Ohlsson Sax, Sieg, Tartaglino-Mazzuchelli 10]

44 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π 6 log 2 2π 1260 λ7 + O(λ 9 ) λ 1 ( ) + O e 2π 2λ λ 1 tests against perturbative results: AdS 4 CP 3 spinning string computation [Mc Loughlin, Roiban; Alday, Arutyunov, Bykov; Krishnan 08 + more]

45 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1 tests against perturbative results: AdS 4 CP 3 worldsheet perturbation theory at two loops [Bianchi, MSB, Bres, Forini, Vescovi 14] confirms predicted anomalous radius shift [Bergman, Hirano 09]

46 The interpolating h-function: expansions weak and strong coupling expansions from conjecture: λ π2 3 λ3 + 5π4 h(λ) = ( 1 λ 1 ) λ5 893π λ7 + O(λ 9 ) λ 1 log 2 ( ) 2π + O e 2π 2λ λ 1 tests against perturbative results: AdS 4 CP 3 worldsheet perturbation theory at two loops [Bianchi, MSB, Bres, Forini, Vescovi 14] confirms predicted anomalous radius shift [Bergman, Hirano 09] mutual consistency of several computations and ingredients: conjecture must be correct, but proof desirable A proof of the exact h(λ) could be derived computing the same observable exactly by two independent methods, e.g. integrability and localization: Bremsstrahlung is a candidate

47 Supersymmetric Wilson loops 1/2 locally supersymmetric WL in N = 4 SYM W 1/2 [Γ] = 1 [ i N Tr P exp dτ Γ : coupling to a scalars ẋ i ẋ n I Φ I (A ) ] (τ)

48 Supersymmetric Wilson loops 1/6 locally supersymmetric WL in N = 6 ABJM: coupling to a scalars [Drukker, Plefka, Young; Chen, Wu; Rey, Suyama, Tamaguchi 09] W 1/6 [Γ] = 1 [ i N Tr P exp dτ (A ẋ i 2πk ) ] ẋ MJI Y I Ȳ J (τ) M = diag ( 1, 1, 1, 1) Γ

49 Supersymmetric Wilson loops 1/6 locally supersymmetric WL in N = 6 ABJM: coupling to a scalars [Drukker, Plefka, Young; Chen, Wu; Rey, Suyama, Tamaguchi 09] Ŵ 1/6 [Γ] = 1 ( [ i N Tr P exp dτ Â ẋ i 2π ) ] k ẋ ˆM I J Ȳ I Y J (τ) ˆM = diag ( 1, 1, 1, 1) companion Ŵ1/6 for the other gauge group Γ

50 Supersymmetric Wilson loops 1/6 locally supersymmetric WL in N = 6 ABJM: coupling to a scalars [Drukker, Plefka, Young; Chen, Wu; Rey, Suyama, Tamaguchi 09] W 1/6 [Γ] = 1 [ i N Tr P exp dτ (A ẋ i 2πk ) ] ẋ MJI Y I Ȳ J (τ) M = diag ( 1, 1, 1, 1) companion Ŵ1/6 for the other gauge group Γ 1/2 locally supersymmetric WLs in ABJM require coupling to both scalars and fermions U(N 1 N 2) supermatrix structure ( ) bosonic N1 N 1 fermionic N 1 N 2 L 1/2 = fermionic N 2 N 1 bosonic N 2 N 2 1/2-BPS WLs are cohomologically equivalent to comb of 1/6-BPS WLs (under supercharge Q) ( ) W1/6 0 W 1/2 = Q V difference is Q-exact 0 Ŵ 1/6 [Drukker, Trancanelli 10]

51 Supersymmetric Wilson loops 1/6 locally supersymmetric WL in N = 6 ABJM: coupling to a scalars [Drukker, Plefka, Young; Chen, Wu; Rey, Suyama, Tamaguchi 09] W 1/6 [Γ] = 1 [ i N Tr P exp dτ (A ẋ i 2πk ) ] ẋ MJI Y I Ȳ J (τ) M = diag ( 1, 1, 1, 1) companion Ŵ1/6 for the other gauge group Γ 1/2 locally supersymmetric WLs in ABJM require coupling to both scalars and fermions U(N 1 N 2) supermatrix structure ( ) bosonic N1 N 1 fermionic N 1 N 2 L 1/2 = fermionic N 2 N 1 bosonic N 2 N 2 1/2-BPS WLs are cohomologically equivalent to comb of 1/6-BPS WLs (under supercharge Q) ( ) W1/6 0 W 1/2 = Q V difference is Q-exact 0 Ŵ 1/6 [Drukker, Trancanelli 10] these WLs were given an interpretation via Higgsing [Lee, Lee 10]

52 1/2 BPS Wilson loop in details W [Γ] = 1 [ ( 2N Tr P exp i A A µẋ µ 2πi A L = 2π i k ẋ ψα I k k η I α ẋ M J I Y I Ȳ J Γ 2π i k ẋ ηα I )] dτ L(τ)  ψ I α [Drukker, Trancanelli 10], M = diag ( 1, +1, +1, +1)  µẋ µ 2πi ẋ M J I Ȳ J Y I Tr denotes the standard matrix trace (and not the super-trace) M are matrices in R-symmetry space controlling coupling to bi-scalars η, η are commuting spinors controlling coupling to fermions

53 1/2 BPS Wilson loop in details W [Γ] = 1 [ ( 2N Tr P exp i A A µẋ µ 2πi A L = 2π i k ẋ ψα I k k η I α ẋ M J I Y I Ȳ J Γ 2π i k ẋ ηα I )] dτ L(τ)  ψ I α [Drukker, Trancanelli 10], M = diag ( 1, +1, +1, +1)  µẋ µ 2πi ẋ M J I Ȳ J Y I Tr denotes the standard matrix trace (and not the super-trace) M are matrices in R-symmetry space controlling coupling to bi-scalars η, η are commuting spinors controlling coupling to fermions global susy contours exist where charges are preserved globally, e.g. line and circle amenable of an exact computation via localization

54 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more]

55 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) Z = N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b ))

56 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) N N Na<b W 1/6 = dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) 1 N e 2πλa N a=1 expectation value of 1/6 -BPS circular WLs W 1/6 = 1+i π N k (1 + 2N 2) π 2 1 k i N (4 + N 2) π 3 1 k

57 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) N N Na<b W 1/2 = dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) 1 N N e 2πλa 2π + e ˆλa 2N a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 = 1+i π N k (1 + 2N 2) π 2 1 k i N (4 + N 2) π 3 1 k , W 1/2 = W 1/6 + Ŵ 1/6 2

58 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) W 1/2 1 = 1 2N N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) N N e 2πλa 2π + e ˆλa a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 1 = 1+i π N k (1 + 2N 2) π 2 1 k (4 6 i N + N 2) π 3 1 W k , W 1/2 1/6 1 + Ŵ 1/6 1 1 = 2 imaginary part in localization, due to framing of WL: removing it [Calugareanu 59; Witten 89] [Kapustin, Willet, Yaakov 09; Closset, Dumitrescu, Festuccia, Komargodski, Seiberg 12] W 1/6 0 = 1 + π2 ( 6 k 2 5N 2 ) , W 1/2 0 = 1 + π2 ( 2N ) 6 k

59 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) W 1/2 1 = 1 2N N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) N N e 2πλa 2π + e ˆλa a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 1 = 1+i π N k (1 + 2N 2) π 2 1 k (4 6 i N + N 2) π 3 1 W k , W 1/2 1/6 1 + Ŵ 1/6 1 1 = 2 imaginary part in localization, due to framing of WL: removing it [Calugareanu 59; Witten 89] [Kapustin, Willet, Yaakov 09; Closset, Dumitrescu, Festuccia, Komargodski, Seiberg 12] W 1/6 0 = 1 + π2 ( 6 k 2 5N 2 ) , W 1/2 0 = 1 + π2 ( ) 2N k 2 perturbative check of 1/6-BPS up to two loops at framing 0 [Drukker, Plefka, Young; Chen, Wu; Rey, Suyama, Tamaguchi 09]

60 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) W 1/2 1 = 1 2N N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) N N e 2πλa 2π + e ˆλa a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 1 = 1+i π N k (1 + 2N 2) π 2 1 k (4 6 i N + N 2) π 3 1 W k , W 1/2 1/6 1 + Ŵ 1/6 1 1 = 2 imaginary part in localization, due to framing of WL: removing it [Calugareanu 59; Witten 89] [Kapustin, Willet, Yaakov 09; Closset, Dumitrescu, Festuccia, Komargodski, Seiberg 12] W 1/6 0 = 1 + π2 ( 6 k 2 5N 2 ) , W 1/2 0 = 1 + π2 ( ) 2N k 2 perturbative check of 1/6-BPS at 3 loops at framing 1 [MSB, Griguolo, Leoni, Mauri, Penati, Seminara 16]

61 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) W 1/2 1 = 1 2N N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) N N e 2πλa 2π + e ˆλa a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 1 = 1+i π N k (1 + 2N 2) π 2 1 k (4 6 i N + N 2) π 3 1 W k , W 1/2 1/6 1 + Ŵ 1/6 1 1 = 2 imaginary part in localization, due to framing of WL: removing it [Calugareanu 59; Witten 89] [Kapustin, Willet, Yaakov 09; Closset, Dumitrescu, Festuccia, Komargodski, Seiberg 12] W 1/6 0 = 1 + π2 ( 6 k 2 5N 2 ) , W 1/2 0 = 1 + π2 ( ) 2N k 2 perturbative check of 1/2-BPS up to two loops at framing 0 [MSB, Giribet, Leoni, Penati; Griguolo, Martelloni, Poggi, Seminara 13]

62 Exact results for ABJM WL N = 2 CS-matter theories on S 3 can be localized [Kapustin, Willet, Yaakov 09; Drukker, Marino, Putrov 10 + many more] ABJM matrix model (not gaussian) W 1/2,n 1 = 1 2N N N Na<b dλ a e iπkλ2 a d ˆλ b e iπk λ 2 sinh 2 (π(λ a λ b )) N b a<b sinh 2 (π(ˆλ a ˆλ b )) Na=1 Nb=1 a=1 cosh b=1 2 (π(λ a ˆλ b )) N N e 2πλan 2π ˆλan + e a=1 a=1 expectation value of 1/6 and 1/2-BPS circular WLs W 1/6 1 = 1+i π N k (1 + 2N 2) π 2 1 k (4 6 i N + N 2) π 3 1 W k , W 1/2 1/6 1 + Ŵ 1/6 1 1 = 2 imaginary part in localization, due to framing of WL: removing it [Calugareanu 59; Witten 89] [Kapustin, Willet, Yaakov 09; Closset, Dumitrescu, Festuccia, Komargodski, Seiberg 12] W 1/6 0 = 1 + π2 ( 6 k 2 5N 2 ) , W 1/2 0 = 1 + π2 ( ) 2N k 2 perturbative check of multiple winding at 2, 3 loops (recursively) [MSB 16] multiple winding n [Klemm, Marino, Soroush 12]

63 Latitude Latitude contour one can deform contour of 1/6- and 1/2-BPS WLs from the equator to a latitude on S 2 partly preserving supersymmetry [Cardinali, Griguolo, Martelloni, Seminara 12; MSB, Griguolo, Leoni, Penati, Seminara 14]

64 Latitude Latitude contour one can deform contour of 1/6- and 1/2-BPS WLs from the equator to a latitude on S 2 partly preserving supersymmetry [Cardinali, Griguolo, Martelloni, Seminara 12; MSB, Griguolo, Leoni, Penati, Seminara 14] 2-parameter deformation: geometric and internal independent angles θ, α

65 Latitude Latitude contour one can deform contour of 1/6- and 1/2-BPS WLs from the equator to a latitude on S 2 partly preserving supersymmetry [Cardinali, Griguolo, Martelloni, Seminara 12; MSB, Griguolo, Leoni, Penati, Seminara 14] 2-parameter deformation: geometric and internal independent angles θ, α one obtains a bosonic 1/12-BPS W B and a fermionic 1/6-BPS W F WLs conjectured to be cohomologically equivalent (similar to undeformed case)

66 Latitude Latitude contour one can deform contour of 1/6- and 1/2-BPS WLs from the equator to a latitude on S 2 partly preserving supersymmetry [Cardinali, Griguolo, Martelloni, Seminara 12; MSB, Griguolo, Leoni, Penati, Seminara 14] 2-parameter deformation: geometric and internal independent angles θ, α one obtains a bosonic 1/12-BPS W B and a fermionic 1/6-BPS W F WLs conjectured to be cohomologically equivalent (similar to undeformed case) computed up to second order in PT: vev is NOT given by a simple rescaling of the coupling (unlike N = 4 SYM) vev depends on a single parameter combination ν = sin 2α cos θ W B (θ, α) = w B (ν) W F (θ, α) = w F (ν)

67 Latitude Latitude contour one can deform contour of 1/6- and 1/2-BPS WLs from the equator to a latitude on S 2 partly preserving supersymmetry [Cardinali, Griguolo, Martelloni, Seminara 12; MSB, Griguolo, Leoni, Penati, Seminara 14] 2-parameter deformation: geometric and internal independent angles θ, α one obtains a bosonic 1/12-BPS W B and a fermionic 1/6-BPS W F WLs conjectured to be cohomologically equivalent (similar to undeformed case) computed up to second order in PT: vev is NOT given by a simple rescaling of the coupling (unlike N = 4 SYM) vev depends on a single parameter combination ν = sin 2α cos θ W B (θ, α) = w B (ν) W F (θ, α) = w F (ν) strong coupling dual description [Aguilera-Damia, Correa, Silva 14] an exact vev via localization is missing, but might be possible

68 Extremal limit The ABJ theory [MSB, Leoni 16] Take the ABJ model, with gauge group U(N 1) k U(N 2) k N 1 N 2 < k

69 The ABJ theory Extremal limit [MSB, Leoni 16] Take the ABJ model, with gauge group U(N 1) k U(N 2) k N 1 N 2 < k Consider 1/6-BPS (bosonic) WL relative to U(N 1): in the N 1 N 2 limit the WL reduces to pure CS exact vev [Witten 89] in the opposite limit N 2 N 1 the leading approximation in N 2 is given by two classes of diagrams to all orders in k (λ i N i /k) ẋ1 ẋ2 = λ 1 f YM (λ 2) = λ x 1 x f O (λ 2) ẋ 1 ẋ 2 x 1 x 2 2

70 The ABJ theory Extremal limit [MSB, Leoni 16] Take the ABJ model, with gauge group U(N 1) k U(N 2) k N 1 N 2 < k Consider 1/6-BPS (bosonic) WL relative to U(N 1): in the N 1 N 2 limit the WL reduces to pure CS exact vev [Witten 89] in the opposite limit N 2 N 1 the leading approximation in N 2 is given by two classes of diagrams to all orders in k (λ i N i /k) ẋ1 ẋ2 = λ 1 f YM (λ 2) = λ x 1 x f O (λ 2) ẋ 1 ẋ 2 x 1 x 2 2 by requiring SUSY (finiteness on the line) and comparing to localization results for the circular 1/6-BPS WL fixes the exact value of f YM and f O f O (λ 2) = f YM (λ 2) = 1 sin πλ2 π

71 conjecture for ABJ exact h-function [Cavagliá, Gromov, Levkovich-Maslyuk 16] The ABJ theory Extremal limit [MSB, Leoni 16] Take the ABJ model, with gauge group U(N 1) k U(N 2) k N 1 N 2 < k Consider 1/6-BPS (bosonic) WL relative to U(N 1): in the N 1 N 2 limit the WL reduces to pure CS exact vev [Witten 89] in the opposite limit N 2 N 1 the leading approximation in N 2 is given by two classes of diagrams to all orders in k (λ i N i /k) = λ 1 sin πλ 2 π ẋ 1 ẋ 2 sin πλ 2 = λ x 1 x π ẋ 1 ẋ 2 x 1 x 2 2 by requiring SUSY (finiteness on the line) and comparing to localization results for the circular 1/6-BPS WL fixes the exact value of f YM and f O In the extremal limit N f O (λ 2) = f YM (λ 2) = 1 2 N 1 sin πλ2 π can compute vev of 1/6-BPS WL on all smooth contours exact cusp anomalous dimension, Bremsstrahlung, h-function can prove integrability of SU(2) sector

72 Supersymmetric cusps in ABJM model

73 Supersymmetric cusps Construct locally BPS cusps in ABJM with: W [Γ] = 1 [ ( )] N Tr P exp i dτ L 1/6 or 1/2 (τ) evaluated along the contour Γ: two rays intersecting at an angle π φ Γ Edge 1 Edge 2 φ 2 φ 2 Γ : x 0 = 0 x 1 = τ cos φ 2 x 2 = τ sin φ 2 τ L 1/6 is the U(N) Lie-algebra connection for 1/6-BPS WL L 1/2 is the U(N N) Lie-superalgebra superconnection for 1/2-BPS WL for the 1/6-BPS we could not find a globally preserved supercharge for the 1/2-BPS...

74 SUSY preserving cusp configuration The 1/2-BPS cusp One can choose the parameters in such a way that 4/24 of supercharges are preserved on the cusped contour [Griguolo, Martelloni, Poggi, Seminara 12] The fermionic couplings on each straight-line factorize ηim α = n im ηi α and η iα M = n i M η iα On the first edge n 1M = ( ) cos θ sin θ 0 0 η α = (e i φ 4 e i φ 4 ) cos θ ( n 1 M 4 = sin θ 4 0 η1α = i 0 e i φ 4 e i φ 4 while on the second edge n 2M = ( ) cos θ sin θ cos θ n 2 M 4 = sin θ η2α = i ( ) e i φ 4 e i φ 4 M 1 = M 1 = η α 2 = (e i φ 4 e i φ 4 ) ) M 2 = M 2 = cos θ 2 sin θ sin θ 2 cos θ cos θ 2 sin θ sin θ 2 cos θ

75 A second angle The 1/2-BPS cusp The angle θ is the counterpart of φ in R symmetry space: angular separation of the two edges in the internal space CP 3 [Griguolo, Martelloni, Poggi, Seminara 12] The fermionic couplings on each straight-line factorize ηim α = n im ηi α and η iα M = n i M η iα On the first edge n 1M = ( ) cos θ sin θ 0 0 η α = (e i φ 4 e i φ 4 ) cos θ ( n 1 M 4 = sin θ 4 0 η1α = i 0 e i φ 4 e i φ 4 while on the second edge n 2M = ( ) cos θ sin θ cos θ n 2 M 4 = sin θ η2α = i ( ) e i φ 4 e i φ 4 M 1 = M 1 = η α 2 = (e i φ 4 e i φ 4 ) ) M 2 = M 2 = cos θ 2 sin θ sin θ 2 cos θ cos θ 2 sin θ sin θ 2 cos θ

76 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 n=1

77 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 B 1/6 = 1 4π 2 n W 1/6,n 1 n=1 tests: n=1

78 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 n=1 B 1/6 = λ2 2 π2 λ π4 λ λ tests: weak coupling 2-loop [MSB, Griguolo, Leoni, Penati, Seminara 14]

79 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 n=1 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 2π2 λ B 1/6 = 1 ( 1 4π 2 4π 2 + 4π 5 ) +... λ tests: strong coupling: up to subleading order [Aguilera-Damia, Correa, Silva 14]

80 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 B 1/6 = 1 4π 2 n W 1/6,n 1 n=1 tests: postulate exact Bremsstrahlung for 1/2-BPS cusp B 1/2 = 1 4π 2 n W 1/2,n 1 n=1 n=1

81 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 B 1/6 = 1 4π 2 n W 1/6,n 1 n=1 tests: postulate exact Bremsstrahlung for 1/2-BPS cusp n=1 B 1/2 = 1 4π 2 n W 1/2,n 1 n=1 = W 1/6,n + ( 1) n W 1/6,n 1 n=1 use cohomological equivalence

82 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 B 1/6 = 1 4π 2 n W 1/6,n 1 n=1 tests: postulate exact Bremsstrahlung for 1/2-BPS cusp n=1 B 1/2 = 1 4π 2 n W 1/2,n 1 n=1 = W 1/6,n + ( 1) n W 1/6,n 1 n=1 use cohomological equivalence B 1/2 is an odd function of k (B 1/2 < 0 for k < 0??)

83 Exact Bremsstrahlung for 1/6-BPS [Lewkowycz, Maldacena 13] Consider small φ limit of cusp constructed from 2 locally 1/6-BPS rays (not a supersymmetric configuration globally, though): connect Bremsstrahlung to stress tensor 1pt-function connect latter to entanglement entropy of sphere with Wilson line insertion in 3d branched sphere squashed sphere Sb 3 S W = (1 n n) log W (S 3 b= n) trade WL on S 3 b with multiply wound W n on S 3 exact formula for 1/6-BPS lines Bremsstrahlung B 1/6 B 1/6 = 1 4π 2 n W 1/6,n 1 n=1 tests: postulate exact Bremsstrahlung for 1/2-BPS cusp n=1 B 1/2 = 1 4π 2 n W 1/2,n 1 n=1 = W 1/6,n + ( 1) n W 1/6,n 1 n=1 use cohomological equivalence B 1/2 is an odd function of k (B 1/2 < 0 for k < 0??)

84 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F (ν, k, N) 0 ν=1

85 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 (ν, k, N) 0 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 4π 2 [ ν log ( W B (ν) + Ŵ B (ν) ) ν=1 + π 2 tgφ B ]

86 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 (ν, k, N) 0 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 4π 2 [ ν log ( W B (ν) + Ŵ B (ν) ) ν=1 + π 2 tgφ B 3) trade deformation parameter with winding number ( ) ( ) ν log WB (ν) ν + Ŵ B ν=1 n(ν) (ν) ν = n log W1/6,n + Ŵ 1/6,n ν ] ν=1

87 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 (ν, k, N) 0 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 4) the former quantity vanishes and can be neglected

88 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 (ν, k, N) 0 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 4) the former quantity vanishes and can be neglected The exact 1/2-BPS Bremsstrahlung B 1/2 = i W1/6 1 Ŵ 1/6 1 8π W 1/6 1 + Ŵ 1/6 1 agrees with weak coupling calculation, and at strong coupling (up to subleading order!) [Aguilera-Damia, Correa, Silva 14; Forini, Giangreco Puletti, Ohlsson Sax 12]

89 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 (ν, k, N) 0 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 4) the former quantity vanishes and can be neglected The exact 1/2-BPS Bremsstrahlung B 1/2 = i W1/6 1 Ŵ 1/6 1 8π W 1/6 1 + Ŵ 1/6 1 agrees with weak coupling calculation, and at strong coupling (up to subleading order!) [Aguilera-Damia, Correa, Silva 14; Forini, Giangreco Puletti, Ohlsson Sax 12] However...

90 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν NOT PROVEN, though recent B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 progress by Bianchi, Griguolo, (ν, k, N) 0 Preti, Seminara 2) use cohomological equivalence to bosonic WL s B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 4) the former quantity vanishes and can be neglected The exact 1/2-BPS Bremsstrahlung B 1/2 = i W1/6 1 Ŵ 1/6 1 8π W 1/6 1 + Ŵ 1/6 1 agrees with weak coupling calculation, and at strong coupling (up to subleading order!) [Aguilera-Damia, Correa, Silva 14; Forini, Giangreco Puletti, Ohlsson Sax 12] However...

91 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν NOT PROVEN, though recent B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 progress by Bianchi, Griguolo, (ν, k, N) 0 Preti, Seminara 2) use cohomological equivalence to bosonic WL s NOT PROVEN B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 4) the former quantity vanishes and can be neglected The exact 1/2-BPS Bremsstrahlung B 1/2 = i W1/6 1 Ŵ 1/6 1 8π W 1/6 1 + Ŵ 1/6 1 agrees with weak coupling calculation, and at strong coupling (up to subleading order!) [Aguilera-Damia, Correa, Silva 14; Forini, Giangreco Puletti, Ohlsson Sax 12] However...

92 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν NOT PROVEN, though recent B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 progress by Bianchi, Griguolo, (ν, k, N) 0 Preti, Seminara 2) use cohomological equivalence to bosonic WL s NOT PROVEN B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 NOT justified 4) the former quantity vanishes and can be neglected The exact 1/2-BPS Bremsstrahlung B 1/2 = i W1/6 1 Ŵ 1/6 1 8π W 1/6 1 + Ŵ 1/6 1 agrees with weak coupling calculation, and at strong coupling (up to subleading order!) [Aguilera-Damia, Correa, Silva 14; Forini, Giangreco Puletti, Ohlsson Sax 12] However...

93 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν NOT PROVEN, though recent B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 progress by Bianchi, Griguolo, (ν, k, N) 0 Preti, Seminara 2) use cohomological equivalence to bosonic WL s NOT PROVEN B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 NOT justified NOT PROVEN, 4) the former quantity vanishes and can be neglected PT and numerics agrees with weak coupling calculation, The exact 1/2-BPS Bremsstrahlung and at strong coupling (up to subleading order!) B 1/2 = i W1/6 1 Ŵ 1/6 1 [Aguilera-Damia, Correa, Silva 14; 8π W 1/6 1 + Ŵ 1/6 1 Forini, Giangreco Puletti, Ohlsson Sax 12] However...

94 exact 1/2-BPS Bremsstrahlung via latitudes? [MSB, Griguolo, Leoni, Penati and Seminara 14] 1) Let s conjecture that the Bremsstrahlung is obtained by deriving the latitude WL wrt ν NOT PROVEN, though recent B 1/2 (k, N) = 1 4π 2 ν log W F ν=1 progress by Bianchi, Griguolo, (ν, k, N) 0 Preti, Seminara 2) use cohomological equivalence to bosonic WL s NOT PROVEN B 1/2 (k, N) = 1 [ ( ) 4π 2 ν log WB (ν) + Ŵ B (ν) + ν=1 π ] 2 tgφ B 3) trade deformation parameter with winding number ( ) ν log WB (ν) ν + Ŵ B ν=1 (ν) ν = 0 NOT justified NOT PROVEN, 4) the former quantity vanishes and can be neglected PT and numerics agrees with weak coupling calculation, The exact 1/2-BPS Bremsstrahlung which is effectively only 1 loop and at strong coupling (up to subleading order!) B 1/2 = i W1/6 1 Ŵ 1/6 1 [Aguilera-Damia, Correa, Silva 14; 8π W 1/6 1 + Ŵ 1/6 1 Forini, Giangreco Puletti, Ohlsson Sax 12] However...

95 Three-loop prediction Still, assuming the conjecture holds true we can derive a three-loop prediction at weak coupling using the localization expectation value for the 1/6-BPS Wilson loops W1/6 1 = 1 + i π N k + 1 ( 1 + 2N 2) π k + 1 ( 2 6 i N 4 + N 2) π 3 1 k + 3 O(k 5 ) Ŵ1/6 1 = W1/6 1 and plugging them in the above conjecture we find B 1/2 (k, N) = N 8 k π2 N ( N 2 3 ) 48 k 3 + O (k 4) 1-loop and (vanishing) 2-loop contributions agree with direct computation [Griguolo, Martelloni, Poggi and Seminara 12] first non-planar correction at three-loops B 1/2 is an odd function of k (B < 0 for k < 0??) from the conjecture B 1/2 comes entirely from the imaginary part of W 1/6 i.e. from the framing of the Wilson loop

96 Three-loop test

97 Computing a three-loop cusp Eight topologies Fill them in with any possible particle (here also fermions can attach to Wilson line)

98 Computing a three-loop cusp Eight topologies Fill them in with any possible particle (here also fermions can attach to Wilson line) In CS theories: CS theory contains ubiquitous ε tensors (e.g. gauge propagator and vertex, γ-algebra) cusp lies on a plane, only contribution with even number of ε tensors can contribute from the Feynman rules at odd loops: completely gluonic graphs + many more are ruled out

99 Computing a three-loop cusp Eight topologies Fill them in with any possible particle (here also fermions can attach to Wilson line) In CS theories: CS theory contains ubiquitous ε tensors (e.g. gauge propagator and vertex, γ-algebra) cusp lies on a plane, only contribution with even number of ε tensors can contribute from the Feynman rules at odd loops: completely gluonic graphs + many more are ruled out Still... it is a massive computation (if done just bruteforce)

100 Computing the cusp at 0 angle ( Use BPS condition: Γ 1/2 = B 1/2 φ 2 θ 2) φ, θ 1 B 1/2 = θ Γ 2 1/2 = 1 2 φ=θ=0 2 φ Γ 2 1/2 φ=θ=0 to extract Bremsstrahlung from cusp at φ = 0. θ φ

101 Computing the cusp at 0 angle ( Use BPS condition: Γ 1/2 = B 1/2 φ 2 θ 2) φ, θ 1 B 1/2 = θ Γ 2 1/2 φ=θ=0 to extract Bremsstrahlung from cusp at φ = 0. θ φ Advantages: reduced number of diagrams diagram algebra is easier integrals are much easier: propagator-type (GPXT) [Chetyrkin, Kataev, Tkachov 80]

102 Computing the cusp at 0 angle ( Use BPS condition: Γ 1/2 = B 1/2 φ 2 θ 2) φ, θ 1 B 1/2 = θ Γ 2 1/2 φ=θ=0 to extract Bremsstrahlung from cusp at φ = 0. θ φ Advantages: reduced number of diagrams diagram algebra is easier integrals are much easier: propagator-type (GPXT) [Chetyrkin, Kataev, Tkachov 80] Can use the Korchemsky-Radyushkin prescription [Korchemsky, Radyushkin 87] log W (θ) = log V (θ) log V (0) φ=0 φ=0 to compute only the 1PI part.

103 Renormalization of the cusp divergences the expectation value of the cusped WL suffers from both IR and UV divergences regulate IR divergences with cutoff L use dimensional regularization for UV divergence d = 3 2ɛ (DRED) renormalize the UV div multiplicatively W R (θ) = Z 1 cusp W (θ) extract the cusp anomalous dimension Γ cusp(k, N) = φ=0 d log Zcusp d log µ φ=0 in our case this is restricted to the φ = 0 limit simple poles in ɛ µ = renormalization scale extract Bremsstrahlung function, taking the double derivative B 1/2 (k, N) = θ Γ 1/2 (k, N, φ = 0, θ) θ=0