Holography, Soft Theorems, and Infinite Dimensional Symmetries of Flat Space

Transcription

1 Holography, Soft Theorems, and Infinite Dimensional Symmetries of Flat Space Clifford Cheung w/ Anton de la Fuente & Raman Sundrum (1607.xxxxx)

2 Asymptotic Symmetries of Yang-Mills Theory Andrew Strominger On BMS Invariance of Gravitational Scattering Andrew Strominger BMS Supertranslations and Weinberg s Soft Graviton Theorem Temple He, Vyacheslav Lysov, Prahar Mitra and Andrew Strominger pril 17, 2014 Evidence for a New Soft Graviton Theorem Freddy Cachazo and Andrew Strominger

3 Some fascinating claims of Strominger et al: flat space gauge and gravity theories enjoy -dimensional symmetries on the boundary soft theorems are the Ward identities for these -dimensional symmetries Generally applicable: QED, YM, gravity.

4 Asymptotic Symmetries of Yang-Mills Theory Andrew Strominger Radcli e Institute for Advanced Study, Harvard University, Cambridge, MA 02138, USA Abstract Asymptotic symmetries at future null infinity (I + ) of Minkowski space for electrodynamics with massless charged fields, as well as non-abelian gauge theories with gauge group G, are considered at the semiclassical level. The possibility of charge/color flux through I + suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a G Kac-Moody symmetry whose generators are large gauge transformations which approach locally holomorphic functions on the conformal two-sphere at I + and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of I +. The current Ward identities include Weinberg s soft photon theorem and its colored extension.!!

5 Soft factors have well-known universal form.

6 Soft factors have well-known universal form. = X n q n k n k k n

7 The -dimensional symmetry parameterized by directions of particle emission.

8 The -dimensional symmetry parameterized by directions of particle emission. emitted in any direction on the celestial sphere

9 However, the story has some fine print: soft gluons must be all (+) or all (-) loop subtlety for subleading soft theorems The actual principle behind these caveats is not remotely obvious from the literature!

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12 Goal: elucidate the underlying origin of miraculous -dimensional symmetries (no abra cadabra allowed) Approach: systematically derive the whole story via the AdS/CFT dictionary (AdS from Minkowski?) Result: explicit construction for 2D CFT dual of 4D amplitudes in gauge and gravity

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18 foilate by proper distance from origin x µ x µ = e 2

19 (Euclidean) AdS3 foilate by proper distance from origin x µ x µ = e 2

20 Minkowski coordinates: x µ =(T,X,Y,Z) ds 2 = dt 2 + dx 2 + dy 2 + dz 2

21 Minkowski coordinates: x µ =(T,X,Y,Z) ds 2 = dt 2 + dx 2 + dy 2 + dz 2 Milne x I =(,,z, z) coordinates: x i =(,z, z) ds 2 = e 2 d (d 2 + dzd z)

22 Minkowski coordinates: x µ =(T,X,Y,Z) ds 2 = dt 2 + dx 2 + dy 2 + dz 2 Milne x I =(,,z, z) coordinates: x i =(,z, z) ds 2 = e 2 d (d 2 + dzd z) Poincare coordinates for AdS3

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25 2D boundary at! 0 is labelled by

26 Boundary of AdS3 is comprised of null rays. x µ!0 = e k µ + O( ) Automatically in CP 1 spinor helicity variables: k µ = µ =(z,1) =( z,1)

27 Use a gauge in which polarizations are simply µ = µ k µ (+) µ = µ z k µ ( ) where we define =(1, 0) and =(0, 1)

28 The usual Lorentz invariants are simply h12i = z 1 z 2 [12] = z 1 z 2 and the Weinberg soft factor takes the form k n = 1 k k n z z n

29 There is an AdS3 slice for each choice of but to which do we apply AdS3 / CFT2?

30 gauge theory

31 To seek clues, just boldly compute the bulk boundary propagator via AdS/CFT dictionary. bulk AdS3 boundary CFT2

32 To seek clues, just boldly compute the bulk boundary propagator via AdS/CFT dictionary. bulk AdS3 (z, z) x boundary CFT2

33 To seek clues, just boldly compute the bulk boundary propagator via AdS/CFT dictionary. bulk K i (z, z; x) AdS3 (z, z) x boundary CFT2

34 Lift propagator to Minkowski coordinates: K I =(K,K i )

35 Lift propagator to Minkowski coordinates: temporal gauge K I =(K,K i )

36 Lift propagator to Minkowski coordinates: temporal gauge K I =(K,K i ) K µ = dxi dx K = x k [ µ] µ I (k x) 2

37 Lift propagator to Minkowski coordinates: temporal gauge K I =(K,K i ) K µ = dxi dx K = x k [ µ] µ I (k x) 2 µ x k x

38 Lift propagator to Minkowski coordinates: temporal gauge K I =(K,K i ) K µ = dxi dx K = x k [ µ] µ I (k x) 2 µ x k x 1) total derivative 2) independent of Propagator is zero mode, i.e. Milne soft!

39 A general Witten diagram takes the form: Z d 4 xk µ (x)j µ (x)

40 A general Witten diagram takes the form: Z d 4 xk µ (x)j µ (x) rest of the Witten diagram

41 A general Witten diagram takes the form: Z d 4 xk µ (x)j µ (x) rest of the Witten diagram = Z d 4 µ x J µ (x) k x Z x = d 4 µ J µ (x) k x

42 A general Witten diagram takes the form: Z d 4 xk µ (x)j µ (x) rest of the Witten diagram = Z d 4 µ x J µ (x) k x Z x = d 4 µ J µ (x) =0 k x

43 Must include non-conservation of charge from in and out states on the boundary. (z, z) K i (z, z; x)

44 Must include non-conservation of charge from in and out states on the boundary. (z, z) K i (z, z; x) J i (x)

45 Must include non-conservation of charge from in and out states on the boundary. on the boundary (z, z) K i (z, z; x) J i µ J µ (x) = X n q n 4 (x x n ) on the boundary

46 Z x = d 4 µ J µ (x) k x = X n q n x n k x n = X n z q n z n bulk boundary propagator = soft factor!

47 Witten diagram Z d 4 xk µ (x)j µ (x)

48 Z Witten diagram d 4 xk µ (x)j µ (x) = leading soft theorem X n q n x n k x n

49 Z Witten diagram d 4 xk µ (x)j µ (x) = leading soft theorem X n q n x n k x n = X n z q n z n

50 Z Witten diagram d 4 xk µ (x)j µ (x) = leading soft theorem X n q n x n k x n = CFT Ward id hj(z)oi X n z q n z n

51 Which AdS gauge theory is this? Clue: the bulk boundary propagator is pure gauge, i.e. non-dynamical. the AdS gauge theory has topological sector To determine the theory, just compute tree Witten diagrams (i.e. solve classical equations of motion). But there is a shortcut

52 Consider sector of (+) soft gauge bosons. (+) (+) (+) (+) (+) E = ib F = i F These soft modes are self-dual on boundary.

53 Away from hard sources, any self-dual gauge configurations remain so. F = i F DF =0 D F =0 So on the classical solutions, we can impose the self-dual condition in the bulk. Z Z d 4 xf 2 i d 4 xf F

54 The bulk action is then a total derivative. Z Z d/d since A =0 and F i =0 A ^ F + 23 A ^ A ^ A d 4 xf F = d 4 xd Z = d 3 xa^ F A ^ A ^ A AdS3 gauge theory Chern-Simons theory. CFT2 dual Wess-Zumino-Witten model.

55 YM (4D) Z d 4 xf 2 = CS (3D) fixed helicity soft limit Z d 3 xa^f A ^ A ^ A AdS/CFT dictionary Z WZW (2D) d 2 z U z U +...

56 WZW has an -dimensional symmetry: j a (z) = X m2z z m 1 j a z j a (z) =0 corresponding to a Kac-Moody algebra. The operator product expansion is: j a (z)o b (w) f abc O c (w) z w ~ Weinberg soft factor

57 CFT structure implies that fixed helicity soft amplitudes are MHV strings! soft n 1 1 n hard

58 CFT structure implies that fixed helicity soft amplitudes are MHV strings! soft n 1 1 n hard 1 (z 1 z 2 )(z 2 z 3 )...(z n 1 z n )

59 CFT structure implies that fixed helicity soft amplitudes are MHV strings! soft n 1 1 n hard 1 h12ih23i...hn 1 ni 1 (z 1 z 2 )(z 2 z 3 )...(z n 1 z n )

60 In hindsight, this structure is obvious: h13i h12ih23i h14i h13ih34i = h14i h12ih23ih34i h24i h23ih34i h14i h12ih24i = h14i h12ih23ih34i

61 There s also a connection to EM memory. R J i (x) Z Z Z I q = d ds i J i = d d`ib i

62 There s also a connection to EM memory. R J i (x) Z Z Z I q = d ds i J i = d d`ib i

63 The hard particle radiation field is either selfdual or anti-self-dual, so ib i = E i A i So electromagnetic memory simplifies to: I q = i / 1 2 dz A z dz j(z) =j 0 ~ Aharonov-Bohm Kac-Moody ~ generator

64 EM memory is measured by Aharonov-Bohm. j 0 R J i (x) Aharonov-Bohm is a Kac-Moody generator.

65 CFT (2D) YM (4D) conserved current non-conserved operator correlation function OPE -dimensional symmetry Kac-Moody generator j a (z) O a (z) hj a (z)o b (w) i j a (z)o b (w) f abc O c (w) z z j a (z) =0 j a 0 soft gluon hard gluon scattering amplitude Weinberg soft factor holomorphy electromagnetic memory

66 gravity

67 The bulk-boundary propagator is: K µ = x2 (k x) 4 (x k [ µ] )(x k [ ] )

68 The bulk-boundary propagator is: K µ = x2 (k x) 4 (x k [ µ] )(x k [ ] ) = x 2 K µ K double copy of gauge

69 The bulk-boundary propagator is: K µ = x2 (k x) 4 (x k [ µ] )(x k [ ] ) = x 2 K µ K double copy of gauge µ µ total derivative

70 The bulk-boundary propagator is: K µ = x2 (k x) 4 (x k [ µ] )(x k [ ] ) = x 2 K µ K double copy of gauge µ µ total derivative Now simply repeat our earlier procedure.

71 Witten diagram Z d 4 xk µ (x)t µ (x)

72 Witten diagram Z d 4 xk µ (x)t µ (x) = subleading soft 3 Z z d 2 z 0 z 0 X n 0 x n k 0 k x [µ 0 0 ] J n µ n

73 Witten diagram Z d 4 xk µ (x)t µ (x) = subleading soft 3 Z z d 2 z 0 z 0 X n 0 x n k 0 k x [µ 0 0 ] J n µ n = X n (z z n ) log 2 n + 1 z n

74 Witten diagram Z d 4 xk µ (x)t µ (x) = subleading soft 3 Z z d 2 z 0 z 0 X n 0 x n k 0 k x [µ 0 0 ] J n µ n = CFT Ward id ht(z)oi X n (z z n ) log 2 n + 1 z n

75 There is a relation between gravitational memory and the Virasoro generators.! X ht(z)o(w)i z!w = m L m (z w) m+2 ho(w)i L 1 i(k 2 + ij 2 ) (K 1 + ij 1 ) L 0 K 3 + ij 3 L 1 i(k 2 + ij 2 )+(K 1 + ij 1 ) (boosts and rotations about hard particle)

76 Gravitational memory relates to local boost. L 0 R T ij (x) This local boost is a Virasoro generator.

77 conclusions Proposed a duality between 4D gauge and gravity in flat space and a 2D CFT on the celestial sphere. Construction comes from faithful application of AdS3/CFT2 to a hyperbolic foliation of flat space. Soft gluons and gravitons are conserved currents while soft theorems are CFT Ward identities. Kac-Moody and Virasoro encode -dimensional symmetry and memory effects in flat space.

78 future directions Translate all aspects of the AdS3/CFT2 dictionary. Construct all operators (hard + soft) in the CFT. Understand higher-loop, higher-dimension, etc. Study implications for the information paradox.

79 thank you!