Double Field Theory Double Fun?

Transcription

1 Double Field Theory Double Fun? Falk Haßler based on..., , , ,... University of North Carolina at Chapel Hill City University of New York March 3, 2016

2 The big picture Phenomenology SUGRA target space Type I SO(32) het. world sheet Type IIA M-theory E 8 E 8 het. Type IIB

3 The big picture Phenomenology SUGRA DFT target space Type IIA Type I T M-theory Type IIB SO(32) het. T E 8 E 8 het. world sheet T = T-duality

4 The big picture Phenomenology SUGRA DFT EFT target space Type I S SO(32) het. world sheet T Type IIA S M-theory S E 8 E 8 het. T = T-duality T Type IIB S S = S-duality

5 What is interesting? interesting rigorous

6 What is interesting? interesting closure constraint generalized Scherk-Schwarz non-geometric backgrounds rigorous strong constraint Closed String Field Theory generalized geometry

7 What is interesting? interesting closure constraint generalized Scherk-Schwarz non-geometric backgrounds rigorous strong constraint Closed String Field Theory generalized geometry

8 SUGRA closed strings in D-dim. flat space with momentum p i truncate all massive excitations match scattering amplitudes of strings with EFT S NS = d D x ge 2φ ( R + 4 i φ i φ 1 12 H ijkh ijk ) p i

9 SUGRA closed strings in D-dim. flat space with momentum p i truncate all massive excitations match scattering amplitudes of strings with EFT S NS = d D x ge 2φ ( R + 4 i φ i φ 1 12 H ijkh ijk ) p i

10 SUGRA closed strings in D-dim. flat space with momentum p i truncate all massive excitations match scattering amplitudes of strings with EFT S NS = d D x ge 2φ ( R + 4 i φ i φ 1 12 H ijkh ijk ) g ij (p k ) φ(p i )

11 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] closed strings on a flat torus with momentum p i and winding w i combine conjugated variables x i and x i into X M = ( x i x i) repeat steps from SUGRA derivation S DFT = d 2D X e 2d R(H MN, d) fields are constrained by strong constraint M M = 0 p i w i

12 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] X M = ( x i x i) S DFT = d 2D X e 2d R(H MN, d)

13 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] X M = ( x i x i) d = φ 1 2 log g S DFT = d 2D X e 2d R(H MN, d)

14 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] X M = ( x i x i) d = φ 1 2 log g S DFT = d 2D X e 2d R(H MN, d) R = 4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK

15 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] M = ( i i ) X M = ( x i x i) d = φ 1 2 log g S DFT = d 2D X e 2d R(H MN, d) R = 4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK

16 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] M = ( i i ) X M = ( x i x i) d = φ 1 2 log g S DFT = d 2D X e 2d R(H MN, d) R = 4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK ( H MN gij B = ik g kl B lj B ik g kj ) g ik B kj g ij O(D, D) T-duality

17 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] ( ) ( ) 0 δ i lower/raise indices with η MN = j δ j and η MN 0 δ j = i i 0 δj i 0 M = ( i i ) X M = ( x i x i) d = φ 1 2 log g S DFT = d 2D X e 2d R(H MN, d) R = 4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK ( H MN gij B = ik g kl B lj B ik g kj ) g ik B kj g ij O(D, D) T-duality

18 Gauge transformations [Hull and Zwiebach, 2009] generalized Lie derivative combines } 1. diffeomorphisms available in SUGRA 2. B-field gauge transformations 3. β-field gauge transformations L λ H MN = λ P P H MN + ( M λ P P λ M )H PN + ( N λ P P λ N )H MP L λ d = λ M M d Mλ M closure of algebra L λ1 L λ2 L λ2 L λ1 = L λ12 with λ 12 = [λ 1, λ 2 ] C only if strong constraint holds

19 Rigorous, more rigorous, generalized geometry trivial solution of SC i = 0 SUGRA x i are coordinated on manifold M interprete components of doubled vector ξ M = ( ξ m ξ m ) as ξ m Γ(TM) and ξ m Γ(T M)

20 Rigorous, more rigorous, generalized geometry trivial solution of SC i = 0 SUGRA x i are coordinated on manifold M interprete components of doubled vector ξ M = ( ξ m ξ m ) as ξ m Γ(TM) and ξ m Γ(T M) generalized Lie derivative = Dorfman bracket [X + ξ, Y + η] = [X, Y ] + L X η i Y dξ on generalized tangent space TM T M O(D, D) metric = bilinear form X + ξ, Y + η = η(x) + ξ(y ) Courant algebroids, generalized complex structure,...

21 M-theory and Exceptional Field Theory winding of F1 string wrapping of M2 brane example T 4 : X I = ( x i x ij ) has /2 = 10 components

22 M-theory and Exceptional Field Theory winding of F1 string wrapping of M2 brane example T 4 : X I = ( x i ) x ij has /2 = 10 components transforms in 10 of U-duality group SL(5) D G SL(5) = E 4(4) SO(5, 5) = E 5(5) E 6(6) E 7(7) X I SC

23 M-theory and Exceptional Field Theory winding of F1 string wrapping of M2 brane example T 4 : X I = ( x i transforms in 10 of U-duality group SL(5) x ij ) has /2 = 10 components D G SL(5) = E 4(4) SO(5, 5) = E 5(5) E 6(6) E 7(7) X I SC SC = section condition generalized Lie derivative K MN IJ M N = 0 L ξ V I = ξ J J V I V J J ξ I + K IJ MN Jξ M V N

24 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003]

25 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes Part I: Introduction to DFT/EFT Part II: Non-geometric backgrounds Part III: CSFT on group manifolds Summary

26 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes T+ DF Part I: Introduction to DFT/EFT Part II: Non-geometric backgrounds SC Part III: CSFT on group manifolds Summary

27 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes T+ DF DFT + CC Part I: Introduction to DFT/EFT Part II: Non-geometric backgrounds SC Part III: CSFT on group manifolds Summary

28 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] CYs SUGRA backgrounds with fluxes DFT + CC DFT + SC [Dabholkar and Hull, 2003,Condeescu, Florakis, Kounnas, and Lüst, 2013,Haßler and Lüst, 2014]

29 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] CYs SUGRA backgrounds with fluxes DFT + CC DFT + SC [Dabholkar and Hull, 2003,Condeescu, Florakis, Kounnas, and Lüst, 2013,Haßler and Lüst, 2014]

30 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] CYs SUGRA backgrounds with fluxes DFT + CC DFT + SC [Dabholkar and Hull, 2003,Condeescu, Florakis, Kounnas, and Lüst, 2013,Haßler and Lüst, 2014]

31 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] String geometry also non-geometry CYs SUGRA backgrounds with fluxes DFT + CC DFT + SC [Dabholkar and Hull, 2003,Condeescu, Florakis, Kounnas, and Lüst, 2013,Haßler and Lüst, 2014]

32 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] String Theory simplification (truncation) matching amplitudes SUGRA CSFT i = 0 Double Field Theory

33 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT i = 0 with fluxes Double Field Theory D d SUGRA gauged SUGRA

34 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT i = 0 with fluxes Double Field Theory D d SUGRA embedding tensor gauged SUGRA

35 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT Double Field Theory i = 0 Scherk-Schwarz strong constraint with fluxes closure constraint? construction twist D d SUGRA embedding tensor gauged SUGRA

36 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT Double Field Theory i = 0 Scherk-Schwarz strong constraint with fluxes closure constraint? construction twist D d SUGRA embedding tensor gauged SUGRA eom vacuum

37 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] hints, but in general unknown String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT Double Field Theory i = 0 Scherk-Schwarz strong constraint with fluxes closure constraint? construction twist uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

38 S 3, the elephant in the room switch on H-flux, solve eom

39 S 3, the elephant in the room switch on H-flux, solve eom no T 3 but S 3 L

40 S 3, the elephant in the room switch on H-flux, solve eom no T 3 but S 3 L

41 S 3, the elephant in the room switch on H-flux, solve eom no T 3 but S 3 L WINDING?

42 S 3, the elephant in the room switch on H-flux, solve eom no T 3 but S 3 DFT WZW DFT strong constraint action L symmetries WINDING?

43 S 3, the elephant in the room switch on H-flux, solve eom no T 3 but S 3 DFT WZW DFT strong constraint action L WINDING? symmetries twist gen. Scherk-Schwarz genuinely non-geometric backgr.

44 DFT WZW = DFT on group manifolds Use group manifold instead of a torus to derive DFT! { T D = U(1) D + includes S 3 = SU(2) + CFT exactly solvable + flux backgrounds with const. fluxes

45 DFT WZW = DFT on group manifolds Use group manifold instead of a torus to derive DFT! { T D = U(1) D + includes S 3 = SU(2) + CFT exactly solvable + flux backgrounds with const. fluxes Double Field Theory = treat left and right mover independently 2D independent coordinates

46 DFT WZW = DFT on group manifolds Use group manifold instead of a torus to derive DFT! { T D = U(1) D + includes S 3 = SU(2) + CFT exactly solvable + flux backgrounds with const. fluxes x i = 1 2 (x i L + x i R) x i = 1 2 (x Li x Ri ) Double Field Theory = treat left and right mover independently 2D independent coordinates

47 DFT WZW = DFT on group manifolds Use group manifold instead of a torus to derive DFT! { T D = U(1) D + includes S 3 = SU(2) + CFT exactly solvable + flux backgrounds with const. fluxes x i = 1 2 (x i L + x i R) x i = 1 2 (x Li x Ri ) Double Field Theory = treat left and right mover independently 2D independent coordinates Questions about DFT WZW What are the covariant objects? How is it connected to DFT? Does it make non-abelian duality manifest? not trivial

48 WZW model & Kač-Moody algebra [Witten, 1983,Walton, 1999] g G, a compact simply connected Lie group S WZW = 1 2πα d 2 zk(g 1 g, g 1 g) + S WZ (g) M

49 WZW model & Kač-Moody algebra [Witten, 1983,Walton, 1999] g G, a compact simply connected Lie group S WZW = 1 2πα d 2 zk(g 1 g, g 1 g) + S WZ (g) M metric and 3-form flux in flat indices η ab := K(t a, t b ) and F abc := K([t a, t b ], t c ) D chiral and D anti-chiral Noether currents (=2D indep. currents) j a (z) = 2 α K( gg 1, t a ) and jā( z) = 2 α K(g 1 g, tā)

50 WZW model & Kač-Moody algebra [Witten, 1983,Walton, 1999] g G, a compact simply connected Lie group S WZW = 1 2πα d 2 zk(g 1 g, g 1 g) + S WZ (g) M metric and 3-form flux in flat indices η ab := K(t a, t b ) and F abc := K([t a, t b ], t c ) D chiral and D anti-chiral Noether currents (=2D indep. currents) j a (z) = 2 α K( gg 1, t a ) and jā( z) = 2 α K(g 1 g, tā) radial quantization j a (z)j b (w) = α 2 1 (z w) 2 η ab + 1 z w F ab c j c (z) +...

51 Action tree level action in CSFT [Zwiebach, 1993] (2κ 2 )S = 2 ( Ψ c α 0 Q Ψ + 1 ) 3 {Ψ, Ψ, Ψ}

52 Action tree level action in CSFT [Zwiebach, 1993] (2κ 2 )S = 2 ( Ψ c α 0 Q Ψ + 1 ) 3 {Ψ, Ψ, Ψ} string field for massless excitations [Hull and Zwiebach, 2009] Ψ = R [ α 4 ɛa b(r)j a 1 j b 1 c 1 c 1 + e(r)c 1 c 1 + ē(r) c 1 c 1 + α ( f a (R)c c 1j a 1 + f b(r)c + 0 c ) ] 1 j b 1 φ R R is highest weight of g g representation

53 Action tree level action in CSFT [Zwiebach, 1993] (2κ 2 )S = 2 ( Ψ c α 0 Q Ψ + 1 ) 3 {Ψ, Ψ, Ψ} string field for massless excitations [Hull and Zwiebach, 2009] Ψ = R [ α 4 ɛa b(r)j a 1 j b 1 c 1 c 1 + e(r)c 1 c 1 + ē(r) c 1 c 1 + α ( f a (R)c c 1j a 1 + f b(r)c + 0 c ) ] 1 j b 1 φ R R is highest weight of g g representation BRST operator (L m from Sugawara construction) Q = m ( : c m L m : : c ml gh m : ) + anti-chiral

54 Geometric representation of primary fields (k ) flat derivative D a = e a i i with e a i = K(g 1 i g, t a ) operator algebra geometry (j a 0 D a ) L 0 φ R = j a 0 j a 0 φ R = h R φ R D a D a Y R (x i ) = h R Y R (x i ) [j a 0, j b 0 ] = F ab c j c 0 [D a, D b ] = F ab c D c e(r) φ R R e(r)y R (x i ) := e(x i ) R

55 Geometric representation of primary fields (k ) flat derivative D a = e a i i with e a i = K(g 1 i g, t a ) operator algebra geometry (j a 0 D a ) L 0 φ R = j a 0 j a 0 φ R = h R φ R D a D a Y R (x i ) = h R Y R (x i ) [j a 0, j b 0 ] = F ab c j c 0 [D a, D b ] = F ab c D c e(r) φ R R ( E I ea i ) 0 A = 0 eā ī e(r)y R (x i ) := e(x i ) R ( ) ηab 0 S AB = 2 0 ηā b ( ) ηab 0 η AB = 2 0 ηā b

56 Weak constraint (level matching), later strong constraint level matched string field (L 0 L 0 ) Ψ = 0 requires (D a D a DāDā) = 0 with {ɛ a b, e, ē, f a, f b} rewritten in terms of η AB and D A = ( D a η AB D A D B = D A D A = 0 ) Dā

57 Weak constraint (level matching), later strong constraint level matched string field (L 0 L 0 ) Ψ = 0 requires (D a D a DāDā) = 0 with {ɛ a b, e, ē, f a, f b} rewritten in terms of η AB and D A = ( D a η AB D A D B = D A D A = 0 change to curved indices using E A M ) Dā ( M M 2 M d M ) = 0 with d = φ 1 2 log g term which is absent in DFT adsorb in cov. derivative M M = 0 with M V N = M V N + Γ MK N V K, Γ MK M = 2 K d

58 Results (leading order k 1 ) calculate quadratic and cubic string functions integrate out auxiliary fields f a and f b perform field redefinition (2κ 2 )S = d 2D X [ 1 H 4 ɛ a b ɛa b ɛ ( a b F ac Dēɛ d bɛ d cē + F b c dd e ɛ a dɛ ) e c 1 ] 12 F ace F b d f ɛ a bɛ c dɛ e f +... terms e.g. potential vanish in abelian limit F abc 0 and Fā b c 0

59 Gauge transformations tree level gauge transformation in CSFT [Zwiebach, 1993] δ Λ Ψ = Q Λ + [Λ, Ψ] 0 + string field for gauge parameter [Hull and Zwiebach, 2009] Λ = R [ 1 2 λa (R)j a 1 c λ b(r)j b 1 c 1 + µ(r)c + 0 ] φ R

60 Gauge transformations tree level gauge transformation in CSFT [Zwiebach, 1993] δ Λ Ψ = Q Λ + [Λ, Ψ] 0 + string field for gauge parameter [Hull and Zwiebach, 2009] Λ = R [ 1 2 λa (R)j a 1 c λ b(r)j b 1 c 1 + µ(r)c + 0 ] φ R after field redefinition and µ gauge fixing δ λ ɛ a b =D bλ a + 1 Da λ 2[ c ɛ c b D c λ a ɛ c b + λ c D c ɛ a b+f d ac λ c ] ɛ d b D a λ b + 2[ 1 D bλ c ɛ a c D c λ bɛ a c + λ c D c ɛ dλ c ] a b+f b c ɛ a d δ λ d = 1 4 D aλ a λ ad a d 1 4 D āλā λ ādād

61 Doubled objects promising results, but bulky Rewrite action/gauge trafo in terms of doubled object + simplifies expressions considerably + extrapolation from cubic to all order in fields

62 Doubled objects object doubled version ( ηab 0 η ab, ηā b η AB = 2 0 ηā b ( e i a, eāī E I ea i ) 0 A = 0 eā ī ) ( ) ηab 0 S AB = 2 0 ηā b D a, Dā D A = ( D a Dā) = EA I I with I = ( ) i ī

63 Doubled objects object doubled version ( ηab 0 η ab, ηā b η AB = 2 0 ηā b ( e i a, eāī E I ea i ) 0 A = 0 eā ī ) ( ) ηab 0 S AB = 2 0 ηā b D a, Dā D A = ( D a Dā) = EA I I with I = ( ) i ī ξ i, ξī ( ) ξ I = ξ i ξī F c ab, Fā b c F c ab F C AB = c Fā b 0 otherw. [D A, D B ] = F AB C D C

64 Gauge transformations doubled version of fluctuations ɛ a b ( ) ɛ AB 0 ɛ a b = ɛāb 0 with ɛ a b = (ɛ T ) ba generate generalized metric [Hohm, Hull, and Zwiebach, 2010] H AB = S AB + ɛ AB ɛac S CD ɛ DB + = exp(ɛ AB ) with the defining property H AC η CD H DB = η AB

65 Gauge transformations doubled version of fluctuations ɛ a b ( ) ɛ AB 0 ɛ a b = ɛāb 0 with ɛ a b = (ɛ T ) ba generate generalized metric [Hohm, Hull, and Zwiebach, 2010] H AB = S AB + ɛ AB ɛac S CD ɛ DB + = exp(ɛ AB ) with the defining property H AC η CD H DB = η AB generalized Lie derivative [Hull and Zwiebach, 2009,Grana and Marques, 2012] L λ H AB = λ C D C H AB + (D A λ C D C λ A )H CB + (D B λ C D C λ B )H AC + F A CDλ C H DB + F B CDλ C H AD

66 δ λ H AB = L λ H AB + O(ɛ 2 ) similar for the generalized dilaton d

67 δ λ H AB = L λ H AB + O(ɛ 2 ) similar for the generalized dilaton d introduce covariant derivative A V B = D A V B F B ACV C generalized Lie derivative, e.g. for vector L λ V A = λ B B V A + ( A λ B B λ A )V B instead of L λ V I = λ J J V I + ( I λ J J λ I )V J in traditional DFT

68 Gauge algebra [λ 1, λ 2 ] A C = λb 1 Bλ A λb 1 A λ 2 B (1 2) algebra closes up to a trivial gauge transformation if 1. fluctuations and parameter fulfill strong constraint D A D A 2. background fulfills Jacobi identity F E[AB F E C]D = 0

69 Gauge algebra [λ 1, λ 2 ] A C = λb 1 Bλ A λb 1 A λ 2 B (1 2) algebra closes up to a trivial gauge transformation if 1. fluctuations and parameter fulfill strong constraint D A D A 2. background fulfills Jacobi identity F E[AB F E C]D = 0 no strong constraint required for background

70 Action X I = ( x i ) x ī S = d 2D X e 2d R(H MN, d)

71 Action X I = ( x i ) x ī S = d 2D X e 2d R(H MN, d) d = d 1 2 log H

72 Action X I = ( x i ) x ī S = d 2D X e 2d R(H MN, d) d = d 1 2 log H R =4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK F MKLF N KL H MN

73 Action X I = ( x i ) x ī S = d 2D X e 2d R(H MN, d) d = d 1 2 log H R =4H MN M N d M N H MN 4H MN M d N d + 4 M H MN N d HMN M H KL N H KL 1 2 HMN N H KL L H MK F MKLF N KL H MN lower indices with η MN = E A ME B Nη AB const. H IJ = E A ME B NS AB background generalized metric M d = M d M H KL = M H KL + Γ MJ K H JL + Γ MJ L H KJ

74 Reminder: Generalized Scherk-Schwarz compactification String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT with fluxes DFT WZW Scherk-Schwarz strong constraint Jacobi identity uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

75 Reminder: Generalized Scherk-Schwarz compactification String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT with fluxes DFT WZW Scherk-Schwarz strong constraint Jacobi identity construction twist uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

76 Reminder: Generalized Scherk-Schwarz compactification String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT with fluxes DFT WZW Scherk-Schwarz strong constraint Jacobi identity construction twist uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

77 Summary DFT is able to satisfy various tastes fancy mathematics pure string theory interesting phenomenology