String Geometry Beyond the Torus

Transcription

1 String Geometry Beyond the Torus Falk Haßler based on arxiv: with Ralph Blumenhagen and Dieter Lüst Arnold Sommerfeld Center LMU Munich October 25, 2014

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

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

4 The big picture Phenomenology SUGRA DFT This talk Type I E 8 E 8 het. T-Duality M-theory Type IIA SO(32) het. target space world sheet T-Duality Type IIB

5 Outline 1. SUGRA & DFT in a nutshell 2. String geometry by violating the strong constraint 3. Deriving DFT WZW from CSFT 4. Applications 5. Summary and outlook

6 SUGRA closed strings in D-dim. flat space 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

7 SUGRA closed strings in D-dim. flat space 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

8 SUGRA closed strings in D-dim. flat space 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 )

9 Manifest & hidden symmetries S NS = action for NS/NS sector of Type IIA and Type IIB manifest invariant under diffeomorphisms gauge transformations g ij = L ξ g ij B ij = L ξ B ij + i α j j α i compactification on circle U(1) isometry Buscher rules implement T -duality [Buscher, 1987] g θθ = 1 g θθ, g θi = 1 g θθ B θi, g ij = g ij + 1 g θθ (g θi g θj B θi B θj ),... from NS/NS sector of IIA to IIB T-duality is a hidden symmetry

10 DFT (Double Field Theory) [Siegel, 1993,Hull and Zwiebach, 2009,Hohm, Hull, and Zwiebach, 2010] closed strings on a flat torus 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

11 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)

12 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)

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) 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

14 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

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 ( H MN gij B = ik g kl B lj B ik g kj ) g ik B kj g ij O(D, D) T-duality

16 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

17 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

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

19 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes SUGRA & DFT String geometry DFTWZW from CSFT Applications Summary

20 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes T+ DF SUGRA & DFT String geometry DFTWZW from CSFT SC Applications Summary

21 A landscape of string backgrounds [Douglas, 2003,Susskind, 2003] SUGRA CYs backgrounds with fluxes DFT + CC SUGRA & DFT String geometry T+ DF DFTWZW from CSFT SC Applications Summary

22 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]

23 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]

24 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]

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

26 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

27 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

28 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

29 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 D d SUGRA embedding tensor gauged SUGRA

30 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 D d SUGRA embedding tensor gauged SUGRA eom vacuum

31 Generalized Scherk-Schwarz compactification [Aldazabal, Baron, Marques, and Nunez, 2011,Geissbuhler, 2011] some 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 uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

32 DFT on group manifolds = DFT WZW Use group manifold instead of a torus to derive DFT! + non-abelian gauge groups + cosmological constant + flux backgrounds with const. fluxes

33 DFT on group manifolds = DFT WZW Use group manifold instead of a torus to derive DFT! + non-abelian gauge groups + cosmological constant + flux backgrounds with const. fluxes Double Field Theory = treat left and right mover independently 2D independent coordinates

34 DFT on group manifolds = DFT WZW Use group manifold instead of a torus to derive DFT! + non-abelian gauge groups + cosmological constant + flux backgrounds with const. fluxes Double Field Theory = treat left and right mover independently 2D independent coordinates Questions about DFT WZW What are the covariant objects? Does it make non-abelian duality manifest? How is it connected to DFT? not trivial

35 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

36 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)

37 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) +...

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

39 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 w R is highest weight of g g representation

40 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 w 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

41 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 L 0 φ R = h R φ R D a D a Y R (x i ) = h R Y R (x i ) j a 0 φ R D a Y R (x i ) [j a 0, j b 0 ] = F c ab j c 0 [D a, D b ] = F c ab D c e(r) φ R e(r)y R (x i ) := e(x i ) R R

42 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 L 0 φ R = h R φ R D a D a Y R (x i ) = h R Y R (x i ) j a 0 φ R D a Y R (x i ) [j a 0, j b 0 ] = F c ab j c 0 [D a, D b ] = F c ab D c e(r) φ R e(r)y R (x i ) := e(x i ) R ( E I ea i ) 0 A = 0 eā ī R ( ) ηab 0 S AB = 0 ηā b ( ηab 0 η AB = 0 ηā b )

43 Weak constraint (level matching) 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 ) Dā η AB D A D B = D A D A = 0 compare with M M = 0

44 Weak constraint (level matching) 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 ) Dā η AB D A D B = D A D A = 0 compare with M M = 0 change to curved indices using E A M ( M M 2 M d M ) = 0 with d = φ 1 2 log g additional 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

45 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 +... additional terms e.g. potential vanish in abelian limit F abc 0 and Fā b c 0

46 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

47 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 + 1 2[ 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

48 Generalized Lie derivative doubled version of fluctuations ɛ a b ( ) ɛ AB 0 ɛ a b = ɛāb 0 generate generalized metric with ɛ a b = (ɛ T ) ba 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

49 Generalized Lie derivative doubled version of fluctuations ɛ a b ( ) ɛ AB 0 ɛ a b = ɛāb 0 generate generalized metric with ɛ a b = (ɛ T ) ba 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 L λ H AB = λ C D C H AB + (D A λ C D C λ A )ɛ CB + (D B λ C D C λ B )H AC + F A CDλ C H DB + F B CDλ C H AD

50 setting δ λ S AB := 0 and using δ λ ɛ AB = 1 2( Lλ S AB + L λ ɛ AB + L λ S (A CS B) Dɛ CD). results in δ λ H AB = 1 2 L λh AB + O(ɛ 2 )

51 setting δ λ S AB := 0 and using δ λ ɛ AB = 1 2( Lλ S AB + L λ ɛ AB + L λ S (A CS B) Dɛ CD). results in δ λ H AB = 1 2 L λh AB + O(ɛ 2 ) introduce covariant derivative A V B = D A V B F B ACV C

52 setting δ λ S AB := 0 and using δ λ ɛ AB = 1 2( Lλ S AB + L λ ɛ AB + L λ S (A CS B) Dɛ CD). results in δ λ H AB = 1 2 L λh AB + O(ɛ 2 ) introduce covariant derivative A V B = D A V B F B ACV C Kǎc-Moody structure coeff. F c ab F C AB = c Fā b 0 otherwise

53 setting δ λ S AB := 0 and using δ λ ɛ AB = 1 2( Lλ S AB + L λ ɛ AB + L λ S (A CS B) Dɛ CD). results in δ λ H AB = 1 2 L λh AB + O(ɛ 2 ) introduce covariant derivative A V B = D A V B F B ACV C generalized Lie derivative of a vector Kǎc-Moody structure coeff. F c ab F C AB = c Fā b 0 otherwise L λ V A = λ C C V A + ( A λ C C λ A )V C

54 Gauge algebra CSFT to cubic order fulfills δ Λ1 δ Λ2 δ Λ2 δ Λ1 = δ Λ12 with Λ 12 = [Λ 2, Λ 1 ] 0 after field redefinition and µ fixing λ A 12 = 1 2 [λ 2, λ 1 ] A C with [λ 1, λ 2 ] A C = λb 1 Bλ A λb 1 A λ 2 B (1 2)

55 Gauge algebra CSFT to cubic order fulfills δ Λ1 δ Λ2 δ Λ2 δ Λ1 = δ Λ12 with Λ 12 = [Λ 2, Λ 1 ] 0 after field redefinition and µ fixing λ A 12 = 1 2 [λ 2, λ 1 ] A C with [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 closure constraint (CC) F E[AB F E C]D = 0 no strong constraint required for background

56 Covariant derivative non-vanishing torsion and Riemann curvature [ A, B ]V C = R D ABC V D T D AB D V C with T A BC = 1 3 F A BC and R D ABC = 2 9 F AB E D F EC

57 Covariant derivative non-vanishing torsion and Riemann curvature [ A, B ]V C = R D ABC V D T D AB D V C with T A BC = 1 3 F A BC and R D ABC = 2 9 F AB E D F EC compatible with E I A, η AB and S AB C E I A = C η AB = C S AB = 0

58 Covariant derivative non-vanishing torsion and Riemann curvature [ A, B ]V C = R ABC D V D T D AB D V C with T A BC = 1 3 F A BC and R ABC D = 2 9 F AB E F EC D compatible with E A I, η AB and S AB C E A I = C η AB = C S AB = 0 compatible with partial integration d 2D Xe 2d U M V M = d 2D Xe 2d M UV M

59 Covariant derivative non-vanishing torsion and Riemann curvature [ A, B ]V C = R ABC D V D T D AB D V C with T A BC = 1 3 F A BC and R ABC D = 2 9 F AB E F EC D compatible with E A I, η AB and S AB C E A I = C η AB = C S AB = 0 compatible with partial integration d 2D Xe 2d U M V M = d 2D Xe 2d M UV M non-vanishing generalized torsion

60 Comparison DFT and DFT WZW DFT DFT WZW background torus group manifold

61 Comparison DFT and DFT WZW DFT DFT WZW background torus group manifold WC / SC M M M M

62 Comparison DFT and DFT WZW DFT DFT WZW background torus group manifold WC / SC M M M M L λ V I = λ J J V I + ( I λ J J λ I )V J λ J J V I + ( I λ J J λ I )V J [λ 1, λ 2 ] I C = λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J

63 Comparison DFT and DFT WZW DFT DFT WZW background torus group manifold WC / SC M M M M L λ V I = λ J J V I + ( I λ J J λ I )V J λ J J V I + ( I λ J J λ I )V J [λ 1, λ 2 ] I C = λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J closure SC fluctuations SC background CC

64 Comparison DFT and DFT WZW DFT DFT WZW background torus group manifold WC / SC M M M M L λ V I = λ J J V I + ( I λ J J λ I )V J λ J J V I + ( I λ J J λ I )V J [λ 1, λ 2 ] I C = λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J λ J [1 Jλ I 2] 1 2 λj [1 I λ 2] J closure SC fluctuations SC background CC abelian limit

65 Reminder: Generalized Scherk-Schwarz compactification String Theory simplification (truncation) matching amplitudes SUGRA without fluxes CSFT Double Field Theory i = 0 Scherk-Schwarz strong constraint with fluxes closure constraint group manifold uplift D d SUGRA embedding tensor gauged SUGRA eom vacuum

66 Embedding tensor fluxes for embedding one [Dibitetto, Fernandez-Melgarejo, Marques, and Roest, 2012] F abc = 2ɛ abc (cos α + sin α) and Fā b c = 2ɛ abc (cos α sin α)

67 Embedding tensor [Dibitetto, Fernandez-Melgarejo, Marques, and Roest, 2012] fluxes for embedding one F abc = 2ɛ abc (cos α + sin α) and Fā b c = 2ɛ abc (cos α sin α) DFT strong constraint holds only for F ABC F ABC = 24 sin(2α) = 0 α = π 2 n n Z closure constraint holds always

68 The landscape again String geometry CYs SUGRA backgrounds with fluxes DFT WZW α = 0 DFT + SC

69 The landscape again String geometry CYs SUGRA backgrounds with fluxes DFT WZW α = 0 T-dual DFT + SC

70 The landscape again String geometry CYs SUGRA backgrounds with fluxes DFT WZW α = 0 T-dual DFT + SC beyond the torus

71 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW

72 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW

73 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry

74 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry Todo action in terms of generalized metric like gauge transformations

75 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry Todo action in terms of generalized metric like gauge transformations loop amplitudes like torus partition function

76 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry Todo action in terms of generalized metric like gauge transformations loop amplitudes like torus partition function non-abelian duality, coset and orbifold CFTs

77 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry Todo action in terms of generalized metric like gauge transformations loop amplitudes like torus partition function non-abelian duality, coset and orbifold CFTs α corrections (here k 2, k 3,... ) [Hohm, Siegel, and Zwiebach, 2013]

78 Summary extended flat space torus group manifold decomp. limit abelian limit SUGRA DFT? DFT WZW covariant instead of partial derivative [Cederwall, 2014] only closure constraint for background string geometry Todo action in terms of generalized metric like gauge transformations loop amplitudes like torus partition function non-abelian duality, coset and orbifold CFTs α corrections (here k 2, k 3,... ) [Hohm, Siegel, and Zwiebach, 2013] phenomenology of non-geometric backgrounds [Hassler, Lust, and Massai, 2014]

79 Thank you for your attention. Are there any questions?