Yoonji Suh. Sogang University, May Department of Physics, Sogang University. Workshop on Higher-Spin and Double Field Theory

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1 U-gravity : SL(N) Department of Physics, Sogang University Yoonji Suh yjsuh@sogang.ac.kr Workshop on Higher-Spin and Double Field Theory Sogang University, May 2015

2 Talk based on works with Jeong-Hyuck Park U-geometry : SL(5) U-gravity : SL(N) arxiv: JHEP arxiv: JHEP

3 Introduction Lorentz symmetry unifies space and time into spacetime Duality requires further extension of the spacetime T-duality in string theory becomes a manifest O(D, D) rotation in doubled spacetime and so do various M-theory U-dualities in extended spacetime

4 Introduction Lorentz symmetry unifies space and time into spacetime Duality requires further extension of the spacetime T-duality in string theory becomes a manifest O(D, D) rotation in doubled spacetime and so do various M-theory U-dualities in extended spacetime

5 Introduction Lorentz symmetry unifies space and time into spacetime Duality requires further extension of the spacetime T-duality in string theory becomes a manifest O(D, D) rotation in doubled spacetime and so do various M-theory U-dualities in extended spacetime

6 M-theory and duality Double Field Theory with T-duality

7 Motivation of U-gravity Study of U-duality in a geometrical manner SL(N) transformation in extended spacetimes Construct the geometry suitable for M-theory U-gravity, Exceptional Field Theory, Extended Geometry, etc.

8 Motivation of U-gravity Study of U-duality in a geometrical manner SL(N) transformation in extended spacetimes Construct the geometry suitable for M-theory U-gravity, Exceptional Field Theory, Extended Geometry, etc.

9 Motivation of U-gravity Study of U-duality in a geometrical manner SL(N) transformation in extended spacetimes Construct the geometry suitable for M-theory U-gravity, Exceptional Field Theory, Extended Geometry, etc.

10 M-theory and duality D d G H 3 8 E 8 SO(16) 4 7 E 7 SU(8) 5 6 E 6 USp(8) 6 5 SO(5, 5) SO(5) SO(5) 7 4 SL(5) SO(5) 8 3 SL(3) SL(2) SO(3) SO(2) 9 2 SL(2) SO(2) 10 1 SO(1, 1) 1 Berman, Godazgar, Perry, West

11 M-theory and duality D d G H 3 8 E 8 SO(16) 4 7 E 7 SU(8) 5 6 E 6 USp(8) 6 5 SO(5, 5) SO(5) SO(5) 7 4 SL(5) SO(5) 8 3 SL(3) SL(2) SO(3) SO(2) 9 2 SL(2) SO(2) 10 1 SO(1, 1) 1 Berman, Godazgar, Perry, West

12 Dynkin diagrams and duality groups A N 1, D N 1, E N 1 and E N Ultimate duality group West

13 E 11 subgroups A 10 sl(11) SL(11) U-gravity D 10 so(10, 10) O(10, 10) DFT E 10 Exceptional Field Theory, Extended Geometry

14 E 11 subgroups A 10 sl(11) SL(11) U-gravity D 10 so(10, 10) O(10, 10) DFT E 10 Exceptional Field Theory, Extended Geometry

15 E 11 subgroups A 10 sl(11) SL(11) U-gravity D 10 so(10, 10) O(10, 10) DFT E 10 Exceptional Field Theory, Extended Geometry

16 E 11 subgroups A 10 sl(11) SL(11) U-gravity D 10 so(10, 10) O(10, 10) DFT E 10 Exceptional Field Theory, Extended Geometry

17 Riemannian geometry Ordinary spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

18 Riemannian geometry Ordinary spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

19 Riemannian geometry Ordinary spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

20 Riemannian geometry Ordinary spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

21 Riemannian geometry Ordinary spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

22 U-gravity Extended-yet-gauged spacetime Symmetry : Diffeomorphism Field contents : Metric Covariant derivative Covariant Riemann curvatures

23 U-gravity Extended-yet-gauged spacetime Symmetry : Generalized diffeomorphism and SL(N) U-duality Field contents : Metric Covariant derivative Covariant Riemann curvatures

24 U-gravity Extended-yet-gauged spacetime Symmetry : Generalized diffeomorphism and SL(N) U-duality Field contents : U-metric (Generalized metric) Covariant derivative Covariant Riemann curvatures

25 U-gravity Extended-yet-gauged spacetime Symmetry : Generalized diffeomorphism and SL(N) U-duality Field contents : U-metric (Generalized metric) Semi-covariant derivative Semi-covariant SL(N) curvatures

26 Extended-yet-gauged spacetime Anti-symmetric SL(N) vector indices x ab = x ba = x [ab] Section condition Weak constraint and strong constraint [ab cd] = 0 (where, a, b, c, = 1, 2,, N.) [ab cd] Φ = 0, [ab Φ cd] Φ = 0

27 Extended-yet-gauged spacetime Anti-symmetric SL(N) vector indices x ab = x ba = x [ab] Section condition Weak constraint and strong constraint [ab cd] = 0 (where, a, b, c, = 1, 2,, N.) [ab cd] Φ = 0, [ab Φ cd] Φ = 0

28 Extended-yet-gauged spacetime Anti-symmetric SL(N) vector indices x ab = x ba = x [ab] Section condition Weak constraint and strong constraint [ab cd] = 0 (where, a, b, c, = 1, 2,, N.) [ab cd] Φ = 0, [ab Φ cd] Φ = 0

29 Extended-yet-gauged spacetime Gauged spacetime Realization of coordinate gauge symmetry x ab x ab + ab Φ(x + ) = Φ(x), ab = 1 (N 4)! ɛabc1 c N 4de φ c1 c N 4 de ϕ Dimension of the extended spacetime is 1 2N(N 1). See also Lee and Park Yet, as we see later the physical dimension is reduced to either N 1 or 3.

30 Extended-yet-gauged spacetime Gauged spacetime Realization of coordinate gauge symmetry x ab x ab + ab Φ(x + ) = Φ(x), ab = 1 (N 4)! ɛabc1 c N 4de φ c1 c N 4 de ϕ Dimension of the extended spacetime is 1 2N(N 1). See also Lee and Park Yet, as we see later the physical dimension is reduced to either N 1 or 3.

31 Extended-yet-gauged spacetime Gauged spacetime Realization of coordinate gauge symmetry x ab x ab + ab Φ(x + ) = Φ(x), ab = 1 (N 4)! ɛabc1 c N 4de φ c1 c N 4 de ϕ Dimension of the extended spacetime is 1 2N(N 1). See also Lee and Park Yet, as we see later the physical dimension is reduced to either N 1 or 3.

32 Extended-yet-gauged spacetime Gauged spacetime Realization of coordinate gauge symmetry x ab x ab + ab Φ(x + ) = Φ(x), ab = 1 (N 4)! ɛabc1 c N 4de φ c1 c N 4 de ϕ Dimension of the extended spacetime is 1 2N(N 1). See also Lee and Park Yet, as we see later the physical dimension is reduced to either N 1 or 3.

33 Generalized Diffeomorphism Generalized Lie derivative ˆL X T a1 ap b 1 b q := 1 2 Xcd cd T a1 ap b 1 b q ( 1 2 p 1 2 q + ω) cdx cd T a1 ap b 1 b q p i=1 T a1 c ap b 1 b q cd X aid + q j=1 b jdx cd T a1 ap b 1 c b q Ordinary Lie derivative generates (ordinary) diffeomorphism (where, ω is a weight.) L ξ T α1 αp β 1 β q = ξ γ γ T α1 αp β 1 β q p i=1 T α1 γ αp β 1 β q γ ξ αi + q j=1 (1) β j ξ γ T α1 αp β 1 γ β q

34 Generalized Diffeomorphism Generalized Lie derivative ˆL X T a1 ap b 1 b q := 1 2 Xcd cd T a1 ap b 1 b q ( 1 2 p 1 2 q + ω) cdx cd T a1 ap b 1 b q p i=1 T a1 c ap b 1 b q cd X aid + q j=1 b jdx cd T a1 ap b 1 c b q (where, ω is a weight.) With the section condition, the commutator of the generalized Lie derivative is closed by a generalized bracket. [ ˆLX, ˆL Y ] = ˆL [X,Y ]G where [X, Y ] ab G = 1 2 Xcd cd Y ab 3 2 X[ab cd Y cd] (X Y )

35 U-metric SL(N) U-metric M ab = M (ab), M = det(m ab ) (where, ω = 4 N.) Duality invariant integral measure M 1 4 N

36 Semi-covariant derivative Semi-covariant derivative cd T a1 ap b 1 b q := cd T a1 ap b 1 b q ( 1 2 p 1 2 q + ω)γ e cde T a1 ap b 1 b q Connection p i=1 T a1 e ap b 1 b q Γ ai cde + q Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) j=1 Γ e cdb j T a1 ap b 1 e b q + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

37 Semi-covariant derivative Semi-covariant derivative cd T a1 ap b 1 b q := cd T a1 ap b 1 b q ( 1 2 p 1 2 q + ω)γ e cde T a1 ap b 1 b q Connection p i=1 T a1 e ap b 1 b q Γ ai cde + q Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) j=1 Γ e cdb j T a1 ap b 1 e b q + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

38 Semi-covariant derivative The connection is the unique solution to the following constraints Connection ab M cd = 0 efgh (ab) = 0 = (ab) ˆL X ( ) = ˆL X ( ) Pabcd Γ efgh = 0 Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

39 Semi-covariant derivative The connection is the unique solution to the following constraints Connection ab M cd = 0 Γ abcd + Γ abdc = 2A abcd efgh (ab) = 0 = (ab) ˆL X ( ) = ˆL X ( ) Pabcd Γ efgh = 0 Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

40 Semi-covariant derivative The connection is the unique solution to the following constraints Connection ab M cd = 0 Γ abcd + Γ abdc = 2A abcd (ab) = 0 Γabc d + Γbac d = 0 ˆL X ( ) = ˆL X ( ) efgh Pabcd Γ efgh = 0 Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

41 Semi-covariant derivative The connection is the unique solution to the following constraints Connection ab M cd = 0 Γ abcd + Γ abdc = 2A abcd (ab) = 0 Γabc d + Γbac d = 0 { ˆL X ( ) = ˆL Γ d X ( ) abc + Γ bca d + Γ cab d = 0 Γcab c + Γ cba c = 0 efgh Pabcd Γ efgh = 0 Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

42 Semi-covariant derivative The connection is the unique solution to the five constraints Connection Γ abcd + Γ abdc = 2A abcd (2) Γabc d + Γbac d = 0 (3) Γabc d + Γbca d + Γcab d = 0 (4) Γcab c + Γcba c = 0 (5) efgh Pabcd Γ efgh = 0 (6) Γ abcd =A abcd (A acbd A adbc + A bdac A bcad ) + 1 N 2 (M aca e (bd)e M ada e (bc)e + M bda e (ac)e M bca e (ad)e ) (where, A abcd := 1 2 abm cd + 1 2(N 4) M cd ab ln M )

43 Projection The connection is uniquely determined by the projection Definition of projection efgh Pabcd Γ efgh = 0 klmn Pabcd = 1 2 δ [k [a δ l] b] δ [m [c δ n] + 1 N 2 d] δ [k [c δ l] d] δ [m [a δ n] b] M c[aδb] m M n[k δ l] d 1 2 M c[aδ [k b] M l]n δd m ( δ n [a M b][c M m[k δ l] d] + δ[c n M d][am m[k δ l] b] M c[a M b]d M m[k M l]n)

44 Projection Basic properties P P pqrs abcd P klmn abcd Definition of projection klmn Pabcd = 1 2 δ [k [a δ l] b] δ [m [c δ n] + 1 N 2 klmn pqrs = P [kl]mn klmn abcd, = P[ab]cd, P klmn ab[cd] = P klmn cd[ab]. d] δ [k [c δ l] d] δ [m [a δ n] b] M c[aδb] m M n[k δ l] d 1 2 M c[aδ [k b] M l]n δd m ( δ n [a M b][c M m[k δ l] d] + δ[c n M d][am m[k δ l] b] M c[a M b]d M m[k M l]n)

45 Projection Trace properties P s asb klmn = 1 2 P s abs klmn = δ m P rs rs klmn and other relations sklmn Pabs = 0 P (N 2)δ m (a M n[k δ l] b) N 2(N 2) M abm m[k M l]n (a M n[k δ l] ( = N 2 2N+2 N 2 b) + N 1 N 2 M abm m[k M l]n ) M m[k M l]n P s asb klmn = 1 2 klmn [klmn] [abc]d = P[abcd] = δ [k [a δ b l δc m δ n] d] klmn (N 2)Psabs M rs klmn abp rs

46 Covariantization The projection dictates the anomalous terms in the diffeomorphic transformations (δ X ˆL X ) ab T c1 cp d 1 d q = p i=1 T c1 e cp d 1 d q Ω ci abe + q j=1 Ω e abd j T c1 cp d 1 e d q Properties of anomalous term klm (where, Ω abcd = Pabcd n kl mex ne ) Ω abcd = Ω [ab][cd] = Ω cdab, Ω [abc]d = 0, efgh Ω abcd = Pabcd Ω efgh, Ωacb c = 0.

47 Covariantization The projection dictates the anomalous terms in the diffeomorphic transformations (δ X ˆL X ) ab T c1 cp d 1 d q = p i=1 T c1 e cp d 1 d q Ω ci abe + q j=1 Ω e abd j T c1 cp d 1 e d q Properties of anomalous term klm (where, Ω abcd = Pabcd n kl mex ne ) Ω abcd = Ω [ab][cd] = Ω cdab, Ω [abc]d = 0, efgh Ω abcd = Pabcd Ω efgh, Ωacb c = 0.

48 Covariantization Complete covariantizations [ab T c1 c q], ab T a, a bt [ca] + a ct [ba], a bt (ca) a ct (ba), ab T [abc1 cq] (divergence), ab [ab T c1 cq] (Laplacian). (where, q = 0, 1, N 2.) Properties of anomalous term Ω abcd = Ω [ab][cd] = Ω cdab, Ω [abc]d = 0, efgh Ω abcd = Pabcd Ω efgh, Ωacb c = 0.

49 Covariantization Complete covariantizations [ab T c1 c q], ab T a, a bt [ca] + a ct [ba], a bt (ca) a ct (ba), ab T [abc1 cq] (divergence), ab [ab T c1 cq] (Laplacian). (where, q = 0, 1, N 2.) Examples of complete covariant quantities ab T ab = ab T ab (ω 1)Γ c ab T ab = ab T ab for ω = 1 abc T ab [ab cd] φ = 0, [ab cd T e] = 0, [ab c]d T d = 0, [ab φ cd T e] = 0

50 SL(N) curvatures Semi-covariant Riemann curvature S abcd := 3 [ab Γ Γ e e][cd] e ab[c Γ f d]ef + 3 [cd Γe][ab] 1 4 Γ e Γ e e cd[a Γ f b]ef Under arbitrary transformation of the connection, δs abcd = 3 [ab δγ e e][cd] Γ abe Γ f cdf 1 2 Γ f abe Γ e cdf f ea[c Γ e d]fb + Γ f eb[c Γ e d]fa + 3 [cd δγ e e][ab]

51 SL(N) curvatures Semi-covariant Riemann curvature S abcd := 3 [ab Γ Γ e e][cd] e ab[c Γ f d]ef + 3 [cd Γe][ab] 1 4 Γ e Γ e e cd[a Γ f b]ef Under arbitrary transformation of the connection, δs abcd = 3 [ab δγ e e][cd] Γ abe Γ f cdf 1 2 Γ f abe Γ e cdf f ea[c Γ e d]fb + Γ f eb[c Γ e d]fa + 3 [cd δγ e e][ab]

52 SL(N) curvatures Semi-covariant Riemann curvature S abcd := 3 [ab Γ Γ e e][cd] e ab[c Γ f d]ef + 3 [cd Γe][ab] 1 4 Γ e Γ e e cd[a Γ f b]ef Γ abe Γ f cdf 1 2 Γ f abe Γ e cdf f ea[c Γ e d]fb + Γ f eb[c Γ e d]fa The semi-covariant Riemann curvature satisfies the same symmetric properties as the ordinary Riemann curvature, including the Bianchi identity, S abcd = S [ab][cd] = S cdab, S [abc]d = 0

53 SL(N) curvatures Semi-covariant Riemann curvature S abcd := 3 [ab Γ Γ e e][cd] e ab[c Γ f d]ef Diffeomorphic transformations + 3 [cd Γe][ab] 1 4 Γ e Γ e e cd[a Γ f b]ef (δ X ˆL X )S abcd = 2 e[a Ω Γ e b][cd] abe Γ f cdf 1 2 Γ f abe Γ e cdf f ea[c Γ e d]fb + Γ f eb[c Γ e d]fa + 2 e[c Ω e d][ab] klm (where, Ω abcd = Pabcd n kl mex ne )

54 SL(N) curvatures Covariant Ricci curvature and scalar curvature S ab := Sacb c = S (ab), S := M ab S ab = Sab ab Action Σ M 1 4 N S (where the integral is taken over a section, Σ.)

55 SL(N) curvatures Einstein equation of motion S ab + 1 2(N 4) M abs = 0 Conservation relation c [a S b]c abs = 0

56 Duality inequivalent sections (N 1)-dimensional (physical) section : Σ N 1 αβ = 0, α := αn 0 (where, α, β = 1, 2,, N 1.) Three-dimensional (physical) section : Σ 3 µi = 0, ij = 0, µν 0 (where, µ, ν = 1, 2, 3 and i, j = 4, 5,, N.)

57 Duality inequivalent sections on Σ N 1 [ab Φ c][d Φ ef] Φ = 0 on Σ 3 [ab Φ c][d Φ ef] Φ 0

58 Duality inequivalent sections on Σ N 1 [Nα Φ β][n Φ γλ] Φ = 0 on Σ 3 [µν Φ ρ][µ Φ ν ρ ]Φ 0 Blair, Malek and JHP, see also Hohm and Samtleben

59 Riemannian reductions Σ N 1 parametrization ( gαβ v α M ab = g v β g ( e φ + v 2 ) ) 1 1, M 4 N = e φ 4 N g, v = C N 2 Parameter of generalized Lie derivative { X αβ = 1 ɛαβρ1 ρ N 3 X ab (N 3)! Λ ρ1 ρ N 3 = X αn = ξ α Generalized Lie derivative L X M ab { δφ = L ξ φ, δg αβ = L ξ g αβ δc N 2 = L ξ C N 2 + dλ N 3

60 Riemannian reductions Σ N 1 parametrization ( gαβ v α M ab = g v β g ( e φ + v 2 ) ) 1 1, M 4 N = e φ 4 N g, v = C N 2 Parameter of generalized Lie derivative { X αβ = 1 ɛαβρ1 ρ N 3 X ab (N 3)! Λ ρ1 ρ N 3 = X αn = ξ α Generalized Lie derivative L X M ab { δφ = L ξ φ, δg αβ = L ξ g αβ δc N 2 = L ξ C N 2 + dλ N 3

61 Riemannian reductions Σ N 1 parametrization ( gαβ v α M ab = g v β g ( e φ + v 2 ) ) 1 1, M 4 N = e φ 4 N g, v = C N 2 Parameter of generalized Lie derivative { X αβ = 1 ɛαβρ1 ρ N 3 X ab (N 3)! Λ ρ1 ρ N 3 = X αn = ξ α Generalized Lie derivative L X M ab { δφ = L ξ φ, δg αβ = L ξ g αβ δc N 2 = L ξ C N 2 + dλ N 3

62 Riemannian reductions Σ N 1 parametrization ( gαβ v α M ab = g v β g ( e φ + v 2 ) ) 1 1, M 4 N = e φ 4 N g U-gravity scalar curvature [ S = 2e φ R g (N 3)(3N 8) 4(N 4) 2 α φ α φ + N 2 N 4 φ + 1 (N 1)! (dc N 2) 2] ΣN 1

63 Riemannian reductions Σ 3 parametrization M ab = ( g µν g ṽ jµ ṽ iν g ( e φ Mij + ṽ iλ ṽ j λ ) ), M 1 4 N = e N 3 4 N φ g, ṽ i = C i 2 Dual coordinates x µ ε µνρ x νρ, µ x ν = δ µ ν (where, a three-dimensional Levi-Civita symbol, ε )

64 Riemannian reductions Σ 3 parametrization M ab = ( g µν g ṽ jµ ṽ iν g ( e φ Mij + ṽ iλ ṽ j λ ) ), M 1 4 N = e N 3 4 N φ g, ṽ i = C i 2 Dual coordinates x µ ε µνρ x νρ, µ x ν = δ µ ν (where, a three-dimensional Levi-Civita symbol, ε )

65 Riemannian reductions Σ 3 parametrization M ab = ( g µν g ṽ jµ ṽ iν g ( e φ Mij + ṽ iλ ṽ j λ ) ), M 1 4 N = e N 3 4 N φ g U-gravity scalar curvature S = 2R g + (N 3)(3N 8) 2(N 4) µ φ µ φ 4(N 3) 2 N 4 φ 1 µ 2 Mij µ Mij + e φ Mij µṽ i µ ν ṽ j ν Σ3 which manifests SL(N 3) S-duality.

66 Summary On the extended-yet-gauged spacetime, we have constructed a duality manifest gravitational theory for the special linear group, SL(N) with N 4. x ab x ab + ab SL(N) U-gravity unifies both M-theory and type IIB theory as different solutions of its section condition, Σ N 1 and Σ 3 section respectively.

67 Summary On the extended-yet-gauged spacetime, we have constructed a duality manifest gravitational theory for the special linear group, SL(N) with N 4. x ab x ab + ab SL(N) U-gravity unifies both M-theory and type IIB theory as different solutions of its section condition, Σ N 1 and Σ 3 section respectively.

68 Comment The common features of SL(N) U-gravity and DFT-geometry Extended-yet-gauged spacetime(section condition) Diffeomorphism generated by a generalized Lie derivative Semi-covariant derivative and semi-covariant Riemann curvature Complete covariantizations of them dictated by a projection operator

69 Comment The common features of SL(N) U-gravity and DFT-geometry Extended-yet-gauged spacetime(section condition) Diffeomorphism generated by a generalized Lie derivative Semi-covariant derivative and semi-covariant Riemann curvature Complete covariantizations of them dictated by a projection operator

70 Comment The common features of SL(N) U-gravity and DFT-geometry Extended-yet-gauged spacetime(section condition) Diffeomorphism generated by a generalized Lie derivative Semi-covariant derivative and semi-covariant Riemann curvature Complete covariantizations of them dictated by a projection operator

71 Comment The common features of SL(N) U-gravity and DFT-geometry Extended-yet-gauged spacetime(section condition) Diffeomorphism generated by a generalized Lie derivative Semi-covariant derivative and semi-covariant Riemann curvature Complete covariantizations of them dictated by a projection operator

72 Future works Taking N = 11, SL(11) U-gravity may provide an SL(11) U-duality manifest reformulation of the ten-dimensional Romans massive type IIA supergravity In U-gravity, invariant measure contains an exponential of scalar term naturally, cosmological constant contributes differently from ordinary cosmology. Σ 1 N 1 e 1 φ 4 N g Λ M 4 N Λ = Σ Σ 3 e N 3 φ 4 N g Λ

73 Future works Taking N = 11, SL(11) U-gravity may provide an SL(11) U-duality manifest reformulation of the ten-dimensional Romans massive type IIA supergravity In U-gravity, invariant measure contains an exponential of scalar term naturally, cosmological constant contributes differently from ordinary cosmology. Σ 1 N 1 e 1 φ 4 N g Λ M 4 N Λ = Σ Σ 3 e N 3 φ 4 N g Λ

74 It has been said that string theory is a piece of 21st century physics that happened to fall into the 20th century Anonymous Italian Our U-gravity may provide a candidate for such a new framework beyond Riemannian geometry. Thank you!

75 It has been said that string theory is a piece of 21st century physics that happened to fall into the 20th century Anonymous Italian Our U-gravity may provide a candidate for such a new framework beyond Riemannian geometry. Thank you!

77 A new differential geometry Part 1 : Geometric constitution of U-gravity Part 2 : Geometric constitution of Double Field Theory

78 Notation Capital Latin alphabet letters denote the O(D, D) vector indices which can be freely raised or lowered by the O(D, D) invariant constant metric, ( ) 0 1 J AB =, A, B, = 1, 2,, D + D. 1 0

79 Doubled-yet-gauged spacetime Coordinate gauge symmetry of doubled spacetime x A x A + φ A ϕ Realization of the coordinate gauge symmetry Φ(x + ) = Φ(x), A = φ A ϕ Section condition A A Φ = 0, A Φ A Φ = 0

80 Diffeomorphism Generalized Lie derivative Siegel, Courant, Grana ˆL X T A1 A n := X B B T A1 A n +ω T B X B T A1 A n + n ( Ai X B B X Ai )T i=1 B A 1 A i 1 A n (where ω T denotes the weight.) In particular, the generalized Lie derivative of the O(D, D) invariant metric is trivial, ˆL X J AB = 0 The commutator of the generalized Lie derivatives is closed by Courant bracket. [ ˆL X, ˆL Y ] = ˆL [X,Y ]C (where, [X, Y ] A C = XB B Y A Y B B X A Y B A X B 1 2 XB A Y B.)

81 Geometric objects Dilaton and a pair of symmetric projectors Orthogonality and complementarity. d, P AB = P BA, PAB = P BA. P B A P C B = 0, (P + P ) AB = J AB Integral measure e 2d

82 Semi-covariant derivative and semi-covariant Riemann curvature Semi-covariant derivative C T A1A 2 A n := C T A1A 2 A n ω T T A1A 2 A n + The (torsionless) connection n B ΓCA i T A1 A i 1B A n, Γ CAB = 2(P C P P ) [AB] + 2( P D [A P E B] P D [A P E B] ) D P EC 4 i=1 D 1 ( P D C[A P B] + P C[A P D B] )( D d + (P E P P ) [ED] ).

83 Semi-covariant derivative and semi-covariant Riemann curvature Semi-covariant derivative C T A1A 2 A n := C T A1A 2 A n ω T T A1A 2 A n + The (torsionless) connection n B ΓCA i T A1 A i 1B A n, Γ CAB = 2(P C P P ) [AB] + 2( P D [A P E B] P D [A P E B] ) D P EC 4 i=1 D 1 ( P D C[A P B] + P C[A P D B] )( D d + (P E P P ) [ED] ).

84 The uniqueness of the torsionless connection The connection is the unique solution to the following constraints: A P BC = 0, A PBC = 0 A d = A d ΓB BA = 0 Γ ABC + Γ ACB = 0 Γ ABC + Γ BCA + Γ CAB = 0 DEF DEF PABC Γ DEF = 0, P ABC Γ DEF = 0.

85 Semi-covariant derivative and semi-covariant Riemann curvature Semi-covariant derivative C T A1A 2 A n := C T A1A 2 A n ω T T A1A 2 A n + The (torsionless) connection n B ΓCA i T A1 A i 1B A n, Γ CAB = 2(P C P P ) [AB] + 2( P D [A P E B] P D [A P E B] ) D P EC 4 i=1 D 1 ( P D C[A P B] + P C[A P D B] )( D d + (P E P P ) [ED] ). Semi-covariant Riemann curvature ( S BACD := 1 2 RABCD + R CDAB Γ E ) ABΓ ECD, Field strength of a connection R CDAB = A Γ BCD B Γ ACD + Γ AC E E Γ BED ΓBC Γ AED,

86 Semi-covariant derivative and semi-covariant Riemann curvature Semi-covariant derivative C T A1A 2 A n := C T A1A 2 A n ω T T A1A 2 A n + The (torsionless) connection n B ΓCA i T A1 A i 1B A n, Γ CAB = 2(P C P P ) [AB] + 2( P D [A P E B] P D [A P E B] ) D P EC 4 i=1 D 1 ( P D C[A P B] + P C[A P D B] )( D d + (P E P P ) [ED] ). Semi-covariant Riemann curvature ( S BACD := 1 2 RABCD + R CDAB Γ E ) ABΓ ECD, Field strength of a connection R CDAB = A Γ BCD B Γ ACD + Γ AC E E Γ BED ΓBC Γ AED,

87 Semi-covariant derivative and semi-covariant Riemann curvature Under arbitrary transformation of the connection, δs ABCD = [A δγ B]CD + [C δγ D]AB Symmetric properties of semi-covariant Riemann curvature Additional identities PI A PJ B P K C S ABCD = S [AB][CD] = S CDAB, S [ABC]D = 0. P L D S ABCD = 0, PI A P J B PK C P L D S ABCD = 0, S AB AB = 0.

88 Six-index projection operators Definitions Projection properties P Properties DEF ABC DEF PCAB := P D DEF P CAB := P C D P GHI DEF C P [E [A P F ] B] + 2 D 1 P C[AP [E B] P F ]D P [E [A GHI = PABC, P P F ] B] + 2 P [E D 1 C[A P B] P F ]D DEF ABC P GHI DEF = P GHI ABC P ABCDEF = P DEF ABC, P ABCDEF = P A[BC]D[EF ], P AB P ABCDEF = 0 P ABCDEF = P DEF ABC, PABCDEF = P A[BC]D[EF ], P AB PABCDEF = 0

89 Diffeomorphic transformation Semi-covariant derivative and curvature (δ X ˆL n X ) C T A1 A n = 2(P + P) BDEF CA i D E X F T A1 A i 1B A n, i=1 (δ X ˆL X )S ABCD =2 [A ((P + P) ) EF G B][CD] E F X G + 2 [C ((P + P) ) EF G D][AB] E F X G.

90 Complete covariantization Complete covariant quantities with projections. P C D B1 Bn D P A 1 P A n D T B1 B n, P C P B1 A 1 P Bn A n D T B1 B n, P AB D P 1 Dn C 1 P C n A T BC1 C n, P AB P D1 C 1 P Dn C n A T BC1 C n, (Divergence) P AB D P 1 Dn C 1 P C n A B T C1 C n, P AB P D1 C 1 P Dn C n A B T C1 C n. (Laplacians) Ricci and scalar curvatures PA C P B D E SCED, (P AC P BD P AC P BD )S ABCD.

91 The action The action of O(D, D) DFT. Σ D e 2d (P AC P BD P AC P BD )S ABCD The dilaton and the projector equations of motion correspond to the vanishing of the scalar curvature and the Ricci curvature respectively.

92 Section Up to O(D, D) duality rotation, the solution to the section is unique. It is a D-dimensional section, Σ D, characterized by the independence of the dual coordinates, x µ 0 The whole doubled coordinates. X A = ( x µ, x ν ) (where, µ, ν are D-dimensional indices.)

93 Reductions Riemannian reduction (P P ) AB = g 1 g 1 B Bg 1 g Bg 1 B, e 2d = g e 2φ. Scalar curvature (P AC P BD P AC P BD ΣD )S ABCD = R g + 4 φ 4 µ φ µ φ 1 12 H λµνh λµν Non-Riemannian background (P P ) AB = J AB c.f. global aspects Berman, Cederwall, Perry, JHP, Marques, Lee and Grana

Einstein Double Field Equations

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