Gravity theory on Poisson manifold with R-flux

Transcription

1 Gravity theory on Poisson manifold with R-flux Hisayoshi MURAKI (University of Tsukuba) in collaboration with Tsuguhiko ASAKAWA (Maebashi Institute of Technology) Satoshi WATAMURA (Tohoku University) References [hep-th] (Int.J.Mod.Phys. A 30 (2015) 17, ) [hep-th] (Fortsch.Phys. 63 (2015) ) 2015/11/12 YITP

2 Space-time Geometry probed with strings String theory would describe quantum gravity Fundamental Objects Space-time Geometry Quantum Gravity General Relativity Point Particles Riemannian String Theory Strings 2015/11/12 YITP 2/21

3 Typical Example of T-duality Compactifying on X 9 X 9 +2 R (periodic) Contribution to KK tower (Mass spectrum of reduced theory) KK momenta Windings P 9 = K R R $ 0 R W 9 = NR = R 0 N Reduced theories are physically equivalent: T-duality [`84 Kikkawa, Yamasaki] 2015/11/12 YITP 3/21

4 Why Generalized Geometry? T-duality implies a physical equivalence between two different background geometries (configurations of space-time metric and NSNS B-field) suggesting the appearances of -Strange metric (T-folds) [`04 Hull],... -Non-geometric fluxes [`05 Shelton, Taylor, Wecht],... -Exotic branes [`10 de Boer, Shigemori],... Ø Machinery treating these as geometry (Poisson) Generalized Geometry 2015/11/12 YITP 4/21

5 Plan of Today s Talk Introduction & Motivations A Little Bit More on T-duality Generalized Geometry -Definitions & Properties -Generalized Riemannian Geometry Poisson Generalized Geometry -Definitions & Properties -Poisson Generalized Riemannian Geometry 2015/11/12 YITP 5/21

6 A Little Bit More on T-duality: Buscher rule T-duality: Physical equiv. between two backgrounds given by Buscher rule ( 0 : isometry) [`87 Buscher] g 00 = 1 g 00, (g, B) ( g, B) g 0i = B 0i g 00, g ij = g ij g 0i g 0j B 0i B 0j g 00, B 0i = g 0i g 00, Bij = B ij g 0i B 0j g 0j B 0i g 00. Metric and B-field should be on equal footing 2015/11/12 YITP 6/21

7 Generalized Geometry [`04 Hitchin, `04 Gualtieri] Consider tangent and cotangent bundles at the same time! l Canonical Inner Product 1 X +,Y + = 2 (i X + i Y ) Ø Invariant under an action of O(D,D) T-duality transf. l Operations corresponding to diffeo. B-field gauge transf. 1 [X +,Y + ] =[X, Y ]+L X L Y 2 d(i X i Y ) 2015/11/12 YITP 7/21

8 Riemannian Geom. based on Gen. Geom. l O(D,D)-invariant inner product Decompose in Positive-/Negative-definite subbundles C ± = {X +(±g + B)(X) X 2 r (TM)} C + l Define a connection on positive-def. subbundle Coefficients Christoffel Symbol H-flux (B-field s field strength) (@ j ) + = g lk 2 kij + H kij (@ l ) /11/12 YITP 8/21

9 Gravity based on Gen. Geom. l Curvature tensor: l Ricci scalar: R(X, Y )u := (r X r Y r Y r X r [X,Y ]C )u R 1 4 Hijk H ijk l Einstein-Hilbert-like action Z S = d D x p g R 14 H2 Ø This is the same as NSNS-sector of SUGRA G.G. would be a good tool to geometrize string theory 2015/11/12 YITP 9/21

10 Variant of Generalized Geometry Slightly modifying structures of GG would be interesting v Vector + 1-form v Inner product (T-dual) v diffeo. B-field gauge trnsf. [X +,Y + ] =[X, Y ]+L X L Y unchanged unchanged changed! 1 2 d(i X i Y ) Then how do we change it??? 2015/11/12 YITP 10/21

11 Hint: A chain of T-duality L g(ẋ X )+BẊX0 String action conjugate momenta [p i,p j ] H ijk w k [p i,w k ] F k ijw j [p i,w k ] Q kj i [w i,w j ] R ijk p k i g ij Ẋ j + B ij X 0j momentum winding p i $ w i = X 0i Fluxes associated with non-geom. background 2015/11/12 YITP 11/21 p j

12 Fluxes in Generalized Geometry H ijk! F k ij! Q ij k! Rijk [@ i,dx k ] [@ [dx i,dx j i,dx ] ] H ijk dx k Fijdx k j Q j R ijk k?? [@ j ] Understood in terms of GG MISSING in GG!! [X, Y ] 6= 0 [, ] =0 [, ] 6= 0 [X, Y ]=0 Variant of GG interchanging roles of Vector and 1-form 2015/11/12 YITP 12/21

13 Poisson Geometry Poisson bi-vector -Poisson bracket -Jacobi id. Lie algebra on 1-forms [, ] = L ( ) [, ] S =0 Lie bracket: = 1 2 i j {f,g} = i f@ j g {{f,g},h} + {{g, h},f} + {{h, f},g} =0 Poisson cond. l jk] =0 Schouten bracket: e.g. [X ^ Y,Z] S =[X, Z] ^ Y Extension of Lie br. to mulg- vector [Y,Z] ^ X L ( ) + d( (, )) 2015/11/12 YITP 13/21

14 Cartan Algebra on Poly-Vector Fields Interior product Exterior derivative -Nilpotent Lie derivative ı ı (X ^ Y )=(i X )Y (i Y )X d =[, ] S d 2 =0 [, ] S =0 L f := ı d f L := [, ] L X := (d ı + ı d )X [, ] = L ( ) i ( ) d Cartan algebra : enables diff. calculus induced by 1-form {ı, ı } =0, {d, ı } = L, [L, ı ]=ı [, ], [L, L ]=L [, ], [d, L ]= /11/12 YITP 14/21

15 Poisson Generalized Geometry l Vector field and 1-form (same as GG) l O(D,D)-invariant inner product (same as GG) Poisson structure l Operations different from GG Vector field 1-form [X +,Y + ] =[, ] + L Y L X 1 2 d (ı Y ı X) cf. [X +,Y + ] C =[X, Y ]+L X L Y 1 2 d(i X i Y ) 2015/11/12 YITP 15/21

16 Physical intuition of New Operation l Operation in GG [X +,Y + ] C =[X, Y ]+L X L Y i = p i + B ij w j 1 2 d(i X i Y ) T-dual ( cf. Marc s talk) e i = w i + ij p j Poisson structure l Operation in PGG [X +,Y + ] =[, ] + L Y L X 1 2 d (i Y i X) 2015/11/12 YITP 16/21

17 Riemann Geom. based on PGG l O(D,D)-inv inner product Decompose in Positive-/Negative-definite subbundles C ± = { +(±G + )( ) 2 (T M)} r l Define a connection on positive-def. subbundle Coefficients Contravariant Levi-Civita R-flux r dx i(dx j ) + = 2 ij k + G lk R kij (dx l ) + R = d =[, ] S C + =( l jk] + l jk] )@ i j k 2015/11/12 YITP 17/21

18 Contravariant Levi-Civita conn. [`00 Fernandes]... Two characteristic tensors G ij ij {ij} k = 1 2 [ mn (@ m G ji ) mi (@ m G jn ) [ij] k = 1 2 (@ k ij ) mj (@ m G in ) G jl (@ l in ) G il (@ l jn )]G nk r dx kg ij =0 Compatible with metric r dx i jk +(cyclic)=0 Respecting Poisson strc. In particular, ij is covariantly constant r i jk =0 LC conn. Extension of the Levi-Civita respecting Poisson structure! 2015/11/12 YITP 18/21

19 Gravity based on PGG l Curvature tensor l Ricci scalar: R(, )u := (r r r r r [, ] )u l Einstein-Hilbert-like action S = Z R 1 4 Rijk R ijk d D x p G R 14 R2 R-flux Curvature defined by Contra. LC 2015/11/12 YITP 19/21

20 Summary We gave l a new geometric framework based on Poisson structure -T-dual counterpart of Generalized Geometry l a well-defined formulation of R-flux -defined as a field strength of local bi-vector gauge potentials l an Riemann geometry compatible with Poisson structure 2015/11/12 YITP 20/21

21 Future directions We established Riemann Geometry compatible with Poisson l Well describable Non-geometric background? l Extension to quasi-poisson, Nambu-Poisson structures l Poisson is semi-classical limit of Non-commutativity Riemann Geometry on Non-commutative space l etc /11/12 YITP 21/21