Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography Generalized Metric Formulation of DFT WZW Pascal du Bosque LMU Munich 31st IMPRS Workshop Munich, March 16th, 2015 based on : [ 1502 . 02428 ] R. Blumenhagen, P. du Bosque, F. Hassler and D. Lüst Pascal du Bosque Generalized Metric Formulation of DFT WZW 1/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography SUGRA Action [1] • Reformulation of SUGRA action for massless string excitations: d D x √− ge − 2 φ � R + 4 ( ∂φ ) 2 − 1 � 12 H ijk H ijk � S NS = • Geometrization of SUGRA action in terms of generalized Ricci scalar? • DFT action: � d 2 D X e − 2 d R ( H , d ) S DFT = • Generalized curvature scalar: R ≡ 4 H MN ∂ M d ∂ N d − ∂ M ∂ N H MN − 4 H MN ∂ M d ∂ N d + 4 ∂ M H MN ∂ N d + 1 8 H MN ∂ M H KL ∂ N H KL − 1 2 H MN ∂ N H KL ∂ L H MK Pascal du Bosque Generalized Metric Formulation of DFT WZW 2/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT Action [2] � ˜ � ˜ � ∂ i � x i • Doubled coordinates and derivative X M = resp. ∂ M = , x i ∂ i along with generalized metric � g ij − g ik B kj � H MN = ∈ O ( D , D ) → T-Duality B ik g kj g ij − B ik g kj B lj • Additionally, identify e − 2 d = e − 2 φ √− g (dilaton density) • How to retrieve SUGRA action from DFT? ˜ ∂ i = 0 − − − → S DFT S NS SUGRA DFT Pascal du Bosque Generalized Metric Formulation of DFT WZW 3/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography T-Duality • Closed strings can wrap non-contractable cycles around compact dimensions: ⇐ ⇒ T-Duality p i ⇐ ⇒ momenta p i winding modes ˜ � � T-Duality coordinates x i ⇐ ⇒ winding coordinates ˜ x i • Connects different background topologies • Manifest symmetry of DFT action Pascal du Bosque Generalized Metric Formulation of DFT WZW 4/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography Generalized Diffeomorphisms [2] • Generalized metric: L ξ H MN = ξ P ∂ P H MN + H PN + ˆ ∂ M ξ P − ∂ P ξ M � ∂ N ξ P − ∂ P ξ N � H MP � � • Dilaton: L ξ d = − 1 2 ∂ M ξ M + ξ M ∂ M d , L ξ e − 2 d = ∂ M ( ξ M e − 2 d ) ˆ ˆ and • O ( D , D ) metric: L ξ η MN = 0 ˆ = ⇒ S DFT invariant when strong constraint imposed Pascal du Bosque Generalized Metric Formulation of DFT WZW 5/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT gauge algebra [2] • Closure: � ˆ L ξ 1 , ˆ = ˆ L ξ 2 � L [ ξ 1 ,ξ 2 ] C modulo strong constraint with C-bracket: 1 − 1 2 + 1 � M C = ξ N 1 ∂ N ξ M 2 − ξ N 2 ∂ N ξ M 2 ξ 1 N ∂ M ξ N 2 ξ 2 N ∂ M ξ N � ξ 1 , ξ 2 1 • Strong constraint: ∂ M ∂ M ( A · B ) = 0 ∀ fields A , B � if M = i ∂ i e.g. solved by 0 else Pascal du Bosque Generalized Metric Formulation of DFT WZW 6/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT on Group Manifolds [3] • Use group manifold instead of torus to derive DFT • Representation for semisimple Lie algebras D a = e ai ∂ i , [ D a , D b ] = F abc D c and commutation relation ⇒ same goes for the anti-chiral flat derivative D ¯ a ··· ⇒ unimodularity of the Lie algebra allows for integration by parts! • Perform CSFT calculations to obtain action and gauge transformations up to cubic order ··· (lengthy formulas) (in terms of ǫ a ¯ b , ˜ d , D a , D ¯ a , F abc , F ¯ c ) a ¯ b ¯ Pascal du Bosque Generalized Metric Formulation of DFT WZW 7/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography Generalized Diffeomorphisms [3, 4] � D a � • Introduce doubled flat derivative : D A = D ¯ a • We can rewrite the gauge transformations as: L ξ H AB = ξ C ∇ C H AB + H CB + ∇ A ξ C − ∇ C ξ A � ∇ B ξ C − ∇ C ξ B � H AC � � with F abc ∇ A V B = D A V B + 1 F ABC = 3 F BAC V C , c ¯ and F ¯ a ¯ b 0 otherwise 2 D A ξ A , δ ξ ˜ d = L ξ ˜ d = ξ A D A ˜ ∇ A d = D A ˜ d − 1 while d δ ξ η AB = L ξ η AB = 0 , however δ ξ S AB = 0 but L ξ S AB � = 0 Pascal du Bosque Generalized Metric Formulation of DFT WZW 8/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT gauge algebra [2] • Closure: � � L ξ 1 , L ξ 2 = L [ ξ 1 ,ξ 2 ] C modulo strong constraint with C-bracket: 1 − 1 2 + 1 � A C = ξ B 1 ∂ B ξ A 2 − ξ B 2 ∂ B ξ A 2 ξ 1 B ∂ A ξ B 2 ξ 2 B ∂ A ξ B � ξ 1 , ξ 2 1 • Strong constraint: ∂ A ∂ A ( f · g ) = 0 ∀ fields f , g Closure Constraint: F E [ AB F E C ] D = 0 (background fields) Pascal du Bosque Generalized Metric Formulation of DFT WZW 9/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT WZW gauge algebra [3] • Closure: � � L ξ 1 , L ξ 2 = L [ ξ 1 ,ξ 2 ] C modulo strong + closure constraint with C-bracket: 1 − 1 2 + 1 � A C = ξ B 1 ∇ B ξ A 2 − ξ B 2 ∇ B ξ A 2 ξ 1 B ∇ A ξ B 2 ξ 2 B ∇ A ξ B � ξ 1 , ξ 2 1 • Strong constraint: D A D A ( f · g ) = 0 ∀ fluctuations f , g • Closure Constraint: F E [ AB F E C ] D = 0 (background fields) Pascal du Bosque Generalized Metric Formulation of DFT WZW 9/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT action [2] • Rewrite action? � d 2 D X e − 2 d R ( H , d ) S DFT = • Generalized curvature scalar: R ≡ 4 H AB ∂ A d ∂ B d − ∂ A ∂ B H AB − 4 H AB ∂ A d ∂ B d + 4 ∂ A H AB ∂ B d + 1 8 H AB ∂ A H CD ∂ B H CD − 1 2 H AB ∂ B H CD ∂ D H AC ? Pascal du Bosque Generalized Metric Formulation of DFT WZW 10/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT WZW action [4] • Rewrite action? � d 2 D X e − 2 d R ( H , d ) S DFT WZW = • Generalized curvature scalar: R ≡ 4 H AB ∇ A d ∇ B d − ∇ A ∇ B H AB − 4 H AB ∇ A d ∇ B d + 4 ∇ A H AB ∇ B d + 1 8 H AB ∇ A H CD ∇ B H CD − 1 2 H AB ∇ B H CD ∇ D H AC ? Pascal du Bosque Generalized Metric Formulation of DFT WZW 10/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography DFT WZW action [4] • Rewrite action? � d 2 D X e − 2 d R ( H , d ) S DFT WZW = • Generalized curvature scalar: R ≡ 4 H AB ∇ A d ∇ B d − ∇ A ∇ B H AB − 4 H AB ∇ A d ∇ B d + 4 ∇ A H AB ∇ B d + 1 8 H AB ∇ A H CD ∇ B H CD − 1 2 H AB ∇ B H CD ∇ D H AC + 1 6 F ACE F BDF H AB S CD S EF ⇒ invariant under generalized diffeomorphisms when s.c. + c.c. = imposed ⇒ additional 2 D -diffeomorphism invariance = Pascal du Bosque Generalized Metric Formulation of DFT WZW 10/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography Transition to toroidal DFT? [4] • Relationship between both DFT formulations? • Extended strong constraint: D A D A ( f · b ) = 0 ∀ fluctuations f , background fields b ⇒ L ξ V M = ˆ L DFT ,ξ V M = = ⇒ S DFT WZW = S DFT SUGRA DFT DFT WZW Pascal du Bosque Generalized Metric Formulation of DFT WZW 11/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography • Found generalized metric formulation of DFT WZW : ⇒ theory invariant under generalized and 2 D -diffeomorphisms • DFT WZW ’generalizes’ original DFT description • Truly non-geometric backgrounds with new physical information? ⇒ Well-defined? • Extension of DFT WZW to arbitrary background geometries possible? • Tool to analyze non-associativity, non-commutativity of non-geometric backgrounds? Pascal du Bosque Generalized Metric Formulation of DFT WZW 12/13
Double Field Theory DFT on Group Manifolds Conclusion and Outlook Bibliography Bibliography C. Hull, and B. Zwiebach. Double Field Theory . JHEP , 0909:099, 2009. O. Hohm, C. Hull, and B. Zwiebach. Generalized Metric Formulation of Double Field Theory . JHEP , 1008:008, 2010. R. Blumenhagen, F. Hassler, and D. Lüst. Double Field Theory on Group Manifolds . JHEP , 1502:001, 2015. R. Blumenhagen, P. d. Bosque, F. Hassler, and D. Lüst. Generalized Metric Formulation of DFT on Group Manifolds . arXiv : 1502.02428, 2015. Pascal du Bosque Generalized Metric Formulation of DFT WZW 13/13
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