T-Duality, fluxes and noncommutativity in closed string theory Athanasios Chatzistavrakidis Rudjer Boškovi´ c Institute, Zagreb Mainly: arXiv:1802.07003 with Larisa Jonke, Fech Scen Khoo, Richard Szabo Also: arXiv:1505.05457 with Larisa Jonke, Olaf Lechtenfeld Matrix Models for Noncommutative Geometry and String Theory @ ESI Vienna 13 July 2018
“Lessons” for the Geometry of Spacetime and Quantum Gravity ✿ Geometry is Generalized (Noncommutativity, String, Matrices) ✿ Geometry is (maybe) Emergent (String, AdS/CFT, Matrices, ...) ✿ Geometry is (maybe) Doubled (Quantum Mechanics/Born Reciprocity, String/T-duality)
Doubling for Closed Strings Circle compactifications � Momentum and Winding modes with mass ∝ 1 / R and R Large radius limit � Only momentum modes probe spacetime, and EFT is supergravity � measure lengths with position operators x √ At QG scales, R ∼ α ′ � both momentum and winding modes become important e.g. in the Brandenberger-Vafa early universe scenario � position operators x and dual (to windings) ˜ x Supergravity is certainly not enough here � need (some kind of) Double Field Theory e.g. proposals by Siegel ’93; Hull, Zwiebach ’09; Freidel, Leigh, Minic ’15; &c.
Symmetries On one hand, we have diffeomorphisms and gauge transformations, as in field theory But for closed strings, also T-duality, exchanging momenta ↔ windings and R ↔ 1 / R T N.B., T-duality is an asymmetric reflection: X ( σ, τ ) = X L + X R → ˜ X ( σ, τ ) = X L − X R When multiple ( d ) circle compactification, the T-duality symmetry group is O ( d , d ; Z ) � The Double Field Theory should enjoy an O ( d , d ; R ) symmetry � The underlying geometric structure should contain/unify these symmetries
Flux, Duality and Open Strings For open strings on D-branes ✤ Turn on B or F � noncommutativity Douglas, Hull ’97; Chu, Ho ’98; Seiberg, Witten ’99 θ 12 = − 2 π i α ′ ( B − F ) [ X 1 ( τ ) , X 2 ( τ )] = i θ 12 , 1 + ( B − F ) 2 . ✿ T-dual frame � commutativity & D-branes at angles. Lesson: New geometries arise in presence of non-trivial flux backgrounds.
Flux, Duality and Closed Strings Left and right movers may experience different geometries (asymmetric strings). T-duality reveals closed string backgrounds which are “non-geometric” (T-folds & co.) e.g. Hull ’04; Shelton, Taylor, Wecht ’05; &c. f k Q jk R ijk H ijk → → → ij i Generic closed string geometries argued to be noncommutative and nonassociative. Lüst ’10; Blumenhagen, Plauschinn ’10; Mylonas, Schupp, Szabo ’12 [ X i , X j ] ∼ Q ij k w k Q -case [ X i , X j , X k ] ∼ R ijk . [ X i , X j ] ∼ R ijk p k R -case Similar to particle in a non-constant magnetic field in QM. Jackiw ’85; Bakas, Lüst ’13
Enter Algebroids ✿ Courant Algebroids: unify Poisson and pre-symplectic structures Courant ’90; Liu, Weinstein, Xu ’95 ◮ Canonical example: TM ⊕ T ∗ M , with a natural O ( d , d ) metric, and fluxes as twists ✿ Generalized Complex Geometry: unify symplectic and complex structures Hitchin ’02; Gualtieri ’04 ◮ g and B on equal footing, Diffs and Gauge trafos as automorphisms of Courant bracket ◮ Main additional player: a generalized metric: g ij − B ik g kl B lj � B ik g kj � H IJ = − g ik B kj g ij . Courant Algebroid vs. Doubling of coordinates ✤ Captures the symmetries, but not the doubling of coordinates ✤ But if the target is doubled, the symmetry would be O ( 2 d , 2 d ) , i.e. too large
Double Field Theory Siegel ’93; Hull, Zwiebach ’09 A proposal for a field theory invariant under O ( d , d ) ; T-duality becomes manifest. It uses doubled coordinates ( x I ) = ( x i , � x i ) , and all fields depend on both. The O ( d , d ) structure is associated to a (constant) O ( d , d ) -invariant metric � 0 � 1 d h t η h = η , η = ( η IJ ) = , h ∈ O ( d , d ) , 1 d 0 used to raise and lower I = 1 , . . . , 2 d indices. Derivatives are also doubled accordingly: ( ∂ I ) = ( ∂ i , � ∂ i ) . The fields are the generalized metric H and invariant dilaton d ( e − 2 d = √− ge − 2 φ ), with Hohm, Hull, Zwiebach ’10 � xe − 2 d � � 2 H IJ ∂ I H KL ∂ L H KJ − 2 ∂ I d ∂ J H IJ + 4 H IJ ∂ I d ∂ J d 1 8 H IJ ∂ I H KL ∂ J H KL − 1 d x d � S = .
DFT symmetries and constraints Gauge transformations are included with a parameter ǫ I = ( ǫ i , � ǫ I ) : ǫ K ∂ K H IJ + ( ∂ I ǫ K − ∂ K ǫ I ) H KJ + ( ∂ J ǫ K − ∂ K ǫ J ) H IK := L ǫ H IJ , δ ǫ H IJ = 2 ∂ K ǫ K + ǫ K ∂ K d , − 1 δ ǫ d = and L ǫ · is called the generalised Lie derivative. But S is not automatically invariant. The theory is constrained. ✤ Weak constraint: ∆ · := ∂ I ∂ I · = 0; stems from the level matching condition. ✿ Strong constraint: ∂ I ⊗ ∂ I ( . . . ) = 0 on products on fields. s.c. Strong constraint eliminates half coordinates � DFT → SUGRA Alternatively, generalized vielbein E formulation H IJ = E A I E B J S AB . Siegel ’93; Hohm, Kwak ’10; Aldazabal et al. ’11; Geissbuhler ’11 ✤ Allows to mildly dispense with the s.c. in generalized Scherk-Schwarz reductions
Questions to address ✿ What is the geometric structure of DFT and its relation to Courant algebroids? ✿ What is the Sigma-Model that captures the flux content of DFT? cf. Heller, Ikeda, Watamura ’16 ✿ What is the origin/role of DFT constraints and how does noncommutativity appear? We want to answer these questions in the context of Membrane Sigma-Models
Membranes for Strings: Why? ✤ Already the familiar NSNS flux (field strength of B ) lives in 3D (open membrane) ✤ Courant Algebroids correspond naturally to 3D Topological Field Theories ✤ Deformation quantization viewpoint acknowledging private communication with Peter Schupp ◮ (“Closed”) Fields Open Strings (Poisson Sigma-Model) � ◮ Closed Strings Open Membranes (Courant Sigma-Model) � ◮ Closed Membranes ? Open Tribranes (LAuth Sigma-Model) �
Plan for the rest of the talk Sigma-Models and Courant Algebroids 1 Doubled Membrane Sigma-Model 2 Universal description of geometric and non-geometric fluxes — NC/NA structure 3 (Almost) Algebroid Structures beyond Courant 4 Epilogue 5
Warm Up: (Twisted) Poisson Sigma-Model Topological action for fields X = ( X i ) : Σ 2 → M and A ∈ Ω 1 (Σ 2 ; X ∗ T ∗ M ) Schaller, Strobl ’94; Ikeda ’94 � � � A i ∧ d X i + 1 2 Π ij ( X ) A i ∧ A j S PSM [ X , A ] = Σ 2 Invariant under the gauge symmetry: δ X i Π ji ǫ j , = d ǫ i + ∂ i Π jk A j ǫ k , δ A i = provided that Π l [ i ∂ l Π jk ] = 0 Π is a Poisson 2-vector → Comments ✿ May be twisted by a 3-form H (Wess-Zumino term) � twisted Poisson structure Klimcik, Strobl ’01 Π l [ i ∂ l Π jk ] = H lmn Π li Π mj Π nk . ✿ 2D case of AKSZ scheme of topological field theories (for H = 0 at least) Alexandrov, Kontsevich, Schwarz, Zaboronsky ’95 ✿ Deformation Quantization of Poisson manifolds ∼ Perturbation theory of PSM Kontsevich ’97; Cattaneo, Felder ’99
Courant Sigma-Model Hofman, Park ’02; Ikeda ’02 Maps X = ( X i ) : Σ 3 → M , 1-forms A ∈ Ω 1 (Σ 3 , X ∗ E ) , and 2-form F ∈ Ω 2 (Σ 3 , X ∗ T ∗ M ) � � 6 T IJK ( X ) A I ∧ A J ∧ A K � F i ∧ d X i + 1 2 η IJ A I ∧ d A J − ρ i I ( X ) A I ∧ F i + 1 S [ X , A , F ] = . E is some vector bundle (here TM ⊕ T ∗ M ), η is the (constant) O ( d , d ) -invariant metric � 0 � 1 d η = ( η IJ ) = . 1 d 0 3D case of AKSZ scheme of topological field theories Roytenberg ’06
Gauge Symmetries of the Courant Sigma-Model The Courant Sigma-Model is invariant under the following gauge transformations Ikeda ‘02 δ X i = ρ i J ǫ J , δ A I = d ǫ I + η IN T NJK A J ǫ K + η IJ ρ i J t i , 2 ǫ J ∂ m T ILJ A I ∧ A L + d t m + ∂ m ρ j δ F m = − ǫ J ∂ m ρ i J F i + 1 J A J t j , where ǫ and t are gauge parameters, provided that η KL ρ i K ρ j L = 0 2 ρ l [ I ∂ l ρ k J ] − ρ k J η JL T LIJ = 0 4 ρ m [ L ∂ m T IJK ] − 3 η MN T M [ IJ T KL ] N = 0 . These three conditions have an interesting relation to both physics and mathematics ✿ Coincide with the fluxes and Bianchi identities in sugra flux compactifications ✿ Coincide with the local form of the axioms of a Courant Algebroid
Courant Algebroid Axioms Liu, Weinstein, Xu ’95 π → M , [ · , · ] , �· , ·� , ρ : E → TM ) , such that for A , B , C ∈ Γ( E ) and f , g ∈ C ∞ ( M ) : ( E Modified Jacobi identity ( D : C ∞ ( M ) → Γ( E ) is defined by �D f , A � = 1 2 ρ ( A ) f . ) 1 N ( A , B , C ) = 1 [[ A , B ] , C ] + c.p. = DN ( A , B , C ) , where 3 � [ A , B ] , C � + c.p. , Modified Leibniz rule 2 [ A , fB ] = f [ A , B ] + ( ρ ( A ) f ) B − � A , B �D f , Compatibility condition 3 ρ ( C ) � A , B � = � [ C , A ] + D� C , A � , B � + � [ C , B ] + D� C , B � , A � , The structures also satisfy the following properties (they follow... Uchino ’02 ): Homomorphism 4 ρ [ A , B ] = [ ρ ( A ) , ρ ( B )] , “Absence of strong constraint” 5 ρ ◦ D = 0 �D f , D g � = 0 . ⇔
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