A Connection for Born Geometry and its Application to DFT String and M-theory Geometries Banff Internation Research Station January 26, 2017 Felix J. Rudolph Ludwig-Maximilian-Universtit¨atM¨unchen to appear 17xx.xxxxx with L. Freidel and D. Svoboda A Connection for Born Geometry Key Points I What is Born Geometry? (idea of dynamical phase space) I Connections for various geometries I Torsion and integrability I Relevance to Double Field Theory A Connection for Born Geometry Outline Born Geometry A Connection for Born Geometry Application to DFT A Connection for Born Geometry Born Geometry Born Geometry A Connection for Born Geometry Born Geometry Born Reciprocity In Quantum Mechanics I symmetry between spacetime and momentum space I freedom to choose a basis of states In General Relativity I this symmetry is broken: spacetime energy-momentum is curved space is flat Max Born (1935): "To unify QM and GR need momentum space to be curved" A Connection for Born Geometry Born Geometry Born Geometry [Freidel, Leigh, Minic] Classical Mechanics I (almost) symplectic structure ! on phase space P Sp(2d) Quantum Mechanics I complex structure I: x ! p; p ! −x with I2 = −1 T I compatibility: I !I = ! I defines a metric on phase space H = !IO (2d; 0) I "quantum" or "generalized" metric A Connection for Born Geometry Born Geometry Born Geometry To split phase space into spacetime and momentum space I bi-Lagrangian (real) structure K: T P = L ⊕ L~ 2 K L = +1;K L~ = −1 with K = +1 T I compatibility: K !K = −! I defines another metric on phase space η = !KO (d; d) I "polarization" or "neutral" metric I spacetime is maximal null subspace w.r.t. η H L = g A Connection for Born Geometry Born Geometry Born Geometry The Born Geometry (P; η; !; H) unifies I symplectic structure of classical mechanics I complex strucuture of quantum mechanics I real structure of general realitivity Quantum gravity needs a dynamical phase space String theory provides a realization of these concepts and geometric structure A Connection for Born Geometry Born Geometry String Theory Tseytlin Action on phase space with X = (x/λ, y/) 1 Z S = dτdσ(η + ! )@ XA@ XB − H @ XA@ XB 2 AB AB τ σ AB σ σ I including topological term [Giveon, Rocek; Hull] String Theory −1 I chiral structure: J = η H I T-duality on target space: X ! J(X) I ! and K not required, but present A Connection for Born Geometry Born Geometry Para-quaternionic Manifold Born Geometry (P; η; !; H) −! para-quaternions (I; J; K) −1 2 I complex structure I = H ! (I = −1) −1 2 I chiral structure J = η H (J = +1) −1 2 I real structure K = η ! (K = +1) All mutually anti-commute I = JK = −KJ; J = IK = −KI;IJK = −1 K = JI = −IJ; A Connection for Born Geometry Born Geometry Integrability Integrability Almost bi-Lagrangian structure K I Splitting into Lagrangian distributions: T P = L ⊕ L~ T I K !K = −! ! Lagrangian eigenspace ! L = ! L~ = 0 T I K ηK = −η ! Null eigenspace η L = η L~ = 0 If K integrable: I [L; L] ⊂ L and [L;~ L~] ⊂ L~ I Induces a polarization I Darboux coordinates (x; x~) spanning L and L~ A Connection for Born Geometry Born Geometry Integrability Projectors of (3; 0)-tensors Lagrangian Subspaces (Polarizations): T P = L ⊕ L~ ± 2 ± + − I (K ) = K ; K K = 0 4K±N(X; Y; Z) := N(X; Y; Z) + N(K(X);K(Y );Z) ± N(X; K(Y );K(Z)) ± N(K(X); Y; K(Z)) Chiral subspaces: T P = C+ ⊕ C− ± 2 ± + − I (J ) = J ; J J = 0 4J ±N(X; Y; Z) := N(X; Y; Z) + N(J(X);J(Y );Z) ± N(X; J(Y );J(Z)) ± N(J(X); Y; J(Z)) A Connection for Born Geometry Born Geometry Torsion, Contorsion and Nijenhuis tensor Torsion For a connection r = @ + Γ Usual Torsion I T (X; Y ) = rX Y − rY X − [X; Y ] k k k I Tij = Γij − Γji Generalized Torsion r @ I T (X; Y ) = LX Y − LX Y k k k k I Tij = Γij − Γji − Γ ji 2 I T 2 Γ(Λ (P) ⊗ X(P)) A Connection for Born Geometry Born Geometry Torsion, Contorsion and Nijenhuis tensor Contorsion For any metric-compatible connection r and Levi-Civita connection r˚: Contorsion tensor Ω I r = r˚ + Ω or Γ = ˚Γ + Ω 1 I Ωijk = 2 (Tijk − Tjki + Tkij) 1 I Tijk = Ωijk + Ωjki + Ωkij = 2 (Tijk + Tjki + Tkij) A Connection for Born Geometry Born Geometry Torsion, Contorsion and Nijenhuis tensor The Nijenhuis Tensor The Nijenhuis Tensor of a tangent bundle structure A 2 NA 2 Γ(Λ (P) ⊗ X(P)) 2 NA(X; Y ) = A [A(X);Y ]+[X; A(Y )] −[A(X);A(Y )]−A [X; Y ] If A is integrable, NA = 0. A Connection for Born Geometry Born Geometry Torsion, Contorsion and Nijenhuis tensor The Nijenhuis Tensor Need it for bi-Lagrangian structure K = η−1! NK (X; Y; Z) = r˚Y !(X; K(Z)) − r˚X !(Y; K(Z)) ˚ ˚ + rK(Y )!(X; Z) − rK(X)!(Y; Z) = d!(K(X);K(Y );K(Z)) + d!(X; Y; K(Z)) ˚ + 2rK(Z)!(X; Y ) A Connection for Born Geometry A Connection for Born Geometry A Connection for Born Geometry A Connection for Born Geometry A Connection for Born Geometry Connections for various geometries Connections for various geometries Levi-Civita Connection I Riemannian geometry: metric η I Unique, compatible, symmetric, torsion-free Fedosov Connection I Symplectic geometry: almost symplectic form ! I Family of compatible, symmetric, torsion-free Bismut Connection I Hermitian geometry: almost complex structure I, metric H I Unique, compatible with H & I, totally skew torsion A Connection for Born Geometry A Connection for Born Geometry Connections for various geometries Connections for various geometries Fedosov + metric = Levi-Civita −1 I Symplectic manifold with metric (real structure K = η !) I Unique, compatible with ! & η, symmetric, torsion-free Doubled space of DFT −1 I Two metrics: η and H (chiral structure J = η H) I DFT connection not fully determined Born Connection I Born geometry (P; η; !; H) with para-quaternions (I; J; K) I Unique, compatible, (generalized) torsion is chiral A Connection for Born Geometry A Connection for Born Geometry The Born Connection The Born Connection Defining Properties I Compatibility with Born geometry (η; !; H) rη = r! = rH = 0 I Generalized torsion is chiral + T ∼ J NK Vanishing generalized torsion T if K integrable NK = 0 )T = 0 A Connection for Born Geometry A Connection for Born Geometry The Born Connection The Born Connection Born Connection r given by η(rX Y; Z) = η(r˚X Y; Z) + η(X; Ω(Y; Z)) where I r˚ is the Levi-Civita connection of η I Ω is the contorsion A Connection for Born Geometry A Connection for Born Geometry The Born Connection The Born Connection Contorsion 1 η(X; Ω(Y; Z)) = r˚ H(Y; J(Z)) 2 X 1 + [r˚ H(J(Z);X) − r˚ H(I(Z);X) 4 Y K(Y ) ˚ ˚ − rJ(Z)H(X; Y ) + rI(Z)H(X; K(Y ))] 1 − [r˚ H(X; J(Y )) − r˚ H(X; I(Y )) 4 Z K(Z) ˚ ˚ − rJ(Y )H(Z; X) + rI(Y )H(K(Z);X)] 1 + [r˚ !(I(Y );Z) + r˚ !(Y; I(Z)) 4 J(X) J(X) − r˚X !(Y; K(Z)) + r˚X !(J(Y );I(Z))] A Connection for Born Geometry A Connection for Born Geometry The Born Connection Properties and Identities e.g. η-compatibility ) skew-symmetry Ω(X; Y ) = −Ω(Y; X) Some identites needed for proofs r˚X H(Y; Z) = Ω(X; Y; J(Z))) − Ω(X; J(Y );Z); −r˚X !(Y; Z) = Ω(X; Y; K(Z)) + Ω(X; K(Y );Z) A Connection for Born Geometry A Connection for Born Geometry The Born Connection Existence and Uniqueness Uniqueness I If such a connection exists, it is unique and given by Ω I Fully determined in terms of (η; !; H) Existence I Constructive proof I Properties of a connection A Connection for Born Geometry A Connection for Born Geometry The Born Connection Vanishing Generalized Torsion Need the following objects I Chiral Nijenhuis tensor for K + J NK (X; Y; Z) measures integrability along the chiral subspaces C± I Polarized component of generalized torsion 1 X K+T (X; Y; Z) = N (X; Y; Z) 2 K cycl(X;Y;Z) A Connection for Born Geometry A Connection for Born Geometry The Born Connection Vanishing Generalized Torsion Can express T in terms of NK T (X; Y; Z) = J +K+T (X; Y; Z) 1 X = J +N (X; Y; Z) 2 K cycl(X;Y;Z) Generalized torsion vanishes if K is integrable NK = 0 )T = 0 A Connection for Born Geometry A Connection for Born Geometry The Born Connection Fluxes No flux: B = 0, d! = 0 I r˚X !(Y; Z) = 0 I Levi-Civita = Fedosov connection Turn on H-flux H = dB I H = H(g; B) and d! ∼ H 6= 0 + I flux appears in torsion (T ∼ J NK with NK ∼ H) I Bismut connection - torsion is totally skew [Ellwood; Gualtieri] More general fluxes (H; f; Q; R) ⊂ F I d! ∼ F 6= 0 A Connection for Born Geometry A Connection for Born Geometry The Born Connection Structure group Have the following groups O(d; d) \ Sp(2d) \ O(2d; 0) = O(d) I Born connection reduces to O(d) connection on Lagrangian submanifold with metric g I Levi-Civita connection of g ? A Connection for Born Geometry Application to DFT Application to DFT A Connection for Born Geometry Application to DFT Coordinate Expression Introduce frame field I Frame field EA A I Local coordinates X = X EA ηAB = η(EA;EB);!AB = !(EA;EB); HAB = H(EA;EB) Covariant derivative of a vector B B B C rAX = r˚AX + ΩAC X A Connection for Born Geometry Application to DFT Coordinate Expression Born connection given by 1 Ω = r˚ H J D + δ[D J E] − K[D IE] r˚ H ABC 2 A BD C [B C] [B C] D EA 1 1 − r˚ ! IE J D − δD KE − J D IE r˚ ! 2 D E[B C] A 4 B C B C A DE A Connection for Born Geometry Application to DFT Application to Double Field Theory DFT Limit of Born Geometry: η and ! flat 0 1 0 1 g 0 η = ;! = ; H = 1 0 −1 0 0 g−1 Connection reduces to (r˚ ! @) 1 1 Ω = @ H J D + δD J E + J D δE @ H ABC 2 A BD C 2 [B C] [B C] D EA 1 − KD IE − KE ID @ H 2 [B C] [B C] D EA A Connection for Born Geometry Application to DFT Application to Double Field Theory [Coimbra, Strickland-Constable, Determined part of DFT connection Waldram; Hohm, Zwiebach; Jeon, Lee, Park] 1 1 Γ = @ H J L + δP J Q + J P δQ @ H MNK 2 M NL K 2