Flux Compactifications and Matrix Models for Superstrings - PowerPoint PPT Presentation

Flux Compactifications and Matrix Models for Superstrings Athanasios Chatzistavrakidis Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at Hannover Based on: A.C., 1108.1107 [hep-th] (PRD 84 (2011)) A.C. and Larisa Jonke, 1202.4310 [hep-th] (PRD 85 (2012)) A.C. and Larisa Jonke, 1207.6412 [hep-th] Edinburgh Mathematical Physics Group 14.11.12 A. Chatzistavrakidis (ITP Hannover) 1 / 33

Introduction and Motivation Main Objective Study properties of string compactifications beyond low-energy sugra. Mainly, unconventional compactifications � related to string length, not captured by vanilla sugra (winding modes, dualities, non-geometric fluxes, non-commutative manifolds etc.). A. Chatzistavrakidis (ITP Hannover) 2 / 33

Introduction and Motivation Main Objective Study properties of string compactifications beyond low-energy sugra. Mainly, unconventional compactifications � related to string length, not captured by vanilla sugra (winding modes, dualities, non-geometric fluxes, non-commutative manifolds etc.). Frameworks: • Doubled formalism - Twisted Doubled Tori • Generalized Complex Geometry • Double Field Theory • CFT - Sigma models ✔ Matrix Models A. Chatzistavrakidis (ITP Hannover) 2 / 33

Introduction and Motivation Main Objective Study properties of string compactifications beyond low-energy sugra. Mainly, unconventional compactifications � related to string length, not captured by vanilla sugra (winding modes, dualities, non-geometric fluxes, non-commutative manifolds etc.). Frameworks: • Doubled formalism - Twisted Doubled Tori Hull; Hull, Reid-Edwards; Dall’Agata et.al. • Generalized Complex Geometry Andriot et.al.; Berman et.al. • Double Field Theory Hohm, Hull, Zwiebach; Aldazabal et.al.; Geissbuhler; Grana, Marques; Dibitetto et.al. • CFT - Sigma models L¨ ust; Blumenhagen, Plauschinn; Mylonas, Schupp, Szabo ✔ Matrix Models Lowe, Nastase, Ramgoolam; A.C., Jonke A. Chatzistavrakidis (ITP Hannover) 3 / 33

Why Matrix Models? Advantages: ✔ Non-perturbative framework. ✔ Non-commutative structures. ✔ Quantization. ✔ Possible phenomenological applications • Particle physics, “matrix model building“. Aoki ’10-’12, A.C., Steinacker, Zoupanos ’11 • Early and late time cosmology. Kim, Nishimura, Tsuchiya ’11-’12 A. Chatzistavrakidis (ITP Hannover) 4 / 33

Why Matrix Models? Advantages: ✔ Non-perturbative framework. ✔ Non-commutative structures. ✔ Quantization. ✔ Possible phenomenological applications • Particle physics, “matrix model building“. Aoki ’10-’12, A.C., Steinacker, Zoupanos ’11 • Early and late time cosmology. Kim, Nishimura, Tsuchiya ’11-’12 Disadvantages: × Sugra limit is not clear. × Less calculability. A. Chatzistavrakidis (ITP Hannover) 4 / 33

Matrix Models as non-perturbative definitions of string/M theory. Banks, Fischler, Shenker, Susskind ’96, Ishibashi, Kawai, Kitazawa, Tsuchiya ’96, ... Matrix Model Compactifications (MMC) on non-commutative tori. Connes, Douglas, A. Schwarz ’97 Constant background B-field ← → Non-commutative deformation CDS θ ij ← → B ij A. Chatzistavrakidis (ITP Hannover) 5 / 33

Matrix Models as non-perturbative definitions of string/M theory. Banks, Fischler, Shenker, Susskind ’96, Ishibashi, Kawai, Kitazawa, Tsuchiya ’96, ... Matrix Model Compactifications (MMC) on non-commutative tori. Connes, Douglas, A. Schwarz ’97 Constant background B-field ← → Non-commutative deformation CDS θ ij ← → B ij What about fluxes? • Geometric (related e.g. to nilmanifolds/twisted tori): f • NSNS (e.g. non-constant B-fields): H • “Non-geometric” (T-duality): Q , R Q: How can they be traced in Matrix Compactifications? A. Chatzistavrakidis (ITP Hannover) 5 / 33

Overview Matrix Models for superstrings 1 Nilmanifolds 2 Matrix Model Compactifications 3 T-duality, Non-associativity and Flux Quantization 4 Work in progress 5 Concluding Remarks 6 A. Chatzistavrakidis (ITP Hannover) 6 / 33

Matrix Models IKKT: non-perturbative IIB superstring, Ishibashi, Kawai, Kitazawa, Tsuchiya ’96 � d X d Ψ e − S , Z = with action � � S IKKT = 1 − 1 2[ X a , X b ] 2 − ¯ ΨΓ a [ X a , Ψ] 2 g Tr . X a : 10 N × N Hermitian matrices (large N ); Ψ: fermionic superpartners. BFSS: non-perturbative M-theory, Banks, Fischler, Shenker, Susskind ’96 � ˙ S BFSS = 1 � � X a − 1 � X a ˙ 2[ X a , X b ] 2 � dt Tr + fermions , 2 g X a ( t ): 9 and time-dependent... A. Chatzistavrakidis (ITP Hannover) 7 / 33

Classical solutions EOM (IKKT; setting Ψ = 0): � [ X b , [ X b , X a ]] = 0 . b • Basic solutions: [ X a , X b ] = i θ ab Rank( θ ) = p + 1 ⇒ Dp brane. A. Chatzistavrakidis (ITP Hannover) 8 / 33

Classical solutions EOM (IKKT; setting Ψ = 0): � [ X b , [ X b , X a ]] = 0 . b • Basic solutions: [ X a , X b ] = i θ ab Rank( θ ) = p + 1 ⇒ Dp brane. • Lie algebra type? [ X a , X b ] = if c ab X c If no deformation � no semisimple. Nilpotent and solvable? Fully classified up to 7D (6D: finite) Morozov ’58, Mubarakzyanov ’63, Patera et.al. ’75 Resulting solutions: 7 nilpotent (3D, 5D(2), 6D(4)) + 2 solvable (4D, 5D). A.C. ’11 A. Chatzistavrakidis (ITP Hannover) 8 / 33

Classical solutions EOM (IKKT; setting Ψ = 0): � [ X b , [ X b , X a ]] = 0 . b • Basic solutions: [ X a , X b ] = i θ ab Rank( θ ) = p + 1 ⇒ Dp brane. • Lie algebra type? [ X a , X b ] = if c ab X c If no deformation � no semisimple. Nilpotent and solvable? Fully classified up to 7D (6D: finite) Morozov ’58, Mubarakzyanov ’63, Patera et.al. ’75 Resulting solutions: 7 nilpotent (3D, 5D(2), 6D(4)) + 2 solvable (4D, 5D). A.C. ’11 Why is this interesting? ✔ Play role in cosmological studies based on IKKT. Kim, Nishimura, Tsuchiya ’11-’12 ✔ Starting point for a class of compact manifolds (nil- and solvmanifolds). A. Chatzistavrakidis (ITP Hannover) 8 / 33

Nilmanifolds Mal’cev ’51 Smooth manifolds M = G / Γ G : Nilpotent Lie group; Γ: Discrete co-compact subgroup of G . Nilpotency � upper triangular matrices... Construction algorithm: α. Find a basis T a of Lie(G) in terms of upper triangular matrices. β. Choose a representative group element g ∈ G . γ. Define the restriction of g for integer matrix entries ( γ ∈ Γ). δ. Γ acts on G by matrix multiplication. Quotient out this action and construct G / Γ. A. Chatzistavrakidis (ITP Hannover) 9 / 33

Some geometry Lie algebra 1-form e = g − 1 dg = e a T a . e a correspond to the vielbein basis and there is a twist matrix such that: e a = U ( x ) a b dx b They satisfy the Maurer-Cartan equations de a = − 1 bc e b ∧ e c , 2 f a f a bc being the structure constants of Lie(G) ∼ geometric fluxes. Certain periodicity conditions render e a globally well-defined. Thus nilmanifolds are (iterated) twisted fibrations of toroidal fibers over toroidal bases. A. Chatzistavrakidis (ITP Hannover) 10 / 33

� � � � � M � T D n � � i D i . . . � M D 1 + D 2 + D 3 T D 3 � � � M D 1 + D 2 T D 2 � � T D 1 → The number of such iterations is set by the nilpotency class. A. Chatzistavrakidis (ITP Hannover) 11 / 33

Prototype example: 3D 3D nilpotent Lie algebra: [ T 1 , T 2 ] = T 3 . Upper triangular basis:       0 1 0 0 0 0 0 0 1  ,  ,  . T 1 = 0 0 0 T 2 = 0 0 1 T 3 = 0 0 0    0 0 0 0 0 0 0 0 0 x 1 x 3   1  , x i ∈ R . x 2 Group element: g = 0 1  0 0 1 γ 1 γ 3   1  , γ i ∈ Z . γ 2 Restriction to Γ: g | Γ = 0 1  0 0 1 A. Chatzistavrakidis (ITP Hannover) 12 / 33

� dx 3 − x 1 dx 2  dx 1  0  . dx 2 Invariant 1-form: e = 0 0  0 0 0 Its components are: e 1 = dx 1 , e 2 = dx 2 , e 3 = dx 3 − x 1 dx 2 .   1 0 0  . Twist matrix: U = 0 1 0  − x 1 0 1 Reading off the required identifications: ( x 1 , x 2 , x 3 ) ∼ ( x 1 , x 2 +2 π R 2 , x 3 ) ∼ ( x 1 , x 2 , x 3 +2 π R 3 ) ∼ ( x 1 +2 π R 1 , x 2 , x 3 +2 π R 1 x 2 ) T 2 � M = ˜ (2 , 3) � � T 3 S 1 (1) A. Chatzistavrakidis (ITP Hannover) 13 / 33

T-duality approach Alternatively, consider a square torus with N units of NSNS flux H = dB , proportional to its volume form: ✔ Metric: ds 2 = δ ab dx a dx b . ✔ B-field: B 23 = Nx 1 . Perform a T-duality along x 3 using the Buscher rules: T i T i T i → B ai → G ab − G ai G bi − B ai B bi 1 − → G ii , − G ii , − , G ii G ai G ab G ii T i T i → G ai → B ab − B ai G bi − G ai B bi − G ii , − B ai B ab G ii In the T-dual frame: ✔ Metric: ds 2 = δ ab e a e b � e a of ˜ T 3 . ✔ B-field: B = 0 . Depicted as: T c → f c H abc ← ab A. Chatzistavrakidis (ITP Hannover) 14 / 33

Matrix Model Compactification-Tori Connes, Douglas, Schwarz ’97 Restriction of the action functional under periodicity conditions. Toroidal T d : U i X i ( U i ) − 1 = X i + 1 , i = 1 , ..., d , U i X a ( U i ) − 1 = X a , a � = i , a = 1 , . . . , 9 , with U i unitary and invertible (gauge transformations of the model). A. Chatzistavrakidis (ITP Hannover) 15 / 33