Flux Compactifications and Matrix Models for Superstrings

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1 Flux Compactifications and Matrix Models for Superstrings Athanasios Chatzistavrakidis Institut für Theoretische Physik, Leibniz Universität Hannover Based on: A.C., [hep-th] (PRD 84 (2011)) A.C. and Larisa Jonke, [hep-th] (PRD 85 (2012)) A.C. and Larisa Jonke, [hep-th] Edinburgh Mathematical Physics Group A. Chatzistavrakidis (ITP Hannover) 1 / 33

2 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

3 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

4 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 Lüst; Blumenhagen, Plauschinn; Mylonas, Schupp, Szabo Matrix Models Lowe, Nastase, Ramgoolam; A.C., Jonke A. Chatzistavrakidis (ITP Hannover) 3 / 33

5 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 A. Chatzistavrakidis (ITP Hannover) 4 / 33

6 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 Disadvantages: Sugra limit is not clear. Less calculability. A. Chatzistavrakidis (ITP Hannover) 4 / 33

7 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 B ij CDS θ ij A. Chatzistavrakidis (ITP Hannover) 5 / 33

8 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 B ij CDS θ 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

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

10 Matrix Models IKKT: non-perturbative IIB superstring, Ishibashi, Kawai, Kitazawa, Tsuchiya 96 Z = dx dψe S, with action S IKKT = 1 ( 2g Tr 1 ) 2 [X a, X b ] 2 ΨΓ a [X a, Ψ]. 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 [ dt Tr ( X a X a 1 2g 2 [X a, X b ] 2) ] + fermions, X a (t): 9 and time-dependent... A. Chatzistavrakidis (ITP Hannover) 7 / 33

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

12 Classical solutions EOM (IKKT; setting Ψ = 0): Basic solutions: Rank(θ) = p + 1 Dp brane. Lie algebra type? [X b, [X b, X a ]] = 0. b [X a, X b ] = iθ ab [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

13 Classical solutions EOM (IKKT; setting Ψ = 0): Basic solutions: Rank(θ) = p + 1 Dp brane. Lie algebra type? [X b, [X b, X a ]] = 0. b [X a, X b ] = iθ ab [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 Starting point for a class of compact manifolds (nil- and solvmanifolds). A. Chatzistavrakidis (ITP Hannover) 8 / 33

14 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

15 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 bdx b They satisfy the Maurer-Cartan equations de a = 1 2 f a bce b e c, 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

16 T Dn M i D i. T D3 M D1+D 2+D 3 T D2 M D1+D 2 T D1 The number of such iterations is set by the nilpotency class. A. Chatzistavrakidis (ITP Hannover) 11 / 33

17 Prototype example: 3D 3D nilpotent Lie algebra: [T 1, T 2 ] = T 3. Upper triangular basis: T 1 = , T 2 = , T 3 = x 1 x 3 Group element: g = 0 1 x 2, x i R Restriction to Γ: g Γ = 1 γ1 γ γ 2, γ i Z A. Chatzistavrakidis (ITP Hannover) 12 / 33

18 Invariant 1-form: e = 0 dx 1 dx 3 x 1 dx dx Its components are: e 1 = dx 1, e 2 = dx 2, e 3 = dx 3 x 1 dx Twist matrix: U = x 1 1 Reading off the required identifications: (x 1, x 2, x 3 ) (x 1, x 2 +2πR 2, x 3 ) (x 1, x 2, x 3 +2πR 3 ) (x 1 +2πR 1, x 2, x 3 +2πR 1 x 2 ) T 2 (2,3) M = T 3 S 1 (1) A. Chatzistavrakidis (ITP Hannover) 13 / 33

19 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: G ii In the T-dual frame: T i 1 G ii, G ai T i B ai G ii, G ab T i Gab G ai G bi B ai B bi G ii, B ai Metric: ds 2 = δ ab e a e b e a of T 3. B-field: B = 0. Depicted as: T i G ai T G ii, B i ab Bab B ai G bi G ai B bi G ii H abc T c f c ab A. Chatzistavrakidis (ITP Hannover) 14 / 33

20 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

21 Solutions Connes, Douglas, Schwarz 97 X i = ir i ˆD i, X m = A m (Û), (m = d + 1,..., 9), U i =e iˆx i, with covariant derivatives ˆD i = ˆ i ia i (Û). The U-algebra is in general: U i U j = λ ij U j U i with complex constants λ ij = e iθij. non-commutative torus. Connes, Rieffel A s depend on a set of operators Û, commuting with U: Brace, Morariu, Zumino 98 Û i = e iˆx i θ ij ˆ j, satisfying dual relations ÛiÛj = e i ˆθ ij Û j Û i, ˆθij = θ ij. A. Chatzistavrakidis (ITP Hannover) 16 / 33

22 Substitution back into the action NCSYM theory on the dual NC torus. Note: the solution involves a quantized phase space of ˆx and ˆp with algebra: [ˆx i, ˆx j ] = iθ ij, [ˆx i, ˆp j ] = iδj i, [ˆp i, ˆp j ] = 0. Interpretation: Deformation parameters θ correspond to moduli of a sugra compactification, i.e. they are reciprocal to a background B field, (θ 1 ) ij dx i dx j B ij A. Chatzistavrakidis (ITP Hannover) 17 / 33

23 Matrix Model Compactification-Nilmanifolds Restrict the action by imposing conditions corresponding to nilmanifolds. Lowe, Nastase, Ramgoolam 03; A.C., Jonke D nilmanifold T 3 (in a more democratic gauge ): U i X i (U i ) 1 = X i + 1, i = 1, 2, 3, U 1 X 3 (U 1 ) 1 = X 3 X 2, U 2 X 3 (U 2 ) 1 = X 3 + X 1, U i X a (U i ) 1 = X a, a i, a = 1,..., 9, (a, i) {(3, 1), (3, 2)}. Solutions: X i = ir i ˆD i, X m = A m (Û), (m = 4,..., 9), Ui =e iˆx i, with covariant derivatives ˆD i = ˆ jk i ia i (Û) + fi A j (Û) ˆ k, f A. Chatzistavrakidis (ITP Hannover) 18 / 33

24 The U-algebra is now given by: U i U j = e iθij if ij k ˆx k U j U i. non-commutative twisted torus Lowe, Nastase, Ramgoolam 03; A.C., Jonke 12; c.f. Rieffel 89 The dual operators are now Û = e iŷ i with: ŷ i = ˆx i iθ ij ˆ j if ij k ˆx k ˆ j. Algebra of phase space: [ˆx i, ˆx j ] = iθ ij + if ij k ˆx k iθ ij (ˆx), [ˆp i, ˆx j ] = iδ j i if jk i ˆp k, [ˆp i, ˆp j ] = 0. The effective action is a NC gauge theory on a dual NC twisted torus. Interpretation: The non-constant deformation is the analog of a geometric flux. Direct generalization for all higher-d nilmanifolds, richer in geometric fluxes. A. Chatzistavrakidis (ITP Hannover) 19 / 33

25 At hand: geometric flux f k ij More fluxes? (nilmanifold). T T-dual to NSNS flux H ijk : H k ijk f k ij. Enlarged chain with unconventional fluxes: H ijk T k f ij k T j Q jk T i i R ijk. A. Chatzistavrakidis (ITP Hannover) 20 / 33

26 At hand: geometric flux f k ij More fluxes? (nilmanifold). T T-dual to NSNS flux H ijk : H k ijk f k ij. Enlarged chain with unconventional fluxes: H ijk T k f ij k T j Q jk T i i R ijk. Q: Matrix Model description? A. Chatzistavrakidis (ITP Hannover) 20 / 33

27 At hand: geometric flux f k ij More fluxes? (nilmanifold). T T-dual to NSNS flux H ijk : H k ijk f k ij. Enlarged chain with unconventional fluxes: H ijk T k f ij k T j Q jk T i i R ijk. Q: Matrix Model description? or Q: Which compactifications correspond to more general phase space algebras? A. Chatzistavrakidis (ITP Hannover) 20 / 33

28 At hand: geometric flux f k ij More fluxes? (nilmanifold). T T-dual to NSNS flux H ijk : H k ijk f k ij. Enlarged chain with unconventional fluxes: H ijk T k f ij k T j Q jk T i i R ijk. Q: Matrix Model description? or Q: Which compactifications correspond to more general phase space algebras? or Q: What is the role of, previously ignored, Ũi = e i ˆp i (esp. when [ˆp, ˆp] 0)? A. Chatzistavrakidis (ITP Hannover) 20 / 33

29 Building Blocks H-block: Consider the phase space algebra c.f. Lüst 10: [ˆx i, ˆx j ] = if ijk ˆp k, [ˆx j, ˆp i ] = iδ j i, [ˆp i, ˆp j ] = 0. If U i = e iˆx i and Ũ i = e i ˆp i, and we make the Ansatz X i = i ˆD i, U i X i (U i ) 1 = X i + 1, Ũ i X i (Ũi) 1 = X i, looks like familiar compactification on torus. A. Chatzistavrakidis (ITP Hannover) 21 / 33

30 Building Blocks H-block: Consider the phase space algebra c.f. Lüst 10: [ˆx i, ˆx j ] = if ijk ˆp k, [ˆx j, ˆp i ] = iδ j i, [ˆp i, ˆp j ] = 0. If U i = e iˆx i and Ũ i = e i ˆp i, and we make the Ansatz X i = i ˆD i, U i X i (U i ) 1 = X i + 1, Ũ i X i (Ũi) 1 = X i, looks like familiar compactification on torus. BUT, the U-algebra is: U i U j = e iθij (ˆp) U j U i, with θ ij = F ijk ˆp k. The Connes-Douglas-Schwarz correspondence suggests a sugra B-field B = x 1 dx 2 dx 3 + x 2 dx 3 dx 1 + x 3 dx 1 dx 2 H ijk where x i are standard toroidal coordinates. A. Chatzistavrakidis (ITP Hannover) 21 / 33

31 The present algebra is related to the f -block one by a canonical transformation : Represent this as a matrix M H f ˆx 3 ˆp 3, ˆp 3 ˆx 3. ( ˆx i acting on ˆp i to the H-solution under the combined action of M H f ). The f -solution is mapped and a grading correction ( 1)ĉi f = diag(1, 1, 1, 1, 1, 1). For the 3D case, this is depicted as: H T 3 f θ(ˆp) M H f ( 1)ĉi f θ(ˆx) A. Chatzistavrakidis (ITP Hannover) 22 / 33

32 Q-block: Consider a different phase space algebra: [ˆx i, ˆx j ] = 0, [ˆp i, ˆx j ] = iδ j i + if j ik ˆx k, [ˆp i, ˆp j ] = if k ij ˆp k. This is motivated by a transformation M f Q on ˆx 2, ˆp 2. If U i = e iˆx i and Ũ i = e ( 1)c i i ˆp i (with c1 = 1, c 2,3 = 0), the Ansatz X i = i ˆD i gives e.g. not a well-defined compactification. U 1 X 2 (U 1 ) 1 = X 2 ˆx 3, A. Chatzistavrakidis (ITP Hannover) 23 / 33

33 Q-block: Consider a different phase space algebra: [ˆx i, ˆx j ] = 0, [ˆp i, ˆx j ] = iδ j i + if j ik ˆx k, [ˆp i, ˆp j ] = if k ij ˆp k. This is motivated by a transformation M f Q on ˆx 2, ˆp 2. If U i = e iˆx i and Ũ i = e ( 1)c i i ˆp i (with c1 = 1, c 2,3 = 0), the Ansatz X i = i ˆD i gives e.g. not a well-defined compactification. Two ways out: U 1 X 2 (U 1 ) 1 = X 2 ˆx 3, Introduce dual elements X i ˆx i, a kind of doubled formalism. This fits well with Twisted Doubled Tori approach to non-geometry Hull, Reid-Edwards 07, 09; Dall Agata, Prezas, Samtleben, Trigiante 07 Change the Ansatz. A. Chatzistavrakidis (ITP Hannover) 23 / 33

34 Different Ansatz: X i = iδ ij ˆ D j, with ˆ Di A=0 = ( 1) c i ˆ i, where i ˆ i = i p i Then: U i X i (U i ) 1 = X i, Ũ i X i (Ũi) 1 = X i + 1, Ũ 2 X 1 (Ũ2) 1 = X 1 X 3, Ũ 3 X 1 (Ũ3) 1 = X 1 + X 2, is the position in the momentum rep. The U-algebra is commutative. But the Ũ one is not: Ũ i Ũ j = e i θ ij (ˆp) Ũ j Ũ i, with θ ij = F k ij ˆp k. A. Chatzistavrakidis (ITP Hannover) 24 / 33

35 What does this compactification correspond to? Comparison to TDT approach; matches with a polarization of a T-fold with a Q-flux. Alternatively: the algebra is obtained by an M-transformation on ˆx 2, ˆp 2. This can be understood as a generalized T-duality Hull; Hull, Reid-Edwards H T 3 T f 2 Q θ(ˆp) M H f ( 1)ĉi f θ(ˆx) M f Q ( 1)ĉi Q θ(ˆp) with iθ ij = [ˆx i, ˆx j ] and i θ ij = [ˆp i, ˆp j ]. A. Chatzistavrakidis (ITP Hannover) 25 / 33

36 R-block: In a similar spirit: [ˆx i, ˆx j ] = 0, [ˆp i, ˆx j ] = iδ j i, [ˆp i, ˆp j ] = if ijk ˆx k. Obtained from the previous via a M Q R on ˆx 3, ˆp 3. Following the Ansatz of the previous case: U i X i (U i ) 1 = X i, Ũ i X i (Ũi) 1 = X i + 1, The Us commute again, unlike the Ũs: Ũ i Ũ j = e i θ ij (ˆx) Ũ j Ũ i with θ ij = F ijk ˆx k. A. Chatzistavrakidis (ITP Hannover) 26 / 33

37 Comparison with TDT approach, and within the generalized T-duality interpretation of M matches with a compactification with R flux. Full Picture: H T 3 T f 2 T Q 1 R θ(ˆp) M H f ( 1)ĉi f θ(ˆx) M f Q ( 1)ĉi Q M Q R θ(ˆp) ( 1)ĉi R θ(ˆx) with iθ ij = [ˆx i, ˆx j ] and i θ ij = [ˆp i, ˆp j ]. A. Chatzistavrakidis (ITP Hannover) 27 / 33

38 There is a correspondence: θ ij f or θ ij H in ˆx-space θ ij Q or θij R in ˆp-space. In position space: MMC with non-constant θ geometric fluxes. In momentum space: MMC with non-constant θ non-geometric fluxes. Similar result in Generalized Complex Geometry approach... Andriot, Larfors, Lüst, Patalong 11 Indication: Just as θ ij (B ij ) 1, also θ ij (β ij ) 1, β: the bivector of GCG. It would be interesting to explore further such relations. A. Chatzistavrakidis (ITP Hannover) 28 / 33

39 Non-Associativity and Flux Quantization All encountered phase space algebras exhibit some non-associativity. E.g. [ˆp i, ˆx j, ˆx k ] f jk i for the f -block, [ˆx i, ˆx j, ˆx k ] F ijk for the H-block, etc. They could induce non-associativity on X i and U i. A. Chatzistavrakidis (ITP Hannover) 29 / 33

40 Non-Associativity and Flux Quantization All encountered phase space algebras exhibit some non-associativity. E.g. [ˆp i, ˆx j, ˆx k ] f jk i for the f -block, [ˆx i, ˆx j, ˆx k ] F ijk for the H-block, etc. They could induce non-associativity on X i and U i. X i : They associate in all cases (in the regime where the compactification is well-defined). A. Chatzistavrakidis (ITP Hannover) 29 / 33

41 Non-Associativity and Flux Quantization All encountered phase space algebras exhibit some non-associativity. E.g. [ˆp i, ˆx j, ˆx k ] f jk i for the f -block, [ˆx i, ˆx j, ˆx k ] F ijk for the H-block, etc. They could induce non-associativity on X i and U i. X i : They associate in all cases (in the regime where the compactification is well-defined). U i : e.g. in H-case: U i (U j U k ) = e i 2 Hijk (U i U j )U k. 3-cocycle; typical in QM systems with fluxes. Jackiw 85 Resolution: The flux has to be quantized, H = 4πn, n Z. Flux Quantization is already built-in. A. Chatzistavrakidis (ITP Hannover) 29 / 33

42 Flux Coexistence work in progress In sugra, metric and NSNS fluxes can coexist. Straightforward implementation in the MMC. Q: Coexistence of all flux types, including non-geometric? In our approach, two ways: Start with an appropriately rich pure geometry; find frame with all fluxes. Combine MM solutions block-diagonally. A. Chatzistavrakidis (ITP Hannover) 30 / 33

43 First approach work in progress Richer chain of duality frames: k f 1 i 1j 1 k f 2 i 2j 2 T i T j k f 3... i 3j 3 k f 4 i 4j 4 H i 1 j 1 k 1 f i 2 j 2 k 2 Q j 3 k 3 i 3 R i 4 j 4 k 4. If the simple chain is understood this is equally well understood. In fact, up to mild requirements, there is a unique nilmanifold able to realize this, M 6 S 1 (6) T 2 (4,5) M 5 T 3 (1,2,3) A. Chatzistavrakidis (ITP Hannover) 31 / 33

44 Second approach work in progress Solutions of MM can be combined block-diagonally. Is it possible to use this property to define a MMC with solution e.g. ( ) X (H) X i = i 0 0 X (R)? i Which are the properties of such a MMC? Are there some associated bound states? A. Chatzistavrakidis (ITP Hannover) 32 / 33

45 Main messages Matrix Models: useful framework for unconventional string compactifications. Fluxes, dualities, non-geometry, non-commutativity. Relations to other frameworks (double field theory, generalized geometry, etc.) Some prospects Analysis of the effective theories with fluxes. in progress, with L. Jonke Full study of possible vacua. Coexistence of all types of fluxes. in progress, with L. Jonke and M. Schmitz Phenomenology of unconventional compactifications? Non-perturbative dualities? A. Chatzistavrakidis (ITP Hannover) 33 / 33

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