Matrix Models and D-Branes in Twistor String Theory Christian S¨amann Dublin Institute for Advanced Studies LMS Durham Symposium 2007 Based on: JHEP 0603 (2006) 002, O. Lechtenfeld and CS. Motivation Extending understanding of topological/super D-branes and mirror symmetry Well-known motivation for studying twistor strings: Alternative description of the AdS/CFT correspondence New tools for calculating gluon scattering amplitudes Alternative descriptions of supergravity My motivation here: Description of super D-branes? Relationship between topological and physical D-branes? Rˆoleof Calabi-Yau supermanifolds in mirror symmetry? ⇒ Study variations of the usual twistor geometries and the associated Penrose-Ward transform. Here: Full dimensional reductions yielding matrix models with interesting interpretations in terms of D-branes. The presented results are only a very preliminary step towards answering the above questions. Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Outline 1 Notation: Twistors and Penrose-Ward transform 2 Construction of the matrix models 3 D-Brane interpretation and completion for ADHM construction Nahm construction 4 Conclusions Christian S¨amann Matrix Models and D-Branes in Twistor String Theory The Twistor Correspondence The twistor correspondence is a relation between subsets of twistor space and spacetime. α αα˙ i α 3 Incidence Relation: ω = x λα˙ , Twistor: Z = (ω , λα˙ ) ∈ CP Twistor Correspondence αα˙ 1 Point x corresponds to sphere CP 3 λα˙ i αα˙ αα˙ α α˙ A twistor Z is incident to a plane of points x = x0 + κ λ . Decompactification C 3 4 4 P is the twistor space of S or Sc CP 1 take out ∞ P3 is the twistor space of R4 or C4 C 1 P ∞ is described by λα˙ = 0, therefore: P3 := O(1) ⊕ O(1) → CP 1 1 α Homog. coords. λα˙ on CP and ω in fibres Moduli of sections of P3: xαα˙ ∈ C4 Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Underlying Idea of Twistor String Theory To make contact with string theory, we need to extend this picture supersymmetrically. Marrying Twistor- and Calabi-Yau geometry ... with supermanifolds: Witten, hep-th/0312171 Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Supertwistor Space The supertwistor space P3|N is a holomorphic vector bundle of rank 3|4N over CP 1. The Supertwistor Space P3|N Start from CP 3|N , take out CP 1|N at infinity: P3|N := C2 ⊗ O(1) ⊕ CN ⊗ ΠO(1) → CP 1 Incidence Relations Double Fibration C4|2N × CP 1 α αα˙ ω = x λα˙ @ α˙ © @R ηi = ηi λα˙ P3|N C4|2N First Chern Class of P3|4 1 T CP 2, O(1) 1, ΠO(1) -1, in total: c1 = 0. Therefore, there exists a holomorphic measure Ω3,0|4,0. Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Outline of the Penrose-Ward Transform on P3|4 The PW-transform takes us from the topological B-model to SDYM theory. topological B-model on P3|4 m holomorphic Chern-Simons theory on E → P3|4: R 3,0|4,0 0,1 ¯ 0,1 2 0,1 0,1 0,1 Ω ∧ tr (A ∧ ∂A + 3 A ∧ A ∧ A ) with eom ∂¯A0,1 + A0,1 ∧ A0,1 = 0 m holomorphic vector bundles over P3|4 m solutions to the N = 4 SDYM equations on C4|8 Field contents: (f , χαi, φ[ij], χ˜[ijk],G[ijkl]) αβ α˙ α˙ β˙ αijk˙ [i jk] fα˙ β˙ = 0 , ∇αα˙ χ˜ − [χα, φ ] = 0 , ∇ χαi = 0 , εα˙ γ˙ ∇ G[ijkl] + ... = 0 . αα˙ αα˙ γ˙ δ˙ ij αi j φ + 2{χ , χα} = 0 , Christian S¨amann Matrix Models and D-Branes in Twistor String Theory 3|4 Penrose-Ward Transform on Pτ Imposing reality conditions simplifies the situation significantly. Introducing a real structure, the double fibration collapses: C4|2N × CP 1 @ 3|N R4|2N © @R −→ Pτ → τ P3|N C4|2N 4|2N (τ±1 related to Kleinian and Euclidean metrics on Rτ .) Now: Field expansion of hCS gauge potential A0,1 available: α˙ i 1 ˆα˙ ij Aα =λ Aαα˙ (x) + ηiχα(x) + γ 2! ηiηj λ φαα˙ (x)+ ˙ ˙ γ2 1 η η η λˆα˙ λˆβ χ˜ijk (x) + γ3 1 η η η η λˆα˙ λˆβ λˆγ˙ Gijkl (x) 3! i j k αα˙ β˙ 4! i j k l αα˙ β˙γ˙ 2 ij 3 α˙ ijk 4 α˙ β˙ ijkl A¯ =γ η η φ (x) − γ η η η λˆ χ˜ (x) + 2γ η η η η λˆ λˆ G (x) λ i j i j k α˙ i j k l α˙ β˙ Popov, CS, ATMP 9 (2005) 931 This field expansion makes the equivalence hCS↔ SDYM manifest. Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Matrix Models Matrix models are obtained by dim. reduction or from spacetime noncommutativity. Two ways of obtaining the matrix models: Dimensionally reducing the moduli space R4|8 → R0|8: Making the moduli space R4|8 noncommutative: Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Matrix Models via Dimensional Reduction Full dimensional reduction yields equivalence between SDYM MM and hCS MQM. Matrix Model from N = 4 SDYM theory: ! α˙ β˙ 1 αβ ε ij αα˙ S := tr G − 2 ε [Aαα˙ , Aββ˙ ] + 2 φ [Aαα˙ , [A , φij]]+... Matrix Model from N = 4 hCS theory (MQM): Z αβ ¯ h 0,1 i S := Ωred ∧ tr ε Xα ∂Xβ + AC 1 , Xβ C 1 P P ch 3,0|4,0 ± ± Ωred := Ω |C 1 Ωred± = ±dλ± ∧ dη ... dη P ch 1 4 Equivalence explicitly via: α˙ i 1 ˆα˙ ij Xα =λ Aαα˙ + ηiχα + γ 2! ηiηj λ φαα˙ + ˙ ˙ γ2 1 η η η λˆα˙ λˆβ χ˜ijk + γ3 1 η η η η λˆα˙ λˆβ λˆγ˙ Gijkl 3! i j k αα˙ β˙ 4! i j k l αα˙ β˙γ˙ 2 ij 3 α˙ ijk 4 α˙ β˙ ijkl A¯ =γ η η φ − γ η η η λˆ χ˜ + 2γ η η η η λˆ λˆ G λ i j i j k α˙ i j k l α˙ β˙ Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Matrix Models from Noncommutativity Functions on the noncommutative moduli space are infinite-dimensional matrices. Noncommutativity on the moduli space ˙ ˙ [ˆxαα˙ , xˆββ] = iθααβ˙ β with: (κ = ±1) θ112˙ 2˙ = −θ221˙ 1˙ = −2iκεθ and θ122˙ 1˙ = −θ211˙ 2˙ = 2iεθ representation space: two oscillator Fock space with |0, 0i 21˙ 12˙ 22˙ 11˙ aˆ1 ∼ xˆ +x ˆ and aˆ2 ∼ xˆ − xˆ derivatives become inner derivations of the above algebra: ∂ f ∼ [ˆx22˙ , f] , etc. ∂xˆ11˙ R 4 2 ˆ integral becomes trace: d x f 7→ (2πθ) tr Hf Christian S¨amann Matrix Models and D-Branes in Twistor String Theory Matrix Models from Noncommutativity Sections ω of the bundle defining supertwistor space are now matrix valued. Noncommutativity on the twistor space Induced algebra: 1 2 1 2 ¯ [ˆω±, ωˆ±] = 2(κ − 1)ελ±θ , [ω¯ˆ±, ω¯ˆ±] = −2(κ − 1)ελ±θ , 1 1 ¯ 1 1 ¯ [ˆω+, ω¯ˆ+] = 2(κε − λ+λ+)θ , [ˆω−, ω¯ˆ−] = 2(κελ−λ− − 1)θ , 2 2 ¯ 2 2 ¯ [ˆω+, ω¯ˆ+] = 2(1 − εκλ+λ+)θ , [ˆω−, ω¯ˆ−] = 2(λ−λ− − εκ)θ , Matrix Models All operators can be seen as infinite dimensional matrices. ⇒ Matrix models from SDYM and hCS theory explict equivalence again via field expansion. Large N limit N: rank of gauge group, limit N → ∞: all MMs equivalent Christian S¨amann Matrix Models and D-Branes in Twistor String Theory D-Brane Interpretation There is an obvious interpretation of the hCS MM in terms of topological B-branes. B-Type Topological Branes D(-1)-, D1-, D3-, and D5-branes stack of N D-branes comes with rank N vector bundle effective action: GL(N, C) holomorphic Chern-Simons theory i.e. F 0,2 = F 2,0 = 0 (stability missing: kd+1 ∧ F 1,1 = γkd) hCS MM: stack of n D1|4-branes wrapping CP 1|4,→P3|4. 0 i ij expand Higgs-fields Xα = Xα + Xαηi + Xα ηiηj + ... 0 0 [X1 , X2 ] = 0 , i 0 0 i [X1, X2 ] + [X1 , X2] = 0 , i j j i ij 0 0 ij {X1, X2 } − {X1 , X2} + [X1 , X2 ] + [X1 , X2 ] = 0 , 0 bodies Xα can be diagonalized: positions of the D1|4-branes Fermionic directions are “smeared out” even classically. Christian S¨amann Matrix Models and D-Branes in Twistor String Theory D-Brane Interpretation Physical D-branes: topological D-branes + stability condition. D-Branes in Type IIB String Theory D(-1)-, D1-, D3-, ... branes stack of N D-branes comes with rank N vector bundle effective action: U(N) SYM reduced from 10 to p + 1 curved spaces: F 0,2 = F 2,0 = 0 and kd+1 ∧ F 1,1 = γkd arising Higgs fields: normal fluctuations of D-branes Christian S¨amann Matrix Models and D-Branes in Twistor String Theory ADHM Construction and D-Brane Bound States There is a nice interpretation of the ADHM construction in terms of D-branes. Bound state of D3-D(-1)-branes (D9-D5-branes + dim. reduction) Perspective of D3-brane D3-D3-strings + BPS condition: SDYM equations D(-1)-brane: instanton, nontrivial ch2 Perspective of D(-1)-brane D(-1)-D(-1)-strings: i N = (0, 1) hypmult., adj. (Aαα˙ , χα) D(-1)-D3-strings: i N = (0, 1) hypmult., bifund. (wα˙ , ψ ) D-flatness condition/ADHM eqns.: i α˙ β˙ ¯αβ˙ 16π2 ~σ β˙ (w ¯ wα˙ + A Aαα˙ ) = 0 Witten, hep-th/9510135, Douglas, hep-th/9512077,..