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Matrix Factorizations, D-Branes and Homological Mirror Symmetry W.Lerche, KITP 03/2009

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Matrix Factorizations, D-Branes and Homological Mirror Symmetry W.Lerche, KITP 03/2009 Matrix Factorizations, D-branes and Homological Mirror Symmetry W.Lerche, KITP 03/2009 • Motivation, general remarks • Mirror symmetry and D-branes • Matrix factorizations and LG models • Toy example: eff. superpotential for intersecting branes, applications much more diverse instantons than for closed strings (world-sheet and D-brane instantons) 1 Part I Motivation: D-brane worlds Typical brane + flux configuration on a Calabi-Yau space closed string (bulk) moduli t open string (brane location + bundle) moduli u 3+1 dim world volume with effective N=1 SUSY theory What are the exact effective superpotential, the vacuum states, gauge couplings, etc ? (Φ, t, u) = ? Weff 2 Quantum geometry of D-branes Classical geometry ("branes wrapping p-cycles", gauge bundle configurations on top of them) makes sense only at weak coupling/large radius! M In fact, practically all of string Classical geometry: phenomenology deals with the cycles, gauge (“bundle”) boundary of the moduli space configurations on them (weak coupling, large radius) 3 Quantum geometry of D-branes Classical geometry ("branes wrapping p-cycles", gauge bundle configurations on top of them) makes sense only at weak coupling/large radius! “Gepner point” (CFT description) typ. symmetry between M 0,2,4,6 cycles ? “conifold point” extra massles states Classical geometry: Quantum corrected geometry: cycles, gauge (“bundle”) (instanton) corrections wipe out configurations on them notions of classical geometry 4 Quantum geometry of D-branes Classical geometry ("branes wrapping p-cycles", gauge bundle configurations on top of them) makes sense only at weak coupling/large radius! Need to develop formalism capable of describing the physics of general D-brane configurations (here: topological B-type D-branes), incl their continuous deformation families over the moduli space ....well developed techniques (mirror symmetry) mostly for non-generic (non-compact, non-intersecting, integrable) brane configurations branes only ! 5 Intersecting branes: eff. potential for quivers Ψ DA Ψ DB C D Ψ boundary changing operator (tachyon) DA (A,B) (A,B) Ψa Ψa B D A B (C,A) D D Ψc (B,C) Ψb (C,A) Ψc (B,C) Ψb DC DC Quiver diagram Disk world sheet in TCFT 6 Intersecting branes: eff. potential for quivers boundary changing operator (tachyon) Quiver diagram Disk world sheet in TCFT Superpotential ~ closed paths in quiver DA (A,B) (A,B) (A,BΨ)a (B,C) (C,A) (A,BΨa ) (B,C) (C,D) (D,A) eff (T, u, t) = TaTbTc Ψ Ψb Ψ +TaTbTcTd Ψ Ψb Ψ Ψd +... W ! a B c " ! a c " D A Cabc(t,u) CabcdB(t,u) (C,A) D D Ψc ! ("#B,C) $ ! "# $ Ψb (C,A) tachyons Ψc (B,C) closed and open string Ψb C t u C moduli D const + (e− , e− ) D ∼ O instanton corrections... how to compute? 7 D-branes: homological mirror symmetry Mirror symmetry acts between full categories descr. A- and B-branes! A-Model B-Model mirror symmetry q p D1 branes on (p,q) cycles (N2,N0) = (r,c1) of “gauge bundle” ... “Fukaya category” of ... “derived category” lagrangian cycles of coherent sheaves There is much more to this than just quantum numbers (K-theory) Homological mirror symmetry also preserves the higher A∞ products (open string correlators) 8 Mirror symmetry and open string TFT correlators (A,B) (B,C) (C,A) Disk amplitude for intersecting branes Cabc(τ ; uA, uB, uC ) = Ψ Ψ Ψ ! a b c " (A,B) Ψa A D DB Σ (B,C) (C,A) Ψ Ψc b C A-Model D B-Model localizes on holomorphic maps: (Landau-Ginzburg model) world-sheet instantons Σ T2 localizes on constant maps: → classical Fukaya products Massey products mirror symmetry SInst m a0 Cabc e− λm(Ψ⊗ ) = Ψa C ∼ 0 a1...am SInst. Area ∼ 9 B-model correlators General structure of 3-pt fct in (holom) Chern-Simons theory: C = Ψ Ψ Ψ = Tr[Ψ Ψ Ψ ] Ω(3,0) abc ! a b c " a ∧ b ∧ c ∧ !X Wedge product of sections Ψ Ψ(A,B) Ext1(X; ¯ A, B) a ≡ a ∈ V V Branes ~ vector bundles: sections are collections of vectors = matrices To perfom mirror map... need flat sections determined by some flatness equations 10 B-model correlators.... For the bulk theory, the flatness equations coincide with the Picard-Fuchs equations for periods, arising from the variation of Hodge structures For open strings theory is not yet well developed (in general no flatness to begin with, obstructions) Physicist’s pragmatic approach: obtain diffeqs from contact terms in Landau-Ginzburg formulation 11 Part II D-branes and Matrix Factorizations Seek: Description of topol. B-type D-branes that captures the mathematical intricacies, while allowing to do explicit computations LG formulation of B-type branes in terms of matrix factorizations Important: explicit moduli dependence of all quantities 12 Landau-Ginzburg description of B-type D-branes Consider bulk d=2 N=(2,2) LG model with superpotential: 2 + d zdθ dθ− WLG(x) + cc. !Σ B-type SUSY variations induce boundary (“Warner”)-term: 2 + 2 + + d zdθ dθ−(Q¯+ + Q¯ ) WLG = d zdθ dθ−(θ ∂+ + θ−∂ )WLG − − !Σ !Σ = dσdθ WLG !∂Σ Restore SUSY by adding boundary fermions Π = (π + θ+#) (... not quite chiral: D ¯ Π = E ( x ) ) |∂Σ via a boundary potential: δS = dσdθ Π J(x) !∂Σ Condition for SUSY: J(x)E(x) = WLG(x) 13 Matrix factorizations J(x) BRST operator: Q(x) = π J(x) + π¯ E(x) = E(x) ! " thus SUSY condition implies a matrix factorization of W: Q(x) Q(x) = W (x) 1 · LG Total BRST operator = Q + Q Q bulk then squares to zero: 2 = 0 Q Generalization for n LG fields: need N=2n boundary fermions, and JN N EN N = EN N JN N = WLG 1N N × · × × · × × B-type Susy D-branes: 1:1 to all possible matrix factorizations of a given W=W(x,t) 14 Physical interpretation _ N... Chan-Paton labels of space-filling DD pairs Boundary potentials J,E form a tachyon profile that describes condensation to given B-type D-brane configuration in IR limit _ _ _ D D DD J,EJ,E n eg. J(x, u) = (x ui) c =n − 1 ! Geometrically: Maps J,E are sections of certain bundles Ker J, Ker E encode bundle data of branes: (r,c1, ... ; u) 15 Open string cohomology Physical open string spectrum is determined by the cohomology of the BRST operator: Ω [QA, ΩA] = 0 , ΩA = [QA, Λ] A ! A boundary preserving D (square matrices) (A,B) (A,B) f (A,B) Ψ QAΨ ( ) Ψ QB = 0 (B,A) − − Ψ boundary changing (rectangular matrices) B D Ω [Q , Ω ] = 0 , Ω = [Q , Λ] B B B B ! B These are the ingredients of a nice category.... 16 Kontsevich’s category CW The LG model provides a concrete physical realization of a certain triangulated Z2-graded category CW _ objects: “complexes” (~composites of DD branes): J (!) D (!) / (!) , J (!)E(!) = W ! ∼= P1 o P0 E(!) ! " maps (boundary Q-cohomology): J (!1) (!1) / (!1) D!1 P1 o P0 E(!1) ! " !1,!2 !1,!2 = φα !1,!2 !1,!2 φα ∼ ψα ψα Ó (! ) J 2 (!2) / (!2) D!2 P1 o P0 E(!2) ! " 17 Kontsevich’s category CW The LG model provides a concrete physical realization of a certain triangulated Z2-graded category CW Category of Matrix factorizations is isomorphic to D(Coh(M)), the derived category of coherent sheaves on M = category of B-type D-branes! (O, HHP) 18 Part III: branes on the elliptic curve complex str modulus Simplest Calabi-Yau: the cubic torus 1 3 3 3 T2 : W = (x1 + x2 + x3 ) a(τ ) x1x2x3 = 0 3 − Mirror map: 3 a (a3 + 8) = J(τ )1/3 , ∆ a3 1 ∆ ≡ − flat coo of complex structure moduli space = Kahler parameter of mirror curve 19 Test example: branes on the elliptic curve B-type D-branes are composites of D2, D0 branes, characterized by N2, N0; u = rank(V ), c1(V ); u ! " ! " ... these are mirror to A-type D1-branes 2 with wrapping numbers p, q = N2, N0 2 2 1 c1 ! " ! " L S L 1 We will consider the ``long-diagonal’’ branes with charges S1 0 2 (N2, N0) A = ( 1, 0), ( 1, 3), (2, 3) L { − − − } 3 1 S r 0 L3 20 3x3 matrix factorization Factorizations corresponding to the long diagonal branes Li (i) (i) (i) α1 x1 α2 x3 α3 x2 (i=1,2,3) (i) (i) (i) Ji = α3 x3 α1 x2 α2 x1 (i) (i) (i) α2 x2 α3 x1 α1 x3 (i) (i) (i) 1 2 α1 1 2 α3 1 2 α2 (i) x1 (i) (i) x2x3 (i) x3 (i) (i) x1x2 (i) x2 (i) (i) x1x3 α1 − α2 α3 α3 − α1 α2 α2 − α1 α3 (i) (i) (i) 1 2 α2 1 2 α1 1 2 α3 Ei = (i) x3 (i) (i) x1x2 (i) x2 (i) (i) x1x3 (i) x1 (i) (i) x2x3 α2 − α1 α3 α1 − α2 α3 α3 − α1 α2 (i) (i) (i) 1 2 α3 1 2 α2 1 2 α1 (i) x2 (i) (i) x1x3 (i) x1 (i) (i) x2x3 (i) x3 (i) (i) x1x2 α α α α α α α α α 3 − 1 2 2 − 1 3 1 − 2 3 W, BHLW These satisfy J iEi = EiJi = WLG1 if the parameters satisfy the cubic equation themselves: W (α ) α 3 + α 3 + α 3 + a(τ ) α α α = 0 LG i ≡ 1 2 3 1 2 3 Thus the parameters parametrize the (jacobian) torus and can be represented by theta-sections: (i) 1 ! 1 1 u, τ ...flat coordinates of α Θ − 3ui, 3τ ! ∼ 3 − 2 − 2 open/closed moduli space ! " # " (Kahler moduli in mirror A-model) " 21 Open string BRST cohomology Solving for the BRST cohomology yields Ω1 explicit t,u-moduli dependent L1 matrix valued maps, eg (a=1,2,3): (a) (a) Ψ12 Ψ31 marginal operators corr.
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