Matrix Factorizations, D- 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 than for closed strings (world-sheet and D- instantons)

1 Part I Motivation: D-brane worlds

Typical brane + flux configuration on a Calabi-Yau space

closed (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 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 ? “ point” extra massles states Classical geometry: Quantum corrected geometry: cycles, gauge (“bundle”) () 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 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 () 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) 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 Superpotential ~ closed paths in quiver

(T, u, t) = T T T Ψ(A,B)Ψ(B,C)Ψ(C,A) +T T T T Ψ(A,B)Ψ(B,C)Ψ(C,D)Ψ(D,A) +... Weff a b c ! a b c " a b c d ! a b c d "

Cabc(t,u) Cabcd(t,u) ! "# $ ! "# $ closed and open string t u moduli const + (e− , e− ) ∼ 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 Z -graded category C 2 W _ 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. to brane locations (a) (a) Φ21 Φ13 1 Ext ( A, A) : ΩA = ∂u Q(uA) (a) A Ω2 L L Φ32 2 3 tachyon operators L (a) L Ψ23 Ω3 0 F (a) Ext1( , ) : Ψ(a) = AB LA LB AB (a) ! GAB 0 "

ζ1 ζ3 ζ2 (1) (2) x1 (1) (2) x2 (1) (2) x3 α α α α α α ζ1 0 0 1 1 1 2 1 3 (1) (1) ζ2 ζ1 ζ3 (2) (1) x2 (2) (1) x3 (1) (2) x1 with eg, F12 = 0 0 ζ2 G12 =  α α α α α α    1 3 2 3 3 3 0 ζ3 0 ζ3 ζ2 ζ1 (2) (1) x3 (1) (2) x1 (1) (2) x2  α α α α α α     1 2 2 2 2 3   

1 ! 1 1 and ζ! Θ − 3u2 3u1, 3τ ∼ 3 − 2 − 2 − ! " # " 22 " Superpotential on brane intersection I

Compute 3-point disk correlators = Yukawa couplings a ΨAB

B DA D = T T T C (τ, u ) + ... Σ Weff a b c abc i Ψc b CA ΨBC C a b c D Cabc(τ, u1, u2, u3) = Ψ13(u1, u3)Ψ32(u3, u2)Ψ21(u2, u1) ! " Use Kapustin-Li super-residue formula (from localization of path integral) for the matrix-valued, moduli-dependent operators:

1 dQ 3 (a) (b) (c) = Str ( )⊗∧ Ψ Ψ Ψ 2πi dW 13 32 21 ! " #

23 (A side remark)

90% of the actual work is not explained here The point is to determine proper “flat cohomology representatives” which includes their t-dependent normalization For the closed string, this is achieved by the Picard-Fuchs differential equations, which are based on the theory of Hodge variations ... the heart of mirror symmetry For open strings, a suitable non-commutative variant is not known Developed an approach based on contact terms that mimics Hodge theory for matrix valued operators, and leads to a matrix diffeq of the form: ∂ + U([∂ Q, ]) ∇t ≡ t t ∗ φ Ψ¯ a dQ, Ψ¯ (t) = d U { ∗} t a ≡ dW ∇ ! dW " ! "+ + (propagator)

24 Superpotential on brane intersection II

Solve differential eqn to determine t-dep of operators

Insert in KL residue formula and make heavy use of theta-function identities such as the addition formula:

θa[u1] θb[u2] = θa b+c[u1 u2] θa+b+c[u1 +u2] · − − (math:! expresses product in Fukaya-category) Final result: theta functions

2 2 6πiξ1ξ2 3ξ2 /2 3m /2 6πimξ C111(τ, ξ) = e q q e m 2 2 6πiξ1ξ2 3ξ2 /2 ! 3(m+1/3) /2 6πi(m+1/3)ξ C123(τ, ξ) = e q q e

2 m 2 6πiξ1ξ2 3ξ2 /2 ! 3(m 1/3) /2 6πi(m 1/3)ξ C132(τ, ξ) = e q q − e − m ! (ξ u + u + u = ξ + τ ξ ) What’s the interpretation of the q-series? ≡ 1 2 3 1 2 25 The topological A-Model: instantons

Interpretation of q-series: In A model mirror language, these are contributions from triangular disk instantons whose world-sheets are bounded by the three D1-branes:

S ∆ C e− inst q abc + ..... abc ∼ ∼ (the u-dependence corresponds to position and Wilson line moduli)

Count maps: Σ T → 2

(31) Ψa C 1 222 3 C D Σ D 132 (12) (23) Ψb Ψc C123 2 D C333

C111

16 Complete effective potential (long diag branes)

6 (a1) (aN ) (N) eff (τ, u, T ) = T . . . T C (τ, u1, . . . , uN ) W a1,...,aN N!=1 tachyons moduli B-model: difficult to compute higher N-point Massey products with N>3! For (flat) elliptic curve, A-model is simpler.... Generically, N-point functions get contributions from N-gonal instantons

General structure: indefinite theta-functions summing over all lattice translates, positive areas

!qmn qmn ≡ − m,n "m,n>0 m,n<0# ! ! ! 27 Polygons and instantons

N=4: trapezoids (3) [b c¯]3 abc¯d¯(τ, ui) = δ ¯ Θtrap − (3τ 3(u1 + u2 + u4), 3(u1 u3)) T a+b,c¯+d [d¯ c¯ + 3/2]3 | − ! − " a 1 (a+3n)(a+3n+2(b+3m)) 2πi (a+3n)(u 1/6)+(b+3m)v Θ (3τ 3u, 3v) = !q 6 e − trap b | ! " m,n $ % N=4: parallelograms # (3) [c ¯b]3 a¯bcd¯(τ, ui) = δ ¯ ¯ Θpara − (3τ 3(u1 u3), 3(u4 u2)) P a+c,b+d [d¯ c]3 | − − ! − " a 1 (a+3n)(b+3m) 2πi (b+3m)u+(a+3n)v Θ (3τ 3u, 3v) !q 3 e para b | ≡ m,n ! " # $ % N=5: pentagons [ b c d]3 (3) − − − [e+c+d] ℘a¯bc¯d¯e¯(τ, ui) = δa,¯b+c¯+d¯+e¯ Θpenta 3 (3τ 3(u5 u2), 3(u1 u4), 3(u3 +u2 +u4))  3  | − − [c d+ 2 ]3 a − 1   1 2 ! (a>+3(n+k))(b>+3(m+k)) (c+3k) 2πi (a>+3(n+k))u+(b>+3(m+k))v+(c+3k)(w 1/6) Θ b (3τ 3u, 3v, 3w) q 3 − 6 e − penta   | ≡ c m,n,k % & '   N=6: hexagons [ b c d] − − − 3 (3) [c +d +e]3 a¯¯bc¯d¯e¯f¯(τ, ui) = δ ¯ ¯ ¯Θhexa  3  (3τ 3(u5 u2), 3(u1 u4), 3(u3 +u2 +u4), 3( u6 u1 u5)) H 0,a¯+b+c¯+d+e¯+f [c d + ]3 | − − − − − − 2  [a f + 3 ]  a  − 2 3    b ! 1 (a+3n)(b+3m) 1 (c+3k)2 1 (d+3l)2 2πi (a+3n)u+(b+3m)v+(c+3k)(w 1/6)+(d+3l)(z+1/6) Θ (3τ 3u, 3v, 3w, 3z) q 3 − 6 − 6 e − hexa  c  | ≡ d m,n,k,l & ' % k >l   ∞ max max −∞ min min ! = − 28 m,n,k,l m,n 0 k 0 l 0 m,n 1 k 1 l 1 ! !≥ !≥ !≥ !≤− !≤− ≤−! Global properties of open string moduli space Indefinite theta-fcts: singularities due to colliding branes eg., rewrite trapezoidal function in terms of Appel function:

1 6 (a+3n)(a+2b+3n) 2πi(a+3n)(u 1/6) a 2πivb q e − Θtrap (3τ 3u, 3v) = e b | 1 qa+3ne6πiv ! " n Z #∈ − analytic continuation Φ Φ Ψ +v1 Ψ Area becomes negative:

u1 +u − 1 resum instantons in terms Φ Φ Ψ Ψ Ψ Ψ of different geometry

Ψ Ψ v − 1 “instanton flop”

29 Global properties of open string moduli space Indefinite theta-fcts: singularities due to colliding branes eg., rewrite trapezoidal function in terms of Appel function:

1 6 (a+3n)(a+2b+3n) 2πi(a+3n)(u 1/6) a 2πivb q e − Θtrap (3τ 3u, 3v) = e b | 1 qa+3ne6πiv ! " n Z #∈ − monodromy

a 6πiv a Θ (3τ 3(u τ ), 3v) = e∓ Θ (3τ 3u, 3v) trap b | ± trap b | ! " ! " 2πi(u 1 )(b 3 3 ) 2πiv(b 3 3 ) 1 (b 3 3 )2 a+b e− − 6 − 2 ± 2 e − 2 ∓ 2 q− 6 − 2 ± 2 Θ (3τ 3u) ∓ 3/2 | ! − " Ψ induces “homotopy transformation”, Φ modular of eff action Φ Φ Φ + (compensate by non-lin field redef) Ψ Ψ Ψ Ψ Ψ Ψ e ¯ ¯ + f ¯ ∆ Tabc¯d → Tabc¯d c¯d abe

G−Φ Φ ! contact term Ψ Ψ 30 Summary and Outlook math: Cat of matrix factorizations D(Coh(M)) phyz: Boundary LG theory Open string B-type top. CFT

Represent all quantities in a quiver diagram (objects and maps) by explicit moduli-dependent, matrix-valued operators Combined with mirror symmetry this allows to explicitly compute instanton-corrected superpotentials (in particular, for intersecting brane configs). Requires solving a flatness DEQ.

Generalization to M = CY 3-folds, eg. for quintic is cumbersome: = C (t) TrXXY + C (t) Tr(XXY )2 + ... Xi Weff XXY XXY XXY

Yij t... interpolates between Gepner-point Xi (BCFT) and large radius ... new results in enumerative geometry

31