Chern-Simons Theory & Topological Strings

Transcription

1 1 Chern-Simons Theory & Topological Strings Bonn August 6, 2009 Albrecht Klemm

2 2 Topological String Theory on Calabi-Yau manifolds A-model & Integrality Conjectures B-model Mirror Duality & Large N-Duality Large N-expansion of Chern-Simons theory Dual Gauge theory String theory backgrounds T S 3 O( 1) O( 1) P 1 Lens spaces L(p, 1) A p -geometries The topological Vertex

3 3 Topological String on Calabi-Yau manifolds String Theory: X : Σ g,h (M, L) Partition function Z (and other correlators) Z(M, L) = DXDhDΨ ± D Ψ ± e S(X,h,Ψ ±, Ψ ±,G,B,A,... ). Dh, w.s. metric integral, becomes finite for M 10 = M M 3,1 (dim C (M) = 3).

4 4 Still the variational integral cannot be performed. Topological string theories are truncations of the theory of physical (supersymmetry) and mathematical (invariants of (M,L)) interest. The relevant cases require M Kähler & c 1 (T M) = 0 Ω hol. (n, 0)-form, L Lagrangian & special Ω L = vol(l). Geometry of a Calabi-Yau manifold with special Lagrangian branes or coherent sheaves.

5 5 For the choice of M one has a (2, 2) w.s. SCFT with U(1) V and U(1) A currents, which can be used to defined two scalar, nilpotent operators: Q 2 A = 0 or Q2 B = 0 Σ µj µ V = 0, U(1) L = U(1) L + U(1) V Q A Σ µj µ A = Σ X (c 1 ), U(1) L = U(1) L + U(1) A Q B This defines two cohomological theories: A model (M,L) Kaehler structure def (M,L) L special Lagrangian B model(w,d) complex structure def (W,D) D coherent sheaf Mirror Symmetry

6 6 A-model: The path integral localizes to holomor. maps z X i = 0. In particular it becomes formal power series in g s, in which every term is a well defined finite dim integral Connected contr. F = Log(Z) = g=0 g2g 2 s F g (q, Q) F = g=0 g 2g 2 s (β,h) (H 2 (M,L,Z),H 1 (L,Z)) q β Q h Gromov Witten Inv. q β = i e βi t i, β Z h 11 degrees, c g vir M g β,h (M,L) β,h }{{} r g β,h

7 7 t i = C (2),i ω + ib complexified Kähler parameter Q h = i e hi u i, h Z h1 (L) windings, u i = C (1),i L ω + ia Open string (Wilson line) moduli. Mathematical theory defining the r g β oriented Σ. Q for closed

8 8 Expected dimension critical for on CY 3-folds vd C M g β (M) = c 1(T M ) β + (d C (M) 3)(1 g). For compl. intersec. M X, X toric, the r g=0/1 β Q can be calculated by localization, w.r.t. induced torus action on M g β (M) Kontsevich, Givental,.... On non-compact toric CY, O( K B ) B, B toric, r g β Q & rg β,h Q on Harvey-Lawson type SLAG L A.K., Zaslow, Graber....

9 9 Symplectic invariants and integrality conjectures: Gromov Witten invariants Gopakumar Vafa invariants (homology indices in moduli space of D0+D2+1 D6 branes count BPS states) Donaldson Thomas invariants (homology indices in moduli space of free sheaves)

10 10 The relation between DT- GV- and GW invariants MNOP 04: Z GV = exp(f ) Z GV (M, q gs, q)m(q gs ) χ(m) 2 = Z DT (M, q gs, q) with q gs = e ig s, Z DT (M, q gs, q) = β,k Z n(k) β qk g s q β and Z GV = e c(t) gs 2 +l(t) β g=1 2g 2 l=0 [ ( ) (1 qg r s q β ) rñ(0) β r=1 (1 q g l 1 g s q β ) ( 1)g+r ( 2g 2 l ] )ñ (g) β where n (k) β Z! and ñ (k) β Z! M McMahon funct.

11 11 B-model: The path integral localizes to constant maps. D i D j D k F 0 = Ω i j k Ω W d 2 τ F 1 = Tr( 1) F F L F R q L0 q L 0, F τ 2 6g 6 F g>1 = (G, µ k ), M g k=1 with (G, µ k ) = Σ g (G zz (µ) z z + G z z ( µ) z z). Natural flat coordinates t i = a i Ω and conjugated

12 12 momenta p i = b i Ω. F 0 calculated by periods. Higher F g hard to obtain in this framework. Open string: Example D-6 brane D = W W supo = Ω Tr[A Ā A A A] W A U(N) valued (1,0)-form on W. In simple situations this integral can be calculated by relative period integrals. u i = C i Ω. Higher genus amplitudes are hard to obtain.

13 13 Type II open closed Mirror duality A model B model 3d Chern Simons theory on L Knot invariants WZW models S duality holomorphic Chern Simons theory on D Donalson Thomas Invariants matrix model Kahler gravity Gromov Witten Gopakumar Vafa invariants Localization Kodaira Spencer Gravity Family Indices Ray SingerTorsion complex structure deform. theory Open/Closed duality Large N duality

14 14 Large N-expansion of Chern-Simons theory: S = k Tr (A Ạ + 23 ) 4π A A A L A U(N) gauge connection in the trivial bundle over L 3 Usual combinatoric of large N expansion (g s = 2π k+n ) N 2 Γ Γ N 0

15 15 With the t Hooft parameter t = g s N one can expand F = Log(Z) as F CS = g=0 h=1 F g,h gs 2g 2 t h If the sum over holes can be performed, this looks like a closed string expansion F CS = g=0 g 2g 2 s F g (t).

16 16 Dual backgrounds: Chern-Simons on L Topological string on CY M. Deg. holomorphic open instantons on T L oriented graphs of CS on L. Witten 95 The geometric transition T S 3 M = O( 1) O( 1) P 1 provides the dual closed string background Gopakumar and Vafa 99 i.e. F T S (M, t) = F CS (S 3, t). t Hooft param. identified with Kähler param. of P 1.

17 17 Generalization by orbifoldisation AKMV 2002 T S 3 M = O( 1) O( 1) P 1 Z p Z p T L(p, 1) A p geometry S 3 : x 2 + y 2 = 1, Z P : (x, y) e 2πi p (x, y), π 1 (L(p, 1)) = Z p the Z p discrete flat connections break U(N) U(N 1 ) U(N p ) t i = g s N i M/Z p = {xy = (e v 1)(e v+pu 1)} resolved geometry ˆM has p (P 1 ) s with p complexified Kähler parameters s i

18 18 F T S ( ˆM, s 1,..., s p ) = F CS (L(p, 1), t 1,..., t p ). Non-trivial identification s i = t i, where the s i = a i Ω are flat coordinates near the small radius orbifold point in ˆM, i.e. Z T S ( ˆM, s 1,..., s p ) is a resummation of the ordinary instanton partition function. Mathematically this calculates orbifold Gromov-Witten invariants Coates, Iritani, Ruan, Chiodo...

19 19 E.g. for p = 2, ˆM = O( KF0 ) F 0. t i log(z i ), s i w i, i = 1, 2. z =0 2 C w =0 2 w =0 1 orbifold point large radius point Seiberg Witten point z =0 1 Schematic complex moduli space for the mirror of O( K F0 ) F 0

20 20 The Topological Vertex The topological vertex is a building block to solve the topological string on any non-compact toric Calabi-Yau using the large N-duality to Chern-Simons Theory AKMV 03. Toric CY backgrounds M T = (C m Z({D i1 D is }))/(C ) r d = m r where the (C ) r action on the coordinates x i of C m is

21 21 specified by charge vectors l (k) x i µ l(k) i k x i, with l (k) i Z, µ (k) C, i = 1,..., m, k = 1,..., r. Canonical symplectic Form ω = i 2 d dx k d x k = 1 2 d d x k 2 dθ k = d du k dv k k=1 k=1 k=1 c 1 (T M ) = 0 i l (k) i = 0, k

22 22 Then in a patch C 3 paramtr. by x 1, x 2, x 3 one has a T 2 R fibration over B 3 generated by flows α x k = {r α, x k } ω of three Hamiltonians r α1 = x 1 2 x 2 2, r α2 = x 3 2 x 1 2, r R = Im(x 1 x 2 x 3 ). Harvey-Lawson SLAGs C S 1 L 1 : r α1 = 0, r α2 = s 1, r R 0, Re(x 1 x 2 x 3 ) = 0 L 2 : r α1 = s 2, r α2 = 0, r R 0, Re(x 1 x 2 x 3 ) = 0 L 3 : r α1 + r α2 = 0, r α1 = s 3, r R 0, Re(x 1 x 2 x 3 ) = 0

23 23 L x r α1 L 1 (1,0) L 2 L 3 s 1 L x L 3 (0,1) r α 2 x 2 1 ( 1, 1) The Vertex calculates the GW invariants for the following maps :

24 24 z 3 U(N ) 3 U(N ) 2 X : Σ g,h z 2 z 1 U(N ) 1 moment map projection of C 3 Z(V i ) = R 1,R 2,R 3 C R1,R 2,R 3 (q gs ) 3 Tr Ri V i i=1

25 25 C R1 R 2 R 3 (q gs ) = N R1 Q 1,R N W Rt 3 Q t R 2 /2+κ R3 /2 R2 t Q (q 1 g s )W R2 Q t(q) 3, 3 Rqκ W R2 (q gs ) R,Q 1,Q 2 where W R1,R 2 are Hopf links, N R 3 R 1 R 2 are tensor product coefficients and κ R = i l i(l i 2i + 1) and l i is the length of the row of the i th line in the Young-Tableaux of the represention R i It is up to transpostion related to the link invariant

26 26 R R R 1 1 W R1 R 2 R(L) = W R 1 R(L 1 )W R2 R(L 2 ) W R (K) Gluing rules: If Γ = Γ 1 Γ 2 and X Γi are the associated toric varieties then Z(X Γ ) = Q Z(X ΓL ) Q ( 1) l(q) e l(q)t Z(X ΓL ) Q t (1)

27 27 Γ... Γ 1 l(q) ( 1) Here t is the Kähler parameter size of the connecting P 1. The quantity ( 1) l(q) e l(q)t, with l(q) the number of boxes in the Young-Tableaux of the intermediate representation, can be viewed as propagator. Here we suppress the data of the framing, which are in general important to patch together arbitrary toric varieties.... e l(q) t Γ 2

28 28 Examples: ( 1,0) Z P 1(V 1, V 2 ) = Z P 2 = O( 1)+O( 1) P 1 (0,1) ( 1, 1) (1,1) (0, 1) (1,0) x =1 2 x =1 4 O( 3) w 3 u 2 u 1 (0, 1) (0,1) w 1 ( 1, 1) ( 1,2) u 3 C Q1 Q 2 R t( 1)l(R) e l(r)t C R Tr(V 1 )Tr(V 2 ) R,Q 1,Q 2 P ( 1) i l(ri) e P i l(ri)t q P i κ R i C R2 R3 t R 1,R 2,R 3 C R1 R t 2 C R 3 R t 1. 2 P w 2 (0,1) (1, 1) v 2 v 3 (0, 1) ( 1,1) v 1 (2, 1) x =1 3

29 29 Conclusions Large N-duality between Chern-Simons theory and topological string on CY backgrounds provides many solvable examples for Gauge-Theorie/String Theory dualities. The vertex solves the topological string on any toric CY in the large readius region. MNOP 04 Moreover the duality applies also to the orbifold phases. In the latter case one can use it also to calculate open orbifold G W invariants BKMP 07.

30 30 The relation provides further many non-trivial combinatrical identitities. For example for 2d gravity integrals MV 01..