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Of Topological Strings Lecture @ 国家理論科学研究中心 15, October 2010 Introduction to Topological Strings on Local Geometry Masato Taki/ YITP, Kyoto University 瀧 雅人 / 京都大学 基礎物理学研究所 Topological Strings Today we will study topological string theory (on the so-called “local geometry”). It is a solvable toy model of string theory, and has many interesting mathematical feature. So, topological strings give a theoretical playground for string theorists. solvable model for string theory ∗ Moreover this theory is a toy model of varied applications. non-perturbative dynamics of gauge theories ∗ dualities relates many physical & mathematical theories ∗ Math Physics curve counting Calabi-Yau compactification knot theory Chern-Simons theory matrix model mirror symmetry topological strings . Seiberg-Witten-Nekrasov theory Dijkgraaf-Vafa theory twister string black holes Applications References ・Hori et al , “Mirror Symmetry” , Clay Mathematics Monographs ・ M. Marino, “Chern-Simons theory and topological strings”, hep-th/0406005 ・M. Marino, “Les Houches lectures on matrix models and topological strings”, hep-th/0410165 ・Vonk, “A mini-course on topological strings”, hep-th/0504147 Plan of the lecture Part 1 ・Worldsheet formulation of topological strings ・Conifold and geometric transition Part 2 ・Geometric engineering of gauge theories ・ Application to AGT relation 1. Worldsheet Formulation of Topological Strings - Why Topological String Theory Calabi-Yau compactification of superstring theory is an important topics in string theory. superstring theory (heterotic, 4 4 =1 , 2 gauge theory on Type II, ...) on M R R × N In general it is very hard to compute stringy corrections on such a curved background. The low-energy effective theory of the Calabi-Yau compactification of Type II A(B) superstring theory ?? Topological sting A(B)-model captures the holomorphic information of the low-energy theory exactly !! So, you will get exact result about stringy correction if you can compute topological string theory. - Calabi-Yau space In this lecture we study topological strings propagating inside a CY. So we define it at the beggining. Let us consider a complex manifold d dimC(= 2d dimR) ¯ Xi,Xi i, ¯i =1, ,d ··· We impose the Kahler condition i ¯j dω =0 for ω = iG ¯ dX dX ij ∧ Then the Calabi-Yau condition requires that the Ricci curvature of a Kahler manifold is flat : Rij =0 c1(M)=0 - Calabi-Yau 1-fold T 2 Torus is a simple example of Calabi-Yau 2-fold. X1 = z X1¯ =¯z z z +1 ∼ - Calabi-Yau 1-fold T 2 β There are two basic 2-cycles αβ . α Generic cycles are spanned by these two cycles as 2 [Σ]=pα + qβ H1(T , Z) ∈ - Calabi-Yau 3-fold The Calabi-Yaus of our interest are 3-fold d =3 . Moreover we deal with only non-compact cases. 4 conifold : hypersurface inside C x2 + y2 + u2 + v2 =0 S2 S3 By changing the coordinates, we have another difiniton eq. xy = uv - String Thoery Let us recall that string theory is a quantum mechanics of one-dimensional objects propagating the target space M. A string sweeps out the two-dimensional surface which we call “world-sheet” t worldsheet Σg string - String Thoery Let us recall that string theory is a quantum mechanics of one-dimensional objects propagating the target space M. A string sweeps out the two-dimensional surface which we call “world-sheet” t M In order to formulate topological string theory propagating inside a target space M, we consider holomorphic embedding of the worldsheet into a Calabi-Yau M. X : Σ M g → - Mathematical “definition” of topological strings X : tΣ Fg = Ng,Σ e− Σ H (M, ) ∈ 2 Z N “number” of holomorphic maps X : Σ Σ M g,Σ g → ⊂ tΣ Kahler parameter (area) of 2-cycle Σ - Mathematical “definition” of topological strings X : Σ area tΣ Ng,Σ maps X : Σ Σ M g → ⊂ tΣ Fg = Ng,Σ e− Σ H (M, ) ∈ 2 Z 2g 2 Z(t) = exp (gs) − Fg(t) g=0 - Physical realization of topological strings = (2, 2) 2D nonlinear sigma model whose target space is M N topological twist (redefinition of the Lorentz charges) topological sigma model whose target space is M [Witten, ’88] we can model the mathematical definition (curve counting). The action of topological sigma model whose target space is M is given by A 2 µν i ¯j µν i ¯j S = d z√gG ¯g ∂ X ∂ X + i ∂ X ∂ X + ij µ ν µ ν ··· Σg i ¯i X (z),X(z) scalars on Σg i ¯i ρµ¯(z), ρµ(z) Grassmannian one-form on Σg i ¯i χ (z), χ (z) Grassmannian scalar on Σg This action is BRST-exact for a topological symmetry Q A S = Q ,V(X, ρ, χ) + X∗(ω) { BRST } Σg 1 2 µν i ¯j ¯j i V = d z√gg Gi¯j ρµ∂ν X + ρµ∂ν X 4 Σ g [Q, Xi]=0 Q, χi =0 { } Q, ρi =2i∂ Xi + Γi ρj χk { z¯} z¯ jk z¯ Then the path-integral is 1 1 tΣ Q,V e− α X χ ρe− α { } [X(Σg )]=Σ D D D Σ H2(M,Z) ∈ topological : independent of α and worldsheet metric Let us evaluate at α 0 → A 2 i ¯j i ¯j Sboson = d zGi¯j (∂zX ∂z¯X + ∂z¯X ∂zX ) Σg 2 i ¯j =2 d zGi¯j ∂z¯X ∂zX + X∗(ω) Σg Σg X∗(ω) ≥ Σg Then the path-integral is dominated by the minimum i ¯j ∂z¯X = ∂zX =0 The path-integral localizes to the integral over the holomorphic maps !! we count“number” of holomorphic maps X : Σ Σ M g → ⊂ In this way the path-integral for genus- g worldsheet implies the counting of the holomorphic maps from the worldsheets to the Calabi-Yau Σ X∗(ω) Fg := e− g R hol.maps X In general it is very hard to compute such a function mathematically. Fortunately, for a wide class of non-compact CY, we can compute these partition function exactly by utilizing string dualities !! So, in the following, we focus on the dual pictures of the topological strings. 2. Conifold and Geometric Transition - Deformed Conifold We can deform the conifold by introducing the (complex structure) deformation xy = uv + R ∈ 2 Let us regard it as a R T fibration. Base is × x x = y = z z = xy | | | | | | | | u = v = z z = uv + | | | | | − | { u v | | | | y | | Re(z) 2 R T x × | | z = xy z = uv + u v { | | | | y iα iα | | x e x, y e− y 2 → → Fiber is R = Im ( z ) and phases T iβ iβ u e u, v e− v { → → These cycles collapse on the locus z =0 or z = α z =0 β z = 2 fiber R T × z =0 We choose a point of the non- z = compact fiber direction : 3 S Im(z)=0 z = xy x = y = z 1/2 x =¯y | | | | | | 1/2 z = uv + u = v = z u =¯v { | | | | | − | By substituting it into the definition eq., we get the form of the minimal line at the locus xy = uv + x 2 + y 2 = . S3 | | | | - Resolved Conifold At =0 , the 3-sphere vanishes, and z =0 we recover the conifold z = xy = uv We can smooth the singular conifold by the resolution 2 1 t S = P - Geometric Transition S3 we can define closed we can wrap topological 3- topological string on it branes on S3 open string theory [Witten, ’93] gauge theory on the branes is the Chern-Simons theory on S3 we will find open/closed duality duality between them ! - A-model topological string on resolved conifold [Gopakumar-Vafa, ’98] 2 1 dgs − dt F (gs,t)= 2sin e− d 2 d 1 ≥ g 1 g 1 1 1 1 ( 1) B 1 1 ( 1) − B2g 2g 2 − 2g 2g 1 = + − z − = + − z − z 2 2 z −z 2g(2g 2)! e 1 z − 2 (2g)! 2sin 2 g=1 − g=1 − By using this formula, we obtain dt g 1 dt 1 e− 1 t 2g 2 ( 1) − B2g e− F = log(1 e− )+ g − − − 2 3 − − s 3 2g gs d 12 g=2 2g(2g 2)! d − d=1 − d=1 - large-N duality as gauge/gravity duality ‘tHooft’s idea amplitudes of closed string genus gauge theory expansion F = log Zgauge 2g 2 2g 2+h = gs − λ − Fg,h λ = Ngs g,h 2g 2 = gs − Fg(λ) = log Zstring !? g Let us study SU(N) Chern-Simons theory as the gauge theory [Witten, `88]: N 1 − CS 3 1 N j jπ Z (S )= 2sin − (k + N)N/2 k + N j=1 2π N 1 − gs = CS 3 1 N j jπ Z (S )= 2sin − k + N (k + N)N/2 k + N j=1 2πiN t = iλ = k + N F = log ZCS N N jπ = log(k + N)+ (N j) log(2 sin ) − 2 − k + N j=1 ∞ z2 By using the formula sin(πz)=πz (1 ) − 2 n=1 n N N 2πj = log(k + N) (N j) log( ) non-pert − 2 − − k + N : F j=1 N ∞ j2 + (N j) log(1 g2 ) pert − − s 2 2 : F n=1 4π n j=1 F pert N 1 k − ∞ ∞ 1 1 g2 F pert = (N j) j2k s − 2k 2 n=1 k n 4π j=1 k=1 ζ(2k) k N 1 ∞ ζ(2k) g2 − = s (N j)j2k k 4π2 − k=1 j=1 k N k [ 2 ] k N k+1 2g B2g j = + k+1C2g N − 2 g=0 k +1 j=1 power-sum formula B0 N 2k+2 (2k + 1)(2k + 2) k 2k+2 2g B2g + 2kC2g 2 N − − g=1 2g F pert N 1 k − ∞ ∞ 1 1 g2 F pert = (N j) j2k s − 2k 2 n=1 k n 4π j=1 k=1 ζ(2k) k N 1 ∞ ζ(2k) g2 − = s (N j)j2k k 4π2 − k=1 j=1 B0 N 2k+2 (2k + 1)(2k + 2) k λ = Ngs 2k+2 2g B2g + 2kC2g 2 N − − g=1 2g constant maps ∞ B2g B2g 2 pert 2 0 2g 2 − F = gs− ( )+gs ( ) + gs − ··· ··· g=2 2g(2g 2) (2g 2)! − − 2g 2 2k+2 2g ζ(2k) B2g + g − λ − 2kC2g 2 s 2k − g=2 k (2π) k=1 `tHooft expansion !! 2g 2 F(g 2) = g − Fg(λ) ≥ s g=2 ζ(2g +2h 2) 2g+2h 2C2g 2 F (λ)=B λ2h − − − g( 2) 2g 2g+2h 2 ≥ (2π) − g(2g +2h 2) h=0 − 1 2g+2h 3C2h − 2g+2h 2 n=1 (2πn) − g(2g 2) − ∞ ∞ B2g 2h 1 Fg ( 2) = 2 λ 2g+2h 2C2h ≥ − 2g+2h 2 2g(2g 2) n=1 (2πn) − − h=0 binomial theorem B2g ∞ 1 1 = + 2g 2 2g 2 2g(2g 2) n=1 (2πn + λ) − (2πn λ) − − − B2g ∞ 1 = 2g 2 2g(2g 2) (2πn + λ) − n Z=0 − ∈ non-pert.
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