Topological vertex and quantum mirror curves

Transcription

1 Topological vertex and quantum mirror curves Kanehisa Takasaki, Kinki University Osaka City University, November 6, 2015 Contents 1. Topological vertex 2. On-strip geometry 3. Closed topological vertex Based on 1. K. T., arxiv: [math-ph] 2. K. T. and T. Nakatsu, arxiv:

2 1. Topological vertex Web diagrams of non-compact toric Calabi-Yau threefolds On-strip geometry Closed topological vertex Topological vertex Topological vertex (Aganagic, Klemm, Mariño and Vafa 2003) is a diagrammatic method to construct the partition functions (or amplitudes) of topological string theory on non-compact toric Calabi-Yau threefolds. 2

3 1. Topological vertex Vertex weight C λµν = q κ(µ)/2 s t ν(q ρ ) η P s t λ/η (q ν ρ )s µ/η (q t ν ρ ) λ = (λ i ) i=1, µ = (µ i) i=1, ν = (ν i) i=1 are partitions representing Young diagrams of arbitrary shapes. t ν denotes the conjugate partition of ν. κ(µ) is the second Casimir invariant ν λ µ κ(µ) = µ i (µ i 2i + 1) = 2 (j i) i=1 (i,j) µ 3

4 1. Topological vertex Vertex weight (cont d) C λµν = q κ(µ)/2 s t ν(q ρ ) η P s t λ/η (q ν ρ )s µ/η (q t ν ρ ) s t ν(q ρ ), s t λ/η (q ν ρ ) and s µ/η (q t ν ρ ) are special values of the Schur/skew Schur functions s t ν(x), s t λ/η (x), s µ/η(x), x = (x 1, x 2, ), at q ρ = (q i 1/2 ) i=1, q ν ρ = (q ν i+i 1/2 ) i=1, q t ν ρ = (q t ν i +i 1/2 ) i=1 Vertex weights are glued together along the internal lines. Edge weights are assigned to those lines. 4

5 1. Topological vertex Examples (local P 1 geometry) 1. Open string amplitudes of X = O( 1) O( 1) CP 1 (resolved conifold): = C α1 α 0 β 1 ( Q) α1 C t α 1 α 2 β 2 Z α 0α 2 β 1 β 2 α 1 P 2. Open string amplitudes of X = O O( 2) CP 1 : Z α 0α 2 β 1 β 2 = α 1 P C α1 α 0 β 1 ( Q) α 1 ( 1) α 1 q κ(α 1)/2 C α2 t α 1 β 2 (A framing factor is inserted in this case.) α 0 β 1 α 1,Q β 2 α 0 α 2 α 1,Q β 1 β 2 α 2 5

6 1. Topological vertex Building blocks of operator formalism Charge-zero sector of fermionic Fock space spanned by the ground states 0, 0 and the excited states λ = ψλ i i+1 ψ λ 2 1 ψ λ 1, λ = ψ λ1 ψ λ2 +1 ψ λi +i 1 Fermion bilinears L 0 = n:ψ n ψn:, K = 1/2) n Z n Z(n 2 :ψ n ψn:, J m = n Z :ψ m n ψ n:, m Z, V (k) m = q km/2 n Z q kn :ψ m n ψ n:, k, m Z. 6

7 1. Topological vertex Vertex operators ( ) ( z k Γ ± (z) = exp k J ±k, Γ ±(z) = exp k=1 and the multi-variable extensions k=1 ) ( z) k J ±k k Γ ± (x) = i 1 Γ ± (x i ), Γ ±(x) = i 1 Γ ±(x i ) Fermionic expression of vertex weight C λµν = s t ν(q ρ ) t λ Γ (q ν ρ )Γ + (q tν ρ )q K/2 µ = s t ν(q ρ ) λ Γ (q ν ρ )Γ +(q tν ρ )q K/2 t µ (Okounkov, Reshetikhin and Vafa 2003) 7

8 Cases where amplitudes are computed explicitly 1. Topological vertex 1. On-strip geometry Iqbal and Kashani-Poor 2004 (by topological vertex) β 2 β 3 2. Closed topological vertex Bryan and Karp 2003, Karp, Liu and Mariño 2005 (by algebraic geometry and topological vertex) Su lkowski 2006 (by crystal model and β 1 β 4 β 5 topological vertex) Our goal: To derive a q-difference equation that may be interpreted as quantization of the mirror curve. 8

9 2. On-strip geometry General setup (a) The toric diagram is a triangulation of a strip (trapezoid of height 1). Q 1 Q 2 Q 3 (b) The web diagram has N vertices, N 1 internal lines and N + 2 external lines (or legs ). Q 4 Example of on-strip geometry (N = 5) (c) The internal lines are assigned with Kähler parameters Q 1,..., Q N 1. 9

10 General setup (cont d) (d) The external lines are assigned with partitions α 0, β 1,..., β N, α N. β 2 β 3 2. On-strip geometry α 5 (e) The sign (or type) σ n of the n-th vertical leg α 0 are defined as +1 if the leg points σ n = 1 if the leg points β 1 β 4 β 5 σ 2 = σ 3 = +1, σ 1 = σ 4 = σ 5 = 1 Let Z α 0α N β 1...β N denote the open string amplitude constructed by topological vertex. 10

11 2. On-strip geometry Infinite-product formula of amplitude If α 0 = α N =, the amplitude can be computed explicitly with the aid of the Cauchy identities for skew Schur functions (Iqbal and Kashani-Poor 2004): Z β 1 β N = s t β 1 (q ρ ) s t β N (q ρ ) 1 m<n N i,j=1 (1 Q mn q t β (m) i β (n) j +i+j 1 ) σ mσ n where β (n) = β n if σ n = +1, t β n if σ n = 1, Q mn = Q m Q m+1 Q n 1 11

12 2. On-strip geometry Fermionic expression of amplitude The infinite-product formula holds only for α 0 = α N =. For general cases, the following fermionic expression is available: Z α 0α N β 1 β N = q (1 σ 1)κ(α 0 )/4 q (1+σ N )κ(α N )/4 s t β 1 (q ρ ) s t β N (q ρ ) t α 0 Γ σ 1 (q β(1) ρ )Γ σ 1 + (q t β (1) ρ )(σ 1 Q 1 σ 2 ) L0 Γ σ N 1 (q β(n 1) ρ )Γ σ N 1 + (q t β (N 1) ρ )(σ N 1 Q N 1 σ N ) L 0 Γ σ N (q β(n) ρ )Γ σ N + (q t β (N) ρ ) α N where Γ σ ± denote Γ ± if σ = +1 and Γ ± if σ = 1 (Eguchi and Kanno 2003, Bryan and Young 2008 for special cases; Nagao 2009 and Su lkowski 2009 for general cases) 12

13 2. On-strip geometry Wave functions Wave functions are defined as Z n,(1 Ψ n (x) = k ) x k, Ψn (x) = Z n, k=0 for n = 0, 1,..., N, N + 1, where k=0 Z n,(k) Z n, x k Z 0,λ = Z λ, Z n,λ = Z λ (1 n N), Z N+1,λ = Z λ Remark: Z n,λ s are the coefficients of Schur function expansion of a KP tau function τ n. In this sense, these wave functions are Baker-Akhiezer functions (at the initial time t = 0) that correspond to free fermion fields ψ( x), ψ (x). 13

14 2. On-strip geometry Ψ n and Ψ n for n = 1,..., N are q-hypergeometric series Ψ n (x) = 1+ k=1 C n (1)C n (q) C n (q k 1 ) B n (1)B n (q) B n (q k 1 )(1 q) (1 q k ) qk/2 x k where B n (y) and C n (y) are Laurent polynomials in y: B n (y) = (1 Q mn y σ n ) (1 Q nm y σ n ), C n (y) = m<n, σ m σ n >0 m<n, σ m σ n <0 (1 Q mn y σ n ) m>n, σ m σ n >0 m>n, σ m σ n <0 (1 Q nm y σ n ) Remark: When N = 1, Ψ reduces to a quantum dilog: q k/2 x k Ψ(x) = 1 + (1 q) (1 q k ) = (1 q i 1/2 x) 1 k=1 i=1 14

15 2. On-strip geometry Ψ 1,..., Ψ N satisfy q-difference equation or, equivalently, Ψ n (x) Ψ n (qx) = q 1/2 x C n(q x x ) B n (q x x ) Ψ n (x) B n (q 1 q x x )(1 q x x )Ψ n (x) = q 1/2 xc n (q x x )Ψ n (x) (Kashani-Poor 2006, Hyun and Yi 2006, Gukov and Su lkowski 2011 for the resolved conifold). Remark: These equations are also studied in the context of the AGT correspondence (Kozçaz, Pasquetti and Wyllard 2010, Taki 2010) and the vortex partition function (Bonelli, Tanzini and Zhao 2011). 15

16 2. On-strip geometry Classical limit As q 1 (ŷ = q x x y), the q-difference equation B n (q 1 q x x )(1 q x x )Ψ n (x) = q 1/2 xc n (q x x )Ψ n (x) reduces to the algebraic equation B n (y)(1 y) = xc n (y) of the the mirror curve. In this sense, the q-difference equation may be thought of as quantization of the mirror curve. Remark: The Newton polygon of B n (y)(1 y) xc n (y) can be mapped to the outline of the toric diagram by an SL(2, Z) transformation. 16

17 2. On-strip geometry How the equations for different n s are related where x = (1 y) B n(y) C n (y) ( ) g(y, ỹ) = ( ) x = (1 ỹ) B n+1(ỹ) C n+1 (ỹ) y = Q σ n n ỹ σ nσ n+1, x = g(y, ỹ) x σ nσ n+1 ỹ 1 if σ n = +1, σ n+1 = +1, 1 if σ n = +1, σ n+1 = 1, yỹ if σ n = 1, σ n+1 = +1, y if σ n = 1, σ n+1 = 1 Birational map ( ) preserves the symplectic structure: d log x d log y = d log x d log ỹ 17

18 2. On-strip geometry What about Ψ 0 and Ψ N+1? They are related to quantum dilogs. E.g., if σ 1 = +1, Ψ 0 (x) is a product of quantum dilogs and satisfies the equation Ψ 0 (qx) = (1 q 1/2 x) 1 N n=2 (1 Q n 1 q 1/2 x) σ n Ψ 0 (x). Its classical limit is the algebraic equation y = (1 x) 1 N n=2 (1 Q n 1 x) σ n. This equation can be transformed to the equation x = (1 ỹ)b 1 (ỹ)/c 1 (ỹ) by a birational symplectic map (x, y) ( x, ỹ): d log y d log x = d log x d log ỹ. 18

19 3. Closed topological vertex Setup (a) Three internal lines are assigned with Kähler parameters Q 1, Q 2, Q 3 (b) Two vertical external lines are assigned with partitions β 1, β 2. All other external lines are given. Q 1 β 1 Q 3 Q 2 β 2 Let Z ctv β 1 β 2 denote the open string amplitude in this setting. 19

20 3. Closed topological vertex Method of computation of Z ctv β 1 β 2 Reconstruct the amplitude Zβ ctv 1 β 2 C t α by gluing a single vertex to the amplitude of an on-strip geometry. Borrow tools from our previous study on integrable structures of the melting crystal models (5D U(1) instanton sum and its variants) (Nakatsu and K.T, since 2007) to compute the sum over α P. Q 3 t α α β 2 β 1 Gluing a vertex (top) to an on-strip geometry (bottom) 20

21 3. Closed topological vertex Result of computation of Z ctv β 1 β 2 Z ctv β 1 β 2 = q κ(β 2)/2 i,j=1 (1 Q 1 Q 2 q β 1i t β 2j +i+j 1 ) 1 t β 1 Γ (q ρ )Γ + (q ρ )( Q 1 ) L 0 Γ (q ρ )Γ +(q ρ )( Q 3 ) L 0 Γ (q ρ )Γ + (q ρ )( Q 2 ) L 0 Γ (q ρ )Γ +(q ρ ) t β 2. The main part Y β1 β 2 := t β 1 t β 2 is essentially the open string amplitude of yet another on-strip geometry. This strange coincidence is a key to derive a q-difference equation. Letting β 1 = β 2 =, this expression reduces to the known result of the closed string amplitudes (Bryan and Karp 2003, Karp, Liu and Mariño 2005). 21

22 3. Closed topological vertex Wave functions Wave functions are defined as Ψ(x) = k=0 Z ctv (1 k ) Z ctv x k, Ψ(x) = along with the auxiliary wave functions k=0 Z(k) ctv Z ctv x k Φ(x) = k=0 Y (1 k ) Y x k, Φ(x) = k=0 Y (k) Y x k obtained from the main part Y β1 β 2 = t β 1 t β 2 of Z ctv β 1 β 2. 22

23 3. Closed topological vertex Wave functions (cont d) The coefficients of Ψ(x) = a 0 = b 0 = 1, are related as a k x k and Φ(x) = k=0 b k x k, k=0 a k = b k k (1 Q 1 Q 2 q i 1 ) 1 i=1 Φ(x) is build from quantum dilogs, and satisfies the q-difference equation Φ(qx) = (1 q 1/2 x)(1 Q 1 Q 3 q 1/2 x) (1 Q 1 q 1/2 x)(1 Q 1 Q 2 Q 3 q 1/2 x) Φ(x). 23

24 3. Closed topological vertex Transforming q-difference equation The q-difference equation (1 Q 1 q 1/2 x)(1 Q 1 Q 2 Q 3 q 1/2 x)φ(qx) = (1 q 1/2 x)(1 Q 1 Q 3 q 1/2 x)φ(x) for Φ(x) is transformed to the q-difference equation (1 Q 1 Q 2 q 2 q x x Q 1 q 1/2 x)(1 Q 1 Q 2 q 1 q x x Q 1 Q 2 Q 3 q 1/2 x)ψ(qx) = (1 Q 1 Q 2 q 2 q x x q 1/2 x)(1 Q 1 Q 2 q 1 q x x Q 1 Q 3 q 1/2 x)ψ(x) for Ψ(x). Let us rewrite this equation as H(x, q x x )Ψ(x) = 0 and examine the q-difference operator H(x, q x x ). 24

25 3. Closed topological vertex Reducing q-difference equation to final form The operator H(x, q x x ) can be factorized as H(x, q x x ) = (1 Q 1 Q 2 q 2 q x x )K(x, q x x ) where K(x, q x x ) = (1 Q 1 Q 2 q 1 q x x )(1 q x x ) (1 + Q 1 Q 3 )q 1/2 x + Q 1 (1 + Q 2 Q 3 )q 1/2 xq x x + Q 1 Q 3 qx 2. Since the prefactor 1 Q 1 Q 2 q 2 q x x is invertible on the space of power series of x, the equation H(x, q x x )Ψ(x) = 0 reduces to K(x, q x x )Ψ(x) = 0 This is the final form of our quantum mirror curve. 25

26 3. Closed topological vertex Classical limit As q 1, the q-difference operator K(x, q x x ) = (1 Q 1 Q 2 q 1 q x x )(1 q x x ) (1 + Q 1 Q 3 )q 1/2 x + Q 1 (1 + Q 2 Q 3 )q 1/2 xq x x + Q 1 Q 3 qx 2 turns into the polynomial K cl (x, y) = (1 Q 1 Q 2 y)(1 y) (1 + Q 1 Q 3 )x + Q 1 (1 + Q 2 Q 3 )xy + Q 1 Q 3 x 2 Its Newton polygon has the same shape as the triangular outline of the toric diagram. 26

27 3. Closed topological vertex What about wave functions obtained from β 2 = (k), (1 k )? The q-difference equations become slightly more complicated because of the framing factor q κ(β2)/2. What about putting β 1 and β 2 on other legs? The open string amplitude can be expressed in a similar form. However q K/2 s remain in the operator product, and they prevent us from doing explicit computation. A similar difficulty takes place when one attempts to prolong the branches of the closed topological vertex. What about web diagrams with cycle(s)? It is an ultimate goal of our project, but we have currently no idea. 27