Refined Topological Branes

Abstract

We study the open refined topological string amplitudes using the refined topological vertex. We determine the refinement of holonomies necessary to describe the boundary conditions of open amplitudes (which in particular satisfy the required integrality properties). We also derive the refined holonomies using the refined Chern–Simons theory.

Appenix A: Useful Identities

In this section, we collect some identities and background materials used in this note. We used the refined topological vertex to compute the open topological string amplitude which has the following explicit form,

$$\begin{aligned} C_{\lambda \mu \nu }(t,q)&=t^{-\frac{\Arrowvert \mu ^{t}\Arrowvert ^{2}}{2}-\frac{\Arrowvert \nu ^{t}\Arrowvert ^{2}}{2}} q^{\frac{\Arrowvert \mu \Arrowvert ^{2} }{2}+\frac{\Arrowvert \nu \Arrowvert ^{2}}{2}}P_{\nu }(t^{-\rho };q,t)\nonumber \\&\quad \sum _{\eta }v^{|\eta |+|\lambda |-|\mu |}s_{\lambda ^{t}/\eta }(q^{-\nu }t^{-\rho })s_{\mu /\eta }(t^{-\nu ^{t}}q^{-\rho }), \end{aligned}$$

(A.1)

where $P_{\nu }(t^{-\rho };q,t)$ is the Macdonald polynomial at the special point $x_{i}=t^{i-1/2}$, and $s_{\lambda /\eta }$ is the skew Schur function, defined below. Moreover, we used the shorthand notation

$$\begin{aligned} \Arrowvert \lambda \Arrowvert ^2\equiv \sum _{i=1}^{\ell (\lambda )}\lambda _{i}^{2}. \end{aligned}$$

(A.2)

v is given by $q^{1/2}t^{-1/2}$. We have also made use of the following identity:

$$\begin{aligned} P_{\nu }(t^{\rho };q,t)=(-1)^{|\nu |}q^{n(\nu ^{t})}t^{-n(\nu )}P_{\nu }(t^{-\rho };q,t). \end{aligned}$$

(A.3)

For completeness, let us remind the reader following identities that are frequently used in our computations,

$$\begin{aligned} n(\lambda )&=\sum _{i=1}^{\ell (\lambda )}(i-1)\lambda _{i}=\sum _{(i,j)\in \lambda }(\lambda ^{t}_{j}-i)=\sum _{(i,j)\in \lambda }(i-1)=\frac{\Arrowvert \lambda ^{t} \Arrowvert ^{2}}{2}-\frac{|\lambda |}{2} \end{aligned}$$

(A.4)

$$\begin{aligned} n(\lambda ^{t})&=\sum _{i=1}^{\ell (\lambda ^{t})}(i-1)\lambda ^{t}_{i}=\sum _{(i,j)\in \lambda ^{t}}(\lambda _{i}-j)=\sum _{(i,j)\in \lambda ^{t}}(j-1)=\frac{\Arrowvert \lambda \Arrowvert ^{2}}{2}-\frac{|\lambda |}{2}. \end{aligned}$$

(A.5)

In this paper, we showed that holonomies which are describing boundary conditions for open topological strings need to be modified in the refined case to ensure the integrality property. We argued that the naive modification from Schur functions to Macdonald functions does not capture the full story, and we need to include the dual Macdonald functions. Let us briefly recall the definition and some properties of them. Macdonald introduced a two-parameter extension of the usual inner product defined for powersum symmetric functions $p_{\mu }$, which we call (q, t)-inner product,

$$\begin{aligned} \langle p_{\lambda },p_{\mu }\rangle _{q,t}=\delta _{\lambda \mu }z_{\lambda }\prod _{i=1}^{\ell (\lambda )}\frac{1-q^{\lambda _{i}}}{1-t^{\lambda _{i}}}, \end{aligned}$$

(A.6)

where $z_{\mu }$ is a combinatorial factor defined by

$$\begin{aligned} z_{\lambda }=\prod _{i\ge 1}i^{m_{i}}m_{i}!, \end{aligned}$$

(A.7)

with $m_{i}=m_{i}(\lambda )$ being the number of rows in $\lambda$ of length i. q and t parameters introduced in refined topological strings are same parameters as the ones introduced by Macdonald, and the unrefined case is when $q=t$, the limit when the (q, t)-inner product reduces to the usual inner product.

The dual Macdonald function $Q_{\mu }(x;q,t)$ is defined as the dual symmetric function of the Macdonald function with respect to the (q, t)-inner product:

$$\begin{aligned} \langle P_{\lambda },Q_{\mu }\rangle _{q,t}=\delta _{\lambda \mu }. \end{aligned}$$

(A.8)

There is a very nice relationship between the Macdonald polynomial $P_{\mu }(x;q,t)$ and its dual $Q_{\mu }(x;q,t)$:

$$\begin{aligned} Q_{\mu }(x;q,t)=b_{\mu }(q,t)P_{\mu }(x;q,t), \end{aligned}$$

(A.9)

where $b_{\mu }(q,t)$ is the inverse of the algebraic norm square of the Macdonald polynomial,

$$\begin{aligned} b_{\mu }(q,t)=\frac{1}{\langle P_{\lambda },P_{\mu }\rangle _{q,t}}. \end{aligned}$$

(A.10)

Another useful approach to understand the duality is obtained by generalizing the involution $\omega$ on the ring of symmetric functions similarly to depend on two parameters. Recall the action of $\omega$ on powersums $p_{\lambda }$,

$$\begin{aligned} \omega \, p_{\lambda }=\epsilon _{\lambda }\,p_{\lambda }, \end{aligned}$$

(A.11)

where $\epsilon _{\lambda }=(-1)^{|\lambda |-\ell (\lambda )}$. The action of $\omega$ on Schur function is particularly interesting,

$$\begin{aligned} \omega \,s_{\lambda }=s_{\lambda ^{t}}. \end{aligned}$$

(A.12)

The action of the two parameter version is defined compatible with the inner product,

$$\begin{aligned} \omega _{q,t}\,p_{\lambda }\equiv \epsilon _{\lambda }\,p_{\lambda }\prod _{i=1}^{\ell (\lambda )}\frac{1-q^{\lambda _{i}}}{1-t^{\lambda _{i}}}. \end{aligned}$$

(A.13)

The two parameter generalization of the involution $\omega _{q,t}$ relates $P_{\mu }$ and $Q_{\mu }$ as

$$\begin{aligned} \omega _{q,t}\,P_{\mu }(x;q,t)&=Q_{\mu ^{t}}(x;t,q),\nonumber \\ \omega _{q,t}\,Q_{\mu }(x;q,t)&=P_{\mu ^{t}}(x;t,q). \end{aligned}$$

(A.14)

The above identities follow from the simple identity $\omega _{t,q}=\omega _{q,t}^{-1}$.

On general grounds, we can easily show that the sum over all Young diagrams takes the following product form,

$$\begin{aligned} \sum _{\lambda }P_{\lambda }(x;q,t)Q_{\lambda }(y;q,t)=\Pi (x,y;q,t), \end{aligned}$$

(A.15)

where we have,

$$\begin{aligned} \Pi (x,y;q,t)=\omega _{t,q}\, \prod _{i,j}(1+x_{i}y_{j})=\exp \left( \sum _{n=1}^{\infty }\frac{1}{n}\frac{1-t^{n}}{1-q^{n}}p_{n}(x)p_{n}(y)\right) . \end{aligned}$$

(A.16)

In [33], the dual Schur function $S_{\lambda }(x;q,t)$ is defined using the (q, t)-inner product

$$\begin{aligned} \langle s_{\lambda },S_{\mu }\rangle _{q,t}=\delta _{\lambda \mu }. \end{aligned}$$

(A.17)

The dual Schur function can also be obtained similar to the case for the Macdonald function with the help of the refined involution $\omega _{t,q}$,

$$\begin{aligned} S_{\lambda }(x;q,t)=\imath \omega _{t,q}\,s_{\lambda }(-x), \end{aligned}$$

(A.18)

and remember that $\imath$ is another involution defined by it action on the powersum symmetric function $\imath p_{\lambda }=-p_{\lambda }$. The Cauchy identity for the Schur and dual Schur function is (sum up to the same product as the Macdonald and the dual Macdonal function),

$$\begin{aligned} \sum _{\mu }s_{\mu }(x)S_{\mu }(y;q,t)=\Pi (x,y;q,t). \end{aligned}$$

(A.19)

Using Littlewood–Richardson coefficients determined by $s_{\mu }s_{\nu }=\sum _{\lambda }N_{\mu \nu }^{\lambda }s_{\lambda }$, the skew (dual) Schur function are defined by,

$$\begin{aligned}&s_{\mu /\nu }(x)=\sum _{\lambda }N_{\nu \lambda }^{\mu }s_{\lambda }(x), \end{aligned}$$

(A.20)

$$\begin{aligned}&S_{\mu /\nu }(x;q,t)=\sum _{\lambda }N_{\nu \lambda }^{\mu }S_{\lambda }(x;q,t). \end{aligned}$$

(A.21)

The dual Schur functions satisfy similar to the Schur functions,

$$\begin{aligned} S_{\mu }(x,y;q,t)=\sum _{\lambda }S_{\mu /\lambda }(x;q,t)S_{\lambda }(y;q,t). \end{aligned}$$

(A.22)

The skew (dual) Macdonald function are defined after refining the Littlewood–Richardson coefficients which we denote by ${\widehat{N}}_{\nu \eta }^{\mu }(q,t)$. There are defined similar to Schur function using the Macdonald polynomials,

$$\begin{aligned} P_{\nu }(x;q,t)P_{\eta }(x;q,t)=\sum _{\mu }{\widehat{N}}_{\nu \eta }^{\mu }(q,t)P_{\mu }(x;q,t). \end{aligned}$$

(A.23)

We list the identities required for our computations without any proofs,

$$\begin{aligned} Q_{\mu /\nu }(x;q,t)&=\sum _{\eta }{\widehat{N}}_{\nu \eta }^{\mu }(q,t)Q_{\eta }(x;q,t), \end{aligned}$$

(A.24)

$$\begin{aligned} P_{\mu /\nu }(x;q,t)&=\sum _{\eta }{\widehat{N}}_{\nu ^{t}\eta ^{t}}^{\mu ^{t}}(t,q)P_{\eta }(x;q,t). \end{aligned}$$

(A.25)

The Cauchy identities are deformed a little bit once we include the skew functions,

$$\begin{aligned}&\sum _{\lambda }P_{\lambda }(x;q,t)Q_{\lambda /\mu }(y;q,t)=P_{\mu }(x;q,t)\Pi (x,y;q,t), \end{aligned}$$

(A.26)

$$\begin{aligned}&\sum _{\lambda }P_{\lambda ^{t}/\mu }(x;t,q)P_{\lambda }(y;q,t)=P_{\mu ^{t}}(y;q,t)\prod _{i,j}(1+x_{i}y_{j}). \end{aligned}$$

(A.27)