A-Polynomial, B-Model, and Quantization

Abstract

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $\hslash \rightarrow 0$, and becomes non-commutative or “quantum” away from this limit. For a classical curve defined by the zero locus of a polynomial A(x, y), we provide a construction of its non-commutative counterpart $\hat{A}(\hat{x},\hat{y})$ using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing $\hat{A}$ that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that “come from geometry,” their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be “quantizable,” and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. The material contained in this chapter was presented at the conference Mirror Symmetry and Tropical Geometry in Cetraro (July 2011) and is based on the work: Gukov and Sułkowski, “A-polynomial, B-model, and quantization”, JHEP 1202 (2012) 070.

Appendices

Appendix 1: A Hierarchy of Differential Equations

In this appendix we provide more details on the hierarchy of differential equations (42) arising from the quantum curve equation $\hat{A}Z = 0$. This hierarchy allows to determine the quantum operator $\hat{A}$, order by order in $\hslash$, from the knowledge of the partition function Z it annihilates, or vice versa. We stress that the hierarchy (42) takes the same form for curves embedded in $\mathbb{C} \times \mathbb{C}$ or $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$, even though its derivation in both cases is much different.

We recall that, in the classical limit, we consider curves embedded either in $\mathbb{C} \times \mathbb{C}$ with coordinates (u, v), or in $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ with coordinates $(x = e^{u},y = e^{v})$. The classical curve is given by the polynomial equation

$$\displaystyle{ 0 = A \equiv A_{0}. }$$

(171)

In the quantum regime we introduce the commutation relation $[\hat{v},\hat{u}] = \hslash$ and use the representation $\hat{u} = u,\hat{v} = \hslash \partial _{u}$. For $\mathbb{C}^{{\ast}}$ coordinates we then have $\hat{x} = x = e^{u},\hat{y} = e^{\hat{v}} = e^{\hslash \partial _{u}}$ and $\hat{y}\hat{x} = q\hat{x}\hat{y}$, where $q = e^{\hslash }$. In what follows we denote derivatives w.r.t u by $' = \partial _{u} = x\partial _{x}$.

To represent the quantum curves corresponding to (171) we use the following expansions, respectively in $\mathbb{C} \times \mathbb{C}$ and $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ case

$$\displaystyle{\hat{A} =\sum _{ j=0}^{d}a_{ j}(u, \hslash )\hat{v}^{j},\qquad \quad \hat{A} =\sum _{ j=0}^{d}a_{ j}(x, \hslash )\hat{y}^{j},}$$

where, respectively,

$$\displaystyle{ a_{j}(u, \hslash ) =\sum _{ l=0}^{\infty }a_{ j,l}(u)\hslash ^{l},\qquad \quad a_{ j}(x, \hslash ) =\sum _{ l=0}^{\infty }a_{ j,l}(x)\hslash ^{l}. }$$

We also reassemble contributions of fixed $\hslash$ order into, respectively,

$$\displaystyle{ A_{l} = A_{l}(u,v) =\sum _{ j=0}^{d}a_{ j,l}(u)v^{j},\qquad \quad A_{ l} = A_{l}(x,y) =\sum _{ j=0}^{d}a_{ j,l}(x)y^{j}. }$$

(172)

Replacing classical variables in these expansions by quantum operators $\hat{u},\hat{v}$ or $\hat{x},\hat{y}$, ordered such that $\hat{v}$ or $\hat{y}$ appear to the right of $\hat{u}$ or $\hat{x}$, defines corrections $\hat{A}_{l}$ to the quantum curve (6). Using the above notation, the quantum curve equation can be written, respectively in $\mathbb{C} \times \mathbb{C}$ and $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ case, as

$$\displaystyle{ \hat{A}Z(u) =\Big (\sum _{j=0}^{d}a_{ j}(u, \hslash )\hat{v}^{j}\Big)Z(u) = 0,\qquad \hat{A}Z(x) =\Big (\sum _{ j=0}^{d}a_{ j}(x, \hslash )\hat{y}^{j}\Big)Z(x) = 0, }$$

(173)

where

$$\displaystyle{ Z =\exp \Big (\frac{1} {\hslash }\sum _{k=0}^{\infty }\hslash ^{k}S_{ k}\Big). }$$

(174)

1.1 Hierarchy in the $\mathbb{C}^{{\ast}}$ Case: q-Difference Equation

The quantum curve equation gives rise to a hierarchy of differential equations which arise as follows. Substituting the partition function (174) into (173) and dividing by $e^{\hslash ^{-1}S_{ 0}}$ results in

$$\displaystyle{ 0 =\sum _{ j,l=0}^{\infty }a_{ j,l}\hslash ^{l}e^{jS'_{0} }\exp \Big(\sum _{n=1}^{\infty }\hslash ^{n}\mathfrak{d}_{ n}(j)\Big), }$$

(175)

where $\mathfrak{d}_{n}(j)$ combine terms with a fixed power of $\hslash$ in the expansion of $\sum _{k}\hslash ^{k}S_{k}\big(e^{u+j\hslash }\big)$

$$\displaystyle{ \mathfrak{d}_{n}(j) =\sum _{ r=1}^{n+1}\frac{j^{r}} {r!} S_{n+1-r}^{(r)}(x). }$$

(176)

For example

$$\displaystyle\begin{array}{rcl} \mathfrak{d}_{1}(j)& =& \frac{j^{2}} {2} S''_{0} + jS'_{1}, {}\\ \mathfrak{d}_{2}(j)& =& \frac{j^{3}} {6} S'''_{0} + \frac{j^{2}} {2} S''_{1} + jS'_{2}, {}\\ \mathfrak{d}_{3}(j)& =& \frac{j^{4}} {4!} S_{0}^{(4)} + \frac{j^{3}} {3!} S'''_{1} + \frac{j^{2}} {2} S''_{2} + jS'_{3}, {}\\ \end{array}$$

and note that for each n we have $\mathfrak{d}_{n}(0) = 0$. Let us now expand the exponent in (175) and collect terms with fixed power of $\hslash$

$$\displaystyle{ \exp \Big(\sum _{n=1}^{\infty }\hslash ^{n}\mathfrak{d}_{ n}(j)\Big) =\sum _{ r=0}^{\infty }\hslash ^{r}\mathfrak{D}_{ r}(j), }$$

(177)

so that, for example,

$$\displaystyle\begin{array}{rcl} \mathfrak{D}_{0}(j)& =& 1, {}\\ \mathfrak{D}_{1}(j)& =& \mathfrak{d}_{1}(j) = \frac{S''_{0}} {2} j^{2} + S'_{ 1}j, {}\\ \mathfrak{D}_{2}(j)& =& \mathfrak{d}_{2}(j) + \frac{1} {2}\mathfrak{d}_{1}(j)^{2} = \frac{(S''_{0})^{2}} {8} j^{4} + \frac{1} {6}\big(S'''_{0} + 3S''_{0}S'_{1}\big)j^{3} {}\\ & & +\,\frac{1} {2}\big(S''_{1} + (S'_{1})^{2}\big)j^{2} + S'_{ 2}j, {}\\ \mathfrak{D}_{3}(j)& =& \mathfrak{d}_{3}(j) + \mathfrak{d}_{1}(j)\mathfrak{d}_{2}(j) + \frac{1} {6}\mathfrak{d}_{1}(j)^{3} {}\\ & =& \frac{(S''_{0})^{3}} {48} j^{6} +\Big (\frac{S''_{0}S'''_{0}} {12} + \frac{(S''_{0})^{2}S'_{1}} {8} \Big)j^{5} {}\\ & & +\, \frac{1} {24}\big(S''''_{0} + 6S''_{0}S''_{1} + 4S'''_{0}S'_{1} + 6S''_{0}(S'_{1})^{2}\big)j^{4} + {}\\ & & +\frac{1} {6}\big(3S''_{1}S'_{1} + (S'_{1})^{3} + S'''_{ 1} + 3S''_{0}S'_{2}\big)j^{3} +\big (\frac{S''_{2}} {2} + S'_{1}S'_{2})j^{2} + S'_{ 3}j, {}\\ \mathfrak{D}_{4}(j)& =& \mathfrak{d}_{4}(j) + \mathfrak{d}_{1}(j)\mathfrak{d}_{3}(j) + \frac{1} {2}\mathfrak{d}_{2}(j)^{2} {}\\ & & +\frac{1} {2}\mathfrak{d}_{1}(j)^{2}\mathfrak{d}_{ 2}(j) + \frac{1} {4!}\mathfrak{d}_{1}(j)^{4} {}\\ & =& \frac{(S''_{0})^{4}} {384} j^{8} + \frac{1} {48}\big((S''_{0})^{2}S'''_{ 0} + (S''_{0})^{3}S'_{ 1}\big)j^{7} {}\\ & & +\ldots + \frac{1} {2}\big((S'_{2})^{2} + S''_{ 3} + 2S'_{1}S'_{3}\big)j^{2} + S'_{ 4}j. {}\\ \end{array}$$

Finally, expanding (175) in total power of $\hslash$ and collecting terms with a fixed such power $\hslash ^{n}$, gives rise to a hierarchy of differential equations

$$\displaystyle{ 0 =\sum _{j}e^{jS'_{0} }\sum _{r=0}^{n}a_{ j,r}\mathfrak{D}_{n-r}(j). }$$

(178)

Now we use the fact that the disk amplitude in $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ case is $S_{0} =\int \log (y)\frac{\mathit{dx}} {x}$, so S′₀ = log(y). Therefore $e^{jS'_{0}} = y^{j}$ and we can write (178) in terms of corrections A _k to the quantum curve (172). In particular the first equation in the hierarchy $0 =\sum _{ j=0}^{d}a_{j,0}y^{j} = A_{0}(x,y)$ coincides with the classical curve equation (171). Now, writing $\mathfrak{D}_{n-r}(j) =\sum _{m}\mathfrak{D}_{n-r,m}j^{m}$, we can rewrite (178) as

$$\displaystyle\begin{array}{rcl} 0& =& \sum _{r=0}^{n}\sum _{ j,m}a_{j,r}\mathfrak{D}_{n-r,m}j^{m}y^{j} =\sum _{ r=0}^{n}\sum _{ j,m}a_{j,r}\mathfrak{D}_{n-r,m}(y\partial _{y})^{m}y^{j} {}\\ & =& \sum _{r=0}^{n}\Big(\sum _{ m}\mathfrak{D}_{n-r,m}(y\partial _{y})^{m}\Big)A_{ r}. {}\\ \end{array}$$

The expression in the last bracket is nothing but the operator $\mathfrak{D}_{n-r}(j)$ from (177) with all j replaced by y∂ _y  = ∂ _v . Therefore we denote this operators by $\mathfrak{D}_{n-r}(\partial _{v})$, or simply $\mathfrak{D}_{n-r}$; for example

$$\displaystyle{\mathfrak{D}_{1} = \frac{S''_{0}} {2} (y\partial _{y})^{2} + S'_{ 1}(y\partial _{y}),}$$

etc. In terms of these new operators, the hierarchy of Eqs. (178) takes a particularly simple form

$$\displaystyle{ 0 =\sum _{ r=0}^{n}\mathfrak{D}_{ n-r}A_{r}, }$$

(179)

as advertised in (42), and with $\mathfrak{D}_{n-r}$ defined as in (177) with j replaced by ∂ _v .

1.2 Hierarchy in the $\mathbb{C}$ Case: Differential Equation

Now we show that the hierarchy of equations which arises for curves in $\mathbb{C} \times \mathbb{C}$ takes the same form (42) as in $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ case, even though the explicit derivation of this hierarchy is much different. Now Eq. (173) takes a form

$$\displaystyle{ 0 =\hat{ A}Z(u) =\sum _{ j=0}^{d}\sum _{ l=0}^{\infty }a_{ j,l}\hslash ^{l+j}\partial _{ u}^{j}Z(u), }$$

and by induction we find that the last term can be written as $\partial _{u}^{j}Z = Z(\partial _{u} + S')^{j}S'$. Then the factor of Z can be factored out of an entire expression, which results in

$$\displaystyle{ 0 =\sum _{ l=0}^{\infty }\Big[a_{ 0,l}\hslash ^{l} +\sum _{ j=0}^{d-1}a_{ j+1,l}\hslash ^{l}\Big(\hslash \partial _{ u} +\sum _{ k=0}^{\infty }\hslash ^{k}S'_{ k}\Big)^{j}\sum _{ r=0}^{\infty }\hslash ^{r}S'_{ r}\Big]. }$$

(180)

Recalling that S′₀ = v, an explicit computation reveals that the last term in this expression can be written as

$$\displaystyle{ \big(\hslash \partial _{u} + \hslash S'\big)^{j}\hslash S' = v^{j+1} + \hslash \Big(S''_{ 0}\frac{j(j + 1)} {2} v^{j-1} + S'_{ 1}(j + 1)v^{j}\Big)+ }$$

(181)

$$\displaystyle{+\hslash ^{2}\Big((S''_{ 0})^{2}\frac{(j - 2)(j - 1)j(j + 1)} {8} v^{j-3} +\big (S'''_{ 0} + 3S''_{0}S'_{1}\big)\frac{(j - 1)j(j + 1)} {6} v^{j-2}+}$$

$$\displaystyle{+\big(S''_{1} + (S'_{1})^{2}\big)\frac{j(j + 1)} {2} v^{j-1} + S'_{ 2}(j + 1)v^{j}\Big) + \mathcal{O}(\hslash ^{3}) =}$$

$$\displaystyle\begin{array}{rcl} & =& \Big[1 + \hslash \Big(\frac{S''_{0}} {2} \partial _{v}^{2} + S'_{ 1}\partial _{v}\Big) + \hslash ^{2}\Big(\frac{(S''_{0})^{2}} {8} \partial _{v}^{4} + \frac{S'''_{0} + 3S''_{0}S'_{1}} {6} \partial _{v}^{3} + \frac{S''_{1} + (S'_{1})^{2}} {2} \partial _{v}^{2} {}\\ & & +\,S'_{2}\partial _{v}\Big) + \mathcal{O}(\hslash ^{3})\Big]v^{j+1}. {}\\ \end{array}$$

We see that a coefficient at each power $\hslash ^{r}$ above is nothing but $\mathfrak{D}_{r}$ introduced in (179), i.e. the operator defined in (177) with j replaced by ∂ _v . Therefore

$$\displaystyle{\big(\hslash \partial _{u} + \hslash S'\big)^{j}\hslash S' =\sum _{ r=0}^{\infty }\hslash ^{r}\mathfrak{D}_{ r}.}$$

Using a definition A _r from (172) we find that (180) takes form

$$\displaystyle{0 =\sum _{r,l=0}\sum _{j=0}^{d}a_{ j,l}\hslash ^{l}\hslash ^{r}\mathfrak{D}_{ r}v^{j} =\sum _{ r,l}\hslash ^{r+l}\mathfrak{D}_{ r}A_{l} =\sum _{ n=0}^{\infty }\hslash ^{n}\Big(\sum _{ r=0}^{n}\mathfrak{D}_{ n-r}A_{r}\Big).}$$

Therefore at order $\hslash ^{n}$ we get

$$\displaystyle{ 0 =\sum _{ r=0}^{n}\mathfrak{D}_{ n-r}A_{r}, }$$

(182)

with $\mathfrak{D}_{n-r}$ defined as in (177) with j replaced by ∂ _v . This is the same equation as in $\mathbb{C}^{{\ast}}\times \mathbb{C}^{{\ast}}$ case (179), and as already advertised in (42).

Appendix 2: Quantum Dilogarithm

In literature several representations of quantum dilogarithm can be found. We use the following one

$$\displaystyle\begin{array}{rcl} \psi (x)& =& \prod _{k=1}^{\infty }(1 - xe^{\hslash (k-1/2)})^{-1} = \\ & =& \exp \Big(-\sum _{k=1}^{\infty } \frac{x^{k}} {k(e^{\hslash k/2} - e^{-\hslash k/2})}\Big) = \\ & =& \sum _{k=0}^{\infty }x^{k}e^{\frac{\hslash k} {2} }\prod _{i=1}^{k} \frac{1} {1 - e^{i\hslash }}, {}\end{array}$$

(183)

which has the following “genus expansion”

$$\displaystyle\begin{array}{rcl} \log \psi (x)& =& \frac{1} {\hslash }S_{0}(x) + S_{1}(x) + \hslash S_{2}(x) + \hslash ^{2}S_{ 3}(x) + \hslash ^{3}S_{ 4}(x) + \hslash ^{4}S_{ 5}(x) +\ldots \\ & \equiv & -\frac{1} {\hslash }\mathrm{Li}_{2}(x) + \frac{\hslash } {24}\mathrm{Li}_{0}(x) - \frac{7\hslash ^{3}} {5760}\mathrm{Li}_{-2}(x) \\ & & +\, \frac{31\hslash ^{5}} {967680}\mathrm{Li}_{-4}(x)+\ldots = {}\end{array}$$

(184)

$$\displaystyle\begin{array}{rcl} & =& \sum _{k=0}^{\infty }\hslash ^{k-1}(1 - 2^{1-k})\frac{B_{k}} {k!} \mathrm{Li}_{2-k}(x)\,.{}\end{array}$$

(185)

Note, all terms with even power of $\hslash$ vanish. For terms $\sim \hslash ^{k-1}B_{k}$ with k = 3, 5, 7, … this is so, because $B_{3} = B_{5} = B_{7} =\ldots = 0$. On the other hand, the term with k = 1 is proportional to $(1 - 2^{1-1}) = 0$, hence it vanishes as well. Further details can be found e.g. in [21].