On $R$-matrix formulation of $qq$-characters

Quantum deformation is one of the central themes of modern ‘‘physical mathematics’’. In particular, quantum deformations of Lie algebras [1, 2] were discovered in connection with lattice integrable models in statistical physics (see e.g. the classic book [3]), but soon found applications in many other areas such as knot theory [4], supersymmetric gauge theories [5], [6], and $2d$ conformal field theory (CFT) [7]–[9].

To capture the intricacies of the representation theory of quantum affine Lie algebras $U_{q}(\widehat{\mathfrak{g}})$ Knight [10] and Frenkel-Reshetikhin [11] have introduced the notion of $q$-characters which can be thought of as generalizations of conventional characters with more parameters. The principal ingredient in the construction of $q$-character is the $R$-matrix of $U_{q}(\widehat{\mathfrak{g}})$. The $q$-characters are partial traces of the $R$-matrix taken in finite-dimensional representations, so that algebraic properties of the $R$-matrix imply identities for the $q$-characters.

It also turned out that there is a second quantum deformation of $q$-characters involving parameter $t$, called $qq$-characters, natural from both geometric [12] and algebraic [11] points of view. The deformation turns the parameters in the character into non-commutative operators so that the characters themselves generate a nontrivial algebra — the $(q,t)$-deformed $W$-algebra, $W_{q,t}(\mathfrak{g})$.

It would be desirable to have an $R$-matrix construction of $qq$-characters similar to that of $q$-characters. Some attempts along these lines have been made in [13], but that paper dealt exclusively with a specific (and complicated) case of $\mathfrak{g}=\widehat{\mathfrak{gl}}_{1}$. In the current paper we propose a more general $R$-matrix formalism for $qq$-characters, which to our knowledge has not been suggested previously.

There are some parallels between our formalism and that for $q$-characters: we also use an $R$-matrix as a central tool. However, the details differ a lot. Firstly, we use one and the same ‘‘master’’ $R$-matrix to obtain all the $qq$-characters for the root systems $\mathfrak{g}=A_{n}$. The $R$-matrix is that of the Ding-Iohara-Miki (DIM) [14] or quantum toroidal algebra $\mathcal{A}=U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1})$. In this sense both the root system $\mathfrak{g}$ and the representation featuring in the $qq$-character are free parameters that can be varied in our approach without changing the underlying algebraic structure.

The second important difference between our formalism and [11] is that we don’t take a trace of our $R$-matrix, but instead a (partial) matrix element thereof. The states in the matrix element are what encodes the representation featuring in the $qq$-character.

The technical framework that we employ to derive the expressions for $qq$-characters is refined topological string. This allows us to interpret various algebraic expressions as refined amplitudes on certain toric Calabi-Yau (CY) three-folds. In fact the toric diagram of the three-fold can be viewed as a lattice statistical model [15] with crossings corresponding to DIM $R$-matrices [16].

Let us note that ours is not the first attempt to obtain $qq$-characters in the refined topological string framework. In [17] (see also [18]) $qq$-characters were identified with specific generators of the algebra $\mathcal{A}$ inserted between combinations of refined topological vertices. However, the generators were not viewed as coming from an $R$-matrix. In [19] $qq$-characters were engineered by certain linear combinations of Lagrangian brane insertions, but no algebraic interpretation was given. In contrast, in the present paper we provide an explicit algorithm to draw a toric diagram whose refined partition function produces a given $qq$-character.

In the remaining part of the introduction we recall briefly the $R$-matrix construction of $q$-characters of quantum affine Lie algebras [11] (sec. 1.1), and then introduce the general idea of our approach to $qq$-characters and their connection with Type IIB 5-branes and refined topological strings (sec. 1.2). In this way we demonstrate both the similarities as well as differences between the two approaches before delving into full technical description.

The rest of the paper is organized as follows. In sec. 2 we introduce the DIM $R$-matrix which can be interpreted as a spectator brane insertion. In sec. 3 we show how the $R$-matrix formulas give rise to the coproduct of the generating current of the DIM algebra and match it with some previous works on the relation between DIM algebra and $qq$-characters. In sec. 4 we show how $qq$-characters are obtained from the $R$-matrix insertion using Wick’s theorem. We apply our formalism to fundamental $A_{n}$ type $qq$-characters in sec. 5 and to higher representations of $A_{n}$ in sec. 6. In sec. 7 we consider the uplift of the formalism to elliptic $A_{n}$ $qq$-characters. We present our conclusions in sec. 8.

In this section we give a more technical description of DIM $R$-matrices and provide an explicit formula for the crossing operator introduced in Eq. (8). According to [16], the $R$-matrix for the tensor product of vertical and horizontal Fock spaces can be written as a composition of two intertwining operators, or refined topological vertices (see Appendix A for definitions of the Fock representations).

The formulas for the intertwiners were obtained in [25]. Up to prefactors they are given by the following vertex operators (see Eqs. (138), (139) in Appendix C for complete expressions):

$$\Psi^{\lambda}(x)=\quad\includegraphics[valign={c}]{figures/top-vert-crop}\quad\\ \sim\,:\mathrel{\mkern 2.0mu\exp\left[\sum_{n\neq 0}\frac{x^{-n}}{n}\left(\frac{1}{1-q^{-n}}-(1-t^{n})\mathrm{Ch}_{\lambda}(q^{-n},t^{-n})\right)a_{n}\right]\mkern 2.0mu}:,$$

(13)

$$\Psi^{*}_{\mu}(y)=\quad\includegraphics[valign={c}]{figures/top-vert-conj-crop}\\ \sim\,:\mathrel{\mkern 2.0mu\exp\left[-\sum_{n\neq 0}\frac{y^{-n}}{n}\left(\frac{t}{q}\right)^{\frac{|n|}{2}}\left(\frac{1}{1-q^{-n}}-(1-t^{n})\mathrm{Ch}_{\mu}(q^{-n},t^{-n})\right)a_{n}\right]\mkern 2.0mu}:,$$

(14)

where

and the free boson modes $a_{n}$ satisfy the $(q,t)$-deformed Heisenberg commutation relations:

Combining the intertwiners Eqs. (13), (14) as drawn in Eq. (8) we get the (the matrix element of) the $R$-matrix [26]:

$$\mathcal{R}^{\lambda}_{\mu}(w,u,N)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Psi^{\lambda}(w)\Psi_{\mu}\left(\sqrt{\frac{t}{q}}w\right)=\includegraphics[valign={c}]{figures/d5-ns5-simpl-selection-2-crop}\quad\\ =(-w)^{-N|\lambda|}u^{|\lambda|}\left(-\sqrt{\frac{t}{q}}w\right)^{N|\mu|}\left(\frac{q}{u}\sqrt{\frac{t}{q}}\right)^{|\mu|}N_{\lambda\mu}\left(\frac{q}{t}\right)\frac{q^{n(\lambda^{\mathrm{T}})+n(\mu^{\mathrm{T}})}}{f_{\lambda}c_{\lambda}c_{\mu}}\left(\frac{f_{\mu}}{f_{\lambda}}\right)^{N}\\ \times:\!\exp\left[-\sum_{n\geq 1}\frac{w^{n}}{n}\frac{\left(1-\left(t/q\right)^{n}\right)}{(1-q^{n})}a_{-n}\right]\times\\ \times\exp\left[-\sum_{n\neq 0}\frac{(1-t^{n})}{n}\left(\mathrm{Ch}_{\lambda}(q^{-n},t^{-n})-\left(\frac{t}{q}\right)^{\frac{|n|-n}{2}}\mathrm{Ch}_{\mu}(q^{-n},t^{-n})\right)w^{-n}a_{n}\right]\!:$$

(17)

where $N_{\lambda\mu}(x)$ is given by Eq. (B) and

Several remarks are in order:

1. 1.

   Eq. (17) is a generalization of Eq. (8) in which the horizontal line ($(1,0)$ Fock representation in Eq. (8)) has a nontrivial slope $(1,N)$, $N\in\mathbb{Z}$.

2. 2.

   DIM algebra has an infinite number of inequivalent coproducts $\Delta^{(s)}$ labelled by an irrational slope $s$ in the $\mathbb{Z}^{2}$ lattice. Each of the coproducts gives rise to its own set of intertwining operators and $R$-matrices. In all of the formulas in this paper the choice of DIM coproduct (also known as the preferred direction in refined topological strings) is understood to be vertical, $s=\infty$.

3. 3.

   The exponent in the third line of Eq. (17) was obtained in [16] as the vacuum matrix element of the $R$-matrix. The rest of the formula is responsible for its generalization to arbitrary states on the vertical brane.

4. 4.

   $\mathcal{R}^{\lambda}_{\mu}$ satisfies the selection rule (9) due to the vanishing of the Nekrasov factor $N_{\lambda\mu}(q/t)$ in the second line of Eq. (17).

5. 5.

   The additional terms that appear in $\mathcal{R}^{\lambda}_{\mu}$ for nontrivial $\lambda$ and $\mu$ have the form of a product of vertex operators sitting at points corresponding to boxes of $\lambda$ and $\mu$, respectively.

6. 6.

   We will sometimes omit the $u$ and $N$ arguments in $\mathcal{R}^{\lambda}_{\mu}(w,u,N)$ when there is no possibility for confusion.

The operator $\mathcal{R}^{\lambda}_{\mu}$ looks complicated, however in our present study we will need to evaluate it only for $\lambda$, $\mu$ equal to $\square$ or $\varnothing$. Due to the selection rule Eq. (9) there are three cases of this type:

1. 1.

   $\lambda=\varnothing$, $\mu=\varnothing$:

   $$\mathcal{R}^{\varnothing}_{\varnothing}(w,u,N)=\exp\left[-\sum_{n\geq 1}\frac{w^{n}}{n}\frac{\left(1-\left(t/q\right)^{n}\right)}{(1-q^{n})}a_{-n}\right]$$

   (22)

   It is important that the exponent in Eq. (22) contains only negative bosonic modes, so it acts trivially on the bra vacuum state $\langle\varnothing|$.

2. 2.

   $\lambda=\square$, $\mu=\varnothing$:

   $$\mathcal{R}^{\square}_{\varnothing}(w,u,N)=-\sqrt{\frac{t}{q}}\frac{1-\frac{q}{t}}{1-t}\mathcal{R}^{\varnothing}_{\varnothing}(w,u,N)\,x^{+}(w)|_{\mathcal{F}^{(1,N)}(u)},$$

   (23)

   where $x^{+}(w)|_{\mathcal{F}^{(1,N)}(u)}$ is the generating current of the DIM algebra in the $(1,N)$ Fock representation given by Eq. (114) in Appendix A. Notice that the product $\mathcal{R}^{\varnothing}_{\varnothing}(w)\,x^{+}(w)$ is automatically normal ordered. We will use this fact when we compute correlators in sec. 4.

3. 3.

   $\lambda=\square$, $\mu=\square$:

   $$\mathcal{R}^{\square}_{\square}(w,u,N)=\frac{1-q}{1-t}\psi^{-}\left(\left(t/q\right)^{\frac{1}{4}}w\right)|_{\mathcal{F}^{(1,N)}(u)}\mathcal{R}^{\varnothing}_{\varnothing}(w,u,N),$$

   (24)

   where $\psi^{-}(w)$ is another generating current of the DIM algebra given by Eq. (117) in Appendix A. The product in Eq. (24) is also automatically normal ordered.

Having the basic ingredients — the crossings (22), (23) and (24) — we can proceed to combine them. Naturally, there are two ways:

1. 1.

   One can stack crossings on top of each other, so that a single vertical line crosses a number of horizontal ones. This will produce the $qq$-characters of the first fundamental representation $\mathbb{C}^{n}$ of $A_{n-1}$ which we investigate in sec. 3.

2. 2.

   The crossings can be joined along the horizontal legs. This will produce $qq$-characters of higher representations as described in sec. 6.

Consider a vertical spectator brane intersecting $n$ horizontal ones as shown in Eq. (10) with external state $|\square\rangle$ on top and $\langle\varnothing|$ at the bottom. Every brane intersection gives rise to the $R$-matrix operator Eq. (17). We claim that such an intersection produces the $qq$-character of the defining representation $\mathbb{C}^{n}$ of $A_{n-1}$ algebra.

To simplify our presentation in this section we restrict ourselves to the case when the branes intersecting the vertical brane are strictly horizontal, i.e. correspond to Fock representations of type $(1,0)$. However, the same arguments work for $(1,N)$ lines too. Let us follow the example of Eq. (10), where there are three horizontal lines and two diagrams on the intermediate edges $\lambda$ and $\mu$.

As we have mentioned in sec. 1.2, selection rules severely restrict the set of Young diagrams on the intermediate vertical edges. There are three possible pairs $(\lambda,\mu)$ contributing to the answer. Let us write down the operators corresponding to each of these pairs:

1. 1.

   $(\lambda,\mu)=(\varnothing,\varnothing)$:

   $$\includegraphics[valign={c}]{figures/crossing-n-00-crop}\quad=\quad\mathcal{R}^{\square}_{\varnothing}(w,u_{1},0)\otimes\mathcal{R}^{\varnothing}_{\varnothing}\left(\sqrt{\frac{t}{q}}w,u_{2},0\right)\otimes\mathcal{R}^{\varnothing}_{\varnothing}\left(\frac{t}{q}w,u_{1},0\right)$$

   (25)

   which can alternatively be expressed using Eq. (23) as

   	$\displaystyle-\sqrt{\frac{t}{q}}\frac{1-\frac{q}{t}}{1-t}\mathcal{R}^{\varnothing}_{\varnothing}(w,u_{1},0)\otimes\mathcal{R}^{\varnothing}_{\varnothing}\left(\sqrt{\frac{t}{q}}w,u_{2},0\right)\otimes\mathcal{R}^{\varnothing}_{\varnothing}\left(\frac{t}{q}w,u_{1},0\right)(x^{+}(w)|_{\mathcal{F}(u_{1})}\otimes 1\otimes 1)=$
   	$\displaystyle=\mathcal{T}^{\varnothing}_{\varnothing}(w)(x^{+}(w)|_{\mathcal{F}(u_{1})}\otimes 1\otimes 1)=u_{1}\mathcal{T}^{\varnothing}_{\varnothing}(w):\mathrel{\mkern 2.0mu\exp\left[-\sum_{n\neq 0}\frac{1-t^{n}}{n}w^{-n}a_{n}^{(1)}\right]\mkern 2.0mu}:,$		(26)

   where $a^{(i)}_{n}$, $i=1,2,3$ are three sets of bosonic operators acting on three horizontal lines and the operator $x^{+}(w)$ is defined in Eq. (114). The ‘‘empty crossing’’ part is given by

   $$\mathcal{T}^{\varnothing}_{\varnothing}(w)=-\sqrt{\frac{t}{q}}\frac{1-\frac{q}{t}}{1-t}\exp\left[-\sum_{n\geq 1}\frac{w^{n}}{n}\frac{1-\left(t/q\right)^{n}}{1-q^{n}}\bar{a}_{-n}\right].$$

   (27)

   where we have defined the ‘‘diagonal part’’ of the bosonic modes

   $$\bar{a}_{-n}=a_{-n}^{(1)}+\left(\frac{t}{q}\right)^{\frac{n}{2}}a_{-n}^{(2)}+\left(\frac{t}{q}\right)^{n}a_{-n}^{(3)}.$$

   (28)

   Notice that $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ has a natural algebraic meaning of its own: it corresponds to the crossing with a vertical spectator brane, but with empty external states on it. In that case the selection rule (9) dictates that the Young diagrams on the intermediate edges should all be empty. Therefore the operator $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ factorizes into a product of operators each acting on its own horizontal brane (see sec. 4.3 of [16]).

2. 2.

   $(\lambda,\mu)=(\square,\varnothing)$:

   $$\includegraphics[valign={c}]{figures/crossing-n-10-crop}\quad=\quad b_{\square}(q,t)\,\mathcal{R}^{\square}_{\square}(w,u_{1},0)\otimes\mathcal{R}^{\square}_{\varnothing}\left(\sqrt{\frac{t}{q}}w,u_{2},0\right)\otimes\mathcal{R}^{\varnothing}_{\varnothing}\left(\frac{t}{q}w,u_{1},0\right),$$

   (29)

   which can be similarly expressed in terms of the operator $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ and another operator acting on the tensor product of three horizontal Fock spaces,

   $$\text{Eq. }\eqref{eq:25}=b_{\square}(q,t)\mathcal{T}^{\varnothing}_{\varnothing}(w)\left(\left.\psi^{-}\left(\left(t/q\right)^{\frac{1}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{1})}\otimes\left.x^{+}\left(\sqrt{\frac{t}{q}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{2})}\otimes 1\right)=\\ =b_{\square}(q,t)u_{2}\mathcal{T}^{\varnothing}_{\varnothing}(w):\mathrel{\mkern 2.0mu\exp\left[\sum_{n\geq 1}\frac{1-t^{-n}}{n}\left(1-\left(\frac{t}{q}\right)^{n}\right)w^{n}a_{-n}^{(1)}-\sum_{n\neq 0}\frac{1-t^{n}}{n}w^{-n}\left(\frac{q}{t}\right)^{\frac{n}{2}}a_{n}^{(2)}\right]\mkern 2.0mu}:,$$

   (30)

   where

   $$b_{\mu}(q,t)=\langle P_{\mu}|P_{\mu}\rangle^{-1}=\prod_{(i,j)\in\lambda}\frac{1-q^{\lambda_{i}-j}t^{\lambda_{j}^{\mathrm{T}}-i+1}}{1-q^{\lambda_{i}-j+1}t^{\lambda_{j}^{\mathrm{T}}-i}}=\frac{c_{\lambda}}{c^{\prime}_{\lambda}}$$

   (31)

   is the inverse of the norm of Macdonald polynomial corresponding to the intermediate state on the vertical brane ($b_{\square}=\frac{1-t}{1-q}$) and $\psi^{-}(w)$ is defined in Eq. (117).

3. 3.

   $(\lambda,\mu)=(\square,\square)$:

   $$\includegraphics[valign={c}]{figures/crossing-n-11-crop}\quad=\quad b_{\square}(q,t)^{2}\,\mathcal{R}^{\square}_{\square}(w,u_{1},0)\otimes\mathcal{R}^{\square}_{\square}\left(\sqrt{\frac{t}{q}}w,u_{2},0\right)\otimes\mathcal{R}^{\square}_{\varnothing}\left(\frac{t}{q}w,u_{1},0\right),$$

   (32)

   which can be rewritten as

   $$b_{\square}^{2}\mathcal{T}^{\varnothing}_{\varnothing}(w)\left(\left.\psi^{-}\left(\left(t/q\right)^{\frac{1}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{1})}\otimes\left.\psi^{-}\left(\left(t/q\right)^{\frac{3}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{2})}\otimes\left.x^{+}\left(\frac{t}{q}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{3})}\right)=\\ =b_{\square}^{2}u_{3}\mathcal{T}^{\varnothing}_{\varnothing}(w):\mathrel{\mkern 2.0mu\exp\left[\sum_{n\geq 1}\frac{1-t^{-n}}{n}\left(1-\left(\frac{t}{q}\right)^{n}\right)w^{n}\left(a_{-n}^{(1)}+\left(\frac{t}{q}\right)^{\frac{n}{2}}a_{-n}^{(2)}\right)-\sum_{n\neq 0}\frac{1-t^{n}}{n}w^{-n}\left(\frac{q}{t}\right)^{n}a_{n}^{(3)}\right]\mkern 2.0mu}:.$$

   (33)

Collecting the terms for each pair of intermediate diagrams $(\lambda,\mu)$ we get the following result for the crossing:

$$\includegraphics[valign={c}]{figures/crossing-n-lm-crop}\quad=\quad\mathcal{T}^{\varnothing}_{\varnothing}(w)\Biggl{\{}x^{+}(w)|_{\mathcal{F}(u_{1})}\otimes 1\otimes 1+\\ +\left.\psi^{-}\left(\left(t/q\right)^{\frac{1}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{1})}\otimes\left.x^{+}\left(\sqrt{\frac{t}{q}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{2})}\otimes 1+\\ +\left.\psi^{-}\left(\left(t/q\right)^{\frac{1}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{1})}\otimes\left.\psi^{-}\left(\left(t/q\right)^{\frac{3}{4}}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{2})}\otimes\left.x^{+}\left(\frac{t}{q}w\right)\right|_{\mathcal{F}^{(1,0)}(u_{3})}\Biggr{\}}=\\ =\boxed{\mathcal{T}^{\varnothing}_{\varnothing}(w)\left.(\Delta\otimes 1)\Delta(x^{+}(w))\right|_{\mathcal{F}^{(1,0)}(u_{1})\otimes\mathcal{F}^{(1,0)}(u_{2})\otimes\mathcal{F}^{(1,0)}(u_{3})}}$$

(34)

where in the last line we have used the formula for the coproduct of the generating current $x^{+}(w)$ which can be found in Eq. (107) in Appendix A.

At this point we would like to use some of the results of [17] summarized in Appendix A. It was shown there that the insertion of the current $x^{+}(w)$ of the algebra $\mathcal{A}$ acting in the tensor product of $m$ horizontal Fock representations $\mathcal{F}^{(1,0)}_{q,t^{-1}}(u_{i})$ generates the fundamental $qq$-character of $A_{m-1}$ type. In Eq. (34) we get almost the same result from the vertical brane insertion, but with an additional operator $\mathcal{T}^{\varnothing}_{\varnothing}(w)$. This factor, however, will not affect most of the formulas that we get for the following reason. We will consider the insertion of the $R$-matrix/$qq$-character into some ‘‘background’’ network of intertwining operators (refined topological vertices). As we will see in sec. 4 the extra operator $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ has very simple commutation relations with refined topological vertices placed elsewhere in the diagram. This will allow us to effectively get rid of it in any given $qq$-character calculation by moving $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ to the very left of the diagram, where it annihilates the bra vacuum state. Thus, we will dismiss the $\mathcal{T}^{\varnothing}_{\varnothing}(w)$ in most of the formulas.

To understand why the coproduct $(\Delta\otimes 1)\Delta(x^{+}(w))$ appears in Eq. (34) we have to recall the Khoroshkin-Tolstoy formula [20] for the universal $\mathcal{R}$-matrix. The formula has the form:

where $P$ is the permutation operator exchanging the two factors in the tensor product. The other three factors are

Let us explain the notations in Eqs. (36)–(38) (see Appendix A for basic definitions related to the DIM algebra):

1. 1.

   $e_{(n,m)}$ are generators of the DIM algebra,

2. 2.

   $c_{1}$, $c_{2}$ are the two central charges of the algebra,

3. 3.

   $d_{1}$, $d_{2}$ are the grading operators,

4. 4.

   $\kappa_{n}=(1-q^{n})(1-t^{-n})(1-(t/q)^{n})$,

5. 5.

   $x^{\pm}(z)=\sum_{n\in\mathbb{Z}}e_{(\pm 1,n)}z^{\mp n}$ are DIM generating currents.

6. 6.

   In Eq. (38) the terms replaced by the ellipsis involve DIM generators $e_{(n,m)}$ with $|n|\geq 2$. As we will see momentarily these terms will not contribute to the crossing operator (34) and therefore to the $qq$-character.

The universal $R$-matrix (35) satisfies the fundamental identity

where $\Delta$ is the coproduct given by Eqs. (109)–(111) and $\mathcal{R}_{12}$ (resp. $\mathcal{R}_{13}$) represents $\mathcal{R}$ acting on the first two (res. first and third) factors in the triple tensor product.

It is the property in Eq. (39) that explains the appearance of the coproduct of $x^{+}(w)$ in Eq. (34). Indeed, consider $\mathcal{R}$ acting in the tensor product of two representations: one is a vertical Fock space $\mathcal{F}^{(0,1)}_{q,t^{-1}}(w)$ and the other is itself a tensor product of three horizontal Fock spaces $\mathcal{F}^{(1,0)}_{q,t^{-1}}(u_{1})\otimes\mathcal{F}^{(1,0)}_{q,t^{-1}}(u_{2})\otimes\mathcal{F}^{(1,0)}_{q,t^{-1}}(u_{3})$ (see Appendix A for the definitions of the relevant Fock representations). Due to the identity (39) the $R$-matrix acting on the tensor product is

Let us compute the matrix element of the two sides of Eq. (40) between the states $\langle\varnothing|$ and $|\square\rangle$ in the vertical Fock representation. The r.h.s. gives the diagram in Eq. (34) while the l.h.s. can be evaluated using the formulas (36)–(38):

Using explicit formulas (109)–(113) for the coproduct and the Fock representations from Appendix A we find that the second line of Eq. (41) is given by

Notice that the product $P\mathcal{R}_{0}\mathcal{R}_{1}$ acts diagonally on the state with empty Young diagram on the vertical line, but exchanges the two tensor factors (due to $P$ operator) and shifts the spectral parameter of the representation from $w(t/q)^{3/4}$ to $w$ (due to the $\mathcal{R}_{0}$ piece involving grading operators). What remains is to evaluate the third line of Eq. (41):

where we have used the explicit action of $x^{-}(z)$ in the vertical representation

which follows from the definition (121). In the second line of Eq. (43) we have also used the fact that the identity term in the definition (38) of $\mathcal{R}_{2}$ does not contribute to the matrix element since it cannot turn $|\square\rangle$ into $|\varnothing\rangle$. A more nontrivial observation is that all the higher terms from Eq. (38) also don’t contribute: they all subtract two or more boxes from the diagram $|\square\rangle$ so that it vanishes.

Collecting the terms from Eqs. (42) and (43) we find that

This explains why we get the coproduct of the $x^{+}(w)$ current from the brane crossing (10) and ensures that the resulting operator is indeed the $qq$-character, or $qW_{m}$-algebra generator as shown in [17] (and recalled in Appendix A).

In the next section we will see that an insertion of the vertical brane (10) into a toric diagram corresponding to a gauge theory gives the Nekrasov formulas for the $qq$-characters.

To connect with the works of Nekrasov on the gauge theory origin of $qq$-characters [24], we need to insert the $qq$-character operator into a toric diagram corresponding to a gauge theory. The resulting expressions will have the form of sums (averages) over Young diagrams, or instanton series with brane insertion producing an extra contribution to the measure.

Since the vertical brane insertion (10) is essentially a sum of several free field vertex operators, its contribution to a refined topological string partition function is easy to calculate using Wick’s theorem. The expressions for the refined topological vertices in terms of free boson generators are written out in Eqs. (13), (14).

The normal ordering of $\mathcal{R}^{\lambda}_{\mu}(w)$ (Eq. (17)) with a product of $n$ $\Psi$-type and $m$ $\Psi^{*}$-type operators is given by:

1. 1.

   $\lambda=\varnothing$, $\mu=\varnothing$:

   $$\mathcal{R}^{\varnothing}_{\varnothing}(w):\mathrel{\mkern 2.0mu\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:\,=\,:\mathrel{\mkern 2.0mu\mathcal{R}^{\varnothing}_{\varnothing}(w)\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:,$$

   (46)

   i.e. the normal ordering coefficient is trivial.

2. 2.

   $\lambda=\square$, $\mu=\varnothing$:

   		$\displaystyle\mathcal{R}^{\square}_{\varnothing}(w):\mathrel{\mkern 2.0mu\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:\,\,$
   		$\displaystyle=\exp\Biggl{[}\sum_{k\geq 1}\sum_{a=1}^{n}\frac{1}{k}\left(\frac{x_{a}}{w}\right)^{k}\left(1-(1-q^{k})(1-t^{-k})\mathrm{Ch}_{\lambda^{(a)}}(q^{k},t^{k})\right)$
   		$\displaystyle-\sum_{k\geq 1}\sum_{b=1}^{m}\frac{1}{k}\left(\sqrt{\frac{t}{q}}\frac{y_{b}}{w}\right)^{k}\left(1-(1-q^{k})(1-t^{-k})\mathrm{Ch}_{\mu^{(b)}}(q^{k},t^{k})\right)\Biggr{]}:\mathrel{\mkern 2.0mu\mathcal{R}^{\square}_{\varnothing}(w)\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:$
   		$\displaystyle=\frac{\mathsf{Y}_{\vec{\mu}}\left(\sqrt{\frac{q}{t}}w\right)}{\mathsf{Y}_{\vec{\lambda}}(w)}\,:\mathrel{\mkern 2.0mu\mathcal{R}^{\square}_{\varnothing}(w)\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:$		(47)

   with the so-called $\mathsf{Y}$-functions defined as

   $$\mathsf{Y}_{\vec{\lambda}}(w)=\prod_{a=1}^{n}\mathsf{Y}_{\lambda^{(a)}}(w)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\prod_{a=1}^{n}\left[\left(1-\frac{x_{a}}{w}\right)\prod_{(i,j)\in\lambda^{(a)}}\frac{\left(1-\frac{x_{a}}{w}q^{j}t^{1-i}\right)\left(1-\frac{x_{a}}{w}q^{j-1}t^{-i}\right)}{\left(1-\frac{x_{a}}{w}q^{j-1}t^{1-i}\right)\left(1-\frac{x_{a}}{w}q^{j}t^{-i}\right)}\right]=\\ =\prod_{a=1}^{n}\frac{N_{\lambda^{(a)}\square}\left(v^{2}\frac{x_{a}}{w}\right)}{N_{\lambda^{(a)}\varnothing}\left(v^{2}\frac{x_{a}}{w}\right)}$$

   (48)

   where $n$ (the number of $\Psi$ or $\Psi^{*}$ intertwiners on a given horizontal brane) corresponds to the rank of the gauge group $U(n)$ on a given node of the quiver theory and $N_{\lambda\mu}(x)$ is the Nekrasov factor given by Eq. (B). $\mathsf{Y}$-functions (48) are natural observables in gauge theory [24].

3. 3.

   $\lambda=\square$, $\mu=\square$:

   $$\mathcal{R}^{\square}_{\square}:\mathrel{\mkern 2.0mu\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:\,=\,:\mathrel{\mkern 2.0mu\mathcal{R}^{\square}_{\square}(w)\prod_{a=1}^{n}\Psi^{\lambda^{(a)}}(x_{a})\prod_{b=1}^{m}\Psi^{*}_{\mu^{(b)}}(y_{b})\mkern 2.0mu}:.$$

   (49)

   In this case again no normal ordering corrections are produced.