Gluing affine Yangians with bi-fundamentals

The affine Yangian of $\mathfrak{gl}_1$ is isomorphic to the universal enveloping algebra of $\mathcal{W}_{1+\infty}$ and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of $\mathcal{N}=2$ supersymmetric $\mathcal{W}_{\infty}$ algebra was constructed by"gluing"two affine Yangians of $\mathfrak{gl}_1$ using operators that transform as $(\square, \overline{\square})$ and $(\overline{\square}, \square)$ w.r.t. the two affine Yangians. In this paper we realize a similar two-parameter gluing construction where the gluing operators transform as $(\square, \square)$ and $(\overline{\square}, \overline{\square})$ w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.

where the W 1+∞ [λ] algebra is the vertex operator algebra that is generated by one field per spin for s = 1, 2, · · · , ∞. This rather non-trivial isomorphism -between a Yangian algebra and a higher spin algebra -has been useful along both directions in various contexts. All these applications are based on the fact that the affine Yangian of gl 1 has a very natural representation in terms of plane partitions. The affine Yangian of gl 1 is constructed by [2] and [3] in an attempt to prove the AGT correspondence [4] and its higher spin generalization [5], i.e. the correspondence between the Nekrasov instanton partition function of 4D N = 2 su(N ) gauge theory and the conformal blocks of W N algebra. The crucial link is the action of the affine Yangian of gl 1 on the intersection cohomology of the instanton moduli space, which is spanned by N -tuple Young diagrams [2]. These are in turn isomorphic to plane partition representations, see e.g. [6]. The correspondence [4] then follows due to the isomorphism.
On the other hand, the W ∞ algebra and its finite truncation W N algebra appear in many interesting 2D CFTs, e.g. as the symmetry algebra of the CFT dual [7] of Vasiliev higher spin gravity in AdS 3 [8], and as the subalgebra of the CFT dual of string theory in AdS 3 in the tensionless limit [9]. By the isomorphism (1.1), one can also use plane partitions to characterize representations of these W algebras. This has both conceptual and computational advantages over the traditional coset representations, see e.g. [10].
Given the usefulness of the isomorphism (1.1) explained above, we would like to generalize it. There are two directions. The first is to find similar isomorphisms for known affine Yangians (l.h.s.) or for known VOAs with higher spin fields (r.h.s.). The second is to have a more systematic method to construct isomorphisms between new affine Yangians and new VOAs. As we will see, useful for both is the construction that "glues" multiple copies of (1.1).
Along the first direction, one obvious question is how to supersymmetrize the isomorphism (1.1), i.e. to construct affine Yangian type algebras isomorphic to supersymmetric W ∞ algebras. The affine Yangian algebra that is isomorphic to the UEA of the N = 2 supersymmetric W ∞ algebra (tensored with a u(1) factor) can be constructed by gluing two copies of affine Yangian of gl 1 , Y andŶ, using additional operators that transform as ( , ) and ( , ) w.r.t. Y andŶ [11,12].
The corresponding representation is a pair of plane partitions (one for each copy of affine Yangian of gl 1 ) connected by a shared "internal" direction. The additional gluing operators create/annihilate building blocks of non-trivial states along the internal leg, whose cross-sections are either at the end of Y and at the end ofŶ, or vice versa. The resulting representation space of the glued algebra is a pair of plane partitions whose asymptotics along the shared directions are correlated as [R,R] or [R,R t ] -called twin plane partitions in [1,12]. (Here R is a Young diagram,R is its conjugate, and R t is its transpose. The shapes of these Young diagrams give the asymptotics of the plane partitions along the "shared" direction.) All the relations of the glued algebra can then be determined by studying the action of the algebra on these twin plane partitions [12]. One can then check that it reproduces the correct W N =2 charges.
In [11,12], the input from the W N =2 ∞ algebra (in particular, its decomposition in terms of two W 1+∞ subalgebras) inspired the gluing construction and helped fix the relevant parameters. However, the gluing construction itself, in particular, the procedure of fixing all the algebraic relations using constraints from their actions on the pair of plane partitions, is independent of the W N =2 ∞ algebra and can be used to construct isomorphisms between new affine Yangians and new VOAs.
In particular, the construction for the affine Yangian corresponding to W N =2 ∞ algebra has four main ingredients [11,12]: 1. It has two copies of affine Yangians of gl 1 -Y andŶ.
3. The parameters of Y andŶ are related by h i =ĥ i . 4. The conformal dimension of the gluing operators are 3 2 , corresponding to that of the supercharges G ± .
To use the gluing procedure of [11,12] to construct isomorphisms between new affine Yangians and new VOAs, one simply modified these ingredients while retaining the essential feature of the procedure, i.e. fixing the glued algebra using constraints from its action on the corresponding set of plane partition type representations.
In [1], the ingredients (3) and (4) were relaxed. The goal was to construct the most general glued algebras under the condition (1) and (2) using the gluing procedure of [11,12]. The result is a two-parameter generalization of the algebra obtained in [11,12]. (The corresponding VOAs are a two-parameter generalization of the W N =2 ∞ algebra.) In particular, for (3), the relation between the parameters of Y andŶ now falls into three cases; and for (4), the conformal dimension of the gluing operators can now take all positive integer or half-integer values. Note that these constraints follow purely from the geometry of the twin plane partitions. We will call the constructions in [1,11,12] box-antibox constructions.
In this paper, we would like to further relax item (2) in the list, namely, to consider gluing operators that transform as other types of bimodules of the two affine Yangians of gl 1 . The most obvious generalizations are gluing operators that transform as [ , ] and [ , ] w.r.t. Y andŶ.
We will first construct the algebra glued by operators that transform as [ , ] w.r.t. Y andŶ. The main motivation is that its corresponding plane partition representations are much easier to visualize than the one in [1,12]. As shown in [12], in term of the plane partitions, the [ , ] gluing operator creates (on vacuum) an internal leg that looks like a long row of single boxes from the view point of Y and a "high wall" from the view point ofŶ. This makes further development of the glued algebra, in particular the matching of the microscopic action of the glued algebra to some physical systems, a bit unwieldy. For example, when trying to connect to topological strings, it is a bit unnatural to interpret the pairs of plane partitions with "walls" as the complements of melting crystals.
In the current new construction, the [ , ] gluing operator simply creates an internal leg that looks like a long row of boxes from both left and right plane partitions, namely, there is no need for the rows of boxes to undergo the transition to the "higher walls" somewhere along the internal leg, as one moves from the left plane partition to the right one. The representations look precisely like (the complement of) a crystal with two corners and the connecting edge partially melted.
The second motivation is that the new choice [ , ] allows parameters that are outside the construction of [1]. In fact, the change in item (2) necessitates the changes in item (3). We will see that the relations between the parameters in the two affine Yangians of gl 1 again fall into three cases, but different from those in [1]. This also allows more possibilities when trying to match the glued algebra with known algebras. For example, if one wants to reproduce the affine Yangian of gl 2 by gluing two affine Yangians of gl 1 , the parameters between the two Y's satisfy one of the three relations dictated by the [ , ] construction, not the one in the box-antibox construction.
We then apply the same procedure of [1] and construct the most general family of glued algebras with the [ , ] gluing operator. The main content of this paper is the derivation of individual relations based on their actions on pairs of plane partitions glued by [ , ] operators. Although the procedure follows closely [1], we see lots of difference in the detailed mechanisms. And certain aspects are simpler than the counterparts in the box-antibox construction in [1].
Then we show that one can extend the algebra glued by [ , ] operators with additional gluing operators that transform as [ , ] w.r.t. the two affine Yangians of gl 1 . This can be achieved without introducing "high walls" connecting the two plane partitions, but by allowing the [ , ] gluing operators to act appropriately on the pairs of plane partitions glued by [ , ] operators. The extended algebra can then be determined by analyzing their actions on these pairs of plane partitions. Finally, we explain the relation between the algebras constructed in this paper with those in the box-antibox construction in [1].
The plan of this paper is as follows. Section 2 is a review on the affine Yangian of gl 1 , its plane partition representations, and the box-antibox gluing construction. In section 3, we explain the ingredients in the gluing construction using [ , ] operators, in particular, the relevant plane partition type representations and their charge functions. In section 4 we use the ingredients in section 3 to fix the algebra glued by [ , ] operators. In section 5 we extend the algebra constructed in section 4 by additional operators that transform as [ , ] w.r.t. the two affine Yangians of gl 1 . The final section explains the relation between the algebras constructed in this paper with those in the box-antibox construction in [1] and discusses future problems. Appendix A contains long computations used in section 5. The affine Yangian of gl 1 is an associative algebra defined by the following OPE-like equations where the three fields {e, ψ, f } have the mode expansions and throughout this paper and ϕ 3 (u) is a cubic rational function where the triplet (h 1 , h 2 , h 3 ) obeys Since the algebra (2.1) is invariant under permutation of (h 1 , h 2 , h 3 ), sometimes it is convenient to use the S 3 invariant parameters The ∼ sign in (2.1) and throughout the paper means identity up to terms of z m w n with m, n ≥ 0. Finally, the relations (2.1) need to be supplemented by the initial conditions and the Serre relations π∈S 3 z π(1) − 2z π(2) + z π(3) e(z π(1) ) e(z π(2) ) e(z π(3) ) ∼ 0 , The relations (2.1) can be translated in terms of modes in the following way. Take the first equation for example. Moving the denominator of ϕ 3 (∆) to the l.h.s., we have The other equations in (2.1) and the Serre relations (2.8) can be translated in a similar way. For the complete list, see e.g. [6,13].
Note that although {e(z), ψ(z), f (z)} are not holomorphic fields in a 2D CFT, the relations (2.1) are similar to OPE relations because (1) they are only defined up to regular terms, and (2) they are merely rewritings of the corresponding mode relations. Therefore, throughout this paper, we will call relations like (2.1) OPE relations, to distinguish them from their corresponding equations in terms of modes. They have the form (2.11) in which the numerator N (∆) and the denominator D(∆) might share some common factor f (∆). 1 We emphasize that one shouldn't cancel these f (∆) factors, since (2.11) should be understood as which can then be translated into an equation in terms of modes a j and b j . Removing a common factor f (∆) (if present) from D(∆) on l.h.s. and N (∆) on r.h.s. of (2.12) would change the corresponding relation in terms of modes.
It is easy to check that the affine Yangian algebra (2.1) has an automorphism (2.13) In terms of the modes, this is In addition, there is an automorphism induced by the spectral shift u → u + a in {e(u), ψ(u), f (u)}. In particular, under this shift, From the initial condition (2.7) and the defining equations (2.1) (i.e. the version in terms of modes), we see the affine Yangian of gl 1 has two central elements ψ 0 and ψ 1 , where ψ 1 is subject to the spectral shift automorphism (2.15). Therefore one can label an affine Yangian of gl 1 by (h 1 , h 2 , h 3 , ψ 0 ) subject to (2.5)  where the W 1+∞ [λ] algebra is the vertex operator algebra that is generated by one field per spin for s = 1, 2, · · · , ∞ and is labeled by two parameters: the central charge c and the 't Hooft coupling λ. These should be mapped to the two parameters of the affine Yangian of gl 1 , i.e. (h 1 , h 2 , h 3 , ψ 0 ) subject to (2.5) and (2.16). The isomorphism (2.17) was suggested in [13] and proven in [6]. Theoretically, one should be able to derive the isomorphism (2.17) as a classical (i.e. q → 1) limit of the isomorphism between the quantum toroidal gl 1 and the q-deformed W 1+∞ [λ] algebra [14,15]. However, since the detailed map from the quantum toroidal gl 1 to the affine Yangian of gl 1 is rather involved (see [16]), one can simply prove (2.17) directly [6].
Recall that the W 1+∞ [λ] algebra above has a very rigid structure: once the OPE relations for the lower spin fields (up to OPEs between two spin-three fields) are known, all the OPEs with higher spin fields are fixed by Jacobi identities [17,18]. Therefore one only needs to know the map (2.17) up to spin-4 field. Moreover, for each spin s, we only need to fix W (s) m with |m| ≤ s to deduce W (s) m for all m ∈ Z using the W 1+∞ algebraic relations. To summarize, we only need to know the W (s) m for 1 ≤ s ≤ 4 and |m| ≤ s in terms of (e j , ψ j , f i ) modes. They are with |m| ≥ 2 are composed of products of (e j , ψ j , f j ) modes. For details see [6]. The map for the modes W (s) m for 1 ≤ s ≤ 4 and |m| ≤ s in terms of (e j , ψ j , f i ) modes then allows us to fix the map between the parameters of the two algebras [6]:

Plane partition representations
The affine Yangian of gl 1 has a natural representation in terms of the plane partitions {Λ}: Each plane partition is an eigenfunction of the operator ψ(u), whose eigenvalue Ψ Λ (u) (called charge function here) receives a contribution from each box in the plane partition Λ: is the vacuum contribution, ϕ 3 is the cubic rational function defined in (2.4), and finally is the coordinate function. Note that knowing (2.21) and (2.22) would allow us to reproduce the algebra (2.1).
For all plane partition representations, the character of affine Yangian of gl 1 is given by the generating function of the plane partition configurations with the given asymptotics (λ 1 , λ 2 , λ 3 ). For example, the character of the vacuum representation is the MacMahon function.
By the isomorphism (2.17), the set of plane partitions also serve as representations for the W 1+∞ algebra. First of all, the traditional coset representation can be translated into plane partitions. In terms of the coset realization su(N ) k ⊕su(N ) 1 su(N ) k+1 (tensored with an additional u(1)), the representations of W 1+∞ algebra are labeled by two Young diagrams (ν + , ν − ), where ν + transforms under the su(N ) k in the numerator and ν − transforms under the su(N ) k+1 in the denominator. In terms of the plane partitions, they are the representations with asymptotics Namely, the coset representations correspond to plane partition representations with a trivial asymptotics along (at least) one direction. Since the W 1+∞ [λ] algebra has an S 3 automorphism [17], one can use it to rotate (ν + , ν − , 0) into (0, ν + , ν − ) etc. Taking the fusion products of these representations can then produce plane partition representations with non-trivial asymptotics along all three directions. Finally, by the isomorphism (2.17), the W 1+∞ character of a representation is also given by the generating function of its plane partition configurations.
The advantage of using the affine Yangian of gl 1 (i.e. Y) to describe the W 1+∞ algebra is due to the plane partition representations. For example, the W 1+∞ characters can be easily computed via generating functions of plane partitions. Moreover, the action of Y on plane partitions (2.21) is very easy to describe and visualize, which makes it an important tool in constructing bigger algebras with individual Y as building blocks.

Gluing construction using [ , ] and [ , ] gluing operators
Given the usefulness of the isomorphism (2.17) explained above, we would like to generalize it. There are two directions. The first is to find similar isomorphisms for known affine Yangians (l.h.s.) or for known VOAs with higher spin fields (r.h.s.). The second is to have a more systematic method to construct isomorphisms between new affine Yangians and new VOAs. As we will see, useful in both is the construction that "glues" multiple copies of (2.17).

N = 2 supersymmetric W ∞ algebra
One example in the first direction is the analogue of isomorphism (2.17) for the N = 2 supersymmetric W ∞ algebra with the spectrum of one N = 2 multiplet {W (s)0 , W (s)+ , W (s)− , W (s)1 } for each spin s = 1, 2, . . . , ∞. As conjectured and partially checked in [11], all the bosonic fields {W (s)0 , W (s)1 } (after adding an additional spin-one field) together form two mutually commuting W 1+∞ algebras and the fermionic fields have non-trivial OPEs with both of them: (2.26) As proposed in [11], this can be translated into a decomposition in terms of the affine Yangian of gl 1 in the following way. First, the two W 1+∞ subalgebras in (2.26) correspond to two affine Yangians of gl 1 , Y given by (2.1) andŶ given by the hatted version of (2.1), with their parameters denoted collectively by q andq, respectively: (2.27) Using the map (2.19), one can show that the parameters of Y andŶ are related by for details see [11]. Second, the fermionic fields in (2.26) transform as bimodules w.r.t. the two W 1+∞ algebras in (2.26). In particular, the pair of supercharges G + and G − transform as the bimodules [(0, ), (0, )] and [(0, ), (0, )], respectively [11]. They can be translated to the plane partition representations using (2. respectively. We will use the shorthand [ , ] and [ , ] to denote these two bimodules of Y andŶ. They interact with both Y andŶ and in this sense can "glue" them together to form a bigger algebra. Finally, since all fermionic fields in (2.26) are G ± descendants of the bottom components W (s)0 (which are bosonic and hence in (2.27)), they can all be generated by OPEs of G ± and W fields in (2.27). In the corresponding Yangian algebra, they can all be obtained by interaction of the gluing operators in (2.29) and the two bosonic subalgebras (2.27). Namely, using (2.29) to glue (2.27) can produce the analogue of isomorphism (2.17) for the N = 2 supersymmetric W ∞ : (2.30)

Gluing operators and twin plane partitions
We now explain how to determine the algebraic relations for the r.h.s. of (2.30), by gluing Y andŶ using the gluing operators [ , ] and [ , ]. The main strategy is to use the constraints from the algebra's action on the pairs of plane partitions. The starting point is a pair of affine Yangians of gl 1 , Y given by (2.1) andŶ given by the hatted version of (2.1). The affine Yangian of gl 1 (2.1) has a natural representation on the set of plane partitions via (2.21). The hatted versionŶ acts on an independent set of plane partitions. To distinguish the two, we use to denote the boxes in the plane partition acted upon by Y (also called "left" plane partition here), and useˆ for the boxes in the "right" plane partitions, acted upon byŶ: : x : creation P : charge y : annihilation , :x : creationP : chargeȳ : annihilation . (2.32) The building block is shown in Figure 1. To the left plane partition, it is the ground state of the module (0, , 0), i.e. a semi-infinitely long row of boxes along the x 2 direction. To the right one, it is the ground state of (0, , 0) module, i.e. the conjugate of (0, , 0); and it is realized as a "high wall" sitting atx 1 = −1 and extending along thex 2 directions, with the height (i.e. along thex 3 direction) reaching one box lower than the "ceiling". For a derivation of this "high wall" configuration and for examples on other "high wall" states, see section 3 of [12]. The building block is given by the left-right mirror image of the one shown in Figure  1. in the box-antibox construction. Note that the internal leg (along the x 2 ∼x 2 direction) is infinitely long, which makes it possible for the same state to transition from a row of boxes on the left to a high wall on the right.
Repeated application of the creation operators x andx generate pairs of plane partitions whose asymptotics are correlated in the following way where the bar denotes conjugate and the t denotes "transpose". (For detailed explanation see [12]). Such a pair of plane partitions whose asymptotics along the x 2 and x 2 directions are correlated as in (2.34) were called twin plane partitions in [1,12]. Those in (2.34) are in the vacuum module, and a generic representation is labeled by Applying the creation operator e andê then creates descendants of (2.34), represented by additional boxes 's and hatted boxesˆ 's on top of (2.34).

Determine glued algebra via twin plane partitions
The glued algebra contains the following three types of operators type c : ψ,ψ, P,P ; type s : e, f,ê,f ; type g : x, y,x,ȳ . Type (c) operators are the charge operators whose eigenstates are the set of all twin plane partitions Type (s) operators create/annihilate single boxes, and type (g) operators create/annihilate non-trivial states along the internal leg. The goal is to find the OPEs between all pairs of operators in (2.35).
The main constraint in building this glued algebra is that it needs to have a natural representation in terms of the set of twin plane partitions. Namely, the OPEs are fixed together with the action of all operators on an arbitrary twin plane partition state: where i runs over all possible final states Λ i that can be created by applying O on the initial state Λ. Generalizing (2.21), we first write down the ansatz for the action of operators (2.35) in the glued algebra. As in (2.21), the eigenvalue function Ψ Λ (z) (resp.Ψ Λ (z)) controls the action of single-box operators e and f (resp.ê andf ) on twin plane partitions: (2.38) and the hatted version. Comparing (2.31) and (2.32), we have the ansatz for the gluing operators Res w=p + ( ) P Λ (w) Res w=p + ( ) P Λ (w) (2.39) where λ andρ were defined in (2.34). And similarly for (x,P ,ȳ). Note that x can not only add in λ but also remove inρ. And similarly for the other three operators, y,x, andȳ. The difference between e.g. the -adding actions of x andȳ are both in their poles, i.e. p + ( ) v.s.p − ( ), and in their coefficients. For details see [12].
We see that once we determine the four charge functions (Ψ Λ (z),Ψ Λ (z)) and (P Λ (z),P Λ (z)) as functions of the twin plane partition Λ, we obtain the full algebra; or vice versa: knowing the algebra would allow us to fix the action of all the operators on the set of twin plane partitions. In practice, these two aspects are fixed together, see details in [1,12].

New VOAs from gluing multiple affine Yangians of gl 1
We have just seen how constraints from twin plane partitions allow us to fix the algebra that is isomorphic to (the UEA) of u(1) ⊕ W N =2 ∞ algebra. We now explain how the same procedure, slightly generalized, can be used to construct new affine Yangian algebras that are isomorphic to new VOAs with higher spin currents.
Recall that there are four important ingredients in the construction reviewed above: 1. Two copies of affine Yangians of gl 1 -Y andŶ. 4. The conformal dimension of the gluing operators is 3 2 , i.e. that of the supercharge G ± . In terms of ψ 0 andψ 0 , this translates into (2.42) In [1], we asked the question of, assuming we keep (1) and (2) unchanged, can we still modify (3) and (4)? And if yes, what are the most general parameters in (3) and (4)? Finally, does a parameter change in (3) or (4) cause any significant effect on the glued algebra?
The answers are the following. First, both (3) and (4) can be modified, resulting in a two-parameter generalization of the isomorphism (2.30), i.e. the corresponding VOAs are a two-parameter generalization of the N = 2 supersymmetric W ∞ algebra. Second, given how the gluing operators transform (as (2.40)), the constraints from twin plane partitions allow us to solve for all possible solutions for the parameters of Y andŶ. Finally, the change in (2.42) only shifts the conformal dimension of the gluing operators; whereas the change in (2.41) switches the statistics of the gluing operators (i.e. from fermionic to bosonic) hence results in more drastic changes of the algebra.

Constraints on parameters from twin plane partitions
A priori, the parameters of Y andŶ are subject to the scaling automorphism (2.16) and its hatted version. Now let's first consider the constraint on (2.43) from twin plane partitions. The main constraints come from the fact that a gluing operator (i.e. one from {x, y,x,ȳ}) affects both left and right plane partitions, hence its pole can be determined from both sides. The requirement is that the values from the two sides need to coincide: since they are from the action of the same gluing operator. Applying this argument on different twin plane partition states generates various constraints on the parameters in (2.43).
For the algebra with gluing operators transforming as (2.40), the crucial constraints are from the fact that the gluing operator x creates a configuration that is in the representation (0, , 0) w.r.t. Y and (0, , 0) w.r.t.Ŷ. We now list all the constraints on (2.43) from twin plane partitions.
1. From the left and right plane partitions separately, i.e. for Y andŶ separately, we have In twin plane partitions, the two corners share a common x 2 direction, hence h 2 andĥ 2 should be directly related. Consider the action of x(z) on the initial state of | . The (Ψ Λ (z),Ψ Λ (z)) charge functions of the final state x(z)| are (2.46) For both sides, we need to fine the pole z * such that the corresponding Ψ charge function corresponds to a sensible plane partition. The results are On the other hand, while the action of x(z) creates a in the asymptotic Young diagram along x 2 direction for the left plane partition, it creates a in the asymptotic Young diagram along thex 2 direction for the right plane partition. In [1,12], the transverse coordinates of the 's are negative 3 Therefore, the pole from the consideration of the right plane partition is The argument (2.44) then gives Repeating the argument on creating a second at the coordinate (x 1 , In summary, the condition on creating a second gives with s 1,3 ∈ Z and s 1 + s 3 = 2 (due to constraint (2.45)).

In [1]
, we further constrained that twin plane partitions do not have buds of negative length, i.e. buds that stick out of the left or right wall. This gives where we have used s 1 + s 3 = 2.
The four constraints above give the following three sets of solutions of (2.43). First, they all haveĥ Then the three solution of (2.55) gives which can also be written as where p ≡ s 3 −s 1 2 , thus p = −1, 0, 1 for the three cases in (2.57). In [1], it was shown that the three solutions (2.57) correspond to toric Calabi-Yau threefolds O(−1−p)⊕ O(−1 + p) → P 1 with p = −1, 0, 1. In particular, the four constraints (a1)-(a4) are in one-to-one correspondence with the four constraints for a toric Calabi-Yau threefold with two vertices in the toric diagram, for details see [1]. It was then shown, by a character analysis, that the gluing operators are bosonic when p = 0 and fermionic when p = ±1 [1].

Constraints on parameters from conformal dimensions of gluing operators
As we have seen, the relations between h i andĥ i are related to the self-statistics of the gluing operators. We now explain that the equation involving ψ 0 andψ 0 are related to the conformal dimension of the gluing operators. From the spin-two part of the map (2.18) we have for the derivation see [6]. The conformal dimension of the gluing operators are identical to that of the first state | created by x out of the vacuum: where we have defined a new parameter which has to be an integer or a half-integer.
For the algebra to correspond to the N = 2 supersymmetric W ∞ algebra, ρ = 1 2 , i.e. the gluing operators' conformal dimension is 3 2 -that of the supercharges G ± . For a generic algebra from the gluing construction that satisfies the condition (1) and (2) on page 13, their parameters ψ 0 andψ 0 only need to be related via (2.62).

( , ) gluing operators and perturbative twin plane partitions
In section 2, we have seen that the gluing construction for the N = 2 supersymmetric W ∞ satisfies four basic conditions (see page 13). Relaxing the last two conditions results in a two-parameter generalization of the N = 2 W ∞ algebra. In this section, we relax the conditions further. In particular, we choose the gluing operators to transform as bimodules w.r.t. the two affine Yangians of gl 1 . The resulting algebra is denoted schematically as where we useỸ to denote the right Yangian, in order to distinguish from the boxantibox construction reviewed in section 2. Note that since the relations (2.57) between the parameters of the two affine Yangians of gl 1 depend on the choices of bimodules, we would have to solve them for the current choice (3.2).

Operators in [ , ] gluing construction
The algebra in (3.2) consists of two affine Yangians of gl 1 , Y given by (2.1) andỸ by the tilde version of (2.1). Correspondingly, they act on the left and right plane partitions, respectively: : e : creation ψ : charge f : annihilation , :ẽ : creationψ : chargef : annihilation . Similar to the box-antibox construction, the building block for the configurations along the internal leg connecting the two plane partitions is 4 which was defined in (3.1). It consists of boxes 's in positions of the left plane partition and tilde boxes˜ 's in positions of the right plane partition. The building block is shown in Figure 2. Compared to its counterpart in the box-antibox construction, shown in Figure 1, this plane partition configuration is more elegant, since the left and right plane partitions can be connected seamlessly. This is one of the main advantages of the current construction over the box-antibox construction in [1,12]. Similar to (3.3), we define the corresponding operators In summary, the algebra from the gluing construction (3.2) contains the following three types of operators type c : ψ,ψ, Q ; type s : e, f,ẽ,f ; type g : r, s . Type c operators are the charge operators whose eigenstates are the set of all relevant twin plane partitions (to be defined in subsection 3.2), type s operators create/annihilate single boxes, and type g operators create/annihilate non-trivial states along internal legs. The goal is to find the OPEs between all pairs of the operators in (3.8).

Perturbative twin plane partitions
As in the box-antibox construction of [1,12], the algebra can be completely determined by the constraints from its action on the relevant plane partition type representations. For the [ , ] construction, they are pairs of plane partitions connected by the internal leg, whose states are built by repeated actions of r, hence their asymptotics along the x 2 andx 2 directions are correlated by Λ : or Λ : (3.10) As we will show later by a character analysis, these two choices, i.e. whether the symmetric direction of the left Young diagram -namely, the asymptotics of the left plane partition along the x 2 direction -is correlated with the symmetric or the anti-symmetric direction of the right Young diagram, are determined by whether the gluing operators are bosonic or fermionic.
Risking confusion with representations in the box-antibox construction of [1,12], we will name these plane partitions "perturbative twin plane partitions", for want of a better name. 5 The "perturbative" is to emphasize that there is no conjugate representations involved -recall that the conjugate representation is characterized by walls of height ∼ N (for W 1+N algebra). Furthermore, when there is no risk of confusion, we will often drop the "perturbative" in the remainder of the paper.

Set of twin plane partitions and coordinate system
To study the constraints on the algebra, it is enough to focus on the vacuum representation of the glued algebra. A vacuum twin plane partition configuration is given by a pair of plane partitions with asymptotics depending on whether the gluing operators are bosonic or fermionic. The descendants of (3.11) and (3.12) have additional boxes on the left and right corners. Namely, a generic twin plane partition representation in the vacuum module has the following building blocks: , generated by r.
• A collection E of individual 's in the left plane partition, generated by e.
• A collectionẼ of individual˜ 's in the right plane partition, generated byẽ.
The coordinate systems for and˜ are simply The coordinate functions of the and˜ are 14) The restriction on the coordinates (3.13) means that we do not allow any box outside the left and right walls. As in the box-antibox construction, this bound imposes strong constraints on the algebra's action on the twin plane partitions, in particular, on the allowed final states.
The coordinate systems for are simpler than in [1], since they are symmetric w.r.t. the two sides. In particular, without loss of generality, we can choose their (x 1 , x 3 ) and (x 1 ,x 3 ) coordinates to be correlated by 6 hence we would only need the (x 1 , x 3 ) coordinates for ; and they take the value of : A in λ has a pair of coordinate functions since it is visible from both corners

Charge functions of perturbative twin plane partitions
The main constraint in building the glued algebra is that it needs to have a natural representation in terms of the perturbative twin plane partition. Namely, the OPEs are fixed together with the action of all operators on an arbitrary twin plane partition state: The pole structure z * is relatively easy to determine, based on the fact that the resulting state |Λ i also needs to be a valid twin plane partition. However, the coefficient O[Λ → Λ i ] can only be determined together with the OPEs.
In [1], we found a procedure of fixing all OPEs together with the action (3.18), for the gluing operator that transforms as [ , ] and [ , ]. In this paper, we will follow this procedure and adapt it for the current case, with the gluing operators that transforms as [ , ].
We first write down the ansatz for all the operators (3.8) on an arbitrary twin plane partition state. First of all, a twin plane partition is the eigenstate of the charge operators (ψ(z),ψ(z)) and Q(z): The charge function Ψ Λ (z) (resp.Ψ Λ (z)) controls the action of single-box operators e and f (resp.ẽ andf ) on twin plane partitions. Namely, the action of {e, ψ, f } on an arbitrary twin plane partition Λ is given by the analogue of (2.38), where Λ is replaced by the perturbative twin plane partition Λ, and the action of (ẽ,ψ,f ) is given by the tilde version. The charge functions Ψ Λ (z) andΨ Λ (z) will be fixed later, see section 3.3. They need to reproduce (2.1) and its tilde version.
Similarly, the charge function Q Λ (z) controls the action of gluing operators r and s on twin plane partitions by (3.20) In principle, if we can determine the charge functions (Ψ Λ (z),Ψ Λ (z)) and Q Λ (z) for arbitrary twin plane partition |Λ in the vacuum module (3.11) or (3.12), we can derive all the OPEs from there. However, as we will see below, some OPEs (especially those between single-box operators and gluing operators) are more easily determined (at least partially) directly using the constraints from twin plane partitions. These OPEs can in turn fix part of the charge functions.

Evaluating (Ψ,Ψ) charge functions
Before we start fixing the entire algebra, some crucial aspects, such as the parameters of the algebra and the conformal dimension of the gluing operators, can already be fixed once we know the (Ψ,Ψ) charge function for a given twin plane partition and therefore OPEs between (ψ,ψ) and all other operators. We will now determine the (Ψ,Ψ) charge function and leave the Q charge function for the next section.
Same as in the box-antibox construction, the (Ψ Λ (z),Ψ Λ (z)) charge functions have the following decomposition (3.12). Note that the 's on the left side are invisible toΨ Λ (u) while the˜ 's on the right side are invisible to Ψ Λ (u). We can immediately fix all the factors in (3.21).
First of all, since it needs to reproduce the bosonic OPEs (2.1) and its tilde version, each (resp.˜ ) again contributes to the Ψ Λ (z) (resp.Ψ Λ (z)) charge function by Secondly, since to the left plane partition, the is composed of infinite number of 's at the coordinate as shown in Figure 2, the contribution of to the Ψ Λ (u) charge function comes from all these 's: .
Similarly for the right plane partition, the is composed of infinite number of˜ 's at the coordinatẽ as shown in Figure 2. Hence each contributes to the charge function ( These are summarized in the (ψ,ψ) part of Table 1.

OPEs between (ψ,ψ) and all other operators
Since to fix some basic properties of the glued algebra, such as the relation between parameters of Y andỸ, we already need to know OPEs between (ψ,ψ) and all other operators, we will fix them in this subsection. All other OPEs will be left for section 4. First of all, the OPEs of (ψ,ψ) and single-box operators s are already given by (2.1) and its tilde version. They can also be derived from the (Ψ Λ (z),Ψ Λ (z)) charge functions of and˜ in (3.22). (See the top and bottom parts of Figure 3.) Using the same logic, the (Ψ Λ (z),Ψ Λ (z)) charge function of in (3.27) determines the OPE relations between the charge operator (ψ,ψ) and all the gluing operators g: (3.28) See the thick blue arrows in Fig. 3. Actually, there is an alternative, more hand-wavy but essentially the same, method to obtain some of the OPEs above. Since r corresponds to the twin plane partition configuration consisting of boxes at (3.5) and tilde boxes at (3.6) (as shown ψ f e rẽψf s in Figure 2), one can think of r as composed of infinitely many e andẽ operators, acting on different coordinates along the x 2 andx 2 axes: Passing the ψ operator over all the e operators in (3.29) immediately gives and similarly forψ r OPE: Moreover, as we explain now, the results from section 3.3 and 3.4 also determine, for any initial state |Λ and gluing operator g, all allowed final states |Λ i (and therefore the associated poles z * i ). (Note that it is harder to fix the coefficients Consider an arbitrary initial state |Λ and an operator O in the process (3.18). (To be generic, we consider any O in the list (3.8), although the case of O = r or s is the most interesting one.) The goal is to fix all the allowed final states |Λ and their associated poles w * . It takes three steps. final state |Λ : (3.34) (To see this, simply apply the OPEs between (ψ,ψ) and O computed in section 3.4 on the initial state |Λ .) 3. Examine the two lines in (3.33) separately. Demanding Ψ Λ (u) to correspond to a sensible left plane partition fixes a set of allowed poles The final states are those with This method would allow us to fix the allowed pole p + ( ) in the ansatz (3.20). However, in selecting the poles that satisfy (3.37), we will need to know the relation between the parameters {h i } of Y and {h i } ofỸ. Therefore the computation will take the following three steps: 1. Analyze the action of r(z) adding the second and determine the "bud condition" for the second .
2. Based on the "bud condition" for the second , determine the relation between the parameters {h i } of Y and {h i } ofỸ.
3. Use this relation to analyze the "bud condition" for generic and fix the allowed pole p + ( ) in the ansatz (3.20).

Constraints on parameters from perturbative twin plane partitions
Recall that in the box-antibox construction, the relations (2.57) between parameters of Y andŶ was solved using constraints from twin plane partitions, in particular, by an analysis of bud conditions. We now repeat this exercise for the [ , ] construction.

Bud condition for first two 's
We first derive the bud condition for the [ , ] construction. As we will see, the crucial differences from the one in the [ , ] ⊕ [ , ] construction are that (1) now the bud has to have precisely the minimal length, and (2) it is now possible to distribute the boxes in the bud between the left and right plane partitions.
Let's start from the vacuum. Since there is no need for additional boxes; the minimal bud has length zero. Now we show that it is also not possible to have any bud longer than this. Take | for example. Similar to (2.46), we now have Now z * has to be chosen such that Ψ(u) corresponds to a valid plane partition on the left andΨ(u) to a plane partition on the right. The left plane partition demands However, the right plane partition demands  With | + s 1 2 as the initial state, the charge functions for the final state is r(z)| + s 1 2 : (3.47) Then demanding that Ψ(u) andΨ(u) correspond to sensible left and right plane partitions, respectively, we have To proceed, we need a relation between h 2 andh 2 . Recall that for the [ , ] ⊕ [ , ] construction, the derivation of (2.49) was based on the process (3.43). However, since now the process (3.43) is replaced by (3.42), we cannot use the old derivation.
This problem is solved by noticing that different from the [ , ] ⊕ [ , ] construction, the gluing operator r(u) transforms as w.r.t. the two affine Yangians of gl 1 , and therefore looks like a long row of boxes from both the left and right plane partitions. As a consequence, there exists another set of bud conditions from the right plane partition. Take 1 for example, the bud can also be composed of˜ 's, sitting at the right plane partition at wheres 1 (resp.s 3 ) is the length of the bud from the right plane partition for 1 (reps. 3 ). Now the 1 from the left plane partition actually corresponds to exactly the same row of boxes as the 3 from the right plane partition, and similarly for the pair with 1 ↔ 3:

Relations between parameters of Y andỸ
We are now ready to solve for the relation between the parameters of Y andỸ. A priori, the parameters of Y andỸ are  where we used s 1 + s 3 = 2.
The four constraints above give the following three sets of solutions of (3.59). First, they all haveh 2 = −h 2 . Then the three solution of (3.63) gives (s 1 , s 3 ) = (2, 0) : the three cases in (3.64) can be written as One can see that if we fix h i , the parameters of the right affine Yangians are related byh i = −ĥ i . We will explain in section 6 how to understand this relation.

Relations to Toric Calabi-Yau
In [1], a close relation between the geometry of the twin plane partition and the toric Calabi-Yau threefold in the T 2 × R fibration was noted. Remarkably, the four constraints (2.45), (2.49), (2.54), and (2.55) match one-to-one to four constraints in defining the toric CY 3 with two vertices. In the current construction, the four constraint (3.60)-(3.63) from twin plane partitions can also match one-to-one to the four constraints from toric CY 3 , once we flip the sign of the (p, q) charge of the second vertex -this correspond to the sign flip (6.4). More specifically, consider the two-vertex diagram, in which the (p, q) charges of the first and second vertex are where all the charges are integers. First of all, the Calabi-Yau condition demands Choosing their second directions to be the joined leg between the two vertices and all directions pointing outwards, we get the second constraint (t2) Thirdly, the smoothness condition means V 1 ∧ V 2 =Ṽ ∧Ṽ 2 , which can be translated to where c 1 ∈ Z. Finally, demanding that the four external directions in (3.68) do not intersect gives the constraint

Bud condition for generic 's
With the result on the relation between the parameters {h i } of Y and {h i } ofỸ, we are now ready to generalize the derivation for the bud condition for the second (see section 3.6.1) to a generic and obtain its pole p + ( ). An r(z) action contributes to the (Ψ,Ψ) charge function On the other hand, a at a generic position contributes to the (Ψ,Ψ) charge function : where g( ) andg( ) are the coordinate functions defined in (3.17). For (3.73) to have a chance to reach (3.75), a priori we might need a bud at either or both ends of the : (n left , n right ) − bud :  respectively, where we have used the following identities and together with their tilde versions. Demanding the two poles in (3.78) to coincide, we obtain the total length of the bud:  The conformal weight of the state (3.84) is computed by which gives where the parameter ρ is defined as which has to be an integer or a half integer, since the conformal dimension h ∈ Z 0 . Later we will show that, quite remarkably, this constraint ρ ∈ 1 2 Z 0 can also come from twin plane partitions.

Character of the algebra
Depending on whether the gluing operators are bosonic or fermionic, the character of the glued algebra is To show that they indeed reproduce the counting of the twin plane partitions, use the expansion for the bosonic case and for the fermionic case, where the sum is over all possible Young diagrams λ, |λ| is the number of boxes in λ, and χ (w) λ (q) (resp.χ (w) λ (q)) is the wedge character -i.e. the full character divided by the vacuum character -of the representation λ of the left (resp. right) affine Yangian of gl 1 . Note that the main difference between the bosonic and fermionic cases is that for the fermionic one, the representations in the two factors are related by a transpose.
Plugging the two expansion (3.90) and (3.91) into the character (3.88) and (3.89), respectively, we have where we have used (3.94) Let's first look at the bosonic case (3.88). When the gluing operators are bosonic, for each state λ along the internal leg, it is viewed by the left and right plane partitions as λ and λ, namely, the asymptotics of the left and right plane partitions are (0, λ, 0) and (0, λ, 0), respectively. Thus the characters of the left and right plane partition are χ λ (q) and theχ λ (q), respectively. Summing over all possible internal states λ, each weighted by y |λ| , we obtain the full vacuum character of the glued algebra given by (3.92). The proof for the fermionic case -namely (3.89) is equivalent to (3.93) -is parallel.
Finally, the character formulae (3.92) and (3.93) suggest that the symmetric direction of the left Young diagram is correlated with the symmetric (resp. antisymmetric) direction of the right Young diagram if the gluing operators are bosonic (resp. fermionic).

Self-statistics of gluing operators
Finally, we explain what determines whether the gluing operators are bosonic or fermionic. To connect the character formulae derived above to the stacking of by repeated action of the creation operator r(z), we expand (3.90) for the bosonic case (1 + y q n+ρ ) n This correspond to the case of (s 1 , s 3 ) = (1, 1) (i.e. p = 0), namely, stacking the second along either direction requires a bud of length one. Therefore we conclude that the relations (3.64) between h i andh i are related to the self-statistics of the gluing operators by

(3.99)
Note that this is also consistent with the fact that on the vacuum, r(z)r(w) vanishes when p = 0 (because r(z) needs a bud of length-one to act), suggesting a fermionic self-statistics; whereas when p = ±1, r(z)r(w) on the vacuum is non-vanishing (because the bud length for r(z)'s action is s 3 =0 for p = −1 and s 1 = 0 for p = 1), suggesting bosonic self-statistics.

Glued algebra from twin plane partitions
In this section, we derive all the OPEs of the glued algebra based on its action on the twin plane partitions. We will follow the procedure in [1]. However, as we will see, in certain crucial steps, e.g. the OPEs between single-box generators and the gluing operators and the OPEs among gluing operators, the detailed mechanisms are rather different from the corresponding ones in the box-antibox construction of [1,12].

OPEs between single-box generators and gluing generators: incomplete
The OPEs between (ψ,ψ) and all other operators were already fixed in section 3.4. We now use the result from section 3.5 to constrain the OPEs between single-box generators {e, f,ẽ,f } and gluing operators {r, s}.
It is easy to see that, out of the 8 OPE relations, there are only two independent functions G(∆) and H(∆) ,  in which (G,H) is obtained by replacing all h i in (G, H) byh i . Namely, we only need to fix the two OPEs in (4.2). As we shall see below, the result of section 3.5 only allows us to fix the two functions in (4.1) up to two constants. They will be determined in the next subsection, together with the single boxes' contributions to the Q Λ (z) charge functions.

e · r OPE andẽ · r OPE
Using the schematic expression (3.29), one immediately has  For a more rigorous derivation, we use the method in [1,12]. The basic idea is to apply the ansatz e(z) r(w) ∼ G(∆) r(w) e(z) (4.7) on various initial states Λ. For some Λ, the different order of applying e and r, i.e. er and re, produce different sets of final states. Then demanding (4.7) to hold on such initial states constrains the function G(∆), since it must contain factors in the numerator and denominator to cancel those final states that only appear on one side of the equation. As we will see, although the logic is the same as in [1,12], the details are rather different. First, apply (4.7) on the vacuum |∅ . The l.h.s. give and the r.h.s., without the factor G(∆), is . (4.10) In the box-antibox construction, (4.9) contains a state | +ˆ top (due to the process (3.43)), which would have given the (∆ − h 2 ) factor in the numerator of G [1,12]. But in the current construction, since r(w) | = 0 , (4.11) to obtain the factors in the numerator of G(∆), we need a different initial state from | . To generate the constraint on the numerator of G(∆), we need an initial state which, when acted on by r(w)e(z), produces some final states that cannot be created by e(z)r(w). The simplest one is  13) and the r.h.s., without the factor G(∆), is (4.14) Compare the l.h.s. (4.13) and the r.h.s. (4.14). The l.h.s. (4.13) contains four final states and the r.h.s. (4.14) contains three. Two states, | 3 + 1 and | 3 +bud 1 , appear on both sides. The two factors in the denominator of G(∆) fixed in (4.10) cancel the two extra states in the l.h.s. (4.13). To cancel the extra state | 1 in (4.14), the numerator of G(∆) must contain a factor ∆ + h 2 . Taking this together with (4.10), and the fact that G is quadratic, we have where a is a constant to be fixed later. The analysis forG is parallel.
Closely related to the e · r OPE is the f · r OPE: Apply the ansatz (4.16) on the initial state | . The l.h.s. annihilates | , whereas the r.h.s. gives Since the state | only appears on the r.h.s., the numerator of H(∆) must contain a factor of ∆ to kill this state. On the other hand, we also know that H is linear, and G and H satisfy the following relation [12]: Therefore The computation forH is similar.

OPEs between Q and single-box operators
In the previous subsection, we fixed all OPEs between single-box generators {e, f,ẽ,f } and gluing operators {r, s} up to a constant a. Now we will show that this constant can be fixed together with the OPEs between Q and single-box operators {e, f,ẽ,f }, since they are both related to the single boxes' contribution to the Q charge function.

Interlude: evaluating Q charge function
An important part in the construction of the glued algebra is to determine the charge function Q Λ (z). Similar to the (Ψ Λ (z),Ψ Λ (z)) charge function (3.21), the Q Λ (z) charge function has the decomposition The only difference is that while Ψ only sees (andΨ only˜ ), Q sees both the and˜ . It takes the following three steps to fix all the components in (4.20): 1. Fix single boxes' contribution to Q Λ (z) 2. Fix the contribution of the first to Q Λ (z) 3. Fix all higher 's contribution to Q Λ (z)

Single boxes' contribution to Q charge function
The single boxes' contribution to the Q Λ (z) charge function, i.e. Q (u) and Q˜ (u), are related to the following two OPEs between single-box creators and gluing operators and Q˜ (u) :ẽ(z) r(w) ∼G(∆) r(w)ẽ(z) , (4.21) respectively. We only need to study the first one, since the second one is its tilde version.
To understand the first equation in (4.21), apply the e · r OPE on a generic twin plane partition |Λ , and we obtain   The derivation is parallel to the one for ψe OPE in section 3.4. Take the Q · e OPE for example. Applying the ansatz Q(z)e(w) ∼ K(z − w) e(w)Q(z) (4.29) on an arbitrary twin plane partition |Λ , and using the action of Q in (3.20), we get which gives where in the last step we have used the result Q (z) (4.25). This fixes the OPE (4.29) to be Q(z) e(w) ∼ ϕ −1 2 (−∆) e(w) Q(z) . (4.32) The other three cases are completely parallel.

OPEs between single-box generators and gluing generators: final
With (4.25), (4.26), and (4.27), we can now write down the OPEs between {e, f,ẽ,f } and the creation operator r: (4.33) and those for the corresponding annihilation operator s: See the thick red arrows in Fig. 6. Figure 6. OPEs between the single-box generators {e, f,ẽ,f } and the gluing operators {r, s}.

OPEs among {r, Q, s}
In this subsection, we fix the OPEs among {r, Q, s}. This will also allow us to fix 's contribution to the Q charge function. Given the similarity between the generators (e, ψ, f ) for in (3.3) and the generators (r, Q, s) for in (3.7), and in particular, the similarity between the action of (e, ψ, f ) on twin plane partitions (2.38) and the action of (r, Q, s) (3.20), we can write down the ansatz for the OPEs among (r, Q, s) in a similar form to those among (e, ψ, f )t in (2.1):

Relation between S p and 's contribution to Q charge function
In the ansatz (4.35), S p plays the role similar to the ϕ 3 function for the OPEs among {e, ψ, f } in (2.1). Recall that ϕ 3 (z − h( )) gives the contribution of to the Ψ charge function, see (3.22). In other words, it records the change to the Ψ charge function after we apply the e operator once: where h( ), as the coordinate function of the added, is also the pole of the e(z)'s action.
For the , applying in (4.35) on an arbitrary twin plane partition |Λ and then using (3.20), we see that S p records the change to the Q charge function after the application of r operator on |Λ : where |[Λ + ] denotes the final configuration after we apply r on Λ, which is not always |Λ + , in particular, the bud has disappeared into the just created. And p + ( ) is the pole for adding in (3.20), given in (3.82) or equivalently in (3.83). Note that unlike the case for , the pole p + ( ) is not always equal to the coordinate function g( ) of (3.17).
The relation (4.39) allows us to fix the contribution of to the Q charge function recursively. First, consider the vacuum |Λ = |∅ . The pole for adding the first is since n left = B( ) = 0 for the first . The relation (4.39) specializes to Using the decomposition of Q in (4.20), in particular, we immediately see that the first created from the vacuum contributes to the Q charge function by To proceed, we need to fix the function S p (∆).

Fixing S p
The strategy for fixing the S p function is to apply the relation (4.39) on various initial states |Λ and obtain constraints by comparing different resulting equations.
Recall that for the [ , ] ⊕ [ , ] construction, a crucial step in deriving S p was to use the process (3.43). Since now in the [ , ] construction, the process (3.43) is replaced by (3.42), we cannot use the old derivation. Instead, we use the fact that in the [ , ] construction, the looks like a long row of boxes from both sides, and the bud for growing higher 's can sit at both corners.
In particular, consider the process that generates the state | 1 starting from | . Since the require bud can sit at both corners, we can consider two initial states, 9 namely | +bud( 1 ) and | +bud( 3 ) , on which applying r can generate the state | 1 . Let's now consider them in turn.

Apply r on |Λ
where k = (1,k,0) denotes the k th in the left bud, at the position (x 1 , x 2 , x 3 ) = (1, k, 0). The pole of the action is Since we have already derived the contribution to the Q charge function from the first (4.43) and from individual 's (4.25), we can immediately write down the charge function of the initial state: 48) where˜ k =˜ (0,k,1) denotes the k th˜ in the right bud, at the position (x 1 ,x 2 ,x 3 ) = (0, k, 1). The charge function of the initial state is and the one for the final state is Now, since the two final states are the same, we have namely Plugging the Q charge functions (4.46) and (4.49) into the relation (4.52) then gives the constraint Simplifying (4.53) and (4.54), we arrive at In addition, examining the r · r OPE in the ansatz (4.35), we see that S p needs to satisfy Combining the two constraints (4.55) and (4.56), we can solve S p : .

(4.57)
Thus we have fixed the OPE among {r, Q, s} in (4.35), where S p is given by (4.57). They are shown by thick red lines in Figure 7. The result of S p also tells us the contribution of the first to the Q charge function in (4.43).

's contribution to Q charge function
It remains to compute the contribution of higher to the Q charge function. Consider the process of creating the at position (x 1 , x 3 ) on top of an initial state |Λ + bud( ) : where without loss of generality we choose the bud configuration such that all boxes are on the left side: whose contribution to the Q charge function is where g( ) is the 's coordinate function defined in (3.17). And the pole p + ( ) for this initial configuration is Applying (4.38) on the initial state |Λ + bud( ) , we have Now using the decomposition of the Q charge function (4.20) and canceling the Q Λ on both sides, we arrive at Using (4.61) and the recursion relation (4.55), the function (4.63) can also be written as which is what we would have obtained directly if we had started with the initial condition where all the boxes in the bud are in the right plane partition. It is reassuring to see that the two expressions (4.63) and (4.64) are indeed equivalent.

Extension of algebra with [ , ] gluing operators
In section 3 and 4, we constructed the algebra with gluing operators transforming as [ , ] w.r.t. the two affine Yangians of gl 1 , using their action on the representation space of a pair of plane partitions. This pair of plane partitions is the complement of a crystal with two corners and their connecting edge partially melted, generalizing the melting crystal picture for the W 1+∞ algebra (in the context of topological strings). Now we will show that the algebra constructed in section 3 and 4 allows an extension with a pair of additional gluing operators transforming as [ , ] w.r.t. the two affine Yangians of gl 1 . Instead of also extending the representation space (by pairs of plane partitions connected by "high walls"), we will still use the space of twin plane partitions defined in section 3.2, which has simpler and more transparent geometric interpretation, and show how the additional operators act on them. Quite remarkably, the extended algebra can still be derived based on constraints from these actions. However, as we show in appendix A, the plane partition configuration that corresponds to the building block is a "high wall" connecting the left and right plane partitions. Since one of the motivations for the current paper is to have a more elegant set of plane partition representations than the box-antibox construction, which requires gluing of "long rows" with "high walls", we restrict ourselves to the nicer perturbative twin plane partition representation constructed in section 3.2, namely, we will not create high walls made of 's. The additional operators will only act on the original twin plane partitions. Similar to (3.7), we again have a triplet of operators {r,Q,s} that transform as [ , ] w.r.t. the two affine Yangians of gl 1 . To write down the ansatz for their action on the twin plane partition, the intuition is that the and should annihilate each other, leaving a few and˜ behind, since both and have positive conformal dimensions. Therefore, eq. (5.3) should map to :r : removeQ : charges : add (5.4) and their action on |Λ take the form

Extension with [ , ] gluing operators
where similar to (4.20), theQ charge function has the decomposition where we have separated the extending operators (5.4) from those in the subalgebra (3.2). As before the type c are the charge operators, type s are the operators that create/annihilate single boxes and˜ , and type g are the operators that affect building blocks .
To construct the extended algebra (5.2), with the list of all operators in (5.7), we only need to determined the following additional OPEs: • OPEs between {ψ,ψ} and {r,s}. They are determined together with the charge functionQ Λ (u).

OPEs between {ψ,ψ} and {r,s}
As shown in [11,12], for (the ground state) of a representation Λ, its Ψ charge function is related to the Ψ charge function of the conjugateΛ by Since (r,s) are the conjugate of (r, s), the OPE relations between the charge operators (ψ,ψ) and the gluing operators (r,s) can be obtained by transforming those between (ψ,ψ) and (r, s) in (3.28): and (5.10)

Pole structure of the extended algebra
Now we can analyze the poles in the actions of {r,s} on the twin plane partition in (5.5). To use the procedure outlined in (3.5), we just need to supplement eq. (3.34) with (5.11)
Finally, since bothn left andn right are non-negative integers, the constraint (5.19) also implies that both h and the parameter ρ have to be non-negative integers or half-integers. Earlier in section 3.7, we arrived at this constraint by invoking the fact that h corresponds to the conformal dimension of the gluing operators. Here we see that the twin plane partition analysis also produces this constraint, independent from the conformal dimension argument.

5.3.2s adding
The process ofs adding is just the inverse of the one forr removing . Recall that for r to add a , one need a bud of length B( ), defined in (3.80), composed of n left number of 's and n right number of˜ 's with n left + n right = B( ). Now, for s to add a , we now need a bud of length n left +n right = B( ) + 2 + 2ρ , (5.22) again withn left number of 's andn right number of˜ 's, distributed between the left and right plane partitions. The pole is still given by (5.20), or equivalently (5.21).

EvaluatingQ charge function
We are now ready to compute theQ charge function of an arbitrary twin plane partition function. First of all, the vacuum piece is Secondly, in the process of computing the OPE relations betweenQ and {e, f,ẽ,f }, we have also fixed the single boxes' contributions to theQ charge function in section A.2: (5.27) Finally, to compute 's contribution to theQ charge function, we imitate the calculation for the Q charge function in section 4.3.3. We consider hows(w) adds a , starting from the configuration where all the boxes in the bud are on the left side. The result is where Again, using (3.87), (5.29), and the recursion relation (4.55), the function (5.28) can be rewritten as ) which is what we would have obtained directly if we had started with the initial state where all the boxes in the bud are placed in the right plane partition. The identity between (5.28) and (5.30) confirms that, for ther ands actions, the boxes in the same bud can be distributed between the left and right plane partitions. This is similar to the r and s actions, see the discussion around (4.63) and (4.64). The contribution (5.27) and (5.28) are collected in the last column of Table 1.

OPEs among {r,Q,s}
The OPEs among gluing operators {r, Q, s} are (4.35) where S p is given by (4.57). Based on this we can write down the ansatz for the relations for the triplet {r,Q,s} asQ (z)r(w) ∼S p (∆)r(w)Q(z) , where p is given by (4.36) and records the self-statistics of the gluing operators. To solve forS p (u), one can repeat the procedure in solving for S p (u), and the result is: .

(5.32)
We see thatS Applying the ansatz (5.34) on an initial state Λ on which r(z) can add a (labeled by 1 ) ands(w) can add a (labeled by 2 ). Using the action of r(z) in (3.20) and the action ofs(w) in (5.5), we have where the long bud( 2 ) consists of B( 2 ) + 2 + 2ρ boxes and the bud( 1 ) consists of B( 1 ) boxes. Plugging in the contributions of ,˜ , and to the Q andQ charge functions (see Table (1)), we get Note that this computation is insensitive to the self-statistics of the gluing operators, so we need to insert the function p (defined in (4.36)) by hand. The final result for the rs OPE is with p given by (4.36). Similarly, we havē Now we look at r(z)r(w) ∼ A 2 (z − w)r(w) r(z) (5.39) Repeating the procedure used for the rs OPE above, we get where "≈" means that we have cancelled the common factors in the numerator and the denominator of A 2 . Similar to the f r OPE, this doesn't mean that rr OPE is trivial. It only means that the factors in the numerator and denominator of A cancel.
(See the discussion around eq. (2.11) and (2.12)). Therefore, the OPE cannot be fixed using the residue formulae alone. Instead, one needs to apply the ansatz (5.39) on various initial states and obtains constraints when there are different final states appearing on both sides -the same method used to partially fix the e r OPE etc.
Applying this procedure and also using (5.40), we get (5.41) We have now fixed all the OPEs in the extended algebra (5.2).
6 Summary and discussion 6.1 Summary In this paper we have constructed a two-parameter family of affine Yangian algebras by gluing two affine Yangians of gl 1 with additional operators that transform as [ , ] w.r.t. the two affine Yangians of gl 1 . The representation space of the glued algebra consists of pairs of plane partitions whose asymptotic boundary conditions along the common direction are correlated as [λ, λ] or [λ, λ t ]. Demanding that the resulting algebra has sensible actions on these representations then fixes the entire algebra. Finally, we have also extended the algebra with additional gluing operators that transform as [ , ] w.r.t. the two affine Yangians of gl 1 .

Relation to the box-antibox construction
First, we would like to compare the glued algebras in this paper : box-box : where we have labeled the second affine Yangian algebra in the current paper asỸ with parameterq, to distinguish it from theŶ(q) in the previous construction. Since The relation (6.4) was derived based on constraints from twin plane partitions. We will now show that it also follows from the relation between the representations of gluing operators in (6.2) and (6.3), i.e. the gluing operators' representation w.r.t. Y is conjugate to their representation w.r.t.Ŷ.
Given a representation R of the W 1+∞ , its conjugate representationR can be characterized by the change of the signs for all odd-spin charges while leaving the even-spin charges fixed: Although there should exist a highly non-trivial outer-automorphism of W 1+∞ under which all its representations are mapped to their conjugate representations, however, we do not know how to express this automorphism explicitly, in terms of W 1+∞ generators.
To see how the conjugation is realized in terms of the affine Yangian of gl 1 , recall the automorphism of the affine Yangian of gl 1 (2.14). First of all, under this automorphism, the W 1+∞ charges transforms as For example, for the state (i.e. the descendant of the vacuum representation), the first few W charges are And for (i.e. the ground state of the (0, ) representation), we have This then explains the relation (6.4) between the parameters of the two gluing constructions (6.3) in [1] and (6.2) in this paper. This also provides another motivation to study the construction (6.1) or its extended version (6.2). The construction (6.3) only covers those algebras whose subalgebras have parameters satisfying (2.57) and (3.87). However, there exist algebras in which the W 1+∞ subalgebra do not belong to these classes. 10 Although theoretically there should be an automorphism that transforms the subalgebra into one under which all the gluing operators transforms as conjugates of their representations under the original subalgebra, it is not known how to carry out this transformation in the basis of the W algebra.
The non-trivialness of this transformation is reflected in the difference between actions of the algebra (6.3) in [1,12] on the pair of plane partitions and the actions of the algebra (6.1) or (6.2) constructed in the current paper. The main difference is in the action of the gluing operators, in particular, in their bud conditions. For example, for the algebra (6.3), the bud needs to have a minimal length, and all the boxes in it need to be grouped together into one bud; whereas for the algebra (6.1) or (6.2), the bud needs to have a precise length, but the boxes in it can be distributed between the left and right plane partitions. This results in the difference between the pole structures of the gluing operators' actions for the algebra (6.1) or (6.2) and the one for (6.3). Another interesting difference is that in (6.3), the action of a gluing operator might shift a box in the left plane partition to the right side, and vice versa. This can never happen in (6.1) or (6.2). Given these microscopic differences in the mechanisms in which the algebras are fixed by their actions on the twin plan partitions, it is quite remarkable that the final results on the algebraic relations of these two different constructions (the one [1] and the current one) can be related.

Future directions
Given their close relation to the toric Calabi-Yau threefolds, it is natural to expect the glued algebras (6.1), (6.2), and (6.3) to serve as symmetry algebras of some physical systems from the toric threefolds, such as topological strings or the systems in [19]. The microscopic difference between the actions (on twin plane partitions) of the algebra (6.3) constructed in [1,12] and that of (6.1) and (6.2) constructed in the current paper then raises the question as to which algebra applies in a particular situation. The algebra (6.1) constructed in the current paper has the advantage of having representations that are geometrically simpler, i.e. just the gluing of two (complements of) crystals -namely, no "high wall" is necessary. Hence it might be easier to relate to topological strings. It would be good to find systems in which one can match in detail the microscopic actions by the glued algebras on twin plane partitions with the more familiar physical or mathematical processes. This certainly deserves future study.
In addition, it is possible to extend the procedure in this paper and construct bigger affine Yangian algebras using the affine Yangian of gl 1 as building blocks, which would correspond to toric diagrams with more than two vertices. Take n copies of the affine Yangian of gl 1 , with parameters q i with i = 1, · · · , n. By definition, they are mutually commuting. Fix a subset of pairs of Y i , and for each pair < ij > introduce a set of gluing operators g ij that interacts both with Y i and with Y j : i Y i ⊕ <ij> g ij · · · , (6.11) and the · · · denotes possible additional operators that interact with a cluster of Y i , e.g. g ijk . By the isomorphism (2.17), the construction (6.11) would produce a vertex operator algebra that contains n (mutually commuting) W 1+∞ subalgebras, with parameters (c, λ) i (that corresponds to the q i ), and additional gluing operators that have non-trivial OPEs with pairs of W 1+∞ algebras. For example, to construct the affine Yangian that corresponds to the large N = 4 supersymmetric W ∞ algebra, one would need to glue 4 affine Yangians of gl 1 in a linear diagram [21]. Finally, for toric diagrams with loops, we need to generalize our techniques. Recall that for a toric diagram with two vertices (shown in Figure 4), the condition (3.72) that the external lines cannot intersect -namely, they cannot join and thus create the third vertex -correspond precisely to the plane partition constraint that the bud has non-negative length (3.63), i.e. the boxes in the initial configuration have to all sit inside the room. To construct the glued algebra for a toric diagram with loops, we need to relax this condition, i.e. |p| can now be greater than two for some of the vertices. What condition can then replace (3.63)? Can the representation space still be characterized as elegantly as the current twin plane partition? We leave these questions for future work.
where we have used the pole of ther action in eq. (5.20) and that g( ) = 0 for the first . The range ofn left has to start from 1 instead of 0: n left = 1, · · · , 2 + 2ρ , (A. 6) in order for the e action to happen. The second choice is that e can add a next to those boxes in the left plane partition along the x 2 direction: w = σ 3 ψ 0 − h 2 +n left h 2 and z =n left h 2 (A.7) withn left = 0, · · · , 2 + 2ρ (A.8) For the r.h.s., there is only one scenario: e adds a next to the initial along the x 1 or x 3 direction, and thenr removes the and leaves some boxes behind. The final states are identical to those in the first scenario of the l.h.s. in (A.5) with (A.6).
Therefore,Ḡ(∆)'s denominator must contain a factor ∆ + σ 3 ψ 0 − h 2 , in order to cancel those states in (A.7), which only appear on the l.h.s. but not on the r.h.s.. And sinceḠ is homogenous, the minimal result forḠ is where b is a constant to be fixed later. where we have used the pole of ther action in eq. (5.20) and that g( ) = 0 for the first . The range ofn left is again given by (A.6). Note thatn left has to start from 1 instead of 0, in order for thef to be able to remove a . On the other hand, the r.h.s. vanishes since f (z)| = 0. Taking the difference between the two poles in (A.11), we see thatH(∆) has to contain a factor of ∆ + σ 3 ψ 0 in the denominator, in order to remove these 2 + 2ρ states that only appear on the l.h.s. but not on the r.h.s..
Next, consider the initial state | + i , where the additional is at the position (x 1 , x 2 , x 3 ) = (1, 0, 0) for i = 1 or (0, 0, 1) for i = 3. Now for the l.h.s., there are two scenarios. In the first case, f (z) removes the initial . w = σ 3 ψ 0 + (n left − 1) h 2 and z = h i , (A. 12) with the range ofn left still given by (A.6). Note that heren left has to start from 1 instead of 0, in order for the initial not to "float" in the plane partition. The second possibility is that f (z) doesn't remove the initial but removes the box furthest to the right of then left 's: withn left given by (A.8). Now we compare the two sides. The 2 + 2ρ states in the second scenario (A.13) only appear on the l.h.s. but not on the r.h.s.. However, they are already taken care of by the factor ∆ + σ 3 ψ 0 in the denominator ofH(∆) that was fixed just now. Now compare the first scenario (A.12) for the l.h.s. with the states (A.14) for the r.h.s. There is only one state that appears only one on side: namely, (A.14) with n left = 0 for the r.h.s. To cancel this state, we need a factor ∆ + σ 3 ψ 0 − h 2 − h i in the numerator ofH(∆). Since i = 1 or i = 3 are on equal footings, we need (∆ + σ 3 ψ 0 + h 1 )(∆ + σ 3 ψ 0 + h 1 ).
To summarize, this analysis shows that As in the cases of G and H in (4.15) and (4.19), there is one constant b that cannot be fixed. It will be determined in the next subsection, together with the single boxes' contributions to theQ charge function.

A.2 Single boxes' contribution toQ charge function
The single boxes' contribution to theQ charge function are related to the following two OPEs between single-box annihilators and gluing operators The other three cases are completely parallel.