On extensions of glmn⏜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$\end{document} Kac-Moody algebras and Calabi-Yau singularities

We discuss a class of vertex operator algebras Wmn×∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{W}}_{\left.m\right|n\kern0.33em \times \kern0.33em \infty } $$\end{document} generated by a super- matrix of fields for each integral spin 1, 2, 3, . . . . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = zmwn. We propose a free-field realization of such truncations generalizing the Miura transformation for WN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{W}}_N $$\end{document} algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

An interesting perspective on W N × gl(1) algebras comes from considering the large N limit and treating the parameter N as a generic complex number [19][20][21][22][23][24][25][26]. The resulting algebra denoted as W 1+∞ is generated by infinitely many fields U 1 , U 2 , U 3 , . . . of spins 1, 2, 3, . . . and depends on two complex parameters 1 N and Ψ. Furthermore, the algebra W 1+∞ is rigid in the sense that it is the only algebra containing a single strong generator of each integral spin and satisfying Jacobi identities [24][25][26]. Algebras W N × gl(1) can be then recovered by specializing the parameter N to a positive integer and moding out the ideal generated by fields U i of spin greater than N .

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It was observed in [27] (based on results of [25,[28][29][30]) that W 1+∞ algebra actually contains a three-parameter family of truncations Y N 1 ,N 2 ,N 3 parametrized by non-negative integers N i . These more general truncations can be furthermore identified with VOAs from [31] defined in terms of the quantum Hamiltonian reduction. This cohomological definition is motivated by an analysis of a particular configuration of interfaces in fourdimensional gauge theory that provides us with a physical realization of the algebras. Using well-known dualities of string theory [27,[32][33][34], one can relate the setup of [31] to a configuration suggesting a natural generalization of the AGT correspondence and incorporating Y N 1 ,N 2 ,N 3 . Such more general truncations correspond to gauge theories supported on the coordinate planes C 2 12 , C 2 13 , C 2 23 inside C 3 with integers N i labeling ranks of U(N i ) gauge groups of the corresponding three theories. This generalized version of the AGT correspondence was proven in [34] (see also [35,36] for related discussion) by showing that algebras Y N 1 ,N 2 ,N 3 emerge from an action of the cohomological Hall algebra [37] on the equivariant cohomology of the moduli space of spiked instantons from [38,39] analogously to the standard W N × gl(1) story [16][17][18].
The identification of Y N 1 ,N 2 ,N 3 with the algebra arising from the action of the cohomological Hall algebra proceeds in two steps. First, one needs to identify the VOA associated to the simplest configurations Y 1,0,0 , Y 0,1,0 and Y 0,0,1 corresponding to rank-one theories supported on one of the three coordinate planes. Secondly, one can use an existence of the coproduct structure on W 1+∞ to fuse 2 such elementary factors into the general algebra Y N 1 ,N 2 ,N 3 .
More concretely, each of the elementary factors Y 1,0,0 , Y 0,1,0 and Y 0,0,1 can be simply identified with the gl(1) current algebra. The coproduct allows us to identify Y N 1 ,N 2 ,N 3 as a subalgebra of N 1 copies of Y 1,0,0 , N 2 copies of Y 0,1,0 and N 3 copies of Y 0,0,1 . Such an embedding can be conveniently carved out by a generalized version of the Miura transformation [2,3] for W N × gl (1). To define the Miura transformation, [40] first had to introduce N 1 copies of a pseudo-differential operator L (1) , N 2 copies of a pseudo-differential operator L (2) and N 3 copies of a pseudo-differential operator L (3) . By Miura transformation, we mean the process of multiplying these pseudo-differential operators and rewriting them in terms of a pseudo-differential operator in the standard form by commuting derivatives to the right. Generators U i of the resulting algebra can be then identified with coefficients of such a pseudo-differential operator.
It is natural to ask how general is the relation between VOAs and gauge theories supported on more general complex surfaces [17,[41][42][43] or collections of such surfaces wrapping various four-cycles inside higher-dimensional varieties [27,38,[44][45][46][47]. It has been expected for a long time that the above-mentioned AGT correspondence only scratches the surface of such a more general story referred to as AGT, 4d/2d or BPS/CFT correspondence by different people. By considering more general configurations, one expects to uncover a rich connection between VOAs and algebraic geometry of the corresponding configuration. 2 The gauge theories can be physically thought of as arising from the low-energy limit of Ni M5-branes supported on the three coordinate planes. Fusion can be intuitively understood as a process of bringing a system of separated M5-branes together. This motivates the existence of the fusion also in the more complicated setups discussed bellow.

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In this work, we illustrate the richness of the correspondence by analyzing a class of VOAs corresponding to gauge theories supported on a special class of divisors inside the Calabi-Yau singularity CY 3 m,n (or its resolutions) given by solutions to the equation for m, n any non-negative integers. The relevant divisors are those that come from zeros of the holomorphic functions of the form Analogously to the above Y N 1 ,N 2 ,N 3 algebras, we expect an existence of a definition of the corresponding x N 3 y N 2 z N 4 w N 1 -algebra in terms of 1. Truncations of the algebra W m|n×∞ generated by an m|n super-matrix of generators for each integral spin 1, 2, 3, . . . .

Generalized
Miura transformation 3 in terms of a subalgebra of a tensor product of N 3 copies of the x-algebra, N 2 copies of the y-algebra, N 1 copies of the z-algebra and N 4 -copies of the w-algebra. The elementary algebras associated to x, y, z, w will be (roughly) identified with gl(m|n) Kac-Moody algebras at specialized levels.
We expect these two definitions to allow an identification of the x N 3 y N 2 z N 4 w N 1 VOA with an algebra coming from the geometric action of the cohomological Hall algebra on the equivariant cohomology of the corresponding moduli space of instantons along the lines of [34].
In particular, W m|n×∞ should emerge from doubling the cohomological Hall algebra 4 associated to the given geometry CY 3 m,n and the generalized Miura transformation from its coproduct structure. Understanding the above-mentioned definitions can be thought of as a first step in proving the AGT correspondence for divisors inside CY 3 m,n that was the main motivation for the project.
Similarly to W 1+∞ , algebras W m|n×∞ depend on two complex parameters Ψ and N [27,49]. As discussed in the main text, the two parameters can be given a natural geometric interpretation. In particular, the singularity CY 3 m,n admits a T 2 action preserving the Calabi-Yau volume form. Parameter Ψ then corresponds to a ratio of equivariant parameters associated to such an action. On the other hand, a specialization of the parameter N can be identified with the charge of the function Finally, note the relation (1.1) in the ring of holomorphic functions. We expect the algebras associated to functions for a detailed discussion of the Miura transformation for W m|0×∞ and an identification of some of the truncations by an analysis of null states at low levels.
4 W m|n×∞ should emerge as a double of the relevant cohomological Hall algebra. Note that we could consider more general divisors associated to sections of non-trivial bundles on a resolution of CY 3 m,n . We expect these to correspond to truncations of different doubles of the cohomological Hall algebra. Investigation of such more general configurations is left for future work since we do not know how to generalize the Miura transformation for these cases at the moment.

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such a relation to be equivalent since they both correspond to the same geometric data. On the other hand, both expressions x N 3 y N 2 z N 4 w N 1 and x N 3 y N 2 z N 4 +m w N 1 +n lead to a different realization of the algebra in terms of a different subalgebra of a different system of elementary factors. Each relation in the ring of holomorphic functions thus leads to a non-trivial relation between various free-field 5 realizations. For example, the simplest relation xy = z for the simplest geometric configuration CY 3 1,0 = C 3 leads to the wellknown bosonization relation that relates the free boson with the β, γ system. The structure of the paper is as follows: section 2.1 discusses the geometry of CY 3 m,n . Specializations m = 0 are known under the name Z n singularities and CY 3 1,1 is known as the conifold singularity. These two cases are extensively discussed at many places in the literature but I have not found a discussion of the more general CY 3 m,n case. We describe the toric action mentioned above and the ring of holomorphic functions. Section 2.2 discusses resolutions of CY 3 m,n and establishes a precise connection to the (p, q)-webs colored by integers from [27,31]. Section 3 reviews some aspects of the W 1+∞ algebra and the free-field realization of its truncations associated to holomorphic functions on CY 3 0,1 = CY 3 1,0 . The algebra W m|n×∞ generalizing W 1+∞ is discussed in section 4. We derive OPEs of weightone and weight-two fields extending the analysis of [48] for the W m|0×∞ case. Section 5.1 introduces the coproduct structure on W m|n×∞ generalizing the W 1+∞ case. In section, 5.2 discusses the elementary building blocks of W m|n×∞ truncations associated to generators of the ring of holomorphic functions on CY 3 m,n , i.e. x, y, z, w. Finally, section 5.3 discusses how to obtain a free-field realization of a general x N 3 y N 2 z N 4 w N 1 -algebra associated to the function x N 3 y N 2 z N 4 w N 1 by composing Miura operators L (x) , L (y) , L (z) , L (w) associated to the elementary functions. We argue that relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations.

Calabi-Yau singularities
The main aim of this work is a discussion of truncations of the algebra W m|n×∞ that is strongly generated by an m|n super-matrix of generators at each spin s = 1, 2, . . . . Such truncations were conjecturally associated to a particular class of divisors inside a particular toric Calabi-Yau three-fold (see [27]). Here, we propose that the relevant geometry can be identified with a resolution of the following singularity This singularity generalizes the well-studied Z m singularity and the conifold singularity. In particular, note that for m = 1, n = 0 or m = 0, n = 1, the constraint in (2.1) becomes simply z = xy or w = xy, one can solve for z or w and the corresponding three-fold is just 5 Note that Kac-Moody algebras are not free algebras and embedding inside a tensor product of Kac-Moody algebras is literally not a free-field realization. On the other hand, Kac-Moody algebras can be themselves expressed as subalgebras of products of free bosons, b, c and β, γ systems that motivates the name "free-field realization".
The three-fold CY 3 m,n admits a natural T 3 action that we parametrize as One could in principle choose any other parametrization of the torus that is consistent with the relation in (2.1). We will see that this choice has a nice interpretation in the (p, q)-web picture discussed bellow and h i correspond to scaling of coordinates in a particular chart on the resolved three-fold. The Calabi-Yau volume form Ω of CY 3 m,n can be expressed in the x = 0 and the y = 0 patch as 3) The subtorus T 2 ⊂ T 3 preserving the Calabi-Yau volume form then corresponds to the above parameters h i specialized to According to the conjecture of [27], one can associate a vertex operator algebra to any toric divisor inside a toric Calabi-Yau three-fold. We will restrict to a particular class of divisors that can be identified with zero sections of holomorphic functions on CY 3 m,n . As discussed in section 2.2, resolutions of CY 3 m,n lead to configurations from [27] that are expected to lead to truncations of shifted versions of algebras W m|n×∞ . The special class of divisors at hand will be identified with configurations with zero shifts and leading to truncations of W m|n×∞ itself. The relevant divisors are thus in one-to-one correspondence with functions 5) for N 1 , N 2 , N 3 , N 4 non-negative integers, modulo the relation xy = z m w n in the coordinate ring of CY 3 m,n . Such functions scale with the power of

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(1, 0) (0, 1) under the T 2 action described above. This constant will play an important role in the discussion of vertex operator algebras below. Furthermore, note that it reduces to a simple expression for CY 3 1,0 = C 3 , where we can use the relation (2.1) to set N 4 = 0. This specialization is the reason for the above convention for the labels N i in the exponents of (2.5).

Relation to (p, q)-webs
We will now introduce resolutions of CY 3 m,n and their toric diagram that will allow us to make a contact with the (p, q)-webs colored by integers from [27]. For each CY 3 m,n , let us introduce m + 1 complex coordinates z i ∈ C and n + 1 complex coordinates w i ∈ C. Different triples of z i , w i will correspond to coordinates on different patches of the resolved variety. In terms of z i , w i , one can solve the constraint in (2.1) and write There is obviously a lot of freedom in parametrizing x, y, z, w in terms of z i , w i . In particular, x, y, z, w are invariant with respect to the scaling of each triple z i , z i+1 , z i+2 or w i , w i+1 , w i+2 with charges +1, −2, +1 and w 1 , w 2 , z n , z n+1 with charges −1, +1, +1, −1.
The corresponding moment maps for such C × actions are 6 (2.9) 6 Note that we have one moment map condition associated to each root of the gl(n) and gl(m) subalgebra of gl(m|n) and one associated to the fermionic root. One can define a different system of moment maps corresponding to a different choice of the root system in gl(m|n) and leading to a different resolution of CY 3 m,n than the one discussed below. The Calabi-Yau singularity CY 3 m,n can be then alternatively defined as a symplectic quotinent i.e. the preimage of µ i = µ = µ j = 0 inside C m+n+2 modulo U(1) rotations of the coordinates with charges described above.
Calabi-Yau singularity CY 3 m,n can be resolved by introducing a real parameter for each moment map l (2.11) Resolved three-folds CY 3 m,n can be thought of as T 2 × R fibrations over R 3 with the T 2 fiber corresponding to the above h 1 , h 2 , h 3 satisfying (2.4) and the base parametrized by corresponding moment maps. The geometry of CY 3 m,n can be then encoded in a (p, q)-web diagram, i.e. a diagram indicating loci in the base where one of the T 2 cycles degenerate [50]. The (p, q)-web is composed of trivalent vertices associated to different coordinate patches and mutual orientation of vertices then indicate how are two patches glued together. Let us start with a discussion of the toric diagram for CY 3 1,0 = C 3 . The variety can be simply parametrized by x = z 1 , y = z 2 , w = w 1 that scale as with the corresponding moment maps Considering CY 3 0,1 = C 3 goes in a similar fashion and produces a diagram with an opposite orientation from the figure 1 on the right.
Toric diagram of a more complicated CY 3 m,n can be obtained by gluing more vertices to the simplest trivalent vertex. Let us first illustrate the derivation of the toric diagram for CY 3 2,0 = CY 3 0,2 = C 2 /Z 2 × C and the resolved conifold CY 3 1,1 in detail. After dealing with these two examples, we can directly draw the corresponding diagram for CY 3 m,n since such a general case is a simple combination of the two examples and corresponds to an iterative use of the moment maps (2.9).
First, in the CY 3 2,0 case, we have with the corresponding moment map constraint Let us first look at the patch where z 3 = 0. We can then identify the triple w 1 , z 2 , z 1 with coordinates on the corresponding patch with the T 2 action as above (2.12). Using the moment map constraint, one can solve for |z 3 | and adjust its phase arbitrarily using the corresponding S 1 action. We can see that the scaling by h 1 , h 2 , h 3 again corresponds to rotations of the three coordinates in this patch and the corresponding toric diagram in this patch is again the trivalent junction from figure 1. Let us now move to the patch z 1 = 0. The coordinates in this patch are (w 1 , z 3 , z 2 ). The moment maps (2.13) can be rewritten using (2.15) as with the corresponding fiber generated by Note first that the corresponding trivalent junction associated to this patch is shifted by (−l 1 ) in the (µ 1 , µ 2 ) plane. Furthermore, we see that in this patch, we get JHEP01(2020)042 degenerations along the lines (1, 0), (1, 1) and (2, 1) with h 1 = 0, h 1 + h 2 = 0, 2h 1 + h 2 = 0 being the degenerating cycles. The corresponding toric diagram of CY 3 2,0 is shown in the figure 2 on the left. We get an analogous diagram for CY 3 0,2 with an opposite orientation similarly to CY 3 0,1 .
For CY 3 1,1 , we have with the corresponding moment map First, the patch w 2 = 0 can be again parametrized by the triple (w 1 , z 2 , z 1 ) leading to one trivalent vertex with h i rotating the three coordinates. In the chart z 1 = 0, we have coordinates (w 1 , z 2 , w 2 ) with the corresponding moment maps and the associated toric action We can see that this leads to a vertex shifted by (−l, −l) in the (µ 1 , µ 2 ) plane with (1, 0), (0, 1), (1, 1) cycles degenerating as shown in the figure 2 on the right. Finally, let us describe a general (p, q)-web associated to a general CY 3 m,n . As in the previous examples, one can again associate one vertex with the (w 1 , z 2 , z 1 ) coordinates corresponding to the patch where all the other z i and w i are non-vanishing. We can then use the moment map conditions µ (z) i to find moment maps on patches parametrized by triples (w 1 , z i+1 , z i ) giving rise to a sequence of vertices from the figure 3. The moment map condition µ attaches a reversed vertex with the (1, 0) line on the left to the last line of figure 3 as in the resolved conifold example. Finally, using moment maps µ (w) i , one can find expressions for µ 1 and µ 2 on patches parametrized by the triple (w i , z n+1 , w i+1 ). The resulting (p, q)-web is shown in the figure 4.
Let us now discuss functions (2.5) and divisors associated to their zero sections from the perspective of the (p, q)-web. First, considering the simplest case CY 3 1,0 = C 3 , the coordinate planes x = 0, y = 0, w = 0 map to the three corners of the toric diagram under the maps (2.13). The divisor of a general function x N 3 y N 2 w N 1 can be then labeled by the colored (p, q) web from the figure 5 with integers N 1 , N 2 , N 3 indicating multiplicities of different smooth components of the divisor.     This discussion has a generalization to the general case of CY 3 m,n . Each function x, y, w, z can be rewritten in terms of z i , w i using (2.8). Restricting to a patch associated to a given vertex, we can use the moment map conditions 2.19 to solve for |z i | and |w i | that are not part of the corresponding triple and remove the phase of the corresponding z i , w i . One can then read off the corresponding multiplicities of the smooth components in the given patch from powers of the corresponding z i , w i and deduce integers for the corresponding vertex in the (p, q)-web. This leads globally to a colored (p, q)-webs with integers associated to each face of the (p, q)-web. Divisors associated to the elementary functions x, y, z, w from the (p, q)-web perspective are shown for the example of CY 3 2,3 in figure 6. Generally, x leads to a diagram with decreasing numbers from the top to the bottom starting with m on the right and n on the left. On the other hand, y corresponds to decreasing numbers from the bottom to the up again starting with m on the right and n on the left. z is associated with ones on the right-hand side of the diagram and w with ones on the left.
Let us conclude this section with two remarks. First, multiplication of holomorphic functions corresponds to summing corresponding divisors and in particular summing multiplicities associated to various faces of the toric diagram. The relation xy = z m w n then JHEP01(2020)042 equates two different ways of producing the same divisor using such sums. On the VOA side, this multiplication will lead to a coproduct encoded in a generalization of the Miura transformation. Secondly, note that [27] conjectured that algebras associated to the web 4 should be truncations of shifted versions of W m|n×∞ with shifting parameters associated to each finite segment of the (p, q)-web. The shift parameters associated to various configurations in our setup are schematically depicted in figure 7. They can be identified with the U(1) charge of the divisor with respect to the action generated by moment maps (2.9). Note that all the configurations associated to x N 3 y N 2 z N 4 w N 1 have zero shift. Furthermore, it is not hard to convince yourself that all zero-shift configurations come from some The special class of divisors discussed here form a subclass of algebras that are independent of the resolution of the (p, q)-web. This is not going to be the case for algebras associated to divisors associated to sections of a non-trivial line bundle. Dependence on the resolution would have to be taken track of in such examples.

Review of the C case
Let us now move to the discussion of vertex operator algebras and review the generalized Miura transformations from [40]. W N × gl(1) algebras admit a well-known realization [2,3,25,51] in terms of a subalgebra of N copies of gl(1). Let us normalize the corresponding strong generators J (3.1) We can then define Miura operators as formal expressions Considering a product of such operators and rewriting it as an N 'th order differential operator in the standard form by commuting all derivatives to the right, one obtains strong generators U i of the W N × gl(1) algebra in so-called U -basis (sometimes also called the quadratic basis since OPEs in this JHEP01(2020)042 basis are quadratic in derivatives of U i 's). In particular, we have that leads to an algebra with first few OPEs of the form Considering the large N limit and treating the parameter N as a generic complex number, one obtains an algebra W 1+∞ generated by fields U i depending on two continuous parameters κ and N with the above OPEs. It was argued in [24][25][26] that this algebra is actually uniquely fixed by requiring the associativity of OPEs and an existence of a single strong generator for each integral spin. We can then look at the algebras W N × gl(1) as truncations of such a two-parameter family of algebras W 1+∞ coming from the specialization N ∈ Z to be an integer and setting U i = 0 for i > N . It was argued in [27] that there actually exists a larger family of truncations Y N 1 ,N 2 ,N 3 parametrized by three integral numbers N 1 , N 2 , N 3 . To describe the specialization of the parameter N associated to such truncations, it is useful to introduce parameters h 1 , h 2 , h 3 such that 7

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We can then write the formula for the specialized N as Note that the numerator equals the charge (2.7) for the transformation of holomorphic functions on CY 3 1,0 = C 3 with respect to the T 2 action. Furthermore, the expression is independent of the overall scale of h i and the resulting algebra in the end depends only on the ratio Ψ. We will see a natural generalization of the formula for CY 3 m,n bellow. It was found in [40] that there actually exist two more Miura operators whose products give rise to Y N 1 ,N 2 ,N 3 truncations. Let us normalize the currents appearing in these two Miura operators as 8 It is a simple calculation to identify combinations of normally-ordered products of derivatives of J (w) and J (y) respectively that satisfy OPEs (3.5) with the parameter N specialized according to (3.7) as (N 1 , N 2 , N 3 ) = (1, 0, 0) in the first case and as (N 1 , N 2 , N 3 ) = (0, 1, 0) in the second case. One gets = (2κ + 1)(3κ + 1) 6(κ + 1) 2 (J (y) J (y) J (y) ) − 3(κ + 1)(J (y) ∂J (y) ) + (κ + 1) 2 ∂ 2 J (y) .
Having identified U -generators of the three elementary truncations, we can then define a generalized Miura operator as 9

11)
8 Note that the labeling x, y, w is in correspondence with generators of the ring of holomorphic functions on CY 3 1,0 = C 3 from the previous section. 9 A simple and uniform expression for L (x) , L (y) , L (w) in terms of a conjugation of ∂ h i h 3 by an exponential operator appeared in [52]. We do not use this representation here since it is not obvious how to generalize such expressions to the matrix case described below. It would be nice to check whether such a generalization exists.

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where α = x, y, w and N α is the correctly-specialized parameter N , i.e. N x = 1 for the xtruncation, N y = h 2 h 3 for the y-truncation and N w = h 1 h 3 for the w-truncation. Multiplying i and commuting derivatives to the right using the Leibniz rule, we recover a pseudo-differential operator whose coefficients U i give the free-field realization of generators of the algebra. Note that N is additive under such a multiplication and its value for a general truncation agrees with (3.7). The above free-field realization can be equivalently characterized in terms of an intersection of kernels of a system of screening charges as discussed in [29,30,40]. In particular, the two-parameter family of algebras W (gl n,m ) from [29] corresponds to our Y m,n,0 or geometrically the divisor of x m y n inside C 3 1,0 = C 3 . After setting the notation for W 1+∞ that is convenient to make contact with the geometric picture from the previous section, we will move to a matrix generalization of the story. As we will see, the interplay between the geometry and properties of VOAs become much more non-trivial in the matrix case.

W m|n×∞ algebra
We will now discuss a matrix generalization of the W 1+∞ algebra, i.e. an algebra W m|n×∞ generated by m|n super-matrices of fields. The relation of the geometry discussed above with the matrix generalization of W 1×∞ was already pointed out in [27] who proposed an existence of a gluing construction of the relevant algebras in terms of an extension of products of Y N 1 ,N 2 ,N 3 . At the level of vacuum characters (in the large N i -limit), the gluing proposal is identical with the topological vertex calculation [27,45,46] of BPS states/Donaldson-Thomas invariants. The class of examples discussed in this work lead to the vacuum character of W m|n×∞ . Finite N i algebras turn out to contain less states states than the infinite limit [47] and it is reasonable to expect that they can be identified with truncations of the universal algebra W m|n×∞ , analogously to the case of W 1+∞ .
Analogously to the W 1+∞ case, we expect that there exists a two-parameter family of algebras W m|n×∞ satisfying the following conditions: 1. W m|n×∞ contains the Kac-Moody algebra gl(m|n)κ at levelκ as a subalgebra.
2. W m|n×∞ is strongly generated by a m|n super-matrix of generators U i,ab at each integral spin i = 1, 2, 3, . . . . Indices a, b label the row and the column index of the super-matrix with a ≤ m bosonic and a > m fermionic directions, such that the n × n and the m × m diagonal block consists of bosonic generators with the off-diagonal blocks being fermionic.
3. The generators U i,ab at given spin i transform in the adjoint representation of the gl(m|n) zero-mode subalgebra of gl(m|n)κ.

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In analogy with the W 1+∞ algebra, it is tempting to speculate that W m|n×∞ is uniquely fixed by Jacobi identities up to the two parameters. Authors of [48] argued that one can easily find OPEs of W m|0×∞ in a generalized U -basis using a matrix generalization of the Miura transformation. In this section, we will use the same method to determine OPEs of low-spin generators of its super-matrix generalization W m|n×∞ .
Our starting point is the gl(m|n) κ Kac-Moody algebra generated by fields J gl(m|n) κ is clearly a truncation of W m|n×∞ in the same way as gl (1) is the simplest truncation of W 1+∞ . In the above OPE, we have introduced a continuous parameter and |a| = 0 for fermionic directions a ≤ m and one otherwise. For the notational convenience, we often omit the brackets |a| in exponents from now on. One can then define the Miura operator i,ab entries. We can then consider a product of N such factors and rewrite them as a super-matrix-valued differential operator in the standard form multiplying as super-matrices and U i being super-matrices as well. In particular, we have in terms of components (4.5)

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Using these expressions, it is straightforward (though technically demanding) to determine OPEs of these fields. For the first few fields, we get 10 These four OPEs are going to be sufficient for testing our proposals for elementary Miura operators and the generalized Miura transformation below.
We can now state the following conjecture: Conjecture 1 There exists a VOA satisfying conditions listed above and containing fields U 1,ab , U 2,ab , U 3,ab satisfying (4.6) for any Ψ, N ∈ C. In analogy with W 1+∞ , we conjecture that the algebra is uniquely fixed by associativity of OPEs.
Let us finish this section by making few comments on the relation to other work. Structure constants of W m|n×∞ are up to the (−1) # factors the same as those in [48]. It should be straightforward to find a proposal for all OPEs of W m|n×∞ by correctly recovering these (−1) # factors in the formulas of [48]. Algebras of type m|n = m|0 have been recently also studied in a different basis for example in [27,49,53,54] who used names rectangular W-algebras or W m+∞ . Furthermore, the algebra W 1+∞ is well-known to be isomorphic to the affine Yangian of gl(1) [16,28]. We expect our W m|n×∞ to be isomorphic to the affine Yangian of the Lie super-algebra gl(m|n) discussed from different perspectives for example in [55][56][57][58]63]. q-deformed analogues of affine Yangians (quantum toroidal algebras) of type m|0 appearing in the connection to gauge theories on C 2 /Z m singularities were also discussed for example in [59,60]. Finally, let me mention a recent appearance of supergroups in a gauge theory context from [61,62]. i (z) is a super-matrix of the gl(m|n) κ Kac-Moody generators to determine OPEs of the W m|n×∞ algebra in the U -basis. In this section, we generalize the above discussion in two ways. First, we introduce more general matrix-valued pseudo-differential Miura operators to define coproduct on W m|n×∞ . The composition of L (x) i can be then interpreted as a successive use of such a coproduct for the simplest truncation of the algebra. Secondly, we use the geometric and gauge-theoretical picture to determine four elementary truncations associated to functions x, y, z, w. All the other truncations associated to x N 3 y N 2 z N 4 w N 1 can be then constructed using the coproduct of W m|n×∞ and the knowledge of the elementary factors.

Coproduct of W m|n×∞
The Miura operator coming from the product of N elementary Miura factors L (x) i has generally the following form

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where U i are m|n super-matrices of generators of spin i. Furthermore, one can immediately identify the order of the differential operator N with the parameter N appearing in the OPE of gl(m|n) currents i.e. the level of the sl(m|n)κ subalgebra equalsκ = N κ. Since N can be an arbitrary complex number for the general W m|n×∞ algebra, one might be tempted to consider a more general class of pseudo-differential operators containing infinitely many super-matrix-valued fields with U i satisfying OPEs of W m|n×∞ with parameter N equal the order of the pseudo-differential operator.
In analogy with the above analysis, one can define a coproduct on W m|n×∞ by multiplying two factors of this form and expressing it in terms of a pseudo-differential operator of order N 1 + N 2 using (3.12).
This gives an expression of the new generatorsŨ i of the algebra W m|n×∞ for N = 1,ij , 1,kj ) + N κ∂U (2) 1,kj , We have checked that these satisfy OPEs of W m|n×∞ from (4.6) leading us to the following conjecture: Conjecture 2 Miura transformation from (5.5) gives rise to a well-defined coproduct structure on W m|n×∞ with N = N 1 + N 2 being additive.

Building blocks of truncations
After allowing more general pseudo-differential operators in the W 1+∞ case, we have seen that there exist two more elementary Miura operators L generator of the gl(1) Kac-Moody algebra associated to y and w. Composition of such more general Miura operators yields a more general class Y N 1 ,N 2 ,N 3 of W 1+∞ truncations.
We will now use the geometric and the gauge-theoretical picture to identify the elementary algebras associated to divisors of x, y, z, w. These will play the role of elementary building blocks for more general truncations analogously to gl(1) for Y N 1 ,N 2 ,N 3 .
The identification of corresponding algebras can be easily done along the lines of [27,31]. Let us briefly review the idea behind such an identification. Consider a general (p, q)-web configuration from figure 4. The corresponding geometry is known to be dual to a junction of (p, q)-branes in type IIB string theory determined by the same (p, q)-web diagram. Original M5-branes supported on four-cycles inside the toric three-fold map to various stacks of D3-branes stretched between (p, q)-branes. The low energy behavior of such D3-branes is captured by N = 4 SYM super Yang-Mills theories divided by (p, q)interfaces descending from the D3-branes ending on (p, q)-branes. Zooming out the (p, q)web, we get a junction of four interfaces dividing the four-dimensional spacetime into four (or three in the special case of n = 0 or m = 0) corners. The path integral of such a configuration localizes to the path integral of the complexified Chern-Simons theory supported on (n, 1) interfaces with supergroups determined by ranks of gauge groups on the right and on the left of the interface along the lines of Mikhaylov and Witten [65]. The two Chern-Simons theories are furthermore connected by an interface descending from (1, 0) branes. Local operators at such an interface give rise to a vertex operator algebra. An analysis of corresponding interface conditions [31,64,65] leads (at least in some cases) to a proposal for the corresponding algebra in terms of a BRST reduction implementing the interface condition. Luckily, the interface conditions and the corresponding algebras are easy to identify for the simple configurations of x, y, z, w and will be discussed bellow. Interested reader should consult [27,31] for further details and more general configurations.
x-algebra. The gauge-theoretical configuration consists of the U(n) gauge theory in the upper-left corner and the U(m) gauge theory in the upper-right corner. These two are divided by an interface labeled as (0, 1) and both terminate on the horizontal boundary. The boundary condition on the horizontal line was identified in [66] with the Dirichlet boundary condition and the (0, 1) interface couple the two theories by an additional threedimensional bi-fundamental hypermultiplet. It was argued in [31,64,65] that such a gaugetheoretical configuration localizes to the complexified GL(m|n) Chern-Simons theory on the (0, 1) interface with the Dirichlet boundary condition descending from the boundary conditions of the four-dimensional theory. Local operators at the boundary of the Chern-Simons theory (corner of the four-dimensional setup) are known to form the Kac-Moody algebra associated to the gauge group of the corresponding gauge theory. The level of the Kac-Moody algebra gl(m|n) κ is then given by 11 κ = Ψ − m + n (5.7) 11 Note that we reproduce the relation Ψ = κ + 1 from the discussion of W1+∞ algebra if we set m = 1, n = 0.

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where Ψ is the coupling constant of the four-dimensional theory. The building block associated to x is then the Kac-Moody algebra with OPE that is exactly the algebra used above in the construction of the W m|n×∞ algebra.
y-algebra. The configuration associated to y is analogous with only two differences. First, the orientation of the diagram is reversed that leads to the opposite sign of the level. Secondly, the vertical interface is now (m − n, 1) that leads to an extra shift of the level by m − n. The resulting algebra is the Kac-Moody algebra gl(m|n) at level The generators J (y) ab thus satisfy relations i.e. note the particular normalization of the diagonal gl(1) generator.
z-algebra. In the z configuration, the path integral localizes to two copies of GL(1) Chern-Simons theories at levels κ and −κ − m + n coupled together at the interface. It was argued in [27,31] that a consistent coupling requires m copies of β, γ ghosts (or symplectic bosons) associated to the (1, 0) lines ending from the right and n copies of b, c ghosts (free fermions) associated to (1, 0) lines ending from the left, all charged under the GL(1) gauge group. Corresponding VOAs can be then identified with the coset gl(1) × S m|n gl(1) where S m|n is a collection of fields (X i , Y i ) that consists of bosonic generators β i , γ i for i = 1, . . . , m and fermionic generators b i , c i for i = m + 1, . . . , m + n with OPEs The fields X i have +1 under the GL(1) action and Y i is are charge −1. More concretely, the above coset denotes the BRST cohomology of two gl(1) currents J,J together with S m|n with respect to the BRST charge

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that imposes the above-described constraint quantum mechanically. In this expression, we have introduced an auxiliary c, b ghost system. The two currents are normalized as (5.14) One can easily check that combinations of the form are generators of the the cohomology at spin one. Moreover, combinations 12 generate the gl(m|n) −1 Kac-Moody algebra at level i.e. satisfying We expect the generators J (z) 1,ab to be actually strong generators of the algebra unless (m|n) = (1|0) or (2|0). This expectation is supported by the the following arguments. The character of (5.12) is given by the contour integral (5.20) For generic values of (m|n) the algebra contains less states than algebra freely generated by an m|n super-matrix of free generators at spin one. One can also explicitly check this expectation at low levels for small numbers m, n.
In general, normally ordered products of derivatives ofJ 1,ab satisfy various relations that follow from the realization in terms of (5.15). The only exception is the m|n = 1|1 case for which the character agrees exactly with the character of gl(1|1). This can be understood as a consequence of the S 4 symmetry of the corresponding (p, q)-web diagram after zooming out. In such a picture, the (p, q)-web consists of a single (1, 0) and (0, 1) line 12 For n = m, the expression simplifies to crossing each other. All the four configurations associated to x, y, z, w then correspond to the same diagram and they should all lead to the same algebra. Let us briefly discuss the two special examples (m|n) = (1|0), (2|0). First, comparing the character of (2|0) with the the character of an algebra generated by four generators at spin one, we can see that we are missing three fields at spin two. One can define the following combinations and show that (at least at low levels) J Moreover, in analogy with (5.21), we can define and argue that combinations J Finally, note that the resulting algebra is actually independent of the parameter κ in all the cases and one can find a more convenient description by simply sending κ → ∞. In all the above expressions (5.15), (5.21) and (5.23), the tail of fields containing the current J disappears and one finds simpler expressions for i > 1. On the other hand, we cannot simply restrict to fields at low spin in this limit since we would miss some of the generators (the simplified generators satisfy more relations). Instead of the above more-complicated expressions in terms of X i , Y i and J, one can equivalently consider the algebra generated by infinitely many simpler generators of the form (5.24). In the following sections, we will use this simpler, κ independent realization of the algebra.

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w-algebra. The situation associated to w is analogous to the z case with (X i , Y i ) now being fermionic for i = 1, . . . , n and bosonic for i = n + 1, . . . , m + n. Normalizing the J,J currents as we have generating the cohomology at spin one with the combination identified with the standard generators of the gl(m|n) 1 Kac-Moody algebra at level κ = 1, (5.28) i.e. satisfying OPE The discussion of strong generators is the same as above just with the role of m and n interchanged and up to some minus signs. For example at spin two, we have the following combination Instead of these generators, we will again consider the realization of the algebra coming from the κ → ∞ limit, that can be realized in terms of as above.
According to the above analysis, the elementary algebras x, y, z, w form particular truncations of the W m|n×∞ algebra. We will now determine the specialization of the parameter N associated to such elementary truncations and find combinations of their fields that satisfy OPEs of W m|n×∞ in the U -basis with correctly specialized parameter N . Using such an explicit realization of U -generators and the coproduct from the above section, one JHEP01(2020)042 can define a general x N 3 y N 2 z N 4 w N 1 -algebra by multiplying the elementary Miura factors associated to x, y, z, w.
Since N is additive under the coproduct, we can immediately identify its specialization associated to the general x N 3 y N 2 z N 4 w N 1 -algebra by determining N of the elementary factors. As discussed bellow, one recovers that is exactly (up to the overall normalization) the charge of x N 3 y N 2 z N 4 w N 1 from (2.6).
The normalization is such that the order of L (x) i is 1. For m|n = 1|0, the above expression simplifies to from [27].
x-algebra. The case of the x-algebra is trivial. Comparing (5.8) with the general expression (5.21), one gets order N x = 1 as expected. The corresponding Miura operator is simply y-algebra. We will now look at the y-algebra as a quotient of the two-parameter family of algebras (4.6) parametrized by κ = Ψ − m + n and N . As discussed above, the y-algebra contains the Kac-Moody algebra sl(m|n)κ at levelκ = −Ψ = −κ − m + n. One can then writeκ A ac A cb (5.37) and all the products are normally ordered. Note that an existence of the realization of U ab is a non-trivial test of our proposal for generalized the Miura transformation for truncations of W m|n×∞ beyond x N -algebras.

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z-algebra. Similarly as above, we have now an algebra containing the Kac-Moody algebra sl(m|n) −1 at levelκ = −1 and we can again writẽ and identify the specialization N z of the parameter N consistent with the formula (5.32) for It is straightforward to identify the first few generators U (z) i,ab satisfying (4.6) with N correctly specialized in terms of the of the fields J (z) 1,ab + 2(κ +κ)δ ab str(∂J w-algebra. The identification of the parameter N is the same as in the above case of the z-algebra up to an overall minus sign consistently with (5.32). Similarly, the expression for U (w) i,ab has the same form 1,ab .
Having determined the elementary Miura factors L (x) , L (y) , L (z) , L (w) , one can define an algebra associated to any function of the form x N 3 y N 2 z N 4 w N 1 determined as a subalgebra inside a tensor product of N 3 copies of the gl(m|n) κ Kac-Moody algebras, N 2 copies of the gl(m|n) −κ−m+n Kac-Moody algebras, N 4 copies of X i Y i containing m pairs of bosonic fields and n pairs of fermionic fiels, and finally N 1 copies of X i Y i containing m pair of fermionic fields and n pairs of bosonic fiels. The subalgebra can be obtained similarly as above in the W 1+∞ case or the x N 3 -algebras from the derivation of the OPEs of W m|n×∞ by multiplying and rewriting it as a matrix-valued pseudo-differential operator in the standard form with U i super-matrices of generators at spin i and L (α) being the following pseudodifferential operator L (α) = (κ∂) Nα + U for α = x, y, z, w. This gives in particular the following expression for the spin-one and the spin-two fields Let us conclude by stating the following conjecture: The U i generators of W m|n×∞ with the parameter N specialized to N α can be realized in terms of the elementary algebra α = x, y, z, w. Coproduct of W m|n×∞ then gives rise to a free-field realization of the x N 3 y N 2 z N 4 w N 1 algebra.

Bosonization-like relations
Remember the relation xy = z m w n in the ring of holomorphic functions on CY 3 m,n that relate functions x N 3 −1 y N 2 −1 z N 4 w N 1 and x N 3 y N 2 z N 4 +m w N 1 +n . On the other hand both x N 3 −1 y N 2 −1 z N 4 w N 1 and x N 3 y N 2 z N 4 +m w N 1 +n corresponds to a different free-field realization using the Miura transformation (5.44). As a consequence of the above discussion, the generators of the algebra satisfy the same relations but one algebra might still be a subalgebra of the second other. Since both functions correspond to the same physical setup, we actually expect this not to be the case and the algebras should be equal on the nose. This leads to the following conjecture: Conjecture 4 The pseudo-differential operators for CY 3 m,n satisfy i.e. algebras on both sides have the same size and satisfy the same OPEs.
Let us illustrate this relation on the simplest example of xy = z for CY 3 1,0 . In particular, we will now show that the free-field realizations of fields U 1 , U 2 can be related by the standard bosonization of β, γ ghosts. The two Miura operators associated to x and y are with normalization (5.50) Multiplying these two L (y) L (x) gives 1 + (J (x) + J (y) )(κ∂) −1 + 1 + 2κ 2 On the other hand, we have for the free-field realization of the z-algebra in terms of the β, γ system where J relates the two free-field realizations and can be thought of as the VOA analogue of the relation xy = z. The general situation is much more complicated but might likely be proved by the use of the Wakimoto realization together with the bosonization of β, γ and b, c systems. The proper analysis is left for future work but let us at least count the number of fields on both sides of such a generalized bosonization corresponding to xy = z m w n .
The left hand side associated to xy can be thus realized in terms of two copies of the above-listed fields.
On the other hand, the right hand side corresponding to z m w n is realized by m copies of • m copies of (β, γ) ghosts, • n copies (b, c) ghosts, together with n copies of • m copies of (b, c) ghosts, • n copies of (β, γ) ghosts.
Putting everything together, we have an algebra realized in terms of m 2 + n 2 copies of the (β, γ) system and 2mn copies of (b, c) ghosts. Comparing with twice the Wakimoto realization above, we see that we have a mismatch of n + m copies of (β, γ) ghosts on one side and 2(m + n) free bosons on the other side. It is likely that these (β, γ) systems can be bosonized and the two free-field realizations related.

JHEP01(2020)042 6 Conclusion
Let us finish by stating the main results of the above discussion: 1. We find OPEs of spin-one and spin-two generators of a two-parameter family of algebras denoted as W m|n×∞ using a matrix generalization of the Miura transformation.
We defined a coproduct structure on the algebra using general pseudo-differential matrix-valued Miura operators.
2. We identify a large class of truncations of W m|n×∞ associated to a specialization of the parameter N of the algebra 3. Using gauge-theoretical considerations, we identify VOAs associated to the four elementary truncations labeled as x, y, z, w. In particular, we identify them with gl(m|n) Kac-Moody algebras together with subalgebras of various copies of (β, γ) and (b, c) systems. We show that U 1 and U 2 generators of W m|n×∞ can be realized in terms of these algebras if we correctly specialize the parameter N .
4. We propose that general truncations associated to general functions x N 3 y N 2 z N 4 w N 1 can be obtained by composing various matrix-valued pseudo-differential operators L (α) for α = x, y, z, w.

5.
We conjecture new bosonisation-like relations between different free-field realizations inherited from the relation xy = z m w n .
6. We find a non-trivial interplay between properties of W m|n×∞ and its truncations and the geometry of CY 3 m,n singularities. Such a correspondence can be schematically summarized as: Geometry Algebra Function x N3 y N2 z N4 w N1 Truncation (L (x) ) N3 (L (y) ) N2 (L (z) ) N4 (L (w) ) N1 Relation xy = z m w n Bosonization-like relations Charge of x N3 y N2 z N4 w N1 Specialization of parameter N Apart from the rigorous proof of the above-stated conjectures 1,2, 3 and 4, the above analysis raises many questions. Let us mention at least some of them. In [52], a simple JHEP01(2020)042 realization of the elementary Miura operators for W 1+∞ was found. Is there such a closedform expression also for the more general W m|n×∞ algebras? What is the physical origin of the Miura operators? The original motivation for the project was understanding VOAs associated to a large class of generalizations of the AGT correspondence. Can we prove such a correspondence beyond the W 1+∞ case? Can we extend the Miura transformation beyond W m|n×∞ and construct truncations of shifted versions of W m|n×∞ ?
It would be even more interesting to extend the above construction to more complicated geometries. The universal infinitely-generated algebra associated to a given geometry (with the vacuum character identical to the square of the topological vertex partition function) would have truncations labeled by a choice of a holomorphic function on the variety. The simplest example going beyond the configurations discussed in this work would be a geometry with S 3 symmetry leading to a D(2, 1; −Ψ) N version of W 1+∞ . Note that in this case, both parameters of the infinite algebra are visible already at level one. One should be able to identify specializations of N by an analysis of null states of its D(2, 1; −Ψ) N subalgebra. Other geometries containing compact four-cycles are expected to lead to new technical issues and remain to be a challenge for the future research.