JHEP01(2020)042 Springer October 8, 2019 January 8, 2020 : December 6, 2019 : December 20, 2019 : : generated by a super- Received Revised Published ×∞ Accepted n | m algebras. Relations in the ring W N W Published for SISSA by https://doi.org/10.1007/JHEP01(2020)042 . The algebras admit a large family of ,... Kac-Moody algebras and 3 . We propose a free-field realization of such , n 2 ) w , m n z | = m ( \ xy gl . 3 1910.00031 The Authors. Conformal and W Symmetry, Conformal Field Models in String Theory, c

We discuss a class of vertex operator algebras , [email protected] Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo,Center Canada for Theoretical Physics, UniversityLeConte Hall, of California, Berkeley, U.S.A. E-mail: Open Access Article funded by SCOAP Keywords: Supersymmetric Gauge Theory, Differential and Algebraic Geometry ArXiv ePrint: truncations generalizing the Miura transformationof for holomorphic functions lead toalizations. bosonization-like The relations discussion between different providesvertex free-field a operator re- concrete algebras, example algebraic geometry of and a gauge non-trivial theory. interplay between Abstract: matrix of fields fortruncations that each are integral in correspondence spingularity with 1 given holomorphic by functions on solutions the to Calabi-Yau sin- Calabi-Yau singularities Miroslav Rapˇc´ak On extensions of JHEP01(2020)042 17 17 ) gauge of spins N (1) can be [ gl ,... 3 × ] or construct a ]. The resulting ,U N 12 2 26 W – ,U 1 19 U (1) algebras [ ]. Algebras [ gl 26 and Ψ. Furthermore, the algebra – × ]) and supersymmetric U( N N 24 11 1 W – . 1 to a positive integer and moding out the N N (1) algebras comes from considering the large – 1 – [ gl × 18 as a generic complex number [ 3 N m,n N (1) on the equivariant cohomology of the moduli space 27 W [ gl CY 4 23 × 6 4 N 11 ×∞ ]. of spin greater than (1) algebras (see e.g. [ is generated by infinitely many fields ]. Concretely, one can identify the Nekrasov partition function W n | i [ gl 18 ] with conformal blocks of m U ∞ )-webs 13 – , case × -algebras 14 14 W 1 1+ 15 3 12 N p, q (1) already depends on one continuous parameter Ψ related to the central charge N [ W C [ gl w 2 W 4 × 1 C N N 29 z 2 algebra W N y 3 and depends on two complex parameters N ×∞ x n | is rigid in the sense that it is the only algebra containing a single strong generator m ,... ∞ 3 An interesting perspective on The algebra 5.4 Bosonization-like relations 5.1 Coproduct of 5.2 Building blocks5.3 of truncations 2.1 Calabi-Yau singularities 2.2 Relation to ( W , limit and treating the parameter 1 1+ 2 , of each integral spin andthen satisfying recovered Jacobi by identities specializing [ ideal the generated parameter by fields of the Virasoro subalgebra. N algebra denoted as 1 W spondence relating theories living on of such gauge theories [ natural geometric action of of ADHM instantons [ 1 Introduction Vertex operator algebras (VOAs)metric are gauge well theories. known to A appear prototypical example in of the such context an of appearance supersym- is the AGT corre- 6 Conclusion 5 Miura transformation for 3 Review of the 4 Contents 1 Introduction 2 Geometry preliminaries JHEP01(2020)042 ) i as N ] to 3 ,N 31 2 ] on the ,N M5-branes 1 i . Such an 37 N 1 N , Y 0 ] analogously ]. It has been , can be simply 0 47 Y 1 39 algebra actually , – , 0 , 0 ∞ 38 44 Y , 1+ ] first had to introduce 38 W labeling ranks of U( , 40 and i copies of 27 0 , N 3 1 , N 0 parametrized by non-negative Y ]) that 3 , . By Miura transformation, we ] or collections of such surfaces 0 30 , and ,N – 0 (3) 2 , 43 corresponding to rank-one theories 0 1 – ], one can relate the setup of [ , L ,N 1 Y 1 1 , 28 , , 34 0 copies of a pseudo-differential operator 0 N , 41 – Y ] for related discussion) by showing that , 0 Y 2 25 with integers Y N 17 32 3 36 , , , C 27 (1) 35 and such elementary factors into the general alge- L – 2 – 2 0 , ]. 1 copies of , 0 inside 2 18 Y – N 2 23 , to fuse 0 , with the algebra arising from the action of the cohomo- 16 , C 0 0 , , , 3 ] (see also [ 0 ∞ 1 , 2 13 ,N 1 Y 34 2 1+ C Y , ,N W 1 2 12 ] (based on results of [ N C Y of the resulting algebra can be then identified with coefficients of (1) story [ 27 (1). To define the Miura transformation, [ i [ gl [ gl U (1) current algebra. The coproduct allows us to identify × × copies of emerge from an action of the cohomological Hall algebra [ [ gl N N . Such more general truncations correspond to gauge theories supported 1 3 3 N W W ,N ,N 2 2 . copies of a pseudo-differential operator ,N ,N 3 1 1 ] for 3 . These more general truncations can be furthermore identified with VOAs i N ,N N 3 2 N , Y Y N ] defined in terms of the quantum Hamiltonian reduction. This cohomological 2 ,N 1 31 N and It is natural to ask how general is the relation between VOAs and gauge theories More concretely, each of the elementary factors The identification of It was observed in [ Y copies of a pseudo-differential operator The gauge theories can be physically thought of as arising from the low-energy limit of 2 1 (2) by different people. Byrich considering connection more between general VOAs and configurations, algebraic one geometry expects of to the uncover corresponding a configuration. supported on the threea coordinate system planes. of separatedcomplicated Fusion setups can M5-branes discussed be together. bellow. intuitively understood This as motivates a the process existence of of bringing the fusion also in the more supported on morewrapping general various four-cycles complex inside higher-dimensional surfaces varietiesexpected [ [ for a long timesurface that of the such above-mentioned a AGT more correspondence general story only referred scratches to the as AGT, 4d/2d or BPS/CFT correspondence L mean the process of multiplyingterms these of pseudo-differential a operators pseudo-differential andthe operator rewriting right. in them Generators the in standardsuch form a by pseudo-differential commuting operator. derivatives to identified with the a subalgebra of embedding can be conveniently carvedmation out [ by a generalizedN version of the Miura transfor- to the simplest configurations supported on one ofthe the coproduct three structure on coordinatebra planes. Secondly, one can use an existence of algebras equivariant cohomology of the modulito space the of standard spiked instantons from [ logical Hall algebra proceeds in two steps. First, one needs to identify the VOA associated Using well-known dualities of stringa theory configuration [ suggesting a naturalporating generalization of the AGTon correspondence the and coordinate incor- planes gauge groups of thecorrespondence was corresponding proven three in [ theories. This generalized version of the AGT contains a three-parameter familyintegers of truncations from [ definition is motivated bydimensional an gauge analysis theory of that a particular provides configuration us of with interfaces a in physical four- realization of the algebras. JHEP01(2020)042 ]. as- 34 . We 4 (1.1) (1.2) will be 3 m,n CY VOA with related by 1 -algebra and n under such a N z + 1 1 action preserv- w 4 x, y, z, w N N 2 N w w that was the main T z 4 m 2 N + N 3 z 4 m,n y 2 N 3 N and an identification of some z N CY y 2 x 3 super-matrix of generators copies of the N N ×∞ admits a y 1 0 n x | 3 | N m N 3 m x m,n W CY and 1 . -algebra, N 1 y N w n 4 w w N 4 m z depend on two complex parameters Ψ and N 1 z z − 2 – 3 – generated by an in terms of a subalgebra of a tensor product of 2 = N ×∞ N algebras, we expect an existence of a definition of 3 y n y and the generalized Miura transformation from its 3 | 3 xy 1 ×∞ ) Kac-Moody algebras at specialized levels. N m . n − ,N | n copies of the x 3 2 | 3 W m m,n 2 N ) in the ring of holomorphic functions. We expect the ,N m -algebra in terms of x 1 ,... N ( W 1 \ 3 CY N 1.1 N gl , (or its resolutions) given by solutions to the equation Y 2 w , 4 N 3 m,n z 2 -algebra. The elementary algebras associated to , algebras should emerge from doubling the cohomological Hall algebra N -algebra, CY w y x ∞ 3 N 1+ ×∞ x n | W m W should emerge as a double of the relevant cohomological Hall algebra. Note that we could can be identified with the charge of the function ]. As discussed in the main text, the two parameters can be given a natural ] for a detailed discussion of the Miura transformation for copies of the any non-negative integers. The relevant divisors are those that come from zeros N -copies of the ×∞ 4 3 48 n | 49 , N (roughly) identified with for each integral spin 1 N m Truncations of the algebra Generalized Miura transformation Finally, note the relation ( Similarly to Analogously to the above In this work, we illustrate the richness of the correspondence by analyzing a class of 27 m, n See [ W action. [ 1. 2. 4 3 2 of the truncations by an analysis of nullconsider states more at general low levels. divisorsexpect associated these to to sections correspond to oftion truncations of non-trivial of such bundles different more on doubles generalMiura a of configurations transformation the resolution is for cohomological of left these Hall for cases algebra. future at Investiga- work the since moment. we do not know how to generalize the rameter T algebras associated to functions motivation for the project. N geometric interpretation. In particular,ing the the singularity Calabi-Yau volume form.parameters associated Parameter Ψ to then such corresponds an to action. a ratio On of the equivariant other hand, a specialization of the pa- ariant cohomology of the correspondingIn moduli particular, space of instantonssociated along to the the lines given ofcoproduct geometry [ structure. Understanding thea above-mentioned first definitions step can in be proving the thought AGT of correspondence as for divisors inside We expect these two definitionsan to algebra allow coming an from the identification geometric of action the of the cohomological Hall algebra on the equiv- of the holomorphic functions of the form the corresponding VOAs corresponding to gaugeCalabi-Yau theories singularity supported on a special class of divisors inside the for JHEP01(2020)042 . 3 5.2 5.1 m,n (2.1) . The that is . Such 0 CY , 3 1 lead to a discusses discusses ) becomes ,... ×∞ CY n is known as n 2 | + , associated to 2.1 5.3 2.2 1 1 = m , 3 ) 1 N 1 w , W 3 = 1 ( w case. Section 0 leads to the well- )-webs colored by CY L m s , case. In section, + 3 ) system. CY 4 ×∞ z case. We describe the p, q C ( 0 ∞ N | L z systems that motivates the m 1+ 2 , algebra and the free-field = 3 β, γ m,n ) -algebra associated to the N y W . 0 W 1 ( ∞ y , β, γ 3 1

3 N CY L . We derive OPEs of weight- n 1+ N , w . Finally, section ) x 4 w 4 CY x and W ( N m z L z discusses the geometry of 2 b, c ] for the and singularities and N = truncations associated to generators y = 1, the constraint in ( 1 48 n 2.1 3 x, y, z, w N Z xy N and the corresponding three-fold is just | realizations. For example, the simplest w , n ×∞ x 4 4 n w singularity and the conifold singularity. In | 5 C N generalizing the , i.e. m z = 0 m ∈ or 2 W Z ) N 3 z ×∞ m,n m y – 4 – n | 3 m N CY x W is discussed in section x, y, z, w = 0 or ]). Here, we propose that the relevant geometry can be ∞ (  27 1+ , n reviews some aspects of the super-matrix of generators at each spin = W 3 n = 1 | 3 m,n m m by composing Miura operators , one can solve for CY 1 xy and establishes a precise connection to the ( N w = ]. Section 4 = 0 are known under the name 3 m,n w N generalizing for the simplest geometric configuration 31 , z m 2 z or CY 27 N ×∞ y n = | 3 xy N m = x xy W z The structure of the paper is as follows: section Note that Kac-Moody algebras are not free algebras and embedding inside a tensor product of Kac- 5 particular, note that for simply Moody algebras isthemselves literally expressed not as a subalgebras free-fieldname of “free-field realization. realization”. products On of the free bosons, other hand, Kac-Moody algebras can be identified with a resolution of the following singularity This singularity generalizes the well-studied 2.1 Calabi-Yau singularities The main aim ofstrongly this generated work by is an truncations a were discussion conjecturally associated of to truncations atoric of particular Calabi-Yau class three-fold the of (see algebra divisors [ inside a particular the elementary functions.lead We to argue bosonization-like that relations between relations different in free-field the realizations. ring of2 holomorphic functions Geometry preliminaries introduces the coproduct structure on discusses the elementary buildingof blocks the of ring of holomorphichow functions to obtain on a free-fieldfunction realization of a general resolutions of integers from [ realization of its truncationsalgebra associated to holomorphic functionsone on and weight-two fields extending the analysis of [ known bosonization relation that relates the free boson withSpecializations the the conifold singularity. Theseliterature two but I cases have not are found extensivelytoric a discussion discussed action of mentioned at the more above many general and places the in ring the of holomorphic functions. Section On the other hand,different realization both of expressions theof algebra in elementary terms factors. ofa a Each non-trivial different relation subalgebra relation in of betweenrelation a the various different ring free-field system of holomorphic functions thus leads to such a relation to be equivalent since they both correspond to the same geometric data. JHEP01(2020)042 . 3 m,n (2.6) (2.3) (2.4) (2.5) (2.2) )-web CY is known p, q ] that are 1) 1 , 3 1 27 − , CY (1 . in the coordinate ) is the well-studied . The special class 1) n w on the right. } , can be expressed in 1 w 3 m ih ×∞ (0 . m C z n 3 z | m,n = 1 = m z, e dw = ) 0 , 2 N CY 3 ∧ W 1 h 1 xy | xy h + x 3 3 dz CY h + C ( ∧ 0) i 4 . Finally, the case , ∈ N C ) dx (1 y, e 1 2 × h = . ih m , − 1 Z lead to configurations from [ x, y, z 6=0 2 N / = 0 x, e ( y 2 ) action that we parametrize as N w { 3 3 Ω 2 4 C 3 h 3 m,n on the left and 0) h N mh , T 3 z = + + + 2 – 5 – CY C 0 (1 2 2 N 3 , h h 3 y = m, ], one can associate a vertex operator algebra to N 1) 3 1 ) + , − dw N 3 2 27 0 CY 1 x m h ∧ h CY +( − y 1 dz correspond to scaling of coordinates in a particular chart 1 1) nh h ∧ i , ( ) i h e (0 m ( dy , resolutions of admits a natural is unconstrained and − = → preserving the Calabi-Yau volume form then corresponds to the 1) itself. The relevant divisors are thus in one-to-one correspondence n w ) , 2.2 3 m,n 3 6=0 (( x specialized to (1 parametrized by the remaining triple of coordinates. More generally, T 6= 0 patch as ×∞ Ω i CY 3 n y ). We will see that this choice has a nice interpretation in the ( | non-negative integers, modulo the relation Toric diagram of h ⊂ C m 4 2 2.1 x, y, z, w = W 1 T ( ,N . Such functions scale with the power of 1 µ 3 , 3 0 3 m,n ,N 2 2 CY µ Figure 1. CY ,N = 6= 0 and the 1 0 According to the conjecture of [ The three-fold , = 0, the coordinate x singularity. We can thus identify 3 N 1 n m for ring of As discussed in section expected to lead toof truncations of divisors shifted at versions handtruncations of will of algebras be identifiedwith with functions configurations with zero shifts and leading to any toric divisor inside aof toric divisors Calabi-Yau three-fold. that can We will be restrict identified to a with particular zero class sections of holomorphic functions on The subtorus above parameters One could in principle choose anythe other relation parametrization in of ( the toruspicture that discussed is consistent bellow with and on the resolved three-fold.the The Calabi-Yau volume form Ω of if Z as the conifold singularity. CY JHEP01(2020)042 . 1. or on C − 1 , let , , ∈ (2.8) (2.9) (2.7) 1 +2 i 3 i 3 m,n +1 w , z 0) , . In par- ] , CY i ) subalgebra +1 CY i (1 +1 m , ). , w ( , z i 1 i gl z z − 2.5 . 1 ) and , 1) n 1 − , ( gl (0 − ]. For each 27 1) in terms of = 0. This specialization is , , . . . n l ) and write 4 , (0 with charges , . . . m n N = 1 , 2.1 j 6 +1 +1 = 1 n + 1 complex coordinates n n , . . . w i ) and leading to a different resolution of 2 , z x, y, z, w | n 1 n in the exponents of ( 1 | n for − i +1 N m 0) n 2 ) to set . . . w , z ( 1 n , N for 2 w 2 3 z and h gl 2 (1 | n 1 2.1 w , w + 2 2 C w actions are − | on the left and the resolved conifold 1 | +2 2 j w 2 m and their toric diagram that will allow us to w | ∈ × C +2 w N n i | +1 i C 2 – 6 – z z × . z | | m h m 3 . . . z + 2 m,n , m 0) 0) 1 Z + , 2 + n , + | / − 2 2 3 CY (1 2 . . . z (1 | | m 2 ^ +1 2 +1 and 2 C 3 N . . . w z j . . . z , 3 +1 z w 1 i | 2 w m 1 2 1 = h | z z z z w | − 2 0 + , 2 , 2 )-webs colored by integers from [ 2 = = = = 3 − | − 1 1) 2 z y x | w 2 , p, q w | j ] CY i (0 , one can solve the constraint in ( z w )-webs will correspond to coordinates on different patches of the resolved | −| | i i ) z = = = , w ( 1 i l p, q , w ) ) z i µ z w z ( i ( j are invariant with respect to the scaling of each triple µ µ + 1 complex coordinates , where we can use the relation ( with charges +1 action described above. This constant will play an important role in the 3 1 m 2 C µ +2 T Toric diagram of i = 1) 2 , x, y, z, w 0 , w ) and one associated to the fermionic root. One can define a different system of moment maps µ , 3 n 1 (2 | than the one discussed below. +1 i m ( CY Note that we have one moment map condition associated to each root of the 3 m,n 6 , w gl i CY of corresponding to a different choice of the root system in There is obviously aticular, lot of freedom inw parametrizing The corresponding moment maps for such We will now introduce resolutionsmake of a contact withus the introduce ( Different triples of variety. In terms of for the reason for the above convention for2.2 the labels Relation to ( under the discussion of vertex operator algebrasexpression below. Furthermore, note that it reduces to a simple Figure 2. the right. JHEP01(2020)042 2 ]. T 50 )-web (2.11) (2.13) (2.10) 1) line. , p, q with the 3 R = 0, one gets the 2 h . The variety can be . m 3 } 0) cycle associated to cycles degenerate [ ) , C 2 w ( j T (2.12) l , = ) . } 1 0 2 = , | z 3 1 = 0, one gets the (1 ) 1 ) 2 z = 0 0, and corresponds to the line w 2 h fibrations over ( j ) CY h + ) and the base parametrized by ≥ w − | 1 R ( j h + l, µ 2 1 ( can be then encoded in a ( µ | 2.4 i × 1 2 = − , µ h 2 z modulo U(1) rotations of the coordi- = | m,n , e T and half of the toric diagram of the gen- µ 3 , µ 2 = +2 ) = 0 ). The diagram indicating degeneration z z n 2 = 2 C ( i 2 1) cycle associated to on the left. The (1 l + that scale as ] ) CY , ih 2.2 µ z × m 1 1 ( i = , e 2 C w µ satisfying ( 1 ) Z { z – 7 – w / ( i 3 = , µ 2 1 µ // 2 ^ | , h C { ih 1 2 e , w +2 z ( can be then alternatively defined as a symplectic 2 // n = , h z + 1 0 +2 − | corresponds to rotations of the three coordinates in our → can be resolved by introducing a real parameter for each m h = n 3 2 = 0 inside ) 3 m, m,n | C + 1 3 1 = 0, i.e. along j 3 , y m , h 1) cycle associated to m,n µ = w , z 1 1 can be thought of as CY 2 , | C 2 ] z z CY = CY , h = , z 3 = 0 and considering quotients = 1 = m,n 1 m,n µ 1 h 3 2 x w > µ z ( = CY m,n ) 3 i w ] CY ( j µ ] CY , l, l ) z ( i l Toric diagram of . )-web is composed of trivalent vertices associated to different coordinate patches m,n 3 p, q Let us start with a discussion of the toric diagram for Calabi-Yau singularity ] CY 0) in the toric diagram.0) line. Similarly for Finally, the for (0 the (1 = 0 degenerates for , , 1 parametrization. This justifiesloci the of choice various in cycles ( h is depicted in the(1 figure (1 with the corresponding moment maps Note that the scaling by The ( and mutual orientation of vertices then indicate how aresimply two parametrized patches by glued together. Resolved three-folds fiber corresponding to thecorresponding above moment maps. The geometrydiagram, of i.e. a diagram indicating loci in the base where one of the nates with charges described above. moment map quotinent i.e. the preimage of Figure 3. eral The Calabi-Yau singularity JHEP01(2020)042 6= 0. with since m (2.14) (2.15) (2.16) (2.17) 1 1 z , z m,n 2 3 , z 1 ). Using the ] CY w ) z 2.12 ( 1 l ) can be rewritten − again corresponds to 2 2.13 | . in detail. After dealing 3 ) 3 1 2 1 z , | , h z 1 w ) 2 3 2 + h . = , h 2 + 1 | . ] CY 1 2 ) h h m,n z z ( 1 3 l (2 i −| , w − action as above ( 3 = ] CY z = 2 , e 2 2 | 3 2 z 3 T z 1 z can be obtained by gluing more vertices . Let us now move to the patch ) | z 2 h 1 and adjust its phase arbitrarily using the + = + | m,n 1 2 3 3 | h , µ z 2 ). The moment maps ( ( ) | i 2 z z | – 8 – ( 1 , z ] l CY , e 2 2 , z 1 3 z − − 2 6= 0. We can then identify the triple 3 w Toric diagram of ). , z 2 z 1 2 3 | 1 | z 3 ih 1 = w z 2.9 e z | and the resolved conifold ( | + C = → ) plane. Furthermore, we see that in this patch, we get 2 ) , y | × 2 ) z 2 2 2 ( 1 2 on the right. z z Figure 4. , µ | µ 2 Z 1 , z 1 1 2 / z 3 µ 2 case, we have goes in a similar fashion and produces a diagram with an opposite − ^ = , z C 0 3 1 , 2 | x 2 C = w 1 3 ( 2 w , | = action. We can see that the scaling by 0 ] CY 3 1 , 1 = 3 0 ) in the ( S 1 ) ] z CY µ ( 1 CY l = ) as − 0 , , ) 2 z 3 ( 2.15 1 l − First, in the Toric diagram of a more complicated ] n CY with the corresponding fiber generated by Note first thatby the ( corresponding trivalent junction associated to this patch is shifted patch is again theThe trivalent junction coordinates from in figure thisusing patch ( are ( Let us first look atcoordinates the on patch the where correspondingmoment map patch constraint, with one the corresponding can solve for rotations of the three coordinates in this patch and the corresponding toric diagram in this with the corresponding moment map constraint for with these two examples, we cansuch directly a draw the general corresponding diagram caseiterative for is use of a the simple moment combination maps of ( the two examples and corresponds to an Considering orientation from the figure to the simplest trivalent vertex. Let us first illustrate the derivation of the toric diagram JHEP01(2020)042 0), = 0 , , the 2 (2.20) (2.21) (2.18) (2.19) 3 ). The h C . As in + +1 . , one can i ) 3 1 = w m,n C h ( i ) leading to , w 3 0 2 , ) coordinates 1 6= 0, we have µ 3 = , 1 1 +1 1 is shown in the . The moment , z 0 n , ] CY 3 z , z 2 1 0 CY 2 3 , = 0 , z 2 , z i ) plane with (1 2 , z 3 1 CY 2 w 1 h l w w in , µ + ] CY − . 1 1 2 . 1 µ 2 N ) | w 2 2 w , h 1 indicating multiplicities of can be then labeled by the 2 w w w with an opposite orientation ) | 1 N 3 2 l. y N = 0 2 h , 3 = + 0 ,N 1 + w = N are non-vanishing. We can then 2 3 1 2 2 | h x 0) line on the left to the last line 2 h i ) in the ( on the right. 1 N | ( , l i 2 w y ,N w 2 ] CY 3 − , w 1 w , e | 2 N 2 −| l, N z x z 1 + and − 1 = z 1) with 2 i ih 3 | , 2 z 1 = − N 2 w , e N 1 . )-web associated to a general − | – 9 – w , z 4 2 2 2 1 | l, µ p, q ih 2 w 1) and (2 N to find moment maps on patches parametrized by z 2 − − , ) and divisors associated to their zero sections from z e ) with integers 2 ( z | (1 − | ( i 2 = 2.5 , 5 = 0 map to the three corners of the toric diagram under 2 µ → | w 0) | 1 )-web specifying divisor of ) w , z 2 | on patches parametrized by the triple ( + , y p, q 2 2 rotating the three coordinates. In the chart 1 = , w | )-web. First, considering the simplest case µ 2 2 i w = 0, µ z 1 h , z z y p, q 1 −| and = w ) with the corresponding moment maps ( 2 1 = 6= 0 can be again parametrized by the triple ( x = 0, µ 1 . A colored ( 2 , w ) giving rise to a sequence of vertices from the figure µ 1 x i 2 , attaches a reversed vertex with the (1 w 0 3 , z , z , we have µ ). The divisor of a general function 1 1 , )-web is shown in the figure +1 w 1 i ] ) web from the figure CY 3 2.13 , z p, q as in the resolved conifold example. Finally, using moment maps 1 1) cycles degenerating as shown in the figure p, q ] , Figure 5. CY 3 w on the left. We get an analogous diagram for 2 Let us now discuss functions ( Finally, let us describe a general ( For 1), (1 , the perspective ofcoordinate the planes ( the maps ( colored ( different smooth components of the divisor. triples ( map condition of figure find expressions for resulting ( (0 the previous examples, one cancorresponding again to associate the one patch vertexuse where with the all the ( the moment other map conditions and the associated toric action We can see that this leads to a vertex shifted by ( First, the patch one trivalent vertex with coordinates ( with the corresponding moment map degenerations along the lines (1 being the degenerating cycles. Thefigure corresponding toric diagram of similarly to JHEP01(2020)042 | i in w | 3 , then with 3 2 n w w and CY | m i z z corresponds )-webs with | 1 = y p, q on the right and xy . Each function . . 2 y w m 3 m,n CY in the resolved three-fold to solve for and deduce integers for the i 2.19 1 x, y, z, w , w i (b) A divisor of (d) A divisor of z ). Restricting to a patch associated 1 1 3 1 2 1 2.8 using ( )-web. Divisors associated to the elementary on the left. On the other hand, i – 10 – n p, q , w i 1 2 1 z 1 . . 1 )-web. This leads globally to a colored ( )-web perspective are shown for the example of x z p, q p, q on the right and leads to a diagram with decreasing numbers from the top to the m x 3 from the ( (a) A divisor of (c) A divisor of is associated with ones on the right-hand side of the diagram and z 2 1 Divisors associated to the elementary functions x, y, z, w can be rewritten in terms of . Generally, . 6 3 . One can then read off the corresponding multiplicities of the smooth components , i Let us conclude this section with two remarks. First, multiplication of holomorphic This discussion has a generalization to the general case of 2 3 on the left. , w i ] n ones on the left. functions corresponds to summing correspondingtiplicities divisors associated and to in various particular faces summing of mul- the toric diagram. The relation corresponding vertex in theintegers ( associated to eachfunctions face of thefigure ( bottom starting with to decreasing numbers from the bottom to the up again starting with x, y, w, z to a given vertex,that we are can not use partz the of moment the map corresponding conditions triplein and the remove given the patch phase from of powers the of corresponding the corresponding Figure 6. CY JHEP01(2020)042 , ). 2 4 (1) 2.9 [ (3.1) (3.2) (3.3) gl × N W N of the L U i + U + ’th order differential N M ··· M N + − have zero shift. Further- 2 − 1 K N N L . They can be identified with K ) − w 7 4 κ∂ . N ( N 2 z 1 ) 2 with shifting parameters associated U w N . y + 1 ) + − 3 (1). Let us normalize the corresponding x ×∞ κ 1 ( i N z M n [ N gl | − ( J x as m N + ) + ∼ W L ) )-web. This is not going to be the case for alge- κ∂ 2 (1) algebras admit a well-known realization [ κ∂ w ( – 11 – ( 1 [ gl ) K M L − p, q = U x copies of ,...,N ] conjectured that algebras associated to the web ( i × ) x J K + ( i 27 N ) N = 1 L z N ( ) i W ) )-web. The shift parameters associated to various config- x ( i ]. κ∂ J p, q 40 = ( L (1) for ) case x [ gl + ( N L 3 N L M of C M -basis (sometimes also called the quadratic basis since OPEs in this ) 2 ... x U ( ) i x − J ( 2 Shift parameters associated to different finite segments of the diagram. L K ) N . The special class of divisors discussed here form a subclass of algebras that x 1 ( 1 N L w 4 N ] in terms of a subalgebra of z Figure 7. 2 51 N , y 3 25 N , operator in the standard form by commuting all derivatives to the right,algebra one in obtains strong so-called generators We can then define Miura operators as formal expressions Considering a product of such operators and rewriting it as an 3 strong generators 3 Review of the Let us now move toMiura the transformations discussion from of [ vertex operator algebras and review the generalized Note that allmore, the it configurations is associated not hardx to to convince yourself thatare all independent zero-shift of the configurations resolution come ofbras from the associated some ( to divisorson associated the to resolution sections would of have a to non-trivial be line taken bundle. track Dependence of in such examples. transformation. Secondly, note that [ should be truncations ofto shifted each versions finite segment of ofurations the in ( our setupthe are U(1) schematically charge depicted of the in divisor figure with respect to the action generated by moment maps ( equates two different waysside, of this producing multiplication the will same lead divisor to using a such coproduct sums. encoded in On a the generalization VOA of the Miura JHEP01(2020)042 (3.6) (3.5) (3.4) 1 U ) 1 ) . ) x 1 x ( + 1) j ( i ∂U . To describe the U J κ 3 2 ) ( 1) x 3 ∂J ∂ ( κ ) i ,N − 2) 1) to be an integer and w 2 2) ∂J − N − − Z − ( ,N i 1)( z 1 . N ( N , N ∈ ( . N ) 1)( − 1 2 1 3 2 ] that this algebra is actually m,n i as a generic complex number, U 1)( h h − κ ( N κN + 1) 26 i CY − 1) − ( N +1 – 2 κ − i N ) ( + 1) X ) ) = − N =1 24 N κ w j x 2 i 1 w X ( κ ( ) i U N depending on two continuous param- 2 − − − 2 1 w N =1 + κ i i 1)( 1 ∂J X z z ( U − U κ ∂U 1) − 1 z ) + + 2(2 4 ( U (1) as truncations of such a two-parameter − ) + 1 = Ψ = + 1) ) N 1)( x ) + κ ] that there actually exists a larger family of i ( [ gl ( ) κ j w ( ( − x , 27 ( k 2 + × + 1) − ∂J N =1 2 – 12 – J κ , κ i ) N X ) κ z 3 N U x x ) ( κ 2) i 2( ( j 2( ’s). In particular, we have W J w = 0 1)( + 1) i J 2 − − ) ) + 1)(1 + 3 3 U + 1) κ x ) + − − 2 ) 2)( ( ( ) h i w N κ associated to such truncations, it is useful to introduce κ z x κ J w ( , ( j N − ( + − ∂U ( J 1) 1)( 1)( N 2 2)( 2 j − 2 ) z 7 ) +1 ( generated by fields ( x h − j z will be modified for more general − − − ( N i − w ( + 1) X = 2 +1 J κ ∞ 2 i + + 1 k N ( coming from the specialization κ N N N U − N X = 1 ( ( ( ( κ ( 2 1+ j ) z h +1 +1 limit and treating the parameter ( i i x . It was argued in [ ∞ N N N N N N − − + ( i W X X = = N =1 i j j J parametrized by three integral numbers N X 1+ such that ∼ ∼ ∼ ∼ 3 κ 3 W ) ) ) ) N N N =1 =1 =1 i i i ,N X X + X i > N w w w w 2 , h ( ( ( ( 2 1 2 3 2 = = = ,N 1 with the above OPEs. It was argued in [ U U U U , h 2 3 1 N ) ) ) ) 1 U U U z z z z Y N h ( ( ( ( = 0 for 1 1 1 2 i U U U U and U κ We can then look at the algebras The relation between Ψ and 7 specialization of the parameter parameters generator for each integral spin. family of algebras setting truncations Considering the large one obtains an algebra eters uniquely fixed by requiring the associativity of OPEs and an existence of a single strong that leads to an algebra with first few OPEs of the form basis are quadratic in derivatives of JHEP01(2020)042 0) , 1 (3.9) (3.7) (3.8) . , (3.10) (3.11)  ) y ( specialized J ) = (0 2  3 ) ∂ N 2 w ( by an exponential bellow. ,N 2 J i 3 2 h h + 1) 3 ∂ ,N ∂ m,n 1 κ N CY ) + ..., ) w ) + ( + ( ) 2 . y ( 2 − ∂J ) α ) ∂J w N w ) ( , ) 1 y 2 J ( − . ) κ∂ J 3 z ) with the parameter ( w ( h ) 1 3 α 2 as − ( 3.5 2 ) + 3( N z ) U + 1)( action. Furthermore, the expression is N ( w κ + ) for the transformation of holomorphic ( κ + 1) , + 2 2 J in terms of a conjugation of  κ 1 3( T ) , h 3 κ ) + 1 ( 2.7 ) − 2 w y h  w ( ( − α κ ( ) N J ) N L w ) ) ) ( , ∂J – 13 – 0) in the first case and as ( ∼ − ∼ y + ) w ( y , ( ( 1 ) ) κ∂ 0 ∂J J J L ( h ) , w w ( ) , 1 y + 1) and the resulting algebra in the end depends only on ( ( ) ( α  ) ) x ( i N 1 κ ) + ( y J w ) ( ( ) h ( L 1) U y w J = ( J ( − ) ) = (1 ) − + 9 J 3 J z ) z ( ) N ( ) α κ ( ) y w  ) N ( y ( ,N is in correspondence with generators of the ring of holomorphic functions ( w ) 2 J ( J J ) with respect to the ( J 1) (2 y κ∂  ( ,N + 1) 3 2 1 -generators of the three elementary truncations, we can then define − J , κ x, y, w C ] that there actually exist two more Miura operators whose products κ ( truncations. Let us normalize the currents appearing in these two ) = ( N U κ respectively that satisfy OPEs (  2 w ]. We do not use this representation here since it is not obvious how to generalize ( − ) 3 40 ( = ) + 1) α 6 1 1 8 y 52 J ( ,N ( 0 κ 2 , L 3 J ) as ( 1 + 1 + 1) + 1)(3 = = = , 6( ,N ) 1 ) ) ) κ κ κ y 3.7 ( N CY w w w 2 from the previous section. and ( ( ( 2( (2 3 1 2 Y J 3 ) U U U C w = = = ( = ) ) ) J y y y 0 ( ( ( , 1 2 3 3 1 U U U Having identified It was found in [ Note that the labeling A simple and uniform expression for CY 8 9 on operator appeared in [ such expressions to the matrix caseexists. described below. It would be nice to check whether such a generalization a generalized Miura operator as and tives of according to ( in the second case. One gets It is a simple calculation to identify combinations of normally-ordered products of deriva- the ratio Ψ. We will see a natural generalizationgive of rise the to formula for Miura operators as Note that the numeratorfunctions equals on the chargeindependent ( of the overall scale of We can then write the formula for the specialized JHEP01(2020)042 - n x ×∞ × and n | n ) or ge- (3.12) x m ( i at each 0 L W ] of BPS . -limit), the m,n, i,ab i ∞ 46 Y U , N = 1 for the 1+ ]. In particular, ] who proposed x 45 W , N 27 40 , 27 ..., as a subalgebra. 30 + , , i.e. g κ -truncation. Multiplying 2 29 N − w elementary factors N 3 f∂ N 2 at level ˜ algebra, i.e. an algebra ∂ ] corresponds to our ˜ κ for the ∞ ) 1) and 29 1 3 n ) | h h 1+ − y ( i m W fermionic directions, such that the ( = L N \ ( super-matrix of generators gl . w . N 3 ˜ label the row and the column index of the κ N ) from [ n ) | C , analogously to the case of + algebras turn out to contain less states states n | that is convenient to make contact with the m a > m i g transform in the adjoint representation of the = n,m a, b was already pointed out in [ 1 – 14 – m ×∞ N 0 i gl , ( − n ∞ ( 3 \ | 1 N gl m C 1+ ×∞ W 1 W W W . Finite N∂f∂ . Indices inside elementary factors bosonic and case, we expect that there exists a two-parameter family of + ×∞ n -truncation and 2 g n y ,... y | m ∞ N 3 N at given spin m m is the correctly-specialized parameter , , ≤ 1+ ) x ] and it is reasonable to expect that they can be identified with 2 f∂ W α w . At the level of vacuum characters (in the large , a ( i W 3 i,ab 47 N diagonal block consists of bosonic generators with the off-diagonal L U ,N give the free-field realization of generators of the algebra. Note that for the ) = 2 = 1 i m 2 3 i ,N super-matrices of fields. The relation of the geometry discussed above U and satisfying the following conditions: fg h h 1 ( × N n | N algebra = is strongly generated by a Y contains the Kac-Moody algebra m ∂ ×∞ m y n | ) zero-mode subalgebra of N x, y, w m ×∞ ×∞ n ×∞ | n n n | | ). | W = m m m ( m α 3.7 super-matrix with and the blocks being fermionic. gl W W integral spin The generators Analogously to the After setting the notation for The above free-field realization can be equivalently characterized in terms of an inter- W elementary factors is additive under such a multiplication and its value for a general truncation agrees 3. 1. 2. 1 algebras states/Donaldson-Thomas invariants. The class ofthe examples vacuum discussed character in of thisthan work lead the to infinite limittruncations [ of the universal algebra generated by with the matrix generalizationan of existence ofof a products gluing of constructiongluing of proposal the is relevant algebras identical in with terms the of topological an vertex extension calculation [ story. As we will see,much the more interplay non-trivial between in the the geometry matrix and case. properties of VOAs4 become We will now discuss a matrix generalization of the section of kernels of a systemthe of two-parameter screening family charges of as discussed algebras ometrically in the [ divisor of geometric picture from the previous section, we will move to a matrix generalization of the whose coefficients N with ( truncation, N commuting derivatives to the rightoperator using the Leibniz rule, we recover a pseudo-differential where JHEP01(2020)042 ) x ( ab J (4.5) (4.3) (4.4) (4.1) (4.2) ) ) x ( j,cb is uniquely J ) . x . ) ( i,ac x ) ×∞ ( i,ab x n in a generalized ( ∂J | ad J ( 2 J m ∂ c cb ... W ×∞ δ X | 2) 0 + c | || 1) 2 m − b (1) is the simplest trun- | − i − [ W gl 1) N i ) ( 1)( − ( κ∂ +1 − i ( N − i 2 X = ( ) j U , x ( ) w cb N =1 x i N X =1 + J ( i i,ab X − cd 1 2 ad , 2 δ κ z − κ δ ) ∂J | n x ab b N ( i δ 1) || ) c J + | entries. We can then consider a product ) + )) + being super-matrices as well. In particular, + − ) + κ∂ 2 ) | + i x ) m i ( ) d x ( cb j,cb ( x 1 in the same way as ( U and one otherwise. For the notational conve- k,db || ( δ i,ab w − c Kac-Moody algebra generated by fields U J | κ∂ J N =1 ∂J – 15 – ad ) i − + X m ) | x + κ ×∞ = x b ( j,cd z κ ) κδ ( n i,ac || | ) ( | ≤ = Ψ J N in exponents from now on. n a c J x | ( | ) m ( i || ( | a ) κ b ) + | a x 1) L m | ) c ( W κ∂ i,ac ( \ x 1) − X J ( j,cb ( gl ( − J = ( 2) ( + ) ) cd x X x ( − i,ac ( N ∼ algebra, it is tempting to speculate that J j L ( +1 ) ( j ∞ N w c X = +1 ( ... i X k . N ) 1+ super-matrix with ) X = x , x ( j cd ) ( 2 W +1 +1 n i i x ×∞ J | N N ( L ] argued that one can easily find OPEs of i,ab ) n X X = = N =1 ) | i j j J m z X x ( 48 m ( 1 ) κ . In the above OPE, we have introduced a continuous parameter L x N N N =1 =1 =1 W ( i i i ab X X + X ∞ J is an 1+ = = = ) multiplying as super-matrices and is clearly a truncation of W x ,ab ,ab ,ab ( ) i = 0 for fermionic directions κ 3 1 2 x J ) ( | i U U U such factors and rewrite them as a super-matrix-valued differential operator in the n a J | | One can then define the Miura operator Our starting point is the Authors of [ N m -basis using a matrix generalization of the Miura transformation. In this section, we ( \ with we have in terms of components where of standard form and nience, we often omit the brackets gl cation of with OPE fixed by Jacobi identities up to the two parameters. U will use the samegeneralization method to determine OPEs of low-spin generators of its super-matrix In analogy with the JHEP01(2020)042 , cd δ ,ad ab 1 δ ) U 1) ,cb db 2 δ ) − U bc 2 ,cb ad κ 1) 1 ) δ U − 3) bc ,ab ( )) ) ) 2 + ad − δ − ,cb ac ,cb ,cd )) w 1 ∂U 1 1 cb + N ,cb U + U U ,cb cd − ab 1 1 δ cd ab ((4 z U ,ad ,ad U 1) δ + 1 2 + ) ad − − ) ) ) 4 ab U ,ad + δ ) cd ( 1 ∂U bc ,ad 1) ,cd ,cd δ cb ,ad ( w 2 U bc δ + ( 1 2 10 ,cb 1 , , + ( − bc ab 1 + U U ( ad − κ δ cd + ,cb U ∂U δ ,ad cd ,ad ,ad ∂U ab ab 3 z bc 2 + 2 2 3 − cd + cb + δ δ ( bc ad ) cb + U ) U U U ab + 3 bd w cb 1) κδ + + ) cd κδ cb ,ad cd ad cb cb κδ bd 1) δ 0. Furthermore, notice that the OPEs are 1 δ 1) δ δ δ δ + bc − | ) w cb − 3 1) − ,cb ,cb + w U ad bc bc 2 ) bc bc − bd ab + ( 1 2 z 1) w ) ,cd ( − δ − w + cb ( w 1 cd 1) − 1) 1) = 1 U U κδ 1) ( − δ ,ab w z will appear bellow when discussing truncations of + − ac − + U 1 + κ ( − z 2 2 2 bc − + 1)( − − − − bc ad ad (( n 4 + ) ) ) ) . ) z ( ( | − ab ab z ) 2 (( cb z 1) n 1) ab δ w w w ∂U κδ κδ cd m − z ( δ κ κ w − − 1) ,ad ) + m, n ( δ ) − − − cb cb 1 w w w cd 1) – 16 – − ) + ( − − ad − 3 − δ + − ab ,ab (( + + ,cd 3 U N ,cb ,cb ) m − ( 2 z z z δ 1 2 ) − − − 2 3 (( 2 ,cd ( κδ z cd cd ( ( ( ,ab ) for w κ bc 1 1)(( ,cb ( U κ ,ab z w z z U U 1 + + bc + 2 2 − ∂U − U 2 − 2) 1) U 3.5 − 2) ad ad U ab ab ,ab ) − U 1) cb cd ( 1 δ δ z − 2 ,ab cd δ δ w − z ,ad cd 1) 1) ( − ( − 1 ((1 δ bc cb cb ( 3 δ ( κ ad κ ∂U U − − + + + (( − − N U κ N Nκ 2 κ + 1) 2 ac cd cd z κδ cb ( ,cd + + + δ 1) 1) 1)( bc 1)(( 1)(( 2)(( 1) 1)( 1)(( 1)( 2 − ,cd . Dependence on ab bc ab ab 2 1) − − − − − − − − − − N U ∂U 1) 1) 1) 1) 2 ( − N N N m, n ab ab − N N − N − N N N − N (( ( ( ( ( ( δ ( N ( δ ( ( ( ( ( ( ( N N N N + + − + + − + + + + + + + + ∼ ∼ ∼ ∼ ) ) ) ) w w w w ( ( ( ( ,cd ,cd ,cd ,cd 2 1 2 3 U U U U ) ) ) ) z z z z ( ( ( ( Using these expressions, it is straightforward (though technically demanding) to de- ,ab ,ab ,ab ,ab Note that the expressions reduce to ( 2 1 1 1 10 U U U U surprisingly independent of the algebra but only in terms of the supertrace termine OPEs of these fields. For the first few fields, we get JHEP01(2020)042 ]. m Z has 48 / (5.2) (5.1) (4.6) ) 2 to be x 0 have i ( C | L m ×∞ n | ] who used = by correctly m . is well-known n | ) 54 , we conjecture W , ∞ m ,cb ×∞ ∞ 1 n | and the knowledge 1+ 53 can be then inter- 1+ U , m 2 ) W W x ∂ ( i W ×∞ 49 N , n ad L | ]. U δ ) discussed from different m 27 + n − 62 | W , m ··· ,ad ,cd 61 ( 1 1 + gl U U 2 2 2 ]. We expect our factors the same as those in [ ∂ ∂ − ]. Finally, let me mention a recent elementary Miura factors w cb N ab ]. Algebras of type # 28 δ ) , , 60 − N ) , κδ 48 1) . In analogy with x κ∂ z 16 ( i . All the other truncations associated to − ( 59 C − + 1) 2 J Kac-Moody generators to determine OPEs 2 w ∈ U ,ab + κ (1) [ 1 ) − + U Nκ gl n . The composition of z ,N 1 | κ∂ 2 – 17 – (( − . Furthermore, the algebra ∂ Ψ -deformed analogues of affine Yangians (quantum 3 x, y, z, w m m,n = bc ×∞ ( q N cd ∞ \ ) n ) + | + gl x ]. κδ ( i cd m m CY κ∂ + L ( 63 W 1 W bd , are up to the ( + 1) U -basis. In this section, we generalize the above discussion 1) 58 ) for any U – + N − 0 appearing in the connection to gauge theories on ( ( ×∞ | ×∞ N 4.6 55 n 1 1 | ) n m | m − − 2 2 m κ∂ factors in the formulas of [ W N N W -algebras or # = ( + + 1) L W − satisfying ( algebra in the There exists a VOA satisfying conditions listed above and containing fields can be then constructed using the coproduct of ,ab 1 3 ) is a super-matrix of the N ×∞ z n w ,U ( | 4 ) m x N ,ab ( i 2 z W J 2 N ,U Let us finish this section by making few comments on the relation to other work. We can now state the following conjecture: y 3 ,ab 1 N 5.1 Coproduct of The Miura operatorgenerally coming the from following the form product of preted as a successiveSecondly, use we of use such a thetary coproduct geometric truncations for and associated the gauge-theoretical to simplestx functions picture truncation to of determine the algebra. of four the elemen- elementary factors. where of the in two ways.operators First, to we define introduce coproduct more on general matrix-valued pseudo-differential Miura appearance of supergroups in a gauge theory context from5 [ Miura transformation for So far, we have used the simple Miura operator of the form to be isomorphic toisomorphic to the the affine affine Yangian Yangianperspectives of of for the example Lie in [ super-algebra toroidal algebras) of type singularities were also discussed for example in [ Structure constants of It should berecovering straightforward these to ( findbeen a recently proposal also for studiednames rectangular all in OPEs a different of basis for example in [ Conjecture 1 U that the algebra is uniquely fixed by associativity of OPEs. These four OPEs are goingoperators to and be the sufficient generalized for Miura testing transformation our below. proposals for elementary Miura JHEP01(2020)042 ). = ×∞ n (5.5) (5.6) (5.3) (5.4) | N 3.12 m   for W by multi- ...... using ( ×∞ + ×∞ + n 2 n | 3 | 3 m − , N − m . appearing in the 2 1 W N case, we have seen + N W ,ad ,kj ) (2) ) 1 1 ... N 1 can be an arbitrary ) ∞ U U N κ∂ + κ∂ 2 ( db ( ,kj 1+ N 3 ∂ (2) δ 2 2 , − expressed in terms of the (2) 3 (1) W U bc 3 κ N ) U ,kj U ) ) leading us to the following 1) ,ik (2) w satisfying OPEs of 1 1) (1) ( i 1 + − i κ∂ L U ) ( − 4.6 2 ( ( U , + + 2 3 ) cd . Since − − 2 y 2 δ N k of the algebra U w ( i ( − N Nκ∂U X ab i 1 L ) ,cb Nκ . Furthermore, one can immediately + − δ N 1 ˜ N i U from ( 2 ) z U κ∂ ) + ) + + = − ( 2 ad ) ) + κ∂ N cb with the parameter ,kj ,kj ×∞ δ κ ) gives rise to a well-defined coproduct ( (2) (2) ) (2) δ 1 1 2 w n | cb ,kj U U U (2) 1 ad (1) N 2 + m κ∂ − 5.5 ( ,ik ,ik U + cd z κδ 2 (1) (1) 1 2 W ∂U algebra, one might be tempted to consider a ( + 1 U bc + U U – 18 – being additive. − ,ik ( ( ab 1 (1) 2 1 1) + 2 − N 1) k k ×∞ U 1 − 1 ) ( n X X N | − − N (( ( ) m k κ∂ N + + + , generators ( ) N X + 1 W κ∂ ,ij ,ij ,ij ( (2) (2) (2) (2) 1) N 1 2 3 1 κ∂ ∼ (2) i ( U U U U (1) 1 ) subalgebra equals ˜ 1 U − = U U w + + + + ˜ κ ( N N ) 2 ( + + ,ij ,ij ,ij n and N ,cd (1) (1) (1) κ | 1 2 3 1 ) 1 N U U U + ) N m U (1) ) i with ( ) κ∂ \ ( = = = z κ∂ U sl ( κ∂  ( equal the order of the pseudo-differential operator. ×∞ ,ij ,ij ,ij ,ab  × super-matrices of generators of spin = ( 1 2 3 n 1 | N ˜ ˜ ˜ Miura transformation from ( U U U n U L m = | ) currents n m | W (2) m L ( \ are in terms of gl i (1) 2 U L N In analogy with the above analysis, one can define a coproduct on This gives an expression of the new generators + 1 structure on 5.2 Building blocksAfter of allowing truncations more general pseudo-differentialthat operators there in exist the two more elementary Miura operators We have checked that these satisfyconjecture: OPEs of Conjecture 2 and expressing it in terms of a pseudo-differential operatorN of order plying two factors of this form containing infinitely many super-matrix-valued fieldswith with parameter i.e. the level ofcomplex the number for themore general general class of pseudo-differential operators identify the order ofOPE the of differential operator where JHEP01(2020)042 , = )- )- 27 m (5.7) p, q p, q )-web ]. The 65 p, q truncations. ∞ 1+ algebra if we set W ] that such a gauge- ] with the Dirichlet . ) gauge theory in the ∞ 3 of 65 66 n , ,N 1+ . Composition of such 3 2 w W ,N 64 ,N , 2 ) Chern-Simons theory on 1 n ,N N 31 | and will be discussed bellow. 1 Y and ] leads (at least in some cases) N m y Y )-branes. Zooming out the ( 65 , p, q (1) for 64 , x, y, z, w )-branes. The low energy behavior of [ gl n 31 p, q + = 0) corners. The path integral of such 11 m m − – 19 – . These will play the role of elementary building + 1 from the discussion of κ ] for further details and more general configurations. = Ψ 1) and both terminate on the horizontal boundary. . The corresponding geometry is known to be dual , κ = 0 or 4 31 , = 4 SYM super Yang-Mills theories divided by ( x, y, z, w n 27 N 1) interface couple the two theories by an additional three- ) gauge theory in the upper-right corner. These two are , is then given by m κ ) n | m ( \ )-branes in type IIB string theory determined by the same ( gl (1) Kac-Moody algebra associated to [ 1) interfaces with supergroups determined by ranks of gauge groups on gl p, q n, The gauge-theoretical configuration consists of the U( 1) interface with the Dirichlet boundary condition descending from the boundary , The identification of corresponding algebras can be easily done along the lines of [ )-web configuration from figure = 0. Note that we reproduce the relation Ψ = 0) branes. Local operators at such an interface give rise to a vertex operator algebra. ]. Let us briefly review the idea behind such an identification. Consider a general , 11 -algebra. , n p, q Kac-Moody algebra 1 theoretical configuration localizes tothe the (0 complexified GL( conditions of the four-dimensionalSimons theory. theory Local (corner operators of atalgebra the the associated boundary four-dimensional to of setup) the the are gauge Chern- known group to of form the corresponding the gauge Kac-Moody theory. The level of the upper-left corner and thedivided by U( an interfaceThe labeled boundary as (0 conditionboundary on condition the and horizontal the (0 linedimensional bi-fundamental was hypermultiplet. identified It was in argued in [ [ the interface condition.are Luckily, easy the to interface identify conditions forInterested and reader the should the simple consult corresponding configurations [ of algebras x the right and on thetwo left of Chern-Simons the theories interface along(1 are the lines furthermore of connected Mikhaylov andAn by Witten analysis [ an of corresponding interface interfaceto descending conditions a [ from proposal for the corresponding algebra in terms of a BRST reduction implementing such D3-branes is capturedinterfaces by descending from the D3-branesweb, ending we get on a ( junction(or of three four interfaces in dividinga the the four-dimensional configuration special spacetime localizes case into four supported to of on the ( path integral of the complexified Chern-Simons theory 31 ( to a junction ofdiagram. ( Original M5-branes supportedvarious stacks on of four-cycles D3-branes inside stretched the between toric ( three-fold map to more general Miura operatorsWe yields will a now more use the general geometricalgebras class and associated the to gauge-theoretical divisors picture of blocks to identify for the more elementary general truncations analogously to generator of the JHEP01(2020)042 for i (5.8) (5.9) (5.11) (5.12) (5.10) (5.13) , γ i ) β x ( ad J ghosts (free cb δ together with b, c bc algebra. ˜ J 1) , J, ) − ×∞ 1. More concretely, y ( ( n ad | ghosts (or symplectic with OPEs − J . − m n a w ) cb w copies of x δ W ( 1) cb β, γ + − bc − ) at level n J c − z n z ( 1) | ad ! δ − ) m i ( cb (1) currents ( Y \ ∼ − + i [ , . . . , m gl − gl cd ) cd w δ ) copies of (1) generator. X is are charge y ( + w n. ( ab [ i cb i gl + 1 ( − δ m i ab J Y + 1) z ˜ κ κ m X 1) ad coupled together at the interface. It was 2 − ) is analogous with only two differences. δ m n ) ( + − | z = n n ( ( cb y w i m i − cb + + =1 + 1) that leads to an extra shift of the level by δ i X + κ − m cd ad m (1) for n, z − + × S ) that consists of bosonic generators cd + ( i i [ – 20 – gl δ ab − ˜ κδ − ˜ J ,Y , c ,Y κ ab bc (1) 1) i i m δ Ψ = b − [ − w gl 1) − X ( 2 − + 1 J − ) − ( +

cb = w z and δ ˜ κ ∼ − dz κ ∼ ad ) z ) ( I κδ 0) lines ending from the right and w w ( , bc ( ) = i 0) lines ending from the left, all charged under the GL(1) gauge y , ( 1) Y cd ) Q J − configuration, the path integral localizes to two copies of GL(1) z ) ( ( z i z ( thus satisfy relations ) ∼ X y ) ( ) ab y ( J ab w ] that a consistent coupling requires ( J ) x have +1 under the GL(1) action and The configuration associated to In the 31 ( cd , and fermionic generators is a collection of fields ( i J is then the Kac-Moody algebra with OPE ) X 27 n | z x ( m ) x S ( ab . The resulting algebra is the Kac-Moody algebra with respect to the BRST charge J , . . . , m n n | − m -algebra. = 1 -algebra. the above coset denotesS the BRST cohomology of two where i The fields group. Corresponding VOAs can be then identified with the coset z Chern-Simons theories at levels argued in [ bosons) associated to the (1 fermions) associated to (1 i.e. note the particular normalization of the diagonal The generators y First, the orientation of theSecondly, diagram the is vertical reversed interface that is leadsm now to ( the opposite sign of the level. ciated to that is exactly the algebra used above in the construction of the where Ψ is the coupling constant of the four-dimensional theory. The building block asso- JHEP01(2020)042 1 | (5.16) = 1 (5.20) (5.14) (5.15) (5.17) (5.18) (5.19) 1) line , n | m ) z ,cc ( 1 )-web diagram 12 J . 0) and (0 c 1). This can be , | p, q ) . 1) z ,ad ( 1 (1 1 − \ ( U gl − n db 2 + n satisfy various relations =1 δ ) c X m bc w − . ,ab ab 1) . 1 − δ n m m ˜ ) J ) ) − z z ,cc ( k k ( ! ( 1 + J ) + + i − κ 1 2 1 2 cd n w J δ 1) q q ) − 1 1 z ,cb − ab − ab − ( + 1) 1 ( δ − − δ ∼ n z U z z m n 1 κ + ) =1 c X ad m 2 1 − + 1 + ). The only exception is the + w δ − ) ( . ab κ κ ˜ cb w , (1 + (1 δ J m cb ) 1 + n δ m 5.15 ) )-web consists of a single (1 z − − ) cd − ( k ad m, n ) + k z + 1) ˜ + b δ κ + J + 1 + 1)( ( + ( = p, q – 21 – 1 2 κ Y ab bc 1 2 κ ( κ κ a 2 ˜ κ ( κ 1) 1) symmetry of the corresponding ( zq 2 X zq − − 4 s , + ( ( − S to be actually strong generators of the algebra unless 2 = ( ) ) − + (1 + z ,ab (1 ) ( ) 1 + 1 w ghost system. The two currents are normalized as z ,ab z ,ab J ( ∼ 1 ( + 1 1

− =0 ∞ J J ) = Y κ k n c, b z w ) ( ( z ,ab z 1 − ( dz 1 ) the algebra contains less states than algebra freely generated ) ∼ U z ,cd 0 n m ( 1 | ) I Kac-Moody algebra at level U w m − ) 1 ( z − ) J ( ) 0). This expectation is supported by the the following arguments. The z ,ab ) | ( ) 1 n z z ,ab | ) is given by the contour integral J ( ( 1 J m U = ( \ 5.12 gl ) , the expression simplifies to z ,ab 0) or (2 ( super-matrix of free generators at spin one. One can also explicitly check this 1 | m U n = | n m ) = (1 In general, normally ordered products of derivatives of We expect the generators One can easily check that combinations of the form n For | 12 m after zooming out. In such a picture, the ( expectation at low levels for small numbers that follow from thecase realization for in which terms of theunderstood ( as character a agrees consequence exactly of the with the character of For generic values of ( by an ( character of ( i.e. satisfying generate the are generators of the the cohomology at spin one. Moreover, combinations have introduced an auxiliary that imposes the above-described constraint quantum mechanically. In this expression, we JHEP01(2020)042 ) Q in z ,bb ( 2 . In κ J , one (5.22) (5.24) (5.21) J − ) form the z ,aa → ∞ ( 2 and ) κ J z ,ab i ( 1 J . ,Y ) i and X b , then correspond to J∂J 6= ) ( 0). First, comparing | a (5.23) 2 JJ )) (2 , ( independent realization , for J 2 1 )) b κ + 1) 0) ) J Y | b z ,ab κ ( are in the cohomology of together with x, y, z, w ( 1 2 ( a Y + 1) J ( ) ) X z ,bb a 22 κ b z , ( ( 3 ( ( 2 Y X J ∂ ) = (1 )) + 1 J ( n − | − i − 2 + 1 YJ )) + ∂ 2 m + 1 ) ( ) a 11 z ,aa κ z , ( κ 3 but they can be expressed in terms of ( 2 X X YJ J ( ( ), the tail of fields containing the current n ,J ∂ − | ) + X ) + b ) b 21 ( b z m , Y Y ( and 2 5.23 2 + 1 2 a ∂Y b , ∂ 2 ,J κ + 1 X a b a – 22 – 1 ) Y 12 6= κ X z , X − a ( ( i 2 ( a ) + J X ∂ ) and ( − − Y ) + ) 2 ) = = for b J, b Y 5.21 ) Y ) ) a z X∂ z a ,ab z ,ab ), we can define ( i,ab ( ( 1 X∂Y 1 ( + 1 3 ( ), ( J X J J κ ∂X 2 also for general − − ∂ 5.21 ) ) Q 5.15 0, the set of strong generators is = ( | ) + = ( ) XY z ,ab 2 ) ( 2 z ,ab XY ∂XY ∂ and can be expressed in terms of fields of lower spin. ( J 3 0) with the the character of an algebra generated by four generators J n | | = ( = ( = ( m ) ) ) z z z ( ( ( 1 2 3 J J J ). In the following sections, we will use this simpler, . ) 1. On the other hand, we cannot simply restrict to fields at low spin in this z ,ab ( 1 5.24 J Finally, note that the resulting algebra is actually independent of the parameter For the special case 1 Let us briefly discuss the two special examples ( i > disappears and one finds simpler expressions limit since we wouldrelations). miss Instead some of of the abovecan the more-complicated equivalently generators consider expressions the in (the algebra terms generated simplified of form by generators infinitely ( many satisfy simpler generators more ofof the the algebra. for for all the other all the cases and oneall can the find above a expressions more ( J convenient description by simply sending Moreover, in analogy with ( and argue that combinations are in the cohomologyfields of and show that (at leastfull at set low of levels) strong generators in this case. These combinations the same diagram and they should all lead tothe the character same of algebra. (2 at spin one, wefollowing can combinations see that we are missing three fields at spin two. One can define the crossing each other. All the four configurations associated to JHEP01(2020)042 ˜ n J J, ) now (5.29) (5.30) (5.31) (5.26) (5.27) (5.28) (5.25) and i . Using ,Y m i N X ) w ( cc J . c ) , 1) w ,ad ( n 1 − ( J + . Normalizing the n 2 case with ( db form particular trunca- ) n + =1 δ m z c X w m bc + − − ab J. 1) b δ z − Y ( ( . ! a + 1 b ) x, y, z, w X κ − Y J n , . . . , m w 1 Kac-Moody algebra at level ab cd ) − i ,cb ∼ − δ δ i − 2 ( 1 + 1) 1 + 1 + 1 ) ) ∂ ab J z m n κ a n n δ w | 1 + 1 ( ad 1 κ X + + ˜ δ 2 + = J m κ ) ) ( b κ cb i − + \ m z w , + b gl ( cb ∂Y is analogous to the ˜ Y − ) + cd J δ − a b a + κ = 1 , z + 1)( Y w ad X ( X b ( – 23 – ab a δ ˜ 1 κ κ κ Y + ( bc X − 1) a i , ( b − 1) 2 X ∂ s Y − ( ) a − -basis with correctly specialized parameter ( w = = + = + 1 U 1 + ) ) ∂X ) − ∼ w w ,ab κ -generators and the coproduct from the above section, one w ,ab

and bosonic for ( − ( i,ab z 1 ( 1 ) ( J U J n J w = ( 1 − ) ) in the w ,ab w ∼ − ( ( 2 cd m , . . . , n ) J U -algebras w ) − ×∞ 1 algebra. We will now determine the specialization of the parameter ( z n = 1 | N ( ) J z ) ,ab i ) m ( 1 w w z ,ab ×∞ ( 4 J ( 1 limit, that can be realized in terms of W n | J N U = m z The situation associated to ) 2 W w ,ab N ( → ∞ 1 y U κ 3 N x associated to such elementary truncations and find combinations of their fields that -algebra. According to the above analysis,tions the of elementary the algebras N satisfy OPEs of such an explicit realization of as above. 5.3 Instead of these generators,from we the will again consider the realization of the algebra coming The discussion of stronginterchanged and generators up is to the somecombination same minus signs. as For above example just at with spin two, the we role have the of following i.e. satisfying OPE identified with the standard generators of the generating the cohomology at spin one with the combination being fermionic for currents as we have w JHEP01(2020)042 ) ) ). N y ,ab ( 2 2.6 U 5.32 (5.35) (5.36) (5.32) (5.33) (5.34) (5.37) -algebra ) with y ) from ( ) . One can y 4.6 n 1 ( N J ) + w y ( 4 κ of the elementary m J N 2 z − N 2 κ. N str( y κ 1 ) with the general ex- cb y 0, the above expression satisfying ( ab N | 3 − N δ A 1 ) N 5.8 . One gets = y ac h x ) ( + i,ab = 1 y κ A ( ab U + ) 2 2 n
Ψ = J | m n 4 h ) =1 + ) − c m N X − y n − 1 ( 1 consistent with the formula ( 1 h = J . As discussed above, the = m N h 2 2 1 ) − N κ h + N ab . h h 2 str( ) m ) κ x ( N − + ( i − 2 is 1. For κ and ab 1 2 AA J h 2 δ ( ) n h 3 n N ( x ) + h ( i 2 + -algebra by determining − + at level ˜ h m 1 L 3 = ) . N κ∂ y ˜ m κ N – 24 – + ) − , n ( ab n n ) ) ) w 3 = J cc y 2 − 4 n n -algebra as a quotient of the two-parameter family − ( ) | + + ( ) h -algebras. N ) N A y x y c 3 z m ( i m N ( ∂J m m − 2 ( h -algebra by multiplying the elementary Miura factors \ x L = Ψ 1) J N 1 1 of the parameter = 1 as expected. The corresponding Miura operator is + sl − − y − h N κ = y x str( 3 κ ( ) Ψ Ψ -algebra is trivial. Comparing ( w str( N m N N ab 4 N m x δ x =1 Ψ + N − c X n 1 2 − z − beyond 2 ab = 0. n − ) N ) 4 ) ) = (( y y y 3 ( ×∞ N Ψ = ( A ab n N J . = | − ) x = ) m y ( str( y N 3 κ∂J = ( ab J W N J − ˜ κ is a non-trivial test of our proposal for generalized the Miura transformation ) parametrized by = = = ( ) ), one gets order The case of the x, y, z, w We will now look at the 1 is additive under the coproduct, we can immediately identify its specializa- y ) ) ( ab 4.6 y y ,ab ,ab J ( ( 1 2 ,N N 5.21 U U ]. = 1 27 2 It is straightforward to identify the first few generators Since N -algebra. -algebra. and all the products are normally ordered.in terms Note of that an existencefor of truncations the realization of of where we have introduced notation correctly specialized in terms of the Kac-Moody generators then write and identify the specialization for simply y of algebras ( contains the Kac-Moody algebra from [ x pression ( that is exactly (upThe to normalization the is overall such normalization) thatsimplifies the the to charge order of of associated to tion associated to the general factors. As discussed bellow, one recovers can define a general JHEP01(2020)042 ) N 5.32 . (5.41) (5.38) (5.39) (5.40)  ) ) with w ,ab ( 1  ) ) 4.6 ) ) ∂J w z ) ( ( 1 2 n J 2 ∂J  − . ab ) ) ) str(  ) m z ) z str( ( ( ) 1 1 ab z J + J ( 1 δ Nκ satisfying ( ab ) ) z δ κ ) or possibly higher-spin ( ) 1 κ 2 = ∂J b str( z J + ( i,ab Y  κ + ˆ ) − a . κ U 2 w ,ab str( κ ( ab 2 X  ). Similarly, the expression for ab h ( δ ) ) ) + ( J )(2ˆ ) ab ) ) − − z  z δ n )) w ( ( 1 = ( 2 ( 1 1 5.32 n consistent with the formula ( n ) κ J − − h J 1 z ,ab . ) ( ) ) ∂J 2 − − h ) z + ˜ ,ab ) N w is the same as in the above case of the n J m ( 1 z ,ab ( 1. One finds n 1 ) ( str( ( κ 1 m J m J str( κ − N − − ab + 2)ˆ i > δ 2 m ab κ∂J m κ ( δ , 1 2  − ) ) 1 + 2 ) − ) − , ) + 2( ) + κ z n + 2)( )  ) 2 ( w 1 ) ) for κ ( w 1 = + ˆ  z – 25 – ) b ( ( J − 2 − + ( ) 1 J z ,ab (5.42) Y κ ( ) J κ 2 n n ) J 1 z ab m ) m J ( 1 ) ( p str( − w − − i str( ) J ( 1) 1 − str( z str( ∂  − + 2( ( = 1 ab J m m a of the parameter n δ − J ab κ ) ( ab ˆ κ str( ab ) X z δ ,ab z κ δ κ z δ 1 ( (  1 (2 ( str( 2 ( 1  − N ˆ κ  Ψ + Ψ + J )  ab  − κ ∂J ab + n δ 1 n κ 1 and we can again write κ − )( ) ) − δ ) b (2ˆ κ ) κ κ κ − 1 κ − ˆ = 0. − κ + ˜ n z ,ab Y + 1 4  2 ( 1 a 3 + ˆ + ˆ 1 κ = κ κ − + m J + 1 = m − + 1 )ˆ X N κ κ κ κ − 1 ) κ n 1 + = = m z ,ab 2( − ( = 1 i κ = − − − 2( ) ) J − +(2 2 ∂ 1 + 2 z z ) ,ab ,ab ˜ κ κ ( ( + ˆ 1 2 N m  w ,ab = = ( 1 U U κ = ( ˆ 2( κ κ J ( = at level ˜ ) ) The identification of the parameter ) z z ,ab ,ab Similarly as above, we have now an algebra containing the Kac-Moody alge- 1 − − − − z ( ( 1 2 1 ( i,ab − U U J = = ) , the above expression simplifies to ,N n n ) ) | w w ,ab ,ab = 1 ( ( = 2 1 m ( has the same form 4 U U \ It is straightforward to identify the first few generators m sl ) N w -algebra. ( i,ab -algebra. -algebra up to an overall minus sign consistently with ( z U w For where we have introduced a constant for correctly specialized in termsgenerators of the of the fields bra and identify the specialization z JHEP01(2020)042 can then α (5.45) (5.46) (5.47) (5.44) N ×∞ . n 1 | N ab m ) copies of the + ) 4 W j 2 by multiplying N α , one can define ( N and some minus 1 + ) ab pair of fermionic 2 ) U w n ) N ×∞ ( ) . i m w n + | L ) − ( α  1 3 specialized to ( pairs of bosonic fields 1 ) w , m J ) ... ( N ) ) m U z w m L ( ( w N W ( 1 + ( 4 1 L 2 N J determined as a subalgebra ... , ∂J ( − + ) ... 1 3 α y being the following pseudo- − ( +1 N N N . Coproduct of +1 + 4 str( containing ) i ) L + ) w ) 2 N 2 i X = , α 4 algebra. w j ( ) − ab + Y N κ∂ ( 2 N x i 2 δ ( 1 L N containing ( + z J ) N ) 1 N 2 X i α L − + ( N ) 2 Y N w 3 ) Kac-Moody algebras, i κ∂ 4 x, y, z, w w str( y 4 U ( ( N w ,ab 3 and N κ ( X N 2 1 ) L = N + . + ab z i U 4 ) 3 n δ 2 x 1 i | (5.43) N α with the parameter N 1 2 N α ,ab ) − + κ∂J ( + ) 1 m y + =1 α copies of 2 1 i ( 2 3 X w + \ − N ( − N 1 1 N N ) ×∞ gl ) ∂U 2 J – 26 – + N + n x j N w ,ab ) copies of | 3  1 ) ( α 2 z κ∂ ) ( N 4 m N ( ) J N ) str( κ∂ L w N W α ( ( 1 + 1 , ( 1 1 =1 ab ) ) − J i i i j ... X δ U + 1) U α α ,ab ,ab 4 ( ( κ 1 2 2 κ 1 + + N +1 str( 2 ( N U U 2 +  α  4 4 N + 3 N copies of the − ) N ) 3 1 together with some shifts by factors of N N N y + ab ) ) 3 ( N ) 3 δ + + κ + =1 z ˆ κ w ,ab − κ∂ 3 3 i 2 L X ( N 2 N ( κ 1 κ∂ 1 N N N κ 4 J ↔ + + =1 =1 generators of + = ( ... i i ×L 2 2 X X − + − 1 = ( i κ -algebras from the derivation of the OPEs of N N N ) 3 U +1 = = + + α ) 3 κ N 1 1 ( y ) ) ( N x + N N L w w ,ab ,ab , this expression simplifies to L ( ( . This gives in particular the following expression for the spin-one and the 2 1 Kac-Moody algebras, The ) n 3 = = x U U n ( N = + L ,ab ,ab pairs of bosonic fiels. The subalgebra can be obtained similarly as above in the m 1 2 m n − U U ... super-matrices of generators at spin κ ) x, y, z, w − x i case or the ) ( 1 pairs of fermionic fiels, and finally U = n L | ∞ n Having determined the elementary Miura factors Let us conclude by stating the following conjecture: α m 1+ ( \ Conjecture 3 be realized in termsgives of rise the to elementary a algebra free-field realization of the for spin-two fields with differential operator and rewriting it as a matrix-valued pseudo-differential operator in the standard form gl and fields and W an algebra associated to any function ofinside the a form tensor product of signs. For up to the exchange of JHEP01(2020)042 are that (5.53) (5.50) (5.51) (5.52) (5.48) (5.49) y . 1 3 κ m,n ..., − and ) + CY . x 3 2 κ∂ − ) ( . In particular, 1 κ  0 w , 3 − 1 1 ) − ... z CY κ∂ ( + , ( 2 2 can be related by the  ! ) − for ) -algebra in terms of the z ) 2 y ( z 1 z ( κ + 1) κ∂ ,U = . On the other hand both ∂J ( κ 1 ∂J n (  satisfy κ  U ) + ) xy − , 1 y x κ ( ( ) N n 3 χ √ y 1 + m,n ∼ −  ( κ w + ∂J ) ∂J ) √ φ J m w )  CY ( w − − y + ( − ( 1 1 + 4 κ L ) ) ) J y y N  z ( ( κ ∂χe z ( 1

J J 2 m J 1 + ) ) ) N =  y κ z corresponds to a different free-field realiza- z ) (  1 + y ( ( 1 z 3 ) κ J ( n 2 J y N ( L +  1 κ − x – 27 – 1 ). As a consequence of the above discussion, the  1 + 2 ) κ N y + ( . , γ = + w 2 ) χ J 5.44 ) z 2 and in the ring of holomorphic functions on m ) ( − y 2 x − ( ,J 1 + 2 + φ ( J β∂γ 1 n 1 κ 4 2 e L  J N )  ) N w − − x ) z = w + + w ( κ m gives 2 4 + 1 ) z L β system by introducing ghosts. The two Miura operators associated to ) N 2 − N z κ∂ 1 ( x 1 κ ( y z ∂βγ ( z − ) 1 + = 3 1 J ( ) y L ( N − = ) β, γ β, γ 2 x = = y J ) ∼ κ∂ xy ( N z ) ) ) ( 2 )( y L + z z ) 1 ( ( J 1 2 w ) 1 y ( and − ( x U U − ) 3 ( J x 1 κ 1 N ( J N and − x + J ) The pseudo-differential operators for + ) w ) z 4 x ( βγ ( κ∂ N ) κ∂ J z x = ( 1 = = ( ) J − z 2 ) ) ( 1 y 1 + ( x N J ( (  y L 1 L system We can bosonize the Let us illustrate this relation on the simplest example of − 3 N where On the other hand,β, we γ have for the free-field realization of the with normalization Multiplying these two standard bosonization of i.e. algebras on both sides have the same size and satisfywe the will same now OPEs. show that the free-field realizations of fields gebra of the second other.actually Since expect both this functions not correspond toleads be to to the the the same case following physical and conjecture: setup, the we algebrasConjecture should be 4 equal on the nose. This Remember the relation relate functions x tion using the Miura transformationgenerators ( of the algebra satisfy the same relations but one algebra might still be a subal- 5.4 Bosonization-like relations JHEP01(2020)042 (5.56) (5.54) (5.55) copies of copies of 2 m ] in terms of n systems. The ) systems can 69 + – ) ghosts on one (1) currents. It 2 [ gl b, c 67 β, γ m β, γ . ) . and n w w is realized by − m n z β, γ z w = copies of ( m log( z  m xy ∼ ) χ y 2 ) ( + ∂ J w n ( − + χ ) that satisfy OPEs of 2 ) ) y ) ( z x ( ( J ∂χ J ∂χ (  κ + 1 − and κ 2 κ + 1 ) , χ ) ghosts. Comparing with twice the Wakimoto – 28 – ) + κ ∂φ w ∂φ ) b, c ∂φ x can be thus realized in terms of two copies of the r ( − ) ghosts associated to positive bosonic roots, J − = = ( z xy ) ) = = z z β, γ ( ( 1 2 log( J J ∂φ ∂χ ) admits the Wakimoto free-field realization [ copies of ( n ∼ − | ) m mn w ( ) ghosts associated to positive fermionic roots. \ ( copies of ( ) ghosts, ) ghosts. gl ) ghosts, φ ) 1) b, c z − b, c β, γ ( 2 ) ghosts, β, γ n ) free bosons on the other side. It is likely that these ( φ ( copies of n n b, c satisfy . n z + + χ ) free bosons associated to Cartan genrators, = n 1) m copies of ( − + 2 and ) system and 2 copies of ( copies of ( xy m copies of ( copies ( ( φ m m m n m n ( mn β, γ On the other hand, the right hand side corresponding to The general situation is much more complicated but might likely be proved by the use First, the algebra • • • • • • • Putting everything together, wethe have ( an algebrarealization realized above, in we see terms thatside of we and have 2( a mismatchbe of bosonized and the two free-field realizations related. together with above-listed fields. The left hand side associated to of the Wakimoto realizationproper together analysis with is the left bosonization forsides of future of work such but a let us generalized at bosonization least corresponding count to the number of fields on both relates the two free-fieldrelation realizations and can be thought of as the VOA analogue of the that are expressions simplyis in then terms easy of to check that transformation A simple calculation leads us to the following identification where JHEP01(2020)042 ) ) 1 n N | b, c w (6.1) m 4 ( \ N gl z 1 2 N ) N ) y ], a simple ) ( ) and ( y ) w 3 L ) ) ( z 52 x ( N w ( L ( β, γ L x ( L L N 4 N ) . ) and its truncations 1 2 z 1 ( h h L N − ( 1 2 ×∞ h N n | ) ) + can be realized in terms of algebra m y . ( 4 L W N N ( ×∞ ×∞ 1 3 Algebra n n with Miura operator | N associated to a specialization of h | ) m n ) m 2 with Miura operator − x + with Miura operator with Miura operator ( W h 1 2 W ×∞ m L κ 1 Parameter Ψ = − n ) − ) N | − ) κ n n 2 n | | m 1 Bosonization-like relations − | h ) h m m Specialization of parameter m ) W n ( ( | ( + \ \ \ gl gl m 3 gl m ( . In particular, we identify them with \ N . − gl ) n 2 n Truncation ( – 29 – w ( h generators of m 2 − 2 z 1 U 1 , h x, y, z, w 1 = h 1 N z y x N w ) h n w n w 4 m xy w singularities. Such a correspondence can be schematically 4 and w N N m using a matrix generalization of the Miura transformation. m 1 z − z z 2 z 2 3 U m,n n N N = = y ×∞ y (( . 3 3 n CY | N xy N xy = x m x : of the algebra Geometry W N 3 N m,n x, y, z, w CY Relation Elementary function Elementary function Elementary function = Elementary function Function Charge of α Equivariant parameters for ) α ( summarized as: inherited from the relation and the geometry of systems. We show that these algebras if we correctly specialize the parameter can be obtainedL by composing various matrix-valued pseudo-differential operators mentary truncations labeled as Kac-Moody algebras together with subalgebras of various copies of ( matrix-valued Miura operators. the parameter bras denoted as We defined a coproduct structure on the algebra using general pseudo-differential We conjecture new bosonisation-like relations between different free-field realizations We find a non-trivial interplay between properties of We propose that general truncations associated to general functions Using gauge-theoretical considerations, we identify VOAs associated to the four ele- We identify a large class of truncations of We find OPEs of spin-one and spin-two generators of a two-parameter family of alge- Apart from the rigorous proof of the above-stated conjectures 1,2, 3 and 4, the above 5. 6. 4. 3. 2. 1. analysis raises many questions. Let us mention at least some of them. In [ Let us finish by stating the main results of the above discussion: 6 Conclusion JHEP01(2020)042 N Ψ) J. − , 1; , \ (2 D ? . Note that in this ]. ×∞ ∞ n | m 1+ SPIRE W W IN [ ]. was found. Is there such a closed- version of ∞ (1988) 507 N algebras? What is the physical origin 1+ SPIRE Ψ) IN W A 3 [ − ×∞ n | 1; , m (2 W case? Can we extend the Miura transformation – 30 – D by an analysis of null states of its ∞ ]. N (1985) 1205 1+ Lie algebras and equations of Korteweg-de Vries type W The models of two-dimensional conformal quantum field 65 SPIRE ), which permits any use, distribution and reproduction in Int. J. Mod. Phys. IN , [ Infinite additional symmetries in two-dimensional conformal quantum CC-BY 4.0 symmetry ) (1984) 1975 symmetry leading to a This article is distributed under the terms of the Creative Commons n and construct truncations of shifted versions of ( 3 Theor. Math. Phys. Z 30 S , ×∞ n | m W field theory theory with Sov. Math. It would be even more interesting to extend the above construction to more compli- V.A. Fateev and S.L. Lukyanov, V.G. Drinfeld and V.V. Sokolov, A.B. Zamolodchikov, [2] [3] [1] any medium, provided the original author(s) and source are credited. References of Innovation, Science and Economic Developmentthe and Ministry by of the Research, Province of & Ontario Innovation through and Science. Open Access. Attribution License ( thank Mina Aganagic, Kevin Costello, ThomasYaping Creutzig, Yang, Tadashi Gufang Okazaki, Zhao Yan Soibelman, andful Yehao Zhou to for Kris discussions on Thielemanssupported related for by topics. NSF his I grant Mathematica 1521446, amPhysics, package NSF thank- the OPEdefs. grant Simons 1820912, Foundation the The and Berkeleyat Perimeter research Center Perimeter Institute of for Institute for Theoretical is MR Theoretical supported Physics. was by Research the Government of Canada through the Department Acknowledgments I am particularly gratefulfor to sharing Davide his Gaiotto unpublished for results many that suggestions triggered and my Tom´aˇsProch´azka interest in the topic. I would like to case, both parameters ofbe the able infinite to algebra identify aresubalgebra. specializations visible Other already of geometries at containingtechnical level compact issues one. four-cycles and are One remain expected to should to be lead a to challenge new for the future research. cated geometries. The universal infinitely-generated(with algebra the associated vacuum to character a identical given to geometry tion) the would square have of truncations the labeledThe topological by vertex simplest a partition example choice func- of goinggeometry a beyond with holomorphic the function configurations on discussed the in variety. this work would be a form expression also forof the the more Miura general operators?associated The original to motivation a for large thesuch class a project correspondence of was understanding beyond generalizations VOAs beyond the of the AGT correspondence. Can we prove realization of the elementary Miura operators for JHEP01(2020)042 B B ] , ]. B 246 ] Duke , , B 356 SPIRE Nucl. Phys. -algebras Phys. Lett. IN , , W ][ Nucl. Phys. , Phys. Lett. arXiv:1211.1287 ]. hep-th/9410013 , , ]. ]. [ . Nucl. Phys. Adv. Theor. Math. , , SPIRE arXiv:0906.3219 SPIRE SPIRE IN [ IN IN ][ [ ][ (1995) 94 arXiv:0907.2189 ]. [ Extensions of the Virasoro algebra Coset construction for extended algebras and the equivariant (2010) 167 ]. B 343 arXiv:1202.2756 algebras with infinitely generated W SPIRE (1986) 105 , 91 2 IN W A [ Instanton moduli spaces and algebra and second Hamiltonian structure of SPIRE algebras using Jacobi identities (2009) 002 103 ]. ]. hep-th/9312049 symmetry in conformal field theories IN ∞ [ hep-th/9405116 [ W ) 11 [ W n Liouville correlation functions from Phys. Lett. ( , SPIRE SPIRE – 31 – SL IN IN A class of (1988) 371 [ Virasoro algebras and coset space models Unitary representations of the Virasoro and JHEP ][ , (1994) 409 (1992) 713 A study of (1996) 113 Lett. Math. Phys. Cherednik algebras, , Hidden B 304 Quantum groups and quantum cohomology 175 (1989) 49 ]. ]. Universal Drinfeld-Sokolov reduction and matrices of complex B 420 B 373 Quantization of the Drinfeld-Sokolov reduction ]. Commun. Math. Phys. 126 conformal Toda field theory correlation functions from conformal , Nonlinearly deformed SPIRE SPIRE hep-th/0206161 ]. 1) IN IN ]. [ [ Nucl. Phys. SPIRE [ − , algebras of negative rank IN Seiberg-Witten prepotential from instanton counting N [ Nucl. Phys. quiver gauge theories Nucl. Phys. ( Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras SPIRE , W , ) Quasi-primary fields and associativity of chiral algebras (1994) 365. A SPIRE IN N IN [ [ 76 ]. ]. ]. (2003) 831 (1991) 740 7 Commun. Math. Phys. (1988) 348 (1985) 88. , = 2 SU( SPIRE SPIRE SPIRE IN IN IN classical limit size [ arXiv:1406.2381 KP hierarchy cohomology of the moduli space of instantons on [ N Phys. Math. J. (1990) 75 four-dimensional gauge theories [ B 354 Commun. Math. Phys. superVirasoro algebras Virasoro algebras (1991) 367 constructed from Kac-Moody algebras using304 higher order Casimir invariants 152 B. Khesin and F. Malikov, K. Hornfeck, A. Braverman, M. Finkelberg and H. Nakajima, F. Yu and Y.-S. Wu, J. de Boer, L. Feher and A. Honecker, O. Schiffmann and E. Vasserot, D. Maulik and A. Okounkov, N.A. Nekrasov, H. Nakajima, L.F. Alday, D. Gaiotto and Y. Tachikawa, N. Wyllard, B. Feigin and E. Frenkel, H.G. Kausch and G.M.T. Watts, F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, P. Bowcock, P. Goddard, A. Kent and D.I. Olive, P. Goddard, A. Kent and D.I. Olive, F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, M. Bershadsky and H. Ooguri, [9] [7] [8] [5] [6] [4] [21] [22] [18] [19] [20] [16] [17] [14] [15] [12] [13] [11] [10] JHEP01(2020)042 ] ]. , (2012) , JHEP SPIRE A 10 , in the , IN 07 [ (1998) 91 Nucl. Phys. B 800 ]. 2 , ]. (2016) 077 (2011) 231 JHEP arXiv:1411.7697 5 10 , ]. [ SPIRE SPIRE IN IN [ ][ (2019) 160 Phys. Lett. SPIRE JHEP , , 01 IN Int. J. Mod. Phys. [ Hilbert schemes of , (2015) 116 (2018) 109 arXiv:1907.13005 , 09 11 JHEP ]. , Adv. Theor. Math. Phys. ]. JHEP JHEP , -algebras ]. SPIRE , , Cohomological Hall algebras, vertex arXiv:1608.07272 arXiv:1711.00390 IN W ]. [ , ][ SPIRE . SPIRE IN ]. arXiv:1705.01645 ]. ]. [ -algebra and vertex algebras of type Commun. Num. Theor. Phys. IN , Plane partitions with a “pit”: generating functions ∞ SPIRE Vertex algebras and 4-manifold invariants -algebras ][ – 32 – IN Birthday Conference. Geometry and Physics W ]. W algebras commuting with a set of screenings ][ SPIRE SPIRE SPIRE (2017) 503 th Triality in minimal model holography IN IN IN W [ 70 21 ][ Cohomological Hall algebra, exponential Hodge structures ][ Spiked instantons from intersecting D-branes SPIRE -algebra modules, free fields and Gukov-Witten defects ]. On IN W Webs of ]. ]. ]. in the quadratic basis ][ arXiv:1002.0888 The Omega deformation, branes, integrability and Liouville ]. [ Vertex algebras at the corner ]. ∞ arXiv:1512.08779 SPIRE -algebras and defects from gauge origami Branes and toric geometry , W IN arXiv:1810.10402 W Coset realization of unifying W algebras SPIRE SPIRE SPIRE , ][ SPIRE arXiv:1808.08837 IN IN IN SPIRE [ IN Nigel Hitchin’s ][ ][ ][ IN arXiv:1611.03478 ][ [ (2010) 092 ][ -symmetry, topological vertex and affine Yangian arXiv:1710.02275 arXiv:1908.04394 arXiv:1609.06271 Universal two-parameter W Exploring 09 [ [ ), BPS/CFT correspondence II: instantons at crossroads, moduli and compactness On quiver hep-th/9406203 [ AGT relations for sheaves on surfaces (2019) 159 Adv. Theor. Math. Phys. ]. , JHEP (2017) 257 05 ,...,N , 3 arXiv:1205.2472 , [ (2016) 138 SPIRE (2 arXiv:1006.2706 arXiv:1703.00982 hep-th/9711013 arXiv:1512.07178 arXiv:1711.06888 IN JHEP proceedingso of the September 55-16, Aarhus, Denmark, (2016), theorem B 914 (2020) 135101 and motivic Donaldson-Thomas invariants [ theory algebras and instantons nonreduced divisors in Calabi-Yau threefolds and [ [ [ and representation theory 11 W [ [ (1995) 2367 127 A. Negut, M. Dedushenko, S. Gukov and P. Putrov, N. Nekrasov, N. Nekrasov and N.S. Prabhakar, T. and Proch´azka M. Rapˇc´ak, P. Koroteev, M. Kontsevich and Y. Soibelman, M. Rapcak, Y. Soibelman, Y. Yang and G. Zhao, W.-y. Chuang, T. Creutzig, D.E. Diaconescu and Y. Soibelman, D. Gaiotto and M. Rapˇc´ak, N.C. Leung and C. Vafa, N. Nekrasov and E. Witten, M. Bershtein, B.L. Feigin and G. Merzon, A. Litvinov and L. Spodyneiko, T. Proch´azka, A.R. Linshaw, T. and Proch´azka M. Rapˇc´ak, M.R. Gaberdiel and R. Gopakumar, T. Proch´azka, R. Blumenhagen et al., [41] [42] [38] [39] [40] [36] [37] [34] [35] [31] [32] [33] [29] [30] [28] [26] [27] [24] [25] [23] JHEP01(2020)042 , JHEP n , JHEP JHEP , , ]. ]. ]. SPIRE Commun. Math. , ]. ]. IN SPIRE ]. affine Yangian ][ arXiv:1905.01513 IN SPIRE , to appear. , ][ (2019) 131 IN = 2 SPIRE SPIRE SPIRE ][ ]. ]. arXiv:1902.07414 N IN 10 IN (2019) 099 IN , [ [ ][ algebra ]. 12 ∞ SPIRE SPIRE JHEP ]. 1+ IN IN [ Negative branes, supergroups and the , SPIRE 1 W ][ JHEP ]. IN gl [ , (2019) 867 arXiv:1906.05868 The topological vertex SPIRE ]. [ IN arXiv:1712.08016 The supersymmetric affine Yangian 223 Branching rules for quantum toroidal gl [ SPIRE ][ arXiv:1603.05665 IN hep-th/0512245 hep-th/0607032 ]. ]. [ [ [ , ]. SPIRE -algebras, extended higher spin gravity and dual IN Branes, black holes and topological strings on toric – 33 – W ][ SPIRE SPIRE , Ph.D. thesis, University Waterloo, Waterloo, Canada Twin-plane-partitions and SPIRE (2018) 192 IN IN ]. arXiv:1806.02470 (2019) 086008 IN arXiv:1812.07149 ][ ][ , Homomorphisms between different quantum toroidal and arXiv:1101.3216 (2018) 050 ] [ Coloured refined topological vertices and parafermion ][ (2006) 018 03 The matrix-extended 4 , Refined black hole ensembles and topological strings M 02 ]. ]. SPIRE 12 One example of a chiral Lie group arXiv:1309.2147 IN Rectangular W algebras and superalgebras and their Rectangular D 100 [ [ Super instanton counting and localization JHEP VOA[ , arXiv:1811.03024 J. Pure Appl. Algebra SPIRE SPIRE JHEP , , (2019) 147 hep-th/0305132 JHEP Gluing two affine Yangians of , IN IN -KZ equations for quantum toroidal algebra and Nekrasov partition [ , ) ][ ][ 02 q, t (2016) 229 Phys. Rev. arXiv:1711.07449 arXiv:1807.11304 arXiv:1210.1865 , Instanton R-matrix and W-symmetry [ [ [ JHEP M-theory in the Omega-background and 5-dimensional non-commutative gauge The vertex algebra vertex 300 Crystals and intersecting branes Fivebranes and knots , (2005) 425 ]. ]. arXiv:1610.04144 254 , (2018) 200 (2018) 192 (2013) 060 SPIRE SPIRE arXiv:1903.10372 IN IN arXiv:1905.03076 [ [ functions on ALE spaces conformal field theories signature of spacetime 05 11 representations Adv. Math. affine Yangian algebras [ theory Phys. [ 01 coset CFTs (2019). Calabi-Yau manifolds T. Kimura and V. Petsun, W. Li and P. Longhi, E. Witten, W. Chaimanowong and O. Foda, R. Dijkgraaf, B. Heidenreich, P. Jefferson and C. Vafa, M.R. Gaberdiel, W. Li, C. Peng and H. Zhang, M.R. Gaberdiel, W. Li and C. Peng, H. Awata et al., ( B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, M. Bershtein and A. Tsymbaliuk, T. Proch´azka, K. Costello, T. Creutzig and Y. Hikida, M. Aganagic, A. Klemm, M. Mari˜noand C. Vafa, A. Linshaw and F. Malikov, M. Aganagic and K. Schaeffer, L. Eberhardt and T. Proch´azka, T. Creutzig and Y. Hikida, M. Rapcak, M. Aganagic, D. Jafferis and N. Saulina, D. Jafferis, B. Feigin and S. Gukov, [62] [63] [64] [60] [61] [57] [58] [59] [55] [56] [52] [53] [54] [50] [51] [47] [48] [49] [44] [45] [46] [43] JHEP01(2020)042 (2015) 340 ]. super Yang-Mills = 4 SPIRE ]. IN N Commun. Math. Phys. ][ SPIRE (1), 1 IN A ][ ]. Commun. Math. Phys. , SPIRE IN [ arXiv:0806.0190 [ arXiv:0804.2902 – 34 – [ Free field realization of current superalgebra (1990) 307 19 ]. (2007) 053514 Representations of affine Kac-Moody algebras, bosonization Branes and supergroups (2009) 789 48 Supersymmetric boundary conditions in SPIRE IN ]. 135 ][ SPIRE IN Lett. Math. Phys. [ Fock representations of the affine lie algebra , J. Math. Phys. ), k )( J. Statist. Phys. , N | arXiv:1410.1175 (1986) 605 [ M ( theory and resolutions gl 699 104 B.L. Feigin and E.V. Frenkel, W.-L. Yang, Y.-Z. Zhang and X. Liu, D. Gaiotto and E. Witten, M. Wakimoto, V. Mikhaylov and E. Witten, [68] [69] [66] [67] [65]