aa r X i v : . [ h e p - t h ] O c t On even spin W ∞ Tomáš Procházka Arnold Sommerfeld Center for Theoretical PhysicsLudwig Maximilian University of MunichTheresienstr. 37, D-80333 München, Germany
Abstract
We study the even spin W ∞ which is a universal W -algebra for orthosymplectic series of W -algebras. We use the results of Fateev and Lukyanov to embed the algebra into W ∞ .Choosing the generators to be quadratic in those of W ∞ , we find that the algebra hasquadratic operator product expansions. Truncations of the universal algebra include principalDrinfeľd-Sokolov reductions of BCD series of simple Lie algebras, orthogonal and symplecticcosets as well as orthosymplectic Y -algebras of Gaiotto and Rapčák. Based on explicitcalculations we conjecture a complete list of co-dimension truncations of the algebra. Email: [email protected] ontents W ∞ and W ∞ . . . . . . . . . . . . . 183.2 Operator product expansions in V j basis . . . . . . . . . . . . . . . . . . . . . 19 and conjecture . . . . . . . . . . . . . . . . . . . . . . 26 W ∞ Introduction W -algebras and their incarnations as affine Yangians [1, 2], degenerate double affine Heckealgebras [3] or cohomological Hall algebras [4] plan an important role in various areas ofmathematical physics. They were originally introduced in the context of integrable hierar-chies of partial differential equations and soon after in conformal field theory. Some morerecent applications include four-dimensional N = 2 gauge theories [5, 6, 7], M -theory [8, 9]or higher spin AdS /CF T dualities [10].One of the most exciting W -algebras is the universal two-parametric family of algebrascalled W ∞ which has one generating field of every integer spin. It interpolates betweenalgebras of W N family which are among the most well studied examples of W -algebras. Thevery distinct property of W ∞ is that it has quantum triality symmetry [11]. The vacuumcharacter of this algebra is given by MacMahon function which connects the representationtheory of W ∞ to combinatorics of plane partition, plane tilings and dimer models.A less studied example of universal interpolating algebra is the even spin W ∞ which isfreely generated by fields of every even spin [12, 13, 14, 15]. Here we want to study thisalgebra in more detail. We first review and slightly extend the primary bootstrap approachto this algebra [14]. A different choice of independent structure constants with respect to[14] eliminates ambiguities arising from square root factors and spurious duality symmetries.Using three elementary minimal representations of the algebra we introduce a convenienttriality covariant parametrization analogous to parametrization of W ∞ introduced in [16].We also introduce another parametrization which is closer to parametrization of Gaiotto andRapčák [7] featuring the Kapustin-Witten parameter. Next we identify our parameters withparameters of well-known truncations of the algebra, including orthogonal and symplecticquotients and principal Drinfeld-Sokolov reductions of BCD families of simple Lie algebras.We also compare even spin W ∞ with orthosymplectic Y -algebras introduced in [7].In the next section we use the results of Fateev and Lukyanov [17] to embed even spin W ∞ into W ∞ . Although in retrospect this is not very surprising, the possibility of doingthis at the level of non-linear quantum algebras it is not at all obvious. Actually the prin-cipal Drinfeľd-Sokolov reductions of all simple Lie algebras are subalgebras of truncationsof W ∞ (except for F where this is not known) so in this sense W ∞ can be thought ofas interpolating algebra for all W -algebras associated to simple Lie algebras via Drinfeľd-Sokolov reduction (except for F ). One would like to start with Miura operator for GL ( N ) W -algebras and fold it to obtain a Miura operator for BCD -type algebras [18]. Unfortunatelyit is not clear how to do this at quantum level. The trick used by Fateev and Lukyanov isinstead to consider D n algebras which have additional Pfaffian generator of dimension n andstudy its operator product expansion with itself. From here we can identify the generatorsof even spin W ∞ as quadratic composites of the generators of W ∞ . We verify by explicitcalculations using OPEdefs [19] that the resulting subalgebra quadratically closes (up to sumof spins ) and we find a map between parameters of W ∞ and those of even spin W ∞ .The operator product expansions in the quadratic basis share many nice properties with3 ∞ or even the matrix valued W ∞ [20], but unlike in those cases it is not clear at themoment how to sum the derivative terms.In the following section we list the known truncation curves and study co-dimension truncations of vacuum representation up to level . The structure seems to be morecomplicated than in the case of W ∞ , but all truncations found agree nicely with a simpleformula for truncation curves and also with Gaiotto-Rapčák Y -algebras. We conjecture ageneral formula for the level of the first singular vector in all of these truncations. In the lastsection we show on an example of so (2 n + 1) k that the gluing procedure of [21] generalizesalso to the orthosymplectic case. Let us review and slightly extend the results of [14] where the authors used the OPE bootstrapto construct even spin W ∞ in the primary basis . We assume to have one generating field ofeach even spin with the following operator product expansions between the fields up to sumof spins W W ∼ C + C W + C W W W ∼ C W + C W + C [44]46 [ W W ] + C W W W ∼ C W + C W + C W + C [44]48 [ W W ]+ C W + C [44] (2) [ W W ] (2) + C [46]48 [ W W ] + C [46] (1) [ W W ] (1) W W ∼ C + C W + C W + C W + C [44]66 [ W W ]+ C W + C [44] (2) [ W W ] (2) + C [46]66 [ W W ] + C [46] (1) [ W W ] (1) (2.1) W W ∼ C , W + C , W + C , W + C [44]4 , [ W W ]+ C , W + C [44] (2) , [ W W ] (2) + C [46]4 , [ W W ] + C [46] (1) , [ W W ] (1) + C , W + C [48]4 , [ W W ] + C [66]4 , [ W W ] + C [444]4 , [ W W W ]+ C [44] (4) , [ W W ] (4) + C [46] (2) , [ W W ] (2) + C [48] (1) , [ W W ] (1) + C [46] (3) , [ W W ] (3) W W ∼ C W + C W + C W + C [44]68 [ W W ]+ C W + C [44] (2) [ W W ] (2) + C [46]68 [ W W ] + C [46] (1) [ W W ] (1) + C W + C [48]68 [ W W ] + C [66]68 [ W W ] + C [444]68 [ W W W ]+ C [44] (4) [ W W ] (4) + C [46] (2) [ W W ] (2) + C [48] (1) [ W W ] (1) + C [46] (3) [ W W ] (3) The bootstrap procedure is mathematically formalized in [22] and recently reviewed in [23]. are givenin appendix A. Looking at these equations we observe the following: apart from the centralcharge c there is essentially one undetermined structure constant which we choose to be thescale-invariant ratio x ≡ C C . (2.2)All other undetermined parameters can be chosen arbitrarily by rescaling the generatingfields. We can choose C to take any value by rescaling W . Afterwards the constant C (assuming it to be generically non-vanishing) can be chosen to take any value if weappropriately rescale W . At dimension we have apart from W a new composite primary [ W W ] so now we may freely choose C and C [44]46 by redefinition of W (i.e. W can beshifted by any multiple of [ W W ] ) etc. The conjecture of [14] proven in [15] is that thiscontinues to hold in all higher OPEs, i.e. that there exists a two-parametric family of algebrasparametrized by the central charge and parameter x .Note that unlike in [14] where the independent structure constant was chosen to be C (cid:0) C (cid:1) = c (5 c + 22)72( c + 24) − c ( c − c + 22)72(2 c − c + 68) x + c ( c − c + 24)(5 c + 22)12(2 c − c + 68) x , (2.3)here we will work instead with x defined by (2.2). The advantage of working with thisparameter is that we avoid the ambiguity of choosing a square root when solving for otherstructure constants. We will see that the group of duality symmetries of the algebra willhave order just like the triality in W ∞ . The other spurious solutions found in [14] arenot symmetries of the algebra because they don’t preserve the invariant ratio (2.2) and thusnot all of the structure constants of even spin W ∞ . In order to study the dualities of the algebra, we need to parametrize the algebra in termsof rank-like parameter. In [14] it was done by applying the method of [24] and comparingthe conformal dimensions of the minimal representations of the algebra obtained by solvingthe Jacobi identities with the Drinfeľd-Sokolov reduction applied to B , C and D series ofsimple Lie algebras. Minimal representations of even spin W ∞ are those representationswhose character is (for generic values of parameters) χ min = q h m − q χ vac . (2.4)5ollowing [14], we assume OPE of minimal primary φ m with W -algebra generators to be ofthe form W φ m ∼ w m φ m (2.5) W φ m ∼ w m φ m + C [4 m ]6 m [ W φ m ] + C [4 m ] (1) m [ W φ m ] (1) (2.6)where the notation is like in the previous section. The Jacobi identity ( W W φ m ) fixesthe OPE and imposes the following constraints on the conformal dimension and higher spincharges: w m C = − h m ( ch m + c + 3 h m − h m + 2)(2 ch m + c + 16 h m − h m )12(2 c h m − c h m − c + 36 ch m − ch m + 120 ch m − c + 24 h m + 10 h m − h m ) w m C ( C ) = ( c − c + 22) h m ( ch m + c + 3 h m − h m + 2)54( c + 24)(2 c − c + 68) × (2.7) × ( ch m + 2 c + 15 h m − h m + 8)(2 ch m + c + 16 h m − h m )(2 ch m + 3 c + 48 h m − h m )(2 c h m − c h m − c + 36 ch m − ch m + 120 ch m − c + 24 h m + 10 h m − h m ) together with the following equation restricting h m : (4 c h m x − c h m − c h m x + 28 c h m − c x + 91 c + 72 c h m x − c h m − c h m x + 988 c h m + 144 c h m x − c h m − c x + 1010 c + 1776 ch m x − ch m − ch m x + 12646 ch m + 5704 ch m x − ch m − cx + 1224 c + 1152 h m x − h m + 480 h m x + 3944 h m − h m x − h m ) ×× (12 c h m x − c h m − c h m x + 14 c h m − c x + 168 c + 216 c h m x − c h m − c h m x + 3287 c h m + 432 c h m x − c h m − c x + 1548 c + 5328 ch m x − ch m − ch m x + 31544 ch m + 17112 ch m x − ch m − cx − c + 3456 h m x + 3264 h m + 1440 h m x − h m − h m x + 14144 h m ) = 0 (2.8)Given h m and c , there are thus two possible values for the structure constant x of the algebra, x = (2 c − c + 68)( h m − ch m + 3 c + 48 h m − h m )6( c + 24)(2 c h m − c h m − c + 36 ch m − ch m + 120 ch m − c + 24 h m + 10 h m − h m ) (2.9) x = (7 c + 68)(4 c h m − c h m − c + 52 ch m − ch m + 141 ch m − c + 72 h m − h m + 4 h m )2( c + 24)(2 c h m − c h m − c + 36 ch m − ch m + 120 ch m − c + 24 h m + 10 h m − h m ) . (2.10)so knowing c and h m still does not determine the algebra uniquely. Solving next the Jacobiidentity ( W W φ m ) determines the charge w m but more importantly also picks the solution62.9) rather than (2.10). This means that the structure constant x of the algebra can beuniquely determined once we know the dimension of the minimal representation (and thecentral charge).Given c and x there are generically three solutions of the equation for the conformaldimension of the minimal representation. This means that there are three minimal represen-tations of even spin W ∞ and as we will see they are all permuted by the triality symmetryof the algebra. This is slightly different than in the case of W ∞ where we have six minimalrepresentations, but there we have to use the triality symmetry together with the chargeconjugation symmetry to find all of these representations. This is all consistent with thefact that while the minimal representations of W ∞ are charged (i.e. transform under chargeconjugation symmetry which changes sign of odd spin fields), the even spin W ∞ has noconjugation symmetry. Analogously to [16] we can introduce a redundant but triality covariant parametrization ofthe structure constants using three parameters permuted by the triality symmetry. Let usintroduce three parameters ( µ , µ , µ ) such that the conformal dimensions of the minimalrepresentations are h m = 1 + µ , h m = 1 + µ , h m = 1 + µ . (2.11)These parameters are not independent but satisfy µ + 1 µ + 1 µ = 0 (2.12)just like in W ∞ . The central charge in terms of these parameters is c = ( µ + 1)( µ + 1)( µ + 1)2 . (2.13)We can also introduce a parameter ψ , µ = η, µ = − ηψ , µ = ηψ − . (2.14)The parameter ψ is natural parameter from point of view of Drinfeľd-Sokolov reductions – isthe DS level shifted such that the critical level is at ψ = 0 . It also agrees with the Kapustin-Witten parameter Ψ in Gaiotto-Rapčák construction [7]. Under triality transformations ittransforms by fractional linear transformations permuting (0 , , ∞ ) . The other parameter η measures the overall scale of µ j (and is equal to one of µ j depending on the triality frame).The condition (2.12) is identically satisfied. The central charge takes a simple form c = ( η + 1)( ψ − η )( η + ψ − ψ − ψ . (2.15)7here is one more parametrization of the algebra used in [15]. The parameter λ used thereis related to x by x = (7 c + 68)(1 + 49 λ − λc )84 λ (1 − c )(24 + c ) . (2.16)which is a fractional linear transformation so at given generic c the correspondence between x and λ is one-to-one. We may now identify the parameters of even spin W ∞ with parameters of orthogonal cosetsexpected to have even spin W ∞ symmetry. Using the formula for the central charge of affineLie algebra k dim g k + h ∨ (2.17)where k is the level of affine Lie algebra ˆ g and h ∨ is the dual Coxeter number, we can calculatethe central charge of the coset so ( n ) k × so ( n ) so ( n ) k +1 (2.18)and we find c = kn (2 n + k − n + k − n + k − (2.19)(which is uniform for both B and D series of cosets). The conformal dimension of the minimalrepresentation is [14, 12] h = 2 n + k − n + k − (2.20)for ( (cid:3) , (cid:3) ; • ) and h = k n + k − (2.21)for ( • , (cid:3) ; (cid:3) ) . Expressing x in terms of c and one of h j , we can find the dimensions of the othertwo minimal representations. The third minimal representation has conformal dimension h = n . (2.22)This is interesting because in even orthogonal case this is exactly the dimension of theadditional Pfaffian generator that we might add to the truncation of even spin W ∞ .To identify the even spin W ∞ corresponding to these cosets, we can use the centralcharge together with one of the minimal dimensions in formula (2.9) and this determines theparameter x uniquely. It is also possible to verify explicitly that (2.9) is the correct branch8and not the one given by (2.10)) by direct evaluation of the k → ∞ limit of these orthogonalcosets. In that case the coset simplifies to so ( n ) so ( n ) (2.23)which is well-known to be realized by the singlet part of VOA of n free fermions with OPE ψ j ( z ) ψ k ( w ) ∼ δ jk z − w . (2.24)The central charge of this is c = n while the invariant ratio of structure constants is x = C C = 49( n − n + 136)6( n + 48)(19 n − . (2.25)This exactly agrees with the first branch (2.9). The expression for the parameter x in termsof n and k is thus x = ( n − k + 6 n − k + 8 n − k + 2 kn − k − n + 2)6( k n + 48 k + 2 kn + 93 kn − k + 48 n − n + 96) ×× (7 k n + 136 k + 14 kn + 251 kn − k + 136 n − n + 272)(19 k n − k + 76 k n − k n + 408 k + 94 k n − k n + 1267 k n − k ++ 36 kn − kn + 857 kn − kn + 744 k + 24 n − n − n + 376 n − . (2.26)This completely determines the map from ( n, k ) parameters to ( c, x ) . The parameters ψ and η are ψ = 2 − n − k, η = n − . (2.27)In fact, ψ is determined only up to a S subgroup of Möbius transformations permuting (0 , , ∞ ) . The other five choices of ψ correspond to 5 other embeddings related by thetriality symmetry. In terms of parameters ( n, k ) , the following values correspond to thesame even spin W ∞ : ( n, k ) , ( n, − n − k ) , (cid:18) kn + k − , nn + k − (cid:19) , (cid:18) kn + k − , n + k − n + k − (cid:19) , (cid:18) n + k − n + k − , kn + k − (cid:19) , (cid:18) n + k − n + k − , − nn + k − (cid:19) . (2.28)9 ymplectic quotients We can use the duality between orthogonal and symplectic algebrasto study the corresponding symplectic quotients. In general, the Grassmannian coset of thetype so ( n ) k × so ( n ) l so ( n ) k + l ≃ so ( k + l ) n so ( k ) n × so ( l ) n (2.29)with central charge kln ( n − n + k + l − n + k − n + l − n + k + l − (2.30)has a triality symmetry if we define three parameters k = k, k = l, k = 4 − n − k − l justlike the unitary cosets. The unitary cosets have also a Z symmetry which changes signs ofall k j parameters. In the case of orthogonal cosets this Z symmetry instead maps the cosetsto symplectic ones , i.e. sp (2 n ) k × sp (2 n ) l sp (2 n ) k + l ≃ so ( − n ) − k × so ( − n ) − l so ( − n ) − k − l . (2.31)This means that the symplectic analogue of the cosets (2.18) are cosets sp (2 n ) k × sp (2 n ) − sp (2 n ) k − (2.32)of the central charge − kn (4 n + 2 k + 3)( n + k + 1)(2 n + 2 k + 1) . (2.33)The dimensions of the minimal representations are h = − n, h = 4 n + 2 k + 34( n + k + 1) , h = k n + 2 k + 1 . (2.34)The simplest level − representation may be realized as singlet part of VOA of n freesymplectic bosons with OPE ξ j ( z ) ξ k ( w ) ∼ ω jk z − w (2.35)where ω jk is a non-degenerate symplectic form. The structure constants of this symplecticquotient algebra exactly agree with those of n free fermions (i.e. level even orthonormalcoset) if we change the sign of n everywhere. The reason that we have half-integer levels is a consequence of the usual convention for normalization ofKilling form such that the length squared of long roots is . In the C n case that we are considering this leadsto dual Coxeter number n + 1 which is half of we would get if we worked in more symmetric conventionswhere the length squared of short roots in C n would be . .4 Drinfeľd-Sokolov reductions Let us summarize the central charges and dimensions of minimal representations in principalDrinfeľd-Sokolov reduction of B n , C n and D n type algebras following [25, 14]. The first mainformula that we will use is the expression for the central charge c = ℓ − (cid:12)(cid:12) α + ρ + α − ρ ∨ (cid:12)(cid:12) (2.36)where ℓ is the rank of the Lie algebra, ρ is the Weyl vector and ρ ∨ the dual Weyl vector.The parameters α ± are defined as α + = 1 √ k + h ∨ , α − = −√ k + h ∨ (2.37)where k is the level of the affine Lie algebra entering the Drinfeľd-Sokolov reduction and h ∨ is the dual Coxeter number. The second useful formula is the formula for the dimen-sion of maximally degenerate representation parametrized by the pair of highest (co)weights (Λ + , Λ − ) where Λ + is integral dominant weight and Λ − integral dominant co-weight: h = 12 h α + Λ + + α − Λ − , α + (Λ + + 2 ρ ) + α − (Λ − + 2 ρ ∨ ) i . (2.38) Odd orthogonal case - so (2 n B + 1) Turning now to Lie algebra B n B , from (2.36) we findthe central charge c B = − n B (4 n B + 2 n B k B + 2 k B − n B + 2 n B k B − n B + k B )2 n B + k B − (2.39)and from (2.38) the minimal weights h B = − n B (2 n B + k B − n B + k B − (2.40)(corresponding to Λ + = ω ) and h B = 12 (cid:0) n B + 2 n B k B − n B + k B (cid:1) . (2.41)(corresponding to Λ − = ω ∨ which here agrees with ω ). These two weights are compatiblewith third minimal weight h B = 4 n B + 2 n B k B + 2 k B − n B + k B − . (2.42)The shifted level ψ B is ψ B = 2 n B − k B (2.43)and the scale parameter η B is η B = 2 n B ψ B − n B + ψ B . (2.44)11 ymplectic case - sp (2 n C ) For Lie algebra C n C we find the central charge c C = − n C (2 n C + 2 n C k C + 2 n C + k C )(4 n C + 4 n C k C − k C − n C + k C + 1 (2.45)and dimensions of minimal representations h C = − n C + 4 n C k C − k C − n C + k C + 1) (2.46)(corresponding to Λ + = ω ) and h C = n C (2 n C + 2 k C + 1) . (2.47)(corresponding to Λ − = ω ∨ which is twice as long as ω ). The third minimal weight com-patible with these is h C = 2 n C + 2 n C k C + 2 n C + k C n C + 2 k C + 1 . (2.48)The shifted level ψ C is ψ C = 2 n C + 2 + 2 k C (2.49)and the scale parameter η C η C = 2 n C ψ C − n C − . (2.50) Even orthogonal case - so (2 n D ) In the case of D n D the calculation is slightly simplerbecause the Lie algebra is simply laced. We find c D = − n D (4 n D + 2 n D k D − n D − k D + 5)(4 n D + 2 n D k D − n D − k D + 4)2 n D + k D − (2.51)and the minimal dimensions are h D = − n D + 2 n D k D − n D − k D + 54 n D + 2 k D − (2.52)(for Λ + = ω ) and h D = 12 (cid:0) n D + 2 n D k D − n D − k D + 4 (cid:1) (2.53)(for Λ − = ω since now the weights and co-weights agree). The third minimal weightcompatible with these is simply h D = n D (2.54)(just like in the case of orthogonal coset, this is compatible with assumption that the Pfaffiangenerating field transforms in the minimal representation of the algebra). Finally we definethe shifted level ψ D to be ψ D = 2 n D − k D (2.55)and the parameter η D is η D = 2 n D ψ D − n D − ψ D + 1 = (2 n D − ψ D − . (2.56)12 ote on Pfaffian generator Let us briefly discuss the Pfaffian generator of dimension n D . There is no corresponding field in even spin W ∞ , although we have just seen that oneof the minimal primaries has exactly the correct conformal dimension. The reason for it isthat it is unstable as we vary n . In this sense the W -algebra of type W D is not a truncationof even spin W ∞ , only its Z projection which removes the Pfaffian generator is [15]. On theother hand, we will see in the next section when we discuss the Miura transformation thatthe Pfaffian generator can be naturally embedded into u (1) × W N truncation of W ∞ andactually this operator plays a crucial role in construction of the embedding of even spin W ∞ into W ∞ . In [7] the authors found an interesting realization of W -algebras in gauge theory setting.The theory they considered was four-dimensional twisted N = 4 super Yang-Mills theorywith three semi-infinite co-dimension one defects meeting at co-dimension two subspace. Thedegrees of freedom living at this co-dimension subspace were found to be organized by acertain truncation of W ∞ algebra determined by the ranks of the gauge groups in threesubsectors of the full four-dimensional space cut out by the co-dimension defects [7, 21].This setup can be modified by introducing an orientifold plane. The unitary gauge groupsare then projected to orthosymplectic groups and one expects the degrees of freedom at co-dimension subspace to be reduced to even spin W ∞ . Here we verify that the central chargeformula derived in [7] is compatible with the form of the central charge in even spin W ∞ andlater that the orthosymplectic Y -algebras can be identified with the truncations of even spin W ∞ .As discussed in [7] there are actually four different ways how to introduce an orientifoldplane in the theory leading to four different families of Y -algebras. They are shown infigure 2.5. Although we expect that the orthosymplectic algebras constructed in [7] shouldbe truncations of even spin W ∞ , to identify the parameters one would need to know thecentral charge and one of the structure constants. Unfortunately only the central charge wascalculated in [7]. On the other hand, the orthosymplectic Y -algebras transform nicely undertriality transformations and the Kapustin-Witten parameter Ψ has exactly the properties ofthe parameter ψ introduced in (2.14) so one can try to identify these Ψ with ψ . Findingrational expressions for minimal dimensions and compatibility with various truncations andrestrictions would already be a big hint of correctness of the proposed identification. Algebra Y − N ,N ,N Starting with the first algebra of the figure 2.5, Y − , the central chargeis given by (2.15) with η − = 1 + 2( N − N ) − (1 + 2( N − N )) ψ. (2.57) Some of the formulas in [7] contain typos. I would like to thank to Miroslav Rapčák for sharing with methe corrected expressions for these. p (2 N ) SO (2 N + 1) SO (2 N ) Y − N ,N ,N Sp ′ (2 N ) SO (2 N ) SO (2 N + 1)˜ Y − N ,N ,N SO (2 N ) Sp ′ (2 N ) Sp (2 N ) Y + N ,N ,N SO (2 N + 1) Sp (2 N ) Sp ′ (2 N )˜ Y + N ,N ,N Figure 1: Gaiotto-Rapčák orthosymplectic Y -algebras14rom this we can immediately find the three µ parameters of the algebra using (2.14). Thenext algebra is ˜ Y − . The central charge calculated in [7] is of the form (2.15) with ˜ η − = 2( N − N ) + (1 − N − N )) ψ. (2.58)The third algebra, Y + has parameter η equal to η + = − N − N ) − N − N ) ψ. (2.59)The last algebra of figure 2.5 is ˜ Y + with η parameter equal to ˜ η + = 2( N − N ) − N − N ) ψ. (2.60)In all four cases we get nice polynomial expressions for η which has the same structure asfor cosets of Drinfeľd-Sokolov reductions. Let’s summarize some of the properties of thesealgebras have:1. The parameters of even spin W ∞ don’t change if we shift all three N j parameters atthe same time by a constant. This is analogous to what happens in W ∞ and is aconsequence of (2.12). This doesn’t mean though that the Y N ,N ,N algebras are thesame: only their simple quotient is expected to be the same. In the case of W ∞ this isdiscussed in [21] and in particular in [26] in connection with free field representations.2. The transformation ψ ↔ − ψ in parametrization (2.14) exchanges µ ↔ µ andtransforms η only by its action on ψ . The effect on orthosymplectic Y -algebras is Y − ( N , N , N ) ↔ ˜ Y − ( N , N , N ) , Y + ( N , N , N ) ↔ Y + ( N , N , N )˜ Y + ( N , N , N ) ↔ ˜ Y + ( N , N , N ) (2.61)which is exactly the claim in [7]. Note that pictorially it exchanges the upper rightand lower right gauge groups in figure 2.5 from where the action on ranks and type of Y -algebra is obvious.3. To see the effect of the transformation ψ → ψ on Y -algebras it’s better to work directlywith µ j parameters. We find Y − ( N , N , N ) ↔ Y − ( N , N , N ) , ˜ Y − ( N , N , N ) ↔ Y + ( N , N , N )˜ Y + ( N , N , N ) ↔ ˜ Y + ( N , N , N ) (2.62)again in agreement with [7].4. The third operation of exchanging two gauge groups corresponds to ψ → ψψ − . Theaction on Y -algebras is Y − ( N , N , N ) ↔ Y + ( N , N , N ) , ˜ Y − ( N , N , N ) ↔ ˜ Y − ( N , N , N )˜ Y + ( N , N , N ) ↔ ˜ Y + ( N , N , N ) . (2.63)The fact that Y − and Y + exchange their roles is again manifest in figure 2.5.15. At the level of the parameters of the universal algebra, all four orthosymplectic Y -algebras are connected formally by half-integer shifts of rank parameters: apart fromthe shift ˜ η + ( N , N , N ) = η + (cid:0) N + , N , N (cid:1) (2.64)used already in [7] and its generalization ˜ η − ( N , N , N ) = η − (cid:0) N , N − , N + (cid:1) (2.65)which also involves the transformation ψ → − ψ we have also formally ˜ η − ( N , N , N ) = η − (cid:0) N − , N − , N (cid:1) η + ( N , N , N ) = η − (cid:0) N − , N − , N (cid:1) (2.66) ˜ η + ( N , N , N ) = η − (cid:0) N − , N − , N (cid:1) which allows to map the parameters of any two orthosymplectic Y -algebras thought ofin terms of the universal even spin algebra. In this section we show how we can use the results of [17] to find a free field representationof even spin W ∞ and embed it in W ∞ . First of all, recall that given N free fields withcurrents satisfying OPE J j ( z ) J k ( w ) ∼ δ jk ( z − w ) (3.1)we can construct Miura operator ( α ∂ + J ( z )) · · · ( α ∂ + J N ( z )) = N X k =0 U k ( z )( α ∂ ) N − k (3.2)and the currents U k ( z ) defined in this way represent algebra d u (1) × W N [27, 28] and moreoverthe operator product expansions are quadratic in this basis [28, 16].An important observation of [17] is that the fields appearing in the OPE of the generatingfield of the highest spin W N in d u (1) × W N generate an even spin subalgebra. Following [25],we can define fields V j ( z ) by U N ( z ) U N ( w ) = a ( N − z − w ) N + N − X k =1 ( − k a ( N − − k )( z − w ) N − k [ V k ( z ) + V k ( w )] (3.3)where we choose the normalization factors as a ( j ) = j Y r =1 (cid:0) − (2 j )(2 j + 1) α (cid:1) . (3.4)16hese are not so easy to calculate explicitly at larger values of N , because even if we areinterested in fields V j with j small, we still need to know the OPE of U N with itself.Fortunately, we can use the result that the OPE can be written in the form [16] U N ( z ) U N ( w ) = X l + m ≤ N C lmNN ( α , N ) U lm ( z, w )( z − w ) N − l − m (3.5)where U lm ( z, w ) are certain bi-local fields of the form ( U l U m )( w ) + derivatives and can beexplicitly written in terms of fields U j ( z ) U k ( w ) with j + k ≤ l + m . More concretely theyare equal to U lm ( z, w ) = X j + k ≤ l + m D jklm U j ( z ) U k ( w )( z − w ) l + m − j − k (3.6)and the matrix of constants D jklm is the inverse of the matrix of structure constants C jklm (considering ( j, k ) and ( l, m ) as bi-indices as explained in [16]). The structure constants forOPE of U N with itself in our situation simplify to C jkNN ( α , N ) = ( − j − k N − j − k − Y r =1 (cid:0) − r (2 r + 1) α (cid:1) = ( − j − k a (cid:18) N − j − k − (cid:19) (3.7)for j + k even and to C jkNN ( α , N ) = ( − j − k − (2 n − j − k − α N − j − k − Y r =1 (cid:0) − r (2 r + 1) α (cid:1) = ( − j − k − (2 n − j − k − α a (cid:18) N − j − k − (cid:19) (3.8)for j + k odd. In both cases, these depend only on the sum j + k (except for an overall sign).Now the problem with extracting lower spin fields V j at larger values of N is solved, becausewe can use the expression (3.5) to directly extract V s fields, a calculation which involvesknowledge of OPE of fields of spin ≤ s only.Let use write a formula that we can use to extract the generators of even spin W ∞ interms of those of W ∞ . For that, we Taylor expand (3.3) at z = w obtaining an ordinaryOPE. The coefficient of pole of order N − s is − s a ( N − − s ) V s ( w ) + s − X r =1 ( − r a ( N − − r )(2( s − r ))! V (2 s − r )2 r ( w ) . (3.9)On the other hand, the coefficient of the same pole in the expansion of the form (3.5) is equal17o X l + m ≤ j + k ≤ s C jkNN D lmjk OPEPole [ l + m − s ][ U l , U m ] (3.10)which is a convenient expression involving only OPE of fields with spins ≤ s . Equatingthese last two expressions, we can recurrently calculate expressions for V s fields in terms of U j fields. The first few fields are given in the next section. It is a non-trivial check of ourcalculations that the OPEs of V j fields close. W ∞ and W ∞ Using the OPE of the Pfaffian field as described in the previous section we can extract firsttwo V j fields: V = U −
12 ( U U ) − ( N − α U ′ V = U − ( U U ) + 12 ( U U ) − ( N − α U ′ + ( N − α ( U U ′ ) − ( N − α ( U ′ U )+ 14 U ′′ − N −
14 ( U ′′ U ) + 4 N α − N α + 12 α −
14 ( U ′ U ′ ) (3.11) + ( N − α (4 N α − N α + 12 α − N − U ′′′ . The third field, V , is given in the appendix. Since V and V generate even spin W ∞ subalgebra, we can identify the parameters of even spin W ∞ in terms of those of W ∞ ( N and α ). We first need to find the stress-energy tensor which is simply − V and the primarycombination of spin fields and spin fields to extract the central charge and the parameter x . The result is c = N (1 − N − N − α ) (3.12)and x = h ( N − − α + 4 α N + 10 α N − α N − α N + 14 α N − N − ×× (4 α N − α N − i.h α N − α N + 2 α N − N − ×× (12 α + 16 α N − α N − α N − α N + 100 α N ++ 302 α N − α N − α N + 19 N − i (3.13)Expressing N and α in terms of parameters λ of W ∞ [16], c ∞ = ( λ − λ − λ − (3.14) Note that the multiplication of the matrix components with the components of the inverse matrix doesnot give the identity matrix because of the restriction on the range of the indices. λ + 1 λ + 1 λ = 0 (3.15)and λ = N we can write even spin W ∞ parameters as h = 1 + λ + λ λ = 1 + µ h = 1 + λ + λ λ = 1 + µ (3.16) h = λ = 1 + µ . In terms of parameters µ we have µ = λ + λ + 2 λ λ λ µ = λ + λ + 2 λ λ λ (3.17) µ = − λ + λ + 2 λ λ λ + λ . For reference, the map between parameters ( N, α ) and µ j is N = µ + µ − µ µ µ + µ ) = µ (cid:18) − µ − µ (cid:19) (3.18) α = − ( µ + µ ) µ µ = − µ µ µ = (1 − ψ ) ψ . (3.19)We see that the embedding of even spin W ∞ in W ∞ breaks the triality symmetry of W ∞ to a Z exchanging λ ↔ λ or µ ↔ µ . This is related to the fact that the Miuratransformation depends on a choice of a preferred direction. So although both algebras havethe triality symmetry, the triality in W ∞ does not restrict to triality in even spin W ∞ .The choice of even spin W ∞ subalgebra in W ∞ breaks the triality symmetry to Z butwhen restricted to this subalgebra, the duality is enhanced to a triality of the subalgebra.This is analogous to enhancement of duality to triality in unitary Grassmannian cosets whenone of the levels is one. We also see that there are at least six ways of embedding even spin W ∞ in W ∞ , each associated to different asymptotic direction (times two because of thecomplex conjugation in W ∞ ). V j basis As a result of our definition of V j fields they are quadratic composites of the U j fields. Since W ∞ is filtered with degree given by the number of U j fields in each term and since the19perator product expansions preserve the degree, we can also expect the operator productexpansions of V j field to satisfy quadratic operator product expansions.To fix these, we can first calculate the OPE of V with V s , V ( z ) V j ( w ) ∼ N ( − j (cid:2) j +2 − (cid:3) a ( N − B j +2 (2 j + 2) a ( N − j − ( z − w ) j +2 + j − X k =1 N − k )( − j − k (cid:2) j − k +2 − (cid:3) a ( N − k − B j − k +2 (2 j − k + 2) a ( N − j − V k ( w )( z − w ) j − k +2 + ( derivatives )( z − w ) ≥ − jV j ( w )( z − w ) − ∂V j ( w ) z − w . (3.20)where B n are the Benoulli numbers. The Jacobi identity ( V V V j ) fixes all the derivativeterms in V V j OPE. The OPE of V with itself is given in the appendix B. With this input(actually only the coefficient of the identity and of V in V V OPE is necessary) the Jacobiidentities determine all the other operator product expansions. The resulting OPEs have thefollowing properties1. the operator product expansions are purely quadratic, i.e. all the operators appearingin the singular part of the OPE are normal ordered products of (at most) two V j fieldsand their derivatives. This is analogous to the case of W ∞ [28, 16].2. All the structure constants are polynomial functions of N and α . This is again anal-ogous to [16, 20].3. Unlike in W ∞ , the derivatives do not seem to be simply summable into bi-local fields(this seems to be the case even after a simple linear redefinition of the fields). Thisis probably related to different form of the Miura operator which is ‘folded’. As aconsequence of this, the calculation of commutation relations between mode operatorsis more involved because one needs to consider terms with derivatives.4. We verified these claims for OPE of fields V j and V k with j + k ≤ . At every stepdetermination of OPE reduces to solution of linear equations for the coefficients of thequadratic composites in the OPE.For later purposes, it is useful to determine at each even spin the primary field W j whose pole of order j with all dimension j fields not involving V j vanishes, i.e. field whichis orthogonal to all lower dimension fields and their derivatives and composites. Actuallywe don’t even need to require this to be primary, it is a consequence of being orthogonalto lower composites. The special property of this field is that it is the field whose two-point function vanishes for truncations of the algebra. We can thus avoid searching for zerosof Kac determinant to find the truncations of the algebra. It is enough to identify these20rimaries and find zeros of their two point functions. We choose the normalization such that W j = V j + . . . . With this choice, the two-point function of these fields is h W W i = − n (cid:0) α + 4 α n − α n − (cid:1) h W W i = − (cid:0) α n − α n + 10 α n − n − (cid:1) × n (2 n − (cid:0) α n + α n − (cid:1) ×× (cid:0) α + 4 α n − α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) (cid:0) − α + 4 α n − α n − (cid:1) h W W i ∼ ( n − n (2 n − (cid:0) α n + α n − (cid:1) (cid:0) α n + 2 α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) ×× (cid:0) α + 4 α n − α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) (cid:0) α n − α n − (cid:1) ×× (cid:0) − α + 4 α n − α n − (cid:1) (cid:0) − α + 4 α n + 2 α n − (cid:1) (3.21) h W W i ∼ ( n − n ( n + 1)(2 n − n − (cid:0) n α + nα − (cid:1) (cid:0) n α + 2 nα − (cid:1) ×× (cid:0) n α + 3 nα − (cid:1) (cid:0) n α − α − (cid:1) (cid:0) n α − nα + 56 α − (cid:1) ×× (cid:0) n α − nα + 30 α − (cid:1) (cid:0) n α − nα + 12 α − (cid:1) ×× (cid:0) n α − nα + 4 α − (cid:1) (cid:0) n α − nα + 2 α − (cid:1) (cid:0) n α − nα − (cid:1) ×× (cid:0) n α − nα − α − (cid:1) (cid:0) n α + 2 nα − α − (cid:1) (cid:0) n α + 6 nα − α − (cid:1) where ∼ means that we didn’t write the denominator (because we are mainly interested inzeros of these two-point functions). One can actually find higher order two-point functionby the following trick: we have C C = C C (3.22)if the field W is chosen to be orthogonal to W [44] which is equivalent to condition C = 0 . (3.23)Similarly at the next level C C , = C , C (3.24)if we choose W to be orthogonal to W [46] and W [44] (2) which means C , = 0 and C , = 0 . (3.25)In this way, we were able to find the zeros of Kac determinant up to level which wouldotherwise require to knowing th order pole of W with itself. The fact that the numeratorof C , and C , obtained in this way factorizes into factors of the form of (3.21) is a nicecheck of consistency of this procedure. 21 Truncations
We will now collect all the results about truncations using various truncations discussed sofar. All the truncation curves will have formally the same form as in W ∞ , N µ + N µ + N µ = 1 (4.1)with non-negative integers N , N and N . Due to redundancy in parametrization (2.12)shifting all N j by a constant does not change the truncation curve, but we can use thesetriples of integers differing by a constant to describe different truncations of the algebra withthe same truncation curve. The expressions for η for Y -algebras discussed in section 2.5 can be immediately translatedinto truncation curves, i.e. curves in µ -parameter space where the universal even spin W ∞ truncates to a smaller algebra. These curves have the form Y − : 2 N + 1 µ + 2 N + 1 µ + 2 N µ = 1˜ Y − : 2 N + 1 µ + 2 N µ + 2 N + 1 µ = 1 Y + : 2 N µ + 2 N + 1 µ + 2 N + 1 µ = 1 (4.2) ˜ Y + : 2 N µ + 2 N µ + 2 N µ = 1 These have a very simple form: if the gauge group associated to face µ j is Sp (2 N ) , thecoefficient of µ − j is N , while if the gauge group is SO ( N ) , the coefficient is N − (bothfor even and odd N ). Whenever the parameters ( µ , µ , µ ) of the even spin W ∞ satisfy oneof these equations for non-negative integer values of ( N , N , N ) , the algebra develops anideal so can be truncated to a smaller subalgebra. Note that in general for a fixed truncationcurve there might be various choices of this ideal corresponding to the fact that the map fromtriples ( N , N , N ) to truncation curves is not one-to-one (in particular an overall shift of allthree ranks by a constant leads to the same truncation curve). In the case of W ∞ there wasalways a maximal ideal corresponding to a truncation where (at least) one of the integers N j was vanishing. One way to understand what is happening is to analyze the characters of Y -algebras and study the level at which the first singular vector appears.22 .2 Truncations from cosets and DS reductions We can similarly translate the value of η for cosets and Drinfeľd-Sokolov reductions to trun-cation curves. The orthogonal cosets (2.18) have simple truncation curves n − µ = 1 (4.3)while the symplectic ones n + 1 µ + 2 n + 1 µ = 1 . (4.4)The Drinfeľd-Sokolov reductions lead to curves B n : 1 µ + 2 n B + 1 µ = 1 C n : 1 µ + 2 n C + 1 µ = 1 (4.5) D n : 2 n D − µ = 1 . Unitary minimal models
These of course don’t exhaust all possible truncations that wemay get by studying cosets and Drinfeľd-Sokolov reductions. For example cosets (2.18) canbe studied at fixed non-negative integer value of k and generic n . The central charges ofthese models are c = 0 ( k = 0 , only the vacuum state), c = 1 ( k = 1 ), c = n − n +1 ( k = 2 ), . . . .These are the unitary minimal models of the corresponding truncated algebras. For each k ,the parameters of the associated even spin W ∞ lie on a curve kµ + k − µ = 1 (4.6)so we can think of this curve as cutting out the unitary minimal models in the parameterspace. This is again very similar to the situation in W ∞ and in fact even the form of thesecurves is the same. Non-unitary minimal models
The class of all minimal models is larger than one withunitary minimal models. Consider following [17] the minimal models of W -algebra associatedto D n via Drinfeľd-Sokolov reduction parametrized by coprime integers ( p ′ , p ) such that thecentral charge is c = n (cid:20) − n − n −
1) ( p ′ − p ) p ′ p (cid:21) . (4.7)Choosing p ′ = p + 1 and p = 2 n − k we get for k = 0 , , . . . the sequence of unitaryminimal models discussed in the previous paragraph. For | p ′ − p | 6 = 1 we still get minimal23odels but no longer unitary. The level of Drinfeľd-Sokolov reduction can be chosen either k D = p ′ − n − pp or p − n − p ′ p ′ . (4.8)Choosing the first one, we can identify the parameters of even spin W ∞ as µ = p (2 n − p ′ − p ) µ = − p ′ (2 n − p ′ − p ) (4.9) µ = 12 n − . These lie on truncation curve p ′ − n + 1 µ + p − n + 1 µ = 1 . (4.10)Choosing p ′ − p = 1 we reduce to the truncation curve of minimal models discussed in theprevious paragraph (although in another triality frame). Let us summarize truncation curves that we see from the explicit calculation of operatorproduct expansions. This is easier to see in the quadratic basis because we have a naturalnormalization of fields such that the OPEs have only polynomial coefficients in this basis.
Truncation to vacuum
Just like the c = 0 truncation of Virasoro algebra where thevacuum representation is one-dimensional, in even spin W ∞ for µ + 1 µ = 1 (4.11)(and permutations of µ ) the dimension two field V is singular so the theory reduces to asingle state. This happens for example in the zeroth unitary minimal model k = 0 wherethere is just the vacuum state and c = 0 . Truncation to W [2] (Virasoro) The Virasoro algebra generated by T = − V is always asubalgebra of even spin W ∞ . If we are interested in quotient algebras and the correspondingideals, the dimension field is singular only if equation of the form µ + 3 µ = 1 (4.12)is satisfied. In this case the singular vector is at level . These truncations of even spin W ∞ admit free field representation in terms of only one free boson.24 runcation to W [2 , Working in primary basis, truncation to W -algebra with additionalspin field is a little bit irregular because our parameter x is not defined. The only conditioncoming from associativity of the algebra is C ( C ) = c (2 c − c + 22)(7 c + 68)216( c + 24)( c − c + 196) (4.13)Translated to truncation curves, we find three curves of the form µ = 1 (4.14)(which corresponds to first unitary minimal models) and we also have an orbit of six curvesof the form (these are associated to W B or W C truncations) µ + 5 µ = 1 . (4.15)All these algebras have level singular vector. Truncation to W [2 , , The bootstrap for algebras of type W [2 , , is consistent if x takes one of the values c − c + 20)(7 c + 68)6( c + 24)(10 c + 47 c − , c + 50)(2 c − c + 68)3( c + 24)(5 c + 309 c − (4.16)as well as one of two roots of the quadratic equation a x + a x + a = 0 (4.17)with a = 18( c + 24) (85 c + 5275 c + 101736 c + 1806268 c − a = 3( c + 24)(7 c + 68)(65 c + 2409 c − c − c + 17536992) (4.18) a = 98( c + 50)(2 c − c + 68) (13 c + 1320) The first solution for x corresponds to truncations µ = 1 (4.19)(and triality images of this), the second solution to µ + 3 µ = 1 (4.20)25nd the pair of algebraic solutions satisfying the quadratic equation for x correspond to sixtruncation curves of the form µ + 7 µ = 1 (4.21)(these are the W B or W C truncations). There are also some spurious co-dimension twospecializations of parameters where the algebra truncates (for example ( c = − , x = ) or ( c = − , x = 0) but we are interested in co-dimension specializations so we don’t discussthese. Truncation to W [2 , , , Here the truncation curves are of the form µ = 1 (4.22)and µ + 9 µ = 1 . (4.23) Truncation to W [2 , , , , At level there are four different types of truncations, µ = 1 , µ + 1 µ = 1 , µ + 3 µ = 1 , µ + 1 µ = 1 . (4.24) and conjecture Let’s summarize the truncations discussed in this section. The following table lists all thetruncations with singular vector up to level : ( N , N , N ) level of singular vector construction of truncation (1 , ,
0) 2 vacuum (3 , ,
0) 4
Virasoro (1 , ,
0) 6 first unitary minimal models (5 , ,
0) 6
W B ≃ W C (2 , ,
0) 8 so (3) coset (7 , ,
0) 8
W B ≃ W C (3 , ,
0) 8 sp (2) coset (3 , ,
0) 10
W D , so (4) coset (9 , ,
0) 10
W B ≃ W C (4 , ,
0) 12 so (5) coset (2 , ,
0) 12 second unitary minimal models (5 , ,
0) 12(11 , ,
0) 12
W B ≃ W C
26o write a general conjecture for a level of a given truncation we need to distinguish threecases depending on the even/odd parity of the parameters N j in (4.1):1. For the truncation curves of the form N + 1 µ + 2 N + 1 µ = 1 (4.25)the truncation has first singular vector at level
12 (2 N + 2) × (2 N + 2) × . (4.26)In Gaiotto-Rapčák picture this corresponds to one of algebras Y − N ,N , , Y − N ,N , , ˜ Y − N , ,N , ˜ Y − N , ,N , Y +0 ,N ,N or Y +0 ,N ,N . In each of these cases we have gauge groups Sp (2 N ) or SO (2 N + 1) and Sp (2 N ) or SO ( N + 1) . The third gauge group is formally SO (0) .2. Second type of truncation curves are those of the form N + 1 µ + 2 N µ = 1 . (4.27)These truncations have their first singular vector at level
12 (2 N + 3) × (2 N + 2) × (4.28)and the Gaiotto-Rapčák algebras are now Y − ,N ,N +1 , Y − N , ,N +1 , ˜ Y − ,N +1 ,N , ˜ Y − N ,N +1 , , Y + N +1 , ,N or Y + N +1 ,N , . The associated gauge groups are SO (2 N +2) , either Sp (2 N ) or SO (2 N + 1) and formally Sp (0) .3. The last type of truncation curves are those of the form N µ + 2 N µ = 0 . (4.29)The Y -algebras are of the form ˜ Y + N ,N , and permutations and the gauge groups areeither Sp (2 N ) or SO (2 N + 1) and either Sp (2 N ) or SO (2 N + 1) . The third gaugegroup is formally Sp (0) . The level of such truncations is
12 (2 N + 2) × (2 N + 2) × (4.30)The level of the truncation is now given uniformly as ρ ( G ) × ρ ( G ) × ρ ( G ) (4.31)27here ρ ( G ) is an independent factor associated to each gauge group, ρ ( G ) = n + 2 , Sp (2 n )2 n + 2 , SO (2 n + 1)2 n + 1 , SO (2 n ) (4.32)or in other words twice the (Dynkin) rank plus the lacity ( for simply laced D n and fordoubly laced algebras B n and C n ).We explicitly verified these truncation curves only by studying the first appearance ofthe singular vector in the universal even spin algebra. In this way we can only detect thetruncations where one of the N j parameters vanishes. This corresponds to simple quotientsof the algebra. The class of Y -algebras introduced in [7] however includes also algebras whichare not simple. These are still interesting for example when one considers the gluing [21]because in general a simple algebra can obtained by gluing of non-simple subalgebras. In theunitary case the free field representations of these non-simple quotients were found in [26].Since in the unitary case which is better understood and also in all examples discussed herethe level of the first singular vector follows a simple uniform factorized formula (4.31) wherethe individual gauge groups don’t interact and which makes good sense even if all parametersparametrizing the truncation curve are non-zero, we conjecture that this correctly describesthe truncation of Y -algebras in the non-simple situation as well. Comparison of truncation curves of even spin W ∞ and W ∞ In general each trun-cation curve N µ + N µ + N µ = 1 (4.33)in even spin W ∞ lies on a curve N λ + N λ + N + 12 λ = 1 . (4.34)in the parameter space of W ∞ . Due to factor of in the denominator there are curvesin the parameter space of W ∞ where the full algebra does not truncate but the even spinsubalgebra can still truncate. Truncations to algebras W B n , W C n and W D n are examples oftruncations which lie on truncation curves in W ∞ , actually they lie on curves correspondingto W n algebras (this is also true for exceptional algebras where the embeddings in W ∞ areknown). In this last section we illustrate how the gluing procedure discussed in [21] applies to orthog-onal affine Lie algebras. Let us first review the case of unitary affine Lie algebras. The gluing28 ( N ) U ( N − U ( N − λ λ λ ′ λ ′ λ λ ′ Figure 2: Part of gluing diagram for u ( N ) k .diagram of u ( N ) k is based on the decomposition u ( N ) k ⊃ u ( N ) k u ( N − k × u ( N − k u ( N − k × · · · × u (2) k u (1) × u (1) . (5.1)Each of the factors on the right hand side is a truncation of W ∞ . Identifying parametersas in figure 2 we can calculate the λ -parameters of the corresponding W ∞ algebras sittingat the vertices [21]. We have λ = ( N − ǫ + N ǫ ǫ = N ( ψ − − ( N − ψψ − , (5.2) λ ′ = ( N − ǫ ′ + ( N − ǫ ′ ǫ ′ = − ( N − ψ ′ − − ( N − ψ ′ ψ ′ (5.3)From the ( p, q ) charges of the five-branes we see that ǫ ′ = ǫ and ǫ ′ = ǫ + ǫ so ψ ′ ≡ − ǫ ′ ǫ ′ = ψ − (5.4)which guarantees that the dimension of the fundamental gluing fields is h = h + h ′ = 1 + λ λ ′ (5.5)This is exactly what we need in order to find dimension fields charged under Cartan u (1) currents coming from the vertices. These correspond in the language of affine Lie algebras tocurrents associated to positive and negative simple roots. The other generators associated toroots that are not simple corresponding to line operators stretched between vertices whichare not neighbouring. 29 O (2 N + 1) SO (2 N ) SO (2 N − µ µ µ ′ µ ′ µ µ ′ Figure 3: Part of gluing diagram for so (2 N + 1) k .Let’s now consider the orthogonal Lie algebra so (2 n + 1) k . We have a similar decompo-sition so (2 N + 1) k ⊃ so (2 N + 1) k so (2 N ) k × so (2 N ) k so (2 N − k × · · · × so (3) k so (2) × so (2) . (5.6)The first coset on the right hand side has parameters compatible with ˜ Y − ,N,N with thetruncation curve Nµ + 2 N + 1 µ = 1 (5.7)while the second term can be identified with Y − ,N − ,N with truncation curve N − µ ′ + 2 N − µ ′ = 1 . (5.8)The first part of the gluing diagram looks like figure 3. Let us verify that the gluing fieldshave compatible dimensions. The upper vertex has parameter µ = ψ − Nψ (5.9)while the corresponding parameter of the lower vertex is µ ′ = − − N + ψ ′ ψ ′ . (5.10)The relative orientation of the two vertices is just like in the unitary case so we still have(5.4). Now we can calculate the conformal dimension of gluing fields and find h + h ′ = 1 (5.11)30hich is exactly what we want in order to find dimension currents.Note that they way the currents appear is slightly different than in the unitary situation.In the unitary case each vertex represented a truncation of W ∞ algebra which by definitioncarried an affine u (1) current. There are as many of these as is the rank of the algebra andall these currents give the Cartan subalgebra of u ( N ) k . As already discussed the elementarygluing fields give rise to simple positive and negative roots. In the orthogonal case thetruncations of even spin W ∞ algebras at vertices do not have any spin fields so we don’tfind any Cartan fields in this way. On the other hand, we have an alternating sequence of Y − and ˜ Y − algebras and associated to each neighbouring pair of these there is an elementarydimension gluing field (which now do not appear in complex conjugate pairs because theminimal representations of even spin W ∞ are real). For example for rank algebra the firstCartan generator can be chosen to correspond to line operator stretched from ˜ Y − to Y − and the second generator to line operator between ˜ Y − to Y − . The understanding of the universal orthosymplectic W ∞ algebra is still much more limitedthan that of W ∞ . In particular1. All the truncations that we found are associated to truncation curves of the form (4.1)and also each of these can be associated to a certain Y -algebra. We conjecture that thishappens at level (4.31), but our calculations give no proof that this is what actuallyhappens. From the from explicit coset and Drinfeľd-Sokolov reduction description of Y -algebras it should be possible to verify this.2. The free field representations of truncations of W ∞ are reasonably well understood[26]. Since the even spin algebra is a subalgebra of W ∞ , we can find many free fieldrepresentations of truncations of even spin algebra from the free field representations of W ∞ . But one should understand if there are any other representations and how arethese related to truncations of the algebra, i.e. if we have a correspondence betweenfree field representations and co-dimension truncations like in the case of W ∞ [26].3. The combinatorial box counting interpretation of characters of even spin W ∞ is notknown. One cannot simply restrict to subset of box configurations in W ∞ becausethe canonical Virasoro generators don’t agree and the higher spin generators of W ∞ seem not to preserve the even spin subalgebra.4. No analogue of Tsymbaliuk presentation of W ∞ as affine Yangian is known. Onecould try to repeat the steps of [29] to find the ladder operators in Yangian but firstthe folding of GL ( N ) Miura operator should be understood. It is very reminiscentto spin chains with boundary where the boundary reflection operator is ∂ or ∂ − .This surely deserves a deeper study. Once this is understood one can try to apply the31echniques of quantum inverse scattering method or algebraic Bethe ansatz to constructYangian operators for even spin W ∞ .5. Although the algebra admits a quadratic basis, the derivatives of fields don’t seem tofollow the same simple pattern as in the case of W ∞ . If one understands this, onemight hope to be able to write a closed-form formulas for OPEs and commutators ineven spin W ∞ just like those in [16, 20].6. In W ∞ and its matrix extension the fusion and its associated coproduct were ex-tremely efficient tools for construction of free field representations or representations interms of affine Lie algebras. Also the space of co-dimension truncations can be seenas a cone generated by elementary Miura transformations [20, 30]. The unitary Miuraoperator immediately allows us to extract the coproduct. On the other hand, becauseof the folding of the Miura operator in the orthosymplectic case, it is not obvious if theorthosymplectic version of W ∞ admits this coproduct structure. Acknowledgements
I would like to thank to Lorenz Eberhardt, Andrew Linshaw and Miroslav Rapčák for usefuldiscussions. This research was supported by the DFG Transregional Collaborative ResearchCentre TRR 33 and the DFG cluster of excellence Origin and Structure of the Universe.
A Structure constants in primary basis
Here is the list of the structure constants in the primary basis for sum of spins up to 12: C = c (5 c + 22)72( c + 24) (cid:0) C (cid:1) − c ( c − c + 22)72(2 c − c + 68) C C + c ( c − c + 24)(5 c + 22)12(2 c − c + 68) (cid:0) C (cid:1) C = 4(5 c + 22)9( c + 24) (cid:0) C (cid:1) C − c − c + 196) c (2 c − c + 68) C C C = − c − c − c + 24)(5 c + 22)( c − c + 196)(2 c − (7 c + 68) (13 c + 516) (cid:0) C (cid:1) C C − c − c + 22)( c − c + 196)(20 c + 24807 c + 765640 c − c + 31)(2 c − (7 c + 68) (13 c + 516)(55 c − C (cid:0) C (cid:1) C C + 4( c − c + 22)(5605 c − c − c − c + 1312613664)9( c + 24)( c + 31)(2 c − c + 68)(13 c + 516)(55 c − (cid:0) C (cid:1) C C C − c − c + 50)(5 c + 22)(715 c + 90933 c + 2851076 c + 21154896 c + 6967008)12( c + 24)( c + 31)(3 c + 46)(5 c + 3)(13 c + 516)(55 c − C [44]46 (cid:0) C (cid:1) C c − c + 22)(65 c + 8637 c + 364470 c + 2897944 c + 36384)12(2 c − c + 46)(5 c + 3)(7 c + 68)(13 c + 516) C [44]46 C C C − ( c − c + 24)(5 c + 22)(65 c + 8637 c + 364470 c + 2897944 c + 36384)2(2 c − c + 46)(5 c + 3)(7 c + 68) (13 c + 516) C [44]46 (cid:0) C (cid:1) C + 140( c − c + 50)(5 c + 22)(11 c + 656)27( c + 24) ( c + 31)(55 c − (cid:0) C (cid:1) C C C = 8(25 c + 615 c − c + 102332)3( c + 24)( c + 31)(55 c − C C C − c − c + 516) C [44]46 C C C + 16(425 c + 15145 c + 233766 c + 6507708 c − c + 31)(7 c + 68)(13 c + 516)(55 c − (cid:0) C (cid:1) C + 7840( c + 50)(2 c − c + 68)9( c + 24) ( c + 31)(55 c − (cid:0) C (cid:1) C − c + 50)(2 c − c + 68)(3 c + 24)( c + 31)(55 c − C [44]46 C C C C = 8(33 c + 1087 c + 11760)(7 c + 68)(13 c + 516) C − (31 c − c + 516) C C [44]46 − C C [44]48 = − c + 46)(5 c + 3)( c − c + 196)( c + 31)(2 c − c + 68) (55 c − (cid:0) C (cid:1) C C + 8(33 c + 1087 c + 11760)(7 c + 68)(13 c + 516) C C [44]46 C + 3136(3 c + 46)(5 c + 3)( c − c + 196)3( c + 24)( c + 31)(2 c − c + 68)(55 c − C C C C − (31 c − c + 516) C (cid:16) C [44]46 (cid:17) C − c + 10763 c + 140036 c + 38568)3( c + 24)( c + 31)(55 c − C C [44]46 C + 448(3 c + 46)(5 c + 3)(11 c + 656)9( c + 24) ( c + 31)(55 c − (cid:0) C (cid:1) C C C [46] (1) = − c + 24) C C + (13 c + 918)(13 c + 516) C C [44]46 C − c − c + 68)(13 c + 516) C C C = − c − c (5 c + 22) (8 c + 1161 c − c + 24)(2 c − (7 c + 68) (cid:0) C (cid:1) C (cid:0) C (cid:1) + 14( c − c ( c + 24)(5 c + 22) ( c − c + 196)9(2 c − (7 c + 68) C (cid:0) C (cid:1) (cid:0) C (cid:1) − c − c ( c + 24) (5 c + 22) ( c − c + 196)3(2 c − (7 c + 68) (cid:0) C (cid:1) (cid:0) C (cid:1) − ( c − c (5 c + 22) (17 c − c + 25330 c − c − (7 c + 68) (cid:0) C (cid:1) (cid:0) C (cid:1) (cid:0) C (cid:1) ( c − c (5 c + 22) (11 c + 656)162( c + 24) (2 c − c + 68) (cid:0) C (cid:1) (cid:0) C (cid:1) C = 28( c − c + 22)( c − c + 196)3(2 c − (7 c + 68) C (cid:0) C (cid:1) (cid:0) C (cid:1) − c − c + 24)(5 c + 22)( c − c + 196)(2 c − (7 c + 68) (cid:0) C (cid:1) (cid:0) C (cid:1) + 4( c − c + 22)(11 c + 656)9( c + 24)(2 c − c + 68) (cid:0) C (cid:1) C (cid:0) C (cid:1) C = − c − c − c + 32168 c + 859328)27( c + 24) (2 c − c + 68) (cid:0) C (cid:1) C + 10(28 c − c − c + 387728 c + 3726976 c − c + 24)(2 c − (7 c + 68) C C C + 20(92 c + 2389 c + 39632 c + 4060 c − c + 193984)(2 c − (7 c + 68) (cid:0) C (cid:1) C C = 4(4 c + 61)(7 c + 68) C C C − (11 c + 656)6( c + 24) C C C C [44]66 = 784( c − c + 196)3( c + 24)(2 c − c + 68) (cid:0) C (cid:1) C (cid:0) C (cid:1) − (11 c + 656)6( c + 24) (cid:0) C (cid:1) C [44]46 C − c − c + 196)(2 c − c + 68) (cid:0) C (cid:1) (cid:0) C (cid:1) + 112(11 c + 656)9( c + 24) (cid:0) C (cid:1) (cid:0) C (cid:1) + 4(4 c + 61)(7 c + 68) C C [44]46 C C [44] (2) = 1960(47 c − c − c + 196)3( c + 24)( c + 31)(2 c − c + 68)(55 c − C C (cid:0) C (cid:1) + 34 C C [44] (2) C + 5( c + 76)(5 c + 22)(11 c + 232)12( c + 24)( c + 31)(55 c − C C [44]46 C − c − c − c + 196)( c + 31)(2 c − c + 68) (55 c − (cid:0) C (cid:1) (cid:0) C (cid:1) + 280(11 c + 656)(47 c − c + 24) ( c + 31)(55 c − (cid:0) C (cid:1) (cid:0) C (cid:1) C = 34 C C C C [46]66 = − c + 24) C C + 48(81 c + 1274)(7 c + 68)(13 c + 516) C C + 3(13 c + 248)2(13 c + 516) C [44]46 + 34 C C [46]48 C [46] (1) = − c + 24) C C − c − c + 68)(13 c + 516) C C + 5(13 c + 918)4(13 c + 516) C [44]46 − C C [44] (1) C B Quadratic basis in even spin W ∞ The field V expressed in terms of U j fields is V = U − ( U U ) + ( U U ) −
12 ( U U ) + 14 (5 − N )( U ′′ U ) − α ( N − U ′ U )+ α ( N − U U ′ ) + 14 ( N − U ′′ U ) + α ( N − U ′ U ) − α ( N − U U ′ ) −
14 ( U U ′′ ) + 196 ( N − α + 8 α N − α N − N )( U (4)1 U )+ 112 α ( N − α + 4 α N − α N − N )( U (3)1 U ) −
116 (30 α + 4 α N − α N − α + 4 α N − α N − U ′′ U ′′ )+ 14 α ( N − α + 4 α N − α N + 2 N − U ′′ U ′ ) − α ( N − α + 4 α N − α N − U ′ U ′′ )+ 12 (30 α + 4 α N − α N − U ′ U ′ ) − α ( N − α + 4 α N − α N − N )( U U (3)2 )+ 14 ( − α − α N + 22 α N + 1)( U ′ U ′ )+ 124 (cid:2) − α − α − α N + 144 α N + 8 α N − α N − α N + 684 α N + 60 α N − N (cid:3) ( U (3)1 U ′ )+ 112 α ( N − α + 4 α N − α N − U (3)3 + 148 ( − α + α N − α N + 24 α N + 1) U (4)2 − α ( N − (cid:2) α − α + 64 α N − α N − α N + 1904 α N + 52 α N − α N + 26 α N + 5 N (cid:3) U (5)1 − α ( N − U ′ OPE of field V with itself is V ( z ) V ( w ) ∼ − n (cid:0) α + 4 α n − α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) × (cid:0) α + 34 α n − α n − n − (cid:1) ( z − w ) + 4 α ( n − n − n − (cid:0) α + 4 α n − α n − (cid:1) V ( z − w ) + 2 α ( n − n − n − (cid:0) α + 4 α n − α n − (cid:1) V ′ ( z − w ) − n − (cid:0) α + 4 α n − α n − (cid:1) ( V V )( z − w ) + ( n − (cid:0) − α + 4 α n − α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) V ′′ ( z − w ) + 6 (cid:0) α + 4 α n − α n + 1 (cid:1) V ( z − w ) − n − (cid:0) α + 4 α n − α n − (cid:1) ( V ′ V )( z − w ) + 13 ( n − (cid:0) − α + 4 α n − α n − (cid:1) (cid:0) α + 4 α n − α n − (cid:1) V (3)2 ( z − w ) + 3 (cid:0) α + 4 α n − α n + 1 (cid:1) V ′ ( z − w ) − ( n − (cid:0) α + 4 α n − α n − (cid:1) ( V ′′ V )( z − w ) + 12 (cid:0) − α − α n + 14 α n + 1 (cid:1) ( V ′ V ′ )( z − w ) − V V )( z − w ) − V ( z − w ) + 12 (cid:0) − α + 4 α n + 1 (cid:1) V ′′ ( z − w ) + 124 (cid:0) α + 4 α n − α n − (cid:1) ×× (cid:0) α + 8 α n − α n − α n − n + 1 (cid:1) V (4)2 ( z − w ) −
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