An analog of the Feigin-Frenkel homomorphism for double loop algebras
aa r X i v : . [ m a t h . QA ] N ov AN ANALOG OF THE FEIGIN-FRENKEL HOMOMORPHISMFOR DOUBLE LOOP ALGEBRAS
CHARLES YOUNG
Abstract.
We prove the existence of a homomorphism of vertex algebras, fromthe vacuum Verma module over the loop algebra of an untwisted affine algebra,whose construction is analogous to that of the Feigin-Frenkel homomorphism fromthe vacuum Verma module at critical level over an affine algebra.
Contents
1. Introduction and overview 12. Realization of g by differential operators on the big cell 123. Cartan involution and doubling trick 264. Vertex algebras and main results 305. Proof of Theorem 28 476. Proof of Theorem 32 59Appendix A. The coefficients c i Introduction and overview
The goal of this paper is to give an analog of the Feigin-Frenkel homomorphism V g , − h ∨ → M ( n ) in the case in which g is of untwisted affine type. To set the scene,we should first recall the situation in finite types.1.1. The Lie algebra sl (over C ) has a realization in terms of first order differentialoperators: E D H
7→ − XD F
7→ −
XXD. (1)Here
E, F, H are the Chevalley-Serre generators and
X, D are generators of a Weylalgebra with commutation relations [
D, X ] = 1. At the heart of the Wakimotoconstruction [Wak86] is the observation that this homomorphism of Lie algebras canbe promoted to a homomorphism of vertex algebras, given by E [ − | i 7→ β [ − | i H [ − | i 7→ − γ [0] β [ − | i F [ − | i 7→− γ [0] γ [0] β [ − | i − γ [ − | i , (2) Date : November 4, 2020. from the vacuum Verma module over b sl at the critical level, to the vacuum Fockmodule for a βγ -system of free fields. Note the new feature, the term − γ [ − | i .This is a special case of a construction which works for any finite-dimensional simpleLie algebra g = C n − ⊕ h ⊕ n . The realization (1) generalizes to a homomorphism ρ : g → Der O ( n ); A X α ∈ ∆ + P αA ( X ) D α (3)from g to the Lie algebra Der O ( n ) of derivations of the algebra O ( n ) = C [ X α ] α ∈ ∆ + of polynomial functions on the unipotent group U = exp( n ) ∼ = n . This realizationarises from the infinitesimal action of g on a flag manifold, B − \ G , whose big cell isdiffeomorphic to U .Recall (from e.g. [Kac98; FB04]) that the vacuum Verma module V g ,k over b g atlevel k ∈ C is generated as a vertex algebra by states { A [ − | i : A ∈ g } , whosenon-zero non-negative products (i.e., whose OPEs) are given by A [ − | i (0) B [ − | i = [ A, B ][ − | i ,A [ − | i (1) B [ − | i = k κ (cid:0) A | B (cid:1) | i . (4)Here κ (cid:0) ·|· (cid:1) is the invariant symmetric bilinear form on g normalized as in [Kac90].With this normalization, the critical level is equal to − h ∨ , where h ∨ is the dualCoxeter number of g .Let M ( n ) be the vacuum Fock module for the βγ -system on n ∼ = U . It is generatedas a vertex algebra by states β α [ − | i and γ α [0] | i , α ∈ ∆ + , obeying β α [ − | i (0) γ β [0] | i = δ βα | i . (See Section 4 for the details.)Both V g ,k and M ( n ) have natural Z ≥ -gradations (by depth ) and the first twograded subspaces are V g ,k [0] ∼ = C M ( n )[0] ∼ = O ( n ) V g ,k [1] ∼ = g M ( n )[1] ∼ = Der O ( n ) ⊕ Ω O ( n ) , (5)where Ω O ( n ) = Hom O ( n ) (Der O ( n ) , O ( n )) is the space of one-forms. One identifies | i ≃ A [ − | i ≃ A ∈ g ; and γ α [0] ≃ X α , γ α [ − ≃ dX α and D α ≃ β α [ − ρ : g → Der O ( n ) gives rise to agraded linear map V g ,k [ ≤ → M ( n )[ ≤ , sending | i → | i and A [ − | i 7→ X α ∈ ∆ + P αA ( γ [0]) β α [ − | i . This map does not preserve the non-negative products, but the result of Feigin andFrenkel [FF90a], [FF88; FF90b] [Fre05b; Fre07] is that, at the critical level k = − h ∨ ,it may be lifted to one which does. Namely, there exists a linear map φ : g → Ω O ( n ) ; A X α ∈ ∆ + Q α,A ( X ) dX α NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 3 such that the graded linear map V g , − h ∨ [ ≤ → M ( n )[ ≤
1] (7a)associated to ρ + φ : g → Der O ( n ) ⊕ Ω O ( n ) , i.e. the one sending | i → | i and A [ − | i 7→ X α ∈ ∆ + P αA ( γ [0]) β α [ − | i + X α ∈ ∆ + Q α,A ( γ [0]) γ α [ − | i , (7b)does preserve the non-negative products. This latter map (7) is the restriction of ahomomorphism of graded vertex algebras, V g , − h ∨ → M ( n ) . The map φ : g → Ω O ( n ) respects the weight gradation. In particular φ ( h ⊕ n ) = 0on grading grounds.For example in the case of sl , φ ( f ) = − dX , φ ( e ) = φ ( h ) = 0, as in (2).Various perspectives on this important result have subsequently appeared in theliterature [BF97], [FF99],[FG08], [ACM11] In particular, see [GMS01] for an inter-pretation the language of vertex algebroids and chiral algebras [MSV99; GMS04],[GMS00; AG02; GMS03; BD04; Mal17]1.2. Now, and for the rest of this paper, let us suppose instead that g is of untwistedaffine type, i.e. that g ∼ = C ˚ g [ t, t − ] ⊕ C k ⊕ C d for some finite-dimensional simple Lie algebra ˚ g . (Here k is central and d = t∂ t is thederivation element corresponding to the homogeneous gradation.)We still have the Cartan decomposition g = C n − ⊕ h ⊕ n . The Lie subalgebra n = L α ∈ ∆ + n α is now of countably infinite dimension, and no longer nilpotent. Butits completion e n = Q α ∈ ∆ + n α is a pro-nilpotent pro-Lie algebra (i.e. a certain inverselimit of nilpotent Lie algebras – see [Kum02], and Section 2.4 below) and there is stilla bijective exponential map exp : e n ∼ −→ U to a group U , which is now a pro-unipotent pro-group. We shall fix (in Section 2.6)a convenient choice of coordinates on U , X a,n : U → C . Here ( a, n ) runs over a countable index set, A , which also indexes a topological basis J a,n of e n . (Recall dim( n α ) can be greater than 1 in affine types other than b sl , so wecannot simply index by the positive roots ∆ + .)We set O ( n ) := C [ X a,n ] ( a,n ) ∈ A and define Der O ( n ) to be the Lie algebra of deriva-tions of O ( n ) consisting of sums of the form X ( a,n ) ∈ A P a,n ( X ) D a,n , P a,n ( X ) ∈ O ( n ) , subject to the constraint that only finitely summands are nonzero. It has a com-pletion, g Der O ( n ) ⊃ Der O ( n ), consisting of sums of the same form but without the CHARLES YOUNG constraint. (Here the new generators D a,n obey [ D a,n , X b,m ] = δ b,ma,n .) As before, wedefine the space of one-forms Ω O ( n ) = Hom O ( n ) (Der O ( n ) , O ( n )).The group U can still be seen as copy of the big cell of a flag manifold B − \ G , ina sense made precise in [Kas89]. For our purpose the important point is that thereis a homomorphism of Lie algebras, ρ : g → g Der O ( n ); A X ( a,n ) ∈ A P a,nA ( X ) D a,n (8)as we show in a concrete fashion in Section 2.11. This is the analog of the homomor-phism (3).Some examples in the case g = b sl are shown in Fig. 1.(The centre of g lies in the kernel, ρ ( k ) = 0, so the homomorphism ρ actuallyfactors through g / C k ∼ = ˚ g [ t, t − ] ⋊ C d .)1.3. One may define the vacuum Verma module V g ,k over b g at level k ∈ C when g isaffine, such that (4) still holds, where κ (cid:0) ·|· (cid:1) is the standard non-degenerate symmetricinvariant bilinear form from [Kac90] (with κ (cid:0) k | d (cid:1) = 1 and so on). It is still a vertexalgebra.The main result of the present paper (Theorem 29) is that the homomorphism ρ can be promoted to a homomorphism of vertex algebras V g , → M . Of course, wehave yet to explain what M is. To motivate its definition, it is instructive to considerwhat happens when one attempts to generalize the construction above in the mostdirect fashion.Thus, let M ( n ) be, again, the vacuum Fock module for the βγ -system on n ∼ = U . Itis a vertex algebra, generated by (now, countably infinitely many) states β a,n [ − | i and γ a,n [0] | i , ( a, n ) ∈ A , obeying β a,n [ − | i (0) γ b,m [0] | i = δ b,ma,n | i . Both V g ,k and M ( n ) are once more Z ≥ -graded by depth, and the identifications in(5) continue to hold. (One now identifies γ a,n [0] ≃ X a,n , γ a,n [ − ≃ dX a,n and D a,n ≃ β a,n [ − M ( n )[1] ∼ = Der O ( n ) ⊕ Ω O ( n ) . Importantly, it is Der O ( n ) and not its completion g Der O ( n ) which appears here.Indeed, by definition, M ( n ) consists of finite linear combinations of states of theform γ a ,n [ − N ] . . . γ a r ,n r [ − N r ] β b ,m [ − M ] . . . β b s ,m s [ − M s ] | i . Thus, in contrast toSection 1.1, the image ρ ( g ) ⊂ g Der O does not naturally embed in M ( n )[1]. We needsome larger space. We shall introduce a completion e M ( n ) of M ( n ) as a vector space,in which certain infinite linear combinations are allowed provided they truncate to It is perhaps worth stressing that B − \ G is not the affine Grassmannian or the affine flag variety in the usual sense of e.g. [G¨or10, § g = b sl , the (set of C -points of the)affine Grassmannian is SL ( C [[ t ]]) (cid:15) SL ( C (( t ))), whereas here B − = SL ( t − C [ t − ]) ˚ B − with ˚ B − theusual lower-triangular Borel subgroup of SL ( C ). See also the discussion in [Fre04, §
5] (in which oneshould swap t ↔ t − to match the present conventions). NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 5 ρ ( J E, ) = D E, + 2 X H, D E, − X F, D H, + (cid:16) X H, − (cid:0) X H, (cid:1) (cid:17) D E, − X F, D H, + (cid:0) X F, (cid:1) D F, + (cid:16) X H, + (cid:0) X H, (cid:1) (cid:17) D E, + (cid:16) − X F, − X E, (cid:0) X F, (cid:1) (cid:17) D H, + 2 (cid:0) X F, (cid:1) X H, D F, + . . .ρ ( J F, ) = D F, + X E, D H, − X H, D F, + (cid:0) X E, (cid:1) D E, + (cid:0) X E, + 2 X E, X H, (cid:1) D H, + (cid:16) − X H, − (cid:0) X H, (cid:1) (cid:17) D F, + . . .ρ ( J E, − ) = (cid:0) X H, + 2 X E, X F, (cid:1) D E, + (cid:16) X H, + 2 (cid:0) X H, (cid:1) (cid:17) D E, + (cid:0) − X F, − X F, X H, (cid:1) D H, − (cid:0) X F, (cid:1) D F, + (cid:16) X H, + 2 X E, X F, − (cid:0) X H, (cid:1) (cid:17) D E, − X F, D H, + 2 (cid:0) X F, (cid:1) X H, D F, + . . .ρ ( J F, ) = − (cid:0) X E, (cid:1) D E, + X E, D H, − X H, D F, − (cid:0) X E, (cid:1) D E, + X E, D H, + (cid:16) − X H, + 2 (cid:0) X H, (cid:1) (cid:17) D F, + (cid:16) X E, − X E, (cid:0) X H, (cid:1) (cid:17) D H, + (cid:16) − X H, + (cid:0) X H, (cid:1) (cid:17) D F, + . . . Figure 1.
In type g = b sl , the first few terms of the images of theChevalley-Serre generators e = J E, , e = J F, , f = J F, , f = J E, − under the homomorphism ρ : g → g Der O ( n ).finite linear combinations when β a,n [ N ] is set to zero for large n . (See Section 4.7.)The definition is chosen to ensure that e M ( n )[1] ∼ = g Der O ( n ) ⊕ Ω O ( n ) . We get the graded linear map V g ,k [ ≤ → e M ( n )[ ≤ | i → | i and A [ − | i 7→ X ( a,n ) ∈ A P a,nA ( γ [0]) β a,n [ − | i , with P a,nA ( X ) the polynomials from (8). CHARLES YOUNG φ : g → Ω O ( n ) ; A X ( a,n ) ∈ A Q a,n ; A ( X ) dX a,n (9)such that the graded linear map V g ,k [ ≤ → M ( n )[ ≤ ρ + φ : g → g Der O ( n ) ⊕ Ω O ( n ) , i.e. the one sending | i → | i and A [ − | i 7→ X ( a,n ) ∈ A P a,nA ( γ [0]) β a,n [ − | i + X ( a,n ) ∈ A Q a,n ; A ( γ [0]) γ a,n [ − | i , does preserve at least the th vertex algebra product (for any k ∈ C ). That is, if wecall this latter map ϑ , we have ϑ ( A ) (0) ϑ ( B ) = ϑ ( A (0) B ) (10)for all A ≃ A [ − | i and B ≃ B [ − | i in g ∼ = V g ,k [1].The map φ again respects the weight gradation, so that φ ( h ⊕ n ) = 0.For example, in type g = b sl , when using the same choice of coordinates on U asin Fig. 1 one finds that φ ( f = J F, ) = − dX E, , φ ( f = J E, − ) = − dX F, , and then φ ( J H, − ) = − dX H, − X F, dX E, φ ( J F, − ) = − dX E, + 4 X H, dX E, φ ( J E, − ) = − dX F, − X H, dX F, + 2( X F, ) dX E, , and so on.While encouraging, this statement skirts around a serious caveat, which is thereason the question above was naive: the completion e M ( n ) is not a vertex algebra.Or, more precisely, the vertex algebra structure on M ( n ) does not extend to e M ( n ). Inparticular, when one tries to extend the definition of the vertex algebra n th products ( n ) : M ( n ) × M ( n ) → M ( n ) by bilinearity to e M ( n ) × e M ( n ), the results are not in generalfinite.Thus, it was a non-trivial fact about the image of ϑ that the expression ϑ ( A ) (0) ϑ ( B )in (10) was even well-defined. And one finds the would-be 1st products ϑ ( A ) (1) ϑ ( B )are not in general finite. NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 7 g = b sl . For this subsection only , we modify the OPEs by introducing a formal variable z which will serve as a regulator: β a,n [ − | i (0) γ b,m [0] | i = z n δ b,ma,n | i . One has ϑ ( J H, ) = − X n ≥ γ E,n [0] β E,n [ − | i + 2 X n ≥ γ F,n [0] β F,n [ − | i . It follows that the regulated 1st product ϑ ( J H, ) (1) ϑ ( J H, ) is the formal series ϑ ( J H, ) (1) ϑ ( J H, ) = (cid:16) − − X n ≥ z n (cid:17) | i . (We recall the standard details of computing such products in Section 4.) As moreintricate examples, one finds ϑ ( J E, ) (1) ϑ ( J F, − ) = (cid:16) − z − X n ≥ z n +3 (cid:17) | i ϑ ( J H, ) (1) ϑ ( J H, − ) = (cid:16) − z − z − X n ≥ z n +3 (cid:17) | i ϑ ( J E, − ) (1) ϑ ( J F, ) = (cid:16) − z − X n ≥ z n +4 (cid:17) | i . These series are divergent when one attempts to remove the regulator by setting z = 1.One might be tempted to treat these divergences by ζ -function regularization. For anintroduction to the formal-variable approach to ζ -function regularization, see [Lep99](and cf. also [Blo96; Lep00; DLM06]). It amounts to the following prescription. First,one notes that each series above is the small- z expansion of some rational expression in z . One substitutes z = e y in that rational expression, to obtain a rational expressionin e y ; then one expands e y as a formal series in y . The result is a quotient of formalseries in y , and hence a well-defined formal Laurent series in y . Finally, one extractsthe constant term in that series.Very suggestively, when one does that, the result is zero in each example above.For instance for ϑ ( J E, − ) (1) ϑ ( J F, ) one obtains − − (cid:0) − (cid:1) = 0. In what follows weshall use a different approach, but it will indeed be the case that the homomorphismwe construct is from the vacuum Verma module at level zero.1.6. To proceed, we need more information about the image of the homomorphism ρ : g → g Der O ( n ) from (8). We illustrate the idea with an example in type g = b sl .Let us consider a term in the infinite sum ρ ( e ) = P ( a,n ) ∈ A P a,ne ( X ) D a,n for the CHARLES YOUNG generator e = J E, : say, the term P E, e ( X ) D E, . One finds P E, e ( X ) = 2 X H, − (cid:0) X H, (cid:1) + 4 (cid:0) X H, (cid:1) X H, − (cid:0) X H, (cid:1) − (cid:0) X H, (cid:1) X H, X H, + (cid:0) X H, (cid:1) (cid:0) X H, (cid:1) + (cid:0) X H, (cid:1) X H, X H, + (cid:0) X H, (cid:1) X H, − (cid:0) X H, (cid:1) (cid:0) X H, (cid:1) − (cid:0) X H, (cid:1) X H, + (cid:0) X H, (cid:1) X H, − (cid:0) X H, (cid:1) + X E, X F, (cid:0) X H, (cid:1) − X E, X F, (cid:0) X H, (cid:1) − X E, X E, (cid:0) X F, (cid:1) X H, − (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) − (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) − X E, X E, (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) + 4 X E, X E, (cid:0) X F, (cid:1) X H, − X E, X E, (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) + 2 (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) X H, − (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) + 4 (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) X H, X H, + 4 (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) X H, − (cid:0) X E, (cid:1) (cid:0) X F, (cid:1) (cid:0) X H, (cid:1) − (cid:0) X E, (cid:1) X E, (cid:0) X F, (cid:1) X F, . Observe that only the first monomial has any factor X a,n with n >
4. This is anexample of a general pattern: for any fixed A ∈ g , in the coefficient polynomials P a,nA ( X ) there is, for large n , always at most one leading monomial X b,m with m ∼ n ;the remaining monomials have only factors X c,p with p . n/ widening gap in Section 2.12.See Theorem 10, which will show that the difference ρ ( J a,n ) − X b,c ∈I f bac X m> max(1 ,n ) X b,m − n D c,m , (12)has widening gap. (Here f bac are structure constants of ˚ g .) Elements of widening gapform a Lie algebra, Der O ( n ), with Der O ( n ) ⊂ Der O ( n ) ⊂ g Der O ( n ).The notion of widening gap goes over to the vertex algebra M ( n ): one can define asubspace M ( n ) of the completion e M ( n ) in which infinite sums are allowed but only ifthey have widening gap. The vertex algebra structure on M ( n ) does extend to M ( n ).(See Lemma 21.)1.7. The question therefore becomes: what to do with the leading terms in ρ ( J a,n )?Our approach, in Section 3, will be to glue together two copies of the realization (8)back-to-back. Let O := O ( g ) := C [ X a,n ] ( a,n ) ∈I× Z denote the algebra of polynomialfunctions on all of g . We shall define g Der O and its subalgebra, Der O , of elementsof widening gap. The Cartan involution σ : g → g , which exchanges n and n − , givesrise to an involution τ : g Der
O → g Der O , which exchanges g Der O ( n ) and g Der O ( n − ).On twisting the homomorphism ρ : g ֒ → g Der O ( n ) ⊂ from (8) by these involutions,we get a homomorphism τ ◦ ρ ◦ σ : g ֒ → g Der O ( n − ). Adding the two, we obtain ahomomorphism ρ := ( ρ + τ ◦ ρ ◦ σ ) : g → g Der O ( n ) ⊕ g Der O ( n − ) ֒ → g Der O . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 9
Of course, having helped oneself to a copy of O = O ( g ), there is an obvious ho-momorphism g → g Der O coming from the coadjoint representation, which sends J a,n P b,c ∈I f bac P m ∈ Z X b,m − n D c,m . So one should keep in mind that what isspecial about the homomorphism ρ is that, by construction, the resulting action of g on O stabilizes O ( n ) and O ( n − ).We then check that the difference ρ ( J a,n ) − X b,c ∈I f bac X m ∈ Z X b,m − n D c,m has widening gap, i.e. belongs to Der O . The advantage of this statement, comparedto (12), is that we can replace the sums, P m ∈ Z X b,m − n D c,m , by an abstract set ofgenerators S bc,n of the loop algebra gl (˚ g )[ t, t − ]. In this way we can, and shall, regard ρ as a homomorphism from g to the Lie algebra D := Der O ⋊ (cid:0) gl (˚ g )[ t, t − ] ⋊ C D (cid:1) (13)(here D is a derivation element, and g ∋ d D ). See Lemma 17 and the discussionfollowing.1.8. We define a vertex algebra M in light of this definition of the Lie algebra D .Namely, we have M := M ( g ), the vacuum Fock module for the βγ -system on g , andwe introduce its completion as a vector space, e M , and the subspace generated byelements of widening gap, M ⊂ e M . Then M is a vertex algebra, and we can take a“semi-direct product of vertex algebras” with the level zero vacuum Verma module V gl (˚ g )[ t,t − ] ⋊C D , to define M ; so, as a vector space, M ∼ = C M ⊗ V gl (˚ g )[ t,t − ] ⋊C D , . See Section 4.9. By construction we have M [0] ∼ = O M [1] ∼ = D ⊕ Ω O . At that point we shall be in a position to state our main result: see Theorem 28and Theorem 29. It says the following: let φ = φ + τ ◦ φ ◦ σ : g → Ω O where φ : g → Ω O ( n ) ⊂ Ω O is the map from (9). Then the graded linear map V g , [ ≤ → M [ ≤ ρ + φ : g → D ⊕ Ω O (in the same fashion as above) preserves thenon-negative products, and is the restriction of a homomorphism of graded vertexalgebras θ : V g , → M . (14)Associated to this homomorphism of vertex algebras is a homomorphism of Liealgebras L g → L ( M [ ≤ from the loop algebra L g := g ⊗ C (( s )) of the affine algebra g to the Lie algebra offormal modes of states in M [ ≤ k ∈ g is in the kernel of ρ , so we actually get a homomorphism LL ˚ g → L ( M [ ≤ LL ˚ g := ˚ g [ t, t − ] ⊗ C (( s )).The homomorphism θ has the property that the non-negative modes of states in V g , [1] ∼ = g stabilize M ( n ) and M ( n − ) inside M = M ( g ). Thus, we get an action of L + g := g ⊗ C [[ s ]], and in fact of L + L ˚ g := ˚ g [ t, t − ] ⊗ C [[ s ]], on M ( n ) and M ( n − ). SeeProposition 31. (This is in contrast to the obvious vertex-algebra homomorphism V g , → M , which sends J a,n [ − | i → P b,c ∈I f bac S bc,n [ − | i ; cf. Section 1.7.)Finally, in Theorem 32, we shall lift the homomorphism θ to a homomorphism V g , → M ⊗ π where π is the vacuum Fock module for a system of dim h free bosons (see Sec-tion 4.14). (For this homomorphism, the state k [ − | i is no longer in the kernel.)1.9. Let us conclude this introduction with some comments about these results.The homomorphism θ is not a free-field realization: we adjoined the copy of thevacuum Verma module V gl (˚ g )[ t,t − ] ⋊C D , in the definition of M , and so the vertex alge-bra M does not have mutually commuting creation operators and mutually commut-ing annihilation operators. This is apparent already at the level of the Lie algebrahomomorphism ρ : g → D : the Lie algebra D defined in (13) had generators S ab,n in addition to the mutually commuting coordinates X a,n and mutually commutingderivatives D a,n . There is some rough intuition that says that is to be expected. Inthis paper, the vertex algebra structure is always associated to the second coordinate, s , appearing in the double loop algebra LL ˚ g := ˚ g [ t, t − ] ⊗ C (( s )). One would like tobe able to say at the same time that X a,n and D b,n are modes in the t -coordinate ofstates “ X a | i ” and “ D b, − | i ”, and then that the S ab,n are merely the modes in the t -coordinate of a composite state, “ X a D b, − | i ”, roughly speaking. To make sense ofsuch statements, one would need a theory of vertex algebras on polydiscs (of complexdimension two, in our case), perhaps following [CG16; GW18; SWW19]. Since in thepresent paper we confine ourselves to the standard definition of vertex algebras, it isperhaps unsurprising that we need to include these S ab,n as generators in their ownright.Relatedly, whereas in finite types M ( n ) gets the structure of a module at the criticallevel over the central extension b g of the full loop algebra L g , here the subspace M ( n ) ⊂ M is stabilized by L + g , as in Proposition 31, but certainly not by all of L g .It might be interesting to study the L g module through M ( n ) inside M . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 11
To illustrate the structure of the image of θ , let us examine an example in type g = b sl . One finds θ ( J F, [ − | i ) = S EH, [ − | i − S HF, [ − | i (16) − γ E, [0] β H, [ − | i − γ E, − [0] β H, [ − | i + 2 γ H, [0] β F, [ − | i− γ E, − [ − | i + β F, [ − | i + 2 γ E, − [0] γ F, [0] β F, [ − | i − γ E, − [0] γ E, − [0] β E, − [ − | i− γ H, − [0] γ E, − [0] β H, − [ − | i + 2 γ H, − [0] γ H, − [0] β F, − [ − | i + 2 γ E, − [0] γ F, − [0] β F, − [ − | i − γ H, − [0] γ E, − [0] γ E, − [0] β E, − [ − | i + γ H, − [0] γ H, − [0] γ H, − [0] β F, − [ − | i + γ E, [0] γ E, [0] β E, [ − | i + 2 γ H, [0] γ E, [0] β H, [ − | i− γ H, [0] γ H, [0] β F, [ − | i + . . . . In the first line there are terms belonging to V gl (˚ g )[ t,t − ] ⋊C D , [1]; in the second, a finitesum of compensating quadratic terms. Then in the remaining lines is the sum of otherterms, which is infinite but with widening gap. The divergences in the 1st products,cf. Section 1.5, are removed because we set the 1st products of the states S ab,n [ − | i to zero. Note that zero, rather than some other finite level, was not a choice: since θ ( k [ − | i ) = 0 and κ (cid:0) k | d (cid:1) = 1, θ could not be a homomorphism from V g ,k at anynonzero level k . Correspondingly, the homomorphism in (15) is from the loop algebra L g , rather than any central extension thereof. It is tempting to say that the criticallevel is zero for untwisted affine algebras. We do not consider deforming to otherlevels in the present paper.One motivation for the present paper comes from Gaudin models. In the case of g of finite type there is a deep connection [FFR94; Fre05a] between the centre ofthe vacuum Verma module at the critical level, V g , − h ∨ , and the (large, commuta-tive) algebra of Gaudin Hamiltonians, sometimes called the Bethe algebra [MTV06;MTV09; Ryb18]. In the approach to the Bethe ansatz for Gaudin models describedin [FFR94], the Feigin-Frenkel homomorphism V g , − h ∨ → M ( n ) ⊗ π plays a key role.Gaudin models of affine type should provide a means of describing the spectra ofintegrals of motion of certain integrable quantum field theories: an idea pioneered in[FF11], and with further progress in [Vic18; FH18; LVY19; LVY20; Lac18; FJM17;Del+19a; Del+19b; Vic19; You20; Gai+20].1.10. The structure of this paper is as follows.In Section 2 we construct the homomorphism ρ : g → g Der O ( n ). Then in Section 3we introduce the homomorphism ρ : g → D . In Section 4 we recall basic facts about βγ -systems and vertex algebras, before going on to state the main results starting inSection 4.12. The proof of the main theorem, Theorem 28, is given in Section 5. Itfollows the strategy due to Feigin and Frenkel and discussed in detail in [Fre07, § In particular, we introduce the bc -ghost system and use it to define a subcomplex, the local complex , of the Chevalley-Eilenberg complex for (in our case) the double loopalgebra LL ˚ g . In Section 6 we give the proof of Theorem 32. Finally, in Appendix A wecompute explicitly the values of coefficients appearing in the images of the Chevalley-Serre generators of g under the homomorphism θ : see Proposition 30.2. Realization of g by differential operators on the big cell Loop realization.
We work over the complex numbers C . Let ˚ g be a finite-dimensional simple Lie algebra, and ˚ g [ t, t − ] the Lie algebra of Laurent polynomials,in a formal variable t , with coefficients in ˚ g . Let κ (cid:0) ·|· (cid:1) : ˚ g × ˚ g → C denote the non-degenerate symmetric ˚ g -invariant bilinear form on ˚ g , with the standard normalizationfrom [Kac90]. Let g ′ denote the central extension of ˚ g [ t, t − ] by a one dimensionalcentre C k , 0 → C k → g ′ → ˚ g [ t, t − ] → , whose commutation relations are given by [ k , · ] = 0 and[ a ⊗ f ( t ) , b ⊗ g ( t )] := [ a, b ] ⊗ f ( t ) g ( t ) − (res t f dg ) κ (cid:0) a | b (cid:1) k . If we write a n := a ⊗ t n for a ∈ ˚ g and n ∈ Z , the commutation relations take the form[ a m , b n ] = [ a, b ] n + m + mδ n + m, κ (cid:0) a | b (cid:1) k . Define the Lie algebra g := g ′ ⋊ C d , by declaring that d obeys [ d , k ] = 0 and [ d , a ⊗ f ( t )] = a ⊗ t∂ t f ( t ) for all a ∈ ˚ g and f ( t ) ∈ C [ t, t − ].The form κ (cid:0) ·|· (cid:1) extends uniquely to a non-degenerate invariant symmetric bilinearform on g , which we also write as κ (cid:0) ·|· (cid:1) , whose nonzero entries are given by κ (cid:0) a n | b m (cid:1) = κ (cid:0) a | b (cid:1) δ n + m, , κ (cid:0) k | d (cid:1) = κ (cid:0) d | k (cid:1) = 1for a, b ∈ ˚ g , n, m ∈ Z .2.2. Kac-Moody data.
Recall that the Lie algebra g is isomorphic to a Kac-Moodyalgebra g ( A ) with indecomposable Cartan matrix A = ( A ij ) i,j ∈ I of untwisted affinetype. Here I = { , , . . . , ℓ } is the set of the labels of the nodes of the Dynkin diagram.Let h ⊂ g ( A ) be the Cartan subalgebra and g = n − ⊕ h ⊕ n the Cartan decomposition, where n (resp. n − ) is generated by e i (resp. f i ), i ∈ I .These e i , f i are the Chevalley-Serre generators of the derived subalgebra g ′ := [ g , g ].By definition they obey[ h, e i ] = h α i , h i e i , [ h, f i ] = −h α i , h i f i , (17a)[ h, h ′ ] = 0 , [ e i , f j ] = ˇ α i δ ij , (17b)for any h, h ′ ∈ h , together with the Serre relations(ad e i ) − A ij e j = 0 , (ad f i ) − A ij f j = 0 . (17c) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 13
Here h· , ·i : h ∗ × h → C is the canonical pairing, α i (resp. ˇ α i ), i ∈ I , are the simpleroots (resp. simple coroots) of g . We have h α i , ˇ α j i = A ji .Let b := h ⊕ n and b − := h ⊕ n − .There exist unique collections of relatively prime positive integers { ˇ a i } i ∈ I and { a i } i ∈ I such that P j ∈ I A ij a j = 0 and P j ∈ I ˇ a i A ij = 0. Whenever g is untwisted(in fact more generally whenever g is not of type A k ) one has ˇ a = 1 and a = 1.The dual Coxeter and Coxeter numbers of g are given respectively by h ∨ = X i ∈ I ˇ a i , h = X i ∈ I a i = 1 + X i ∈ ˚ I a i . The central element k ∈ h and imaginary root δ ∈ h ∗ are given by k = X i ∈ I ˇ a i ˇ α i ∈ h , δ = X i ∈ I a i α i ∈ L h (18)Let ˚ I = I \{ } = { , . . . , ℓ } . The matrix ( A ij ) i,j ∈ ˚ I obtained by removing the zerothrow and column of A is the Cartan matrix of ˚ g . Let ˚ h := span C { ˇ α i } i ∈ ˚ I ⊂ g denoteits Cartan subalgebra and ˚ h ∗ = span C { α i } i ∈ ˚ I its dual.2.3. Root lattice and basis of root vectors.
Let Q := L i ∈ I Z α i denote the rootlattice of g . We have the decomposition of g into root spaces, g = M α ∈ Q g α , g α := { x ∈ g : [ h, x ] = x h α, h i for all h ∈ h } , and this is a Q -gradation of the the Lie algebra g . Let ∆ := { α ∈ Q : dim g α = 0 } be the set of roots of g . We write Q ≥ := L i ∈ I Z ≥ α i and Q > := Q ≥ \ { } . Thepositive roots of g are ∆ + := Q > ∩ ∆.Let ˚ Q := L i ∈ ˚ I Z α i denote the root lattice of ˚ g and ˚∆ ⊂ ˚ Q its set of roots. Let˚∆ + := ˚∆ ∩ ∆ + be the positive roots of ˚ g .Recall that the roots of g are given by∆ = { α + nδ : α ∈ ˚∆ , n ∈ Z } ∼ = ˚∆ × Z and the positive roots are given by∆ + = ˚∆ + ⊔ { α + nδ : α ∈ ˚∆ , n ∈ Z ≥ } ∼ = ˚∆ + ⊔ ˚∆ × Z ≥ Let { H i } i ∈ ˚ I ⊂ ˚ h be a basis of ˚ h . Choose root vectors E ± α ∈ ˚ g ± α for each α ∈ ˚∆ + ,normalized such that h α, [ E α , E − α ] i = 2. Let J α := E α , J i := H i , and also let I := (cid:16) ˚∆ \ { } (cid:17) ∪ ˚ I , so that { J a } a ∈I = { E ± α } α ∈ ˚∆ + ∪ { H i } i ∈ ˚ I is a Cartan-Weyl basis of ˚ g . Strictly speaking, in the examples in type b sl in the introduction, we wrote for example J E,n ratherthan J α ,n and J H,n rather than J ,n . Let B denote the basis of g given by B = { k , d } ∪ { J a,n } a ∈I ,n ∈ Z where J a,n := J a ⊗ t n . We fix a total ordering ≺ on B as follows. Pick any totalordering ≺ of the positive roots ˚∆ + of ˚ g such that α ≺ β whenever α − β ∈ Q > .and any total ordering ≺ of the Cartan generators { H i } of ˚ g . Then declare that forevery n ∈ Z , E − β,n ≺ E − α,n ≺ H i,n ≺ H j,n ≺ E α,n ≺ E β,n ≺ E − β,n +1 . . . (19a)whenever i ≺ j and α ≺ β , and finally that E − α, ≺ d ≺ k ≺ H i, . We can identify the Chevalley-Serre generators of g ′ as follows: e i = E α i , and f i = E − α i , , for i ∈ ˚ I , while e = E − δ + α , and f = E δ − α , − , where δ − α = X i ∈ ˚ I a i α i ∈ ˚∆ + (20)is the highest root of ˚ g .Let wgt : I × Z → Q ; ( wgt ( α, n ) = α + nδ α ∈ ˚∆ \ { } wgt ( i, n ) = nδ i ∈ ˚ I The pro-nilpotent pro-Lie algebra e n . Define e n := Y α ∈ ∆ + g α . An element of e n is a (possibly infinite) sum of the form P α ∈ ∆ + x α with x α ∈ g α foreach α ∈ ∆ + . (It lies in n ⊂ e n if and only if all but finitely many of the x α are zero.)The Lie bracket is well-defined on e n because, for a given positive root α ∈ ∆ + , thereare only finitely many positive roots β, γ ∈ ∆ + such that β + γ = α . Similarly, theLie bracket is well-defined on e b := h ⊕ e n , e g := n − ⊕ h ⊕ e n . Let ht( α ) denote the grade of a root α ∈ Q in the homogeneous Z -gradation of g ,i.e. ht( nδ + α ) = n for n ∈ Z and α ∈ ˚∆. Define n ≥ k := M α ∈ ∆ + ht( α ) ≥ k n α , k ≥ . These are Lie ideals in n and we have embeddings n ≥ k +1 ֒ → n ≥ k for each k ∈ Z ≥ .For each k , the quotient n (cid:14) n ≥ k is a nilpotent Lie algebra, and these nilpotent Liealgebras form an inverse system . . . ։ n (cid:14) n ≥ k +1 ։ n (cid:14) n ≥ k ։ . . . . The inverse limit of this inverse system is isomorphic to e n : e n ∼ = lim ←− k n (cid:14) n ≥ k . (21) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 15
Indeed, elements of this inverse limit are by definition (possibly infinite) sums of theform P α ∈ ∆ + x α , x α ∈ g α , which truncate to finite sums modulo n ≥ k for any k ; inother words, they are nothing but elements of e n .In this way, e n becomes a topological Lie algebra, with the linear topology in whicha base of the open neighbourhoods of 0 is given by the ideals e n ≥ k := Y α ∈ ∆ + ht( α ) ≥ k n α , k ≥ . (This topology is Hausdorff since T ∞ k =1 e n ≥ k = { } .)In fact, the Lie algebra e n gets the structure of a pro-nilpotent pro-Lie algebra ; seee.g. [Kum02, § § e n consists of the family F consisting of all Lie ideals a ⊂ e n such that a ⊃ e n ≥ k for some k , and the pro-topology is the linear topology in which these ideals form a base of the open neighbourhoodsof 0.2.5. The group U . For each k ≥
1, let exp (cid:0) n (cid:14) n ≥ k (cid:1) denote a copy of the vectorspace n / n ≥ k and let m exp( m ) ≡ e m be the map into this copy. We may endowthis copy with a group structure, given by the Baker-Campbell-Hausdorff formula,exp( x ) exp( y ) := exp (cid:18) x + y + 12 [ x, y ] + . . . (cid:19) , from which only finitely many terms contribute since n / n ≥ k is nilpotent. For all m, k with m ≥ k , there is a commutative diagram n / n ≥ m n / n ≥ k exp( n / n ≥ m ) exp( n / n ≥ k ) ∼ ∼ where the horizontal maps are the canonical projections. The inverse limit U := lim ←− k exp( n / n ≥ k ) (22)is then a group, and the diagram above defines an exponential map exp : ˆ n + ∼ −→ U .Recall that, by definition of the inverse limit, we have the commutative diagram U. . . exp( n / n ≥ ) exp( n / n ≥ ) exp( n / n ≥ ) π π π in which the maps are surjective group homomorphisms, and the group element g in(26) is equivalent to the sequence ( π i ( g )) ∞ i =1 of its truncations.In fact, to any pro-Lie algebra one can canonically associate a pro-algebraic group (or pro-group , for short). If the pro-Lie algebra is pro-nilpotent, then this pro-groupwill be pro-unipotent . For the definitions, see e.g. [Kum02, § U is thepro-unipotent pro-group algebra associated to the pro-nilpotent pro-Lie algebra e n , just as the quotients exp( n / n ≥ k ) are the unipotent affine algebraic groups associatedto the nilpotent Lie algebras n / n ≥ k .2.6. Polynomial functions on U . Now let us choose coordinates on U . Recall ourordered basis ( B , ≺ ) of g from (19). We get an ordered basis of n , B + := B ∩ n = { J α, } α ∈ ˚∆ + ∪ { J a,n } a ∈I ,n ∈ Z ≥ = { J a,n } ( a,n ) ∈ A , (23)where we introduced a notion for the index set, A := { ( α, } α ∈ ˚∆ + ∪ I × Z ≥ . (24)It will also be useful to introduce the set − A := { ( α, } α ∈ ˚∆ − ∪ I × Z ≤− . (25)Any element of g ∈ U can be uniquely written in the form g = −→ Y ( a,n ) ∈ A exp( x a,n J a,n ) (26)for some x a,n ∈ C , where we use −→ Q to denote the product with factors ordered sothat exp (cid:0) x b,m J b,m (cid:1) stands to the left of exp( x c,p J c,p ) if J b,m ≺ J c,p in our basis.For each ( a, n ) ∈ A let X a,n : U → C be the function on U such that X a,n ( g ) = x a,n .Then these are good coordinates on U . We get the C -algebra O ( n ) := C [ X a,n ] ( a,n ) ∈ A of polynomial functions on U . Inside O ( n ) we have for each k ∈ Z ≥ the subalgebra O ( n ) Let H ≥ k denote the left ideal in H generated by D ( a,n ) with n ≥ k . It consists ofthose polynomial differential operators that annihilate O ( n ) A, B ] := AB − BA makes H into a Q -graded Lie algebra. It has the Lie subalgebra Der O ( n ) consistingthe (finite) sums of the form X ( a,n ) ∈ A P a,n ( X ) D a,n , (30)where the coefficients P a,n ( X ) ∈ C [ X b,m ] ( b,m ) ∈ A are nonzero for only finitely ( a, n ) ∈ A , i.e. it the free O ( n )-module with O ( n )-basis consisting of the derivative operators D a,n , ( a, n ) ∈ A . It inherits the Q -grading of H . Lemma 1. The commutator bracket makes the completion e H into a topological Liealgebra. Proof. The associative algebra e H is certainly a Lie algebra with respect to the com-mutator. Let us check the commutator [ · , · ] : e H × e H → e H is continuous, where e H × e H has the product topology. The open neighbourhoods of zero in the product topologyinclude the subspaces e H ≥ k × e H ≥ ℓ for all k, ℓ ∈ Z ≥ . Suppose ( A n , B n ), n = 1 , , . . . ,is a sequence in e H × e H which converges to zero. Then, for any k , the terms in thissequence are eventually in e H ≥ k × e H ≥ k . It follows that, for any k , the terms of thesequence [ A n , B n ], n = 1 , , . . . , of commutators eventually lie in e H ≥ k . But the latterare a base of open neighbourhoods of zero in e H , so the sequence of commutators con-verges to zero. Since these are all linear topologies (i.e. vector addition is continuous)this is enough to show that the commutator is a continuous map, as required. (cid:3) Define g Der O ( n ) ⊂ e H to be the vector subspace of e H topologically generated bythe monomials of the form P a,n ( X ) D a,n . In other words, g Der O ( n ) consists of (nowpossibily infinite) sums of the form (30). Lemma 2. g Der O ( n ) is a topological Lie subalgebra of e H .Proof. This is really immediate, given the previous lemma: all that has to be checkedis that terms linear in D ’s close under the commutator, which is obvious. But let uswrite out the full argument nonetheless. Let P ( a,n ) ∈ A P a,n ( X ) D a,n and P ( a,n ) ∈ A Q a,n ( X ) D a,n be two elements of g Der O ( n ). Their commutator is equal to X ( b,m ) ∈ A X ( a,n ) ∈ A P a,n ( X ) [ D a,n , Q b,m ( X )] − X ( a,n ) ∈ A Q a,n ( X ) h D a,n , P b,m ( X ) i D b,m (31)For each ( b, m ), the polynomial Q b,m ( X ) ∈ O ( n ) lies in O ( n ) The action of e n on e g exponentiates to yield a well-defined action of U on e g from the right, e g × U → e g , given by B.e A = ∞ X k =0 ( − k k ! ad kA B = B + [ B, A ] + 12 [[ B, A ] , A ] + . . . for B ∈ e g , A ∈ e n . (cid:3) Infinitesimal transformations. Unlike the action of e n ⊂ e g on e g , the action ofthe full Lie algebra e g on itself does not exponentiate to an action of a group. Indeed,for A, B ∈ e g , the sum P ∞ k =1 1 k ! ad kA B may not be an element of e g ≥ m for any m ∈ Z ,which is to say it may not be a well-defined element of e g . (Consider for example A = H i, − and B = E α, .)Nonetheless, it is convenient for computations in what follows to be able to treatinfinitesimal transformations (by general elements of e g ) on the same footing as finitegroup transformations. That is, we would like to make sense of group elements of theform exp( ǫA ) = 1 + ǫA , working to first order in an (“infinitesimal”) parameter ǫ .To that end, let ǫ be a formal variable and let C ǫ denote the ring C ǫ := C [ ǫ ] (cid:14) ǫ C [ ǫ ] . For any Lie algebra p (over C ), we have the Lie algebra p ( C ǫ ) := C ǫ ⊗ C p . Considerthe Lie algebra e g ( C ǫ ) := C ǫ ⊗ C e g . We shall write its elements as A + ǫB with A, B ∈ e g . It has a nilpotent Lie ideal ǫ e g .The vector subspace f ǫ := ǫ e g ⊕ e n , (32)forms a Lie subalgebra f ǫ ⊂ e g ( C ǫ ) . It is another pro-nilpotent pro-Lie algebra. We get the corresponding pro-unipotentpro-group exp( f ǫ ). Let us write ad A ( B ) := [ A, B ]. Lemma 4. For all A ∈ e n and B ∈ e g , we have the following relations in exp( f ǫ ) , e A e ǫB = e ǫB ∞ Y k =1 e ǫ k ! ad kA ( B ) ! e A , e ǫB e A = e A ∞ Y k =1 e ǫ ( − kk ! ad kA ( B ) ! e ǫB , together with e ǫA e ǫB = e ǫB e ǫA , and e xA e ǫA = e ( x + ǫ ) A = e ǫA e xA for all x ∈ C .Proof. These follow from the Baker-Campbell-Hausdorff formula. (Note that theorder of the terms in the products Q ∞ k =1 is unimportant by virtue of the equalitye ǫA e ǫB = e ǫB e ǫA .) (cid:3) There is another vector-space decomposition of f ǫ , namely f ǫ = C ǫ b − ⊕ e n ( C ǫ ), andthis gives a useful way to factorize elements of the group exp( f ǫ ). Let U ( C ǫ ) :=exp( e n ( C ǫ )). Then the multiplication mapexp( ǫ b − ) × U ( C ǫ ) ∼ −→ exp( f ǫ ) (33)is a bijection.Let us introduce the right coset space U ( C ǫ ) := exp( ǫ b − ) / exp( f ǫ ) . The action from the right of the subgroup U ( C ǫ ) ⊂ exp( f ǫ ) is both transitive andfree, in view of (33), and we get a bijection U ( C ǫ ) ∼ −→ U ( C ǫ ); g exp( ǫ b − ) g. Let U := exp( ǫ b − ) U denote the orbit of exp( ǫ b − ) under the right action of the subgroup U ⊂ U ( C ǫ ). Thebijection above restricts to a bijection U ∼ −→ U ; g exp( ǫ b − ) g. By means of this bijection, we can regard X a,n , ( a, n ) ∈ A as coordinates on U , and O ( n ) as the ring of polynomial functions on U .For us, the motivation for these constructions is contained in the following lemma. Lemma 5. Let A ∈ e g . Then there exist polynomials n P b,mA ( X ) ∈ O ( n ) o ( b,m ) ∈ A (de-pending linearly on A ) such that e ǫ b − −→ Y ( b,m ) ∈ A e x b,m J b,m e ǫA = e ǫ b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫP b,mA ( x ) (cid:17) J b,m in U ( C ǫ ) , for every element −→ Q ( b,m ) ∈ A e x b,m J b,m of the group U ⊂ U ( C ǫ ) .If A ∈ g α and J b,m ∈ g β then P b,mA ( X ) ∈ O ( n ) α − β . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 21 Proof. This is true by construction, but let us go through the details to explain howin practice one computes these P b,mA . Recall that the group element g = −→ Y ( b,m ) ∈ A e x b,m J b,m ∈ U is defined by the sequence ( g i ) ∞ i =1 of its truncations, g i := π i ( g ) = −→ Y ( b,m ) ∈ A m
The linear map ρ : e g → g Der O ( n ) given by ρ : A X ( b,m ) ∈ A P b,mA ( X ) D b,m is a homomorphism of Lie algebras. It respects the Q -gradation, and is thereforecontinuous.Proof. Let C ǫ,η := C [ ǫ, η ] (cid:14) (cid:0) ǫ C [ ǫ, η ] ⊕ η C [ ǫ, η ] (cid:1) . Replacing C ǫ by C ǫ,ν in the defini-tions above, we get the Lie algebra e g ( C ǫ,η ) := C ǫ,η ⊗ e g and its Lie subalgebra f ǫ,η = C ǫ e g ⊕ η e g ⊕ e n = C ǫ b − ⊕ η b − ⊕ e n ( C ǫ,η ). Define U ( C ǫ,η ) := exp( ǫ b − ⊕ η b − ) / exp( f ǫ,η ).We have U ( C ǫ,η ) ∼ = U ( C ǫ,η ). We can identify U as defined above with the orbit exp( ǫ b − ⊕ η b − ) U of exp( ǫ b − ⊕ η b − ) under the right action of U ⊂ U ( C ǫ,η ). Let A, B ∈ e g and consider the cosete ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e x b,m J b,m e ǫA e ηB e − ǫA e − ηB . (35)On the one hand e ǫA e ηB e − ǫA e − ηB = e ǫη [ A,B ] and hence (35) is equal toe ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫηP b,m [ A,B ] ( x ) (cid:17) J b,m = e ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫη P γ P γ [ A,B ] ( x ) D γ .x b,m (cid:17) J b,m . On the other hand, we see thate ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e x b,m J b,m e ǫA e ηB = e ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫP b,mA ( x )+ ηP b,mB ( x + ǫP A ( x )) (cid:17) J b,m = e ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫP b,mA ( x )+ ηP b,mB ( x )+ ǫη P ( a,n ) (cid:16) P a,nA ( x )( D a,n P b,mB )( x ) (cid:17)(cid:17) J b,m and hence (35) is equal toe ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫη P ( a,n ) (cid:16) P a,nA ( x )( D a,n P b,mB )( x ) − P a,nB ( x )( D a,n P b,mA )( x ) (cid:17)(cid:17) J b,m = e ǫ b − ⊕ η b − −→ Y ( b,m ) ∈ A e( x b,m + ǫη [ P ( a,n ) P a,nA ( x ) D a,n , P γ P γB ( x ) D γ ] .x b,m ) J b,m (36)This is true for all x b,m . We conclude that X ( a,n ) P a,nA ( X ) D a,n , X ( b,m ) P b,mB ( X ) D b,m = X ( a,n ) P a,n [ A,B ] ( X ) D a,n , so that the map ρ is indeed a homomorphism of Lie algebras. The polynomial P b,mA be-longs to O ( n ) wgt ( b,m ) − wgt ( a,n ) whenever A ∈ g wgt( a,n ) , so ρ respects the Q -gradation.Hence it respects the principal Z -gradation and is therefore continuous. (cid:3) Remark . Just as in the case of g of finite type, there is also another realization ρ L : e n → g Der O ( n ) of e n , coming from the left action of U on itself. The image ρ L ( e n ) ⊂ g Der O ( n ) lies in the centralizer of ρ ( e n ), but not of ρ ( e g ). In the case of g offinite type, ρ L plays an important role in the definition of screening operators [FF92;FB04; Fre07].In what follows, we get more information about image of the homomorphism ρ . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 23 Bounded grade and widening gap. Let us say a collection of polynomials { P a,n ( X ) } ( a,n ) ∈ A in O ( n ) has widening gap if, for every K ≥ 1, we have P a,n ( X ) ∈ C h X b,m : m < n − K, b ∈ I i (37)(for all a ∈ I ) for all but finitely many n ∈ Z ≥ . Let us say that an element v = X ( a,n ) ∈ A P a,n ( X ) D a,n ∈ g Der O ( n )has widening gap if the collection of coefficient polynomials { P a,n ( X ) } ( a,n ) ∈ A haswidening gap. Let us say that v has bounded grade if there exists M ∈ Z such that P ( a,n ) ( X ) ∈ M M k = − M O ( n ) [ n + k ] for all ( a, n ) ∈ A , where we denote by O ( n ) = L k ∈ Z O ( n ) [ k ] the Z -gradation of O ( n )in which X ( a,n ) ∈ O ( n ) [ − n ] . Example 8. For any a, b, c ∈ I :(i) P n ∈ Z ≥ X a,n X b,n D c, n has bounded grade and widening gap,(ii) P n ∈ Z ≥ X a, D b,n has widening gap but not bounded grade,(iii) P n ∈ Z ≥ X a,n D b,n has bounded grade but not widening gap.Define Der O ( n ) to be the subset of g Der O ( n ) consisting of elements with boundedgrade and widening gap. It is a Lie subalgebra (and an O ( n )-submodule) of g Der O ( n ).(See Lemma 15 and its proof, below.)The following lemma says that any element of bounded grade “almost” has widen-ing gap. Lemma 9. Suppose P ( a,n ) ∈ A P a,n ( X ) D a,n ∈ g Der O ( n ) has bounded grade. For every K , we eventually (i.e. for all but finitely many n ) have P a,n ( X ) ∈ O ( n ) Suppose to the contrary that for some K there is, for every N , always an n > N such that P a,n ( X ) (for some a ∈ I ) has a nonzero monomial m ( X ) with X b,r X c,s as a factor, for some r, s ≥ n − K , b, c ∈ I . The grade of m ( X ) D a,n isbounded above by − r − s + n ≤ K − n < K − N . So for every N we find terms ofgrade less than 2 K − N . (cid:3) This applies in particular to the image ρ ( J a,n ) ∈ g Der O ( n ) of one of the generatorsof g ′ (which is certainly of bounded grade: it is in grade n , since homomorphism ρ : e g → g Der O ( n ) respects the Q -gradation and hence the homogeneous Z -gradation). It is natural to try to isolate the terms in ρ ( J a,n ) which fail to have widening gap.This is the content of Theorem 10 below. Let f abc denote the structure constants of˚ g in our basis { J a } a ∈I : [ J a , J b ] = X c ∈I f abc J c . For every ( a, n ) ∈ I × Z , define the element + J a,n ∈ g Der O ( n ) by + J a,n := X b,c ∈I f bac X m> max(1 ,n ) X b,m − n D c,m (38) Theorem 10. For all ( a, n ) ∈ I × Z , ρ ( J a,n ) − + J a,n ∈ Der O ( n ) , i.e. this difference has widening gap.Proof. For the entirety of this proof, pick and fix some ( a, n ) ∈ A . The element ρ ( J a,n ) − + J a,n is in grade n . The non-trivial thing we have to check is that it haswidening gap. We have ρ ( J a,n ) = X ( b,m ) ∈ A P b,m ( X ) D b,m , for some polynomials P b,m ( X ). As in Lemma 5, these polynomials are given by thedemand that, for any group element g = −→ Y ( b,m ) ∈ A e x b,m J b,m ∈ U ⊂ U ( C ǫ ) , we have, for some B ∈ b − , the following equality in the group exp( f ǫ ): −→ Y ( b,m ) ∈ A e x b,m J b,m e ǫJ a,n = e ǫB −→ Y ( b,m ) ∈ A e( x b,m + ǫP b,m ( x ) ) J b,m . (39)Given any positive integer K , we can always find a positive integer N large enoughthat 2( N − K ) + min( n, > N . Consider any m > N , so that we have2( m − K ) + min( n, > m. (40)To compute P b,m ( X ) in (39) it is enough to compute the truncation, π m +1 (cid:0) π p ( g )e ǫJ a,n (cid:1) , (41)where p = m + 1 − min( n, π p ( g ) as follows: π p ( g ) = g Definition of D . Let us introduce the polynomial algebra O := C [ X a,n ] ( a,n ) ∈I× Z and define the vector spacesDer O := M ( a,n ) ∈I× Z O D a,n , g Der O := Y ( a,n ) ∈I× Z O D a,n . Let us say a collection of polynomials { P a,n ( X ) } ( a,n ) ∈I× Z in O has widening gap if,for every K ≥ 1, we have, P a,n ( X ) ∈ C h X b,m : | m | < | n | − K, b ∈ I i . (45)(for all a ∈ I ) for all but finitely many n ∈ Z . Lemma 11. Equivalently, the polynomials { P a,n ( X ) } ( a,n ) ∈I× Z have widening gap ifand only if, for every K ≥ there is some B ( K ) such that P a,n ( X ) ∈ C h X b,m : | m | < max( | n | − K, B ( K )) , b ∈ I i (46) for all ( a, n ) ∈ I × Z .Proof. Suppose { P a,n ( X ) } ( a,n ) ∈I× Z have widening gap. Then for all K ≥ m ( K ) such that whenever | n | > m ( K ), P a,n has no factor X c,m with | m | > | n | − K . But that leaves only finitely many polynomials P a,n , and we can find B ( K ) such that none of them has any factor X c,m with | m | > B ( K ). Conversely,suppose { P a,n ( X ) } ( a,n ) ∈I× Z obeys the condition in the lemma. Then for all K ≥ | n | > B ( K ) for all but finitely many n , so { P a,n ( X ) } ( a,n ) ∈I× Z has widening gap. (cid:3) Corollary 12. Suppose we are given collections { P a,n } , { Q a,n } , { R a,n } , . . . , ofpolynomials with widening gap. Then X ( b,m ) ∈I× Z P b,m ∂Q a,n ∂X b,m , ( a, n ) ∈ I × Z , NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 27 is a collection of polynomials with widening gap (and, hence, so is X ( b,m ) , ( c,p ) ∈I× Z P b,m ∂Q c,p ∂X b,m ∂R a,n ∂X c,p , ( a, n ) ∈ I × Z , and so on). (cid:3) For future use, let us note also the following. Corollary 13. More generally, if { P a,ni } ( a,n ) ∈I× Z , i = 1 , . . . , N and { Q a,n } are col-lections of polynomials with widening gap then X ( b ,m ) ,..., ( b N ,m N ) ∈I× Z P b ,m . . . P b N ,m N N ∂ N Q a,n ∂X b ,m . . . ∂X b N ,m N , ( a, n ) ∈ I × Z , is a collection of polynomials with widening gap. (cid:3) Corollary 14. If { Q a,n } is a collection of polynomials with widening gap then thesum X ( a,n ) ∈I× Z ∂Q a,n ∂X a,n has only finitely many non-zero terms, and hence is a well-defined polynomial in O = C [ X a,n ] ( a,n ) ∈I× Z . (cid:3) Let Der O ⊂ g Der O denote the subspace consisting of elements of the form X ( a,n ) ∈I× Z P a,n ( X ) D a,n such that the polynomials { P a,n ( X ) } ( a,n ) ∈I× Z have widening gap. (We shall alsorefer to elements of Der O themselves as having widening gap.) Evidently, we haveDer O ⊂ Der O ⊂ g Der O and the Lie algebra Der O ( n ) from Section 2.12 embeds in Der O . Lemma 15. g Der O is a Lie algebra and Der O is a Lie subalgebra.Proof. The Lie bracket of the Lie algebra Der O , given by X ( a,n ) ∈I× Z P a,n ( X ) D a,n , X ( b,m ) ∈I× Z Q b,m ( X ) D b,m = X ( a,n ) , ( b,m ) ∈I× Z (cid:18) P b,m ∂Q a,n ∂X b,m − Q b,m ∂P a,n ∂X b,m (cid:19) D a,n , extends to a well-defined Lie bracket on g Der O : for each ( a, n ) on the right, thesum on ( b, m ) contains only finitely many non-zero terms since Q a,n and P a,n arepolynomials. Corollary 12 then implies Der O is a Lie subalgebra. (cid:3) Let { S ab,n } a,b ∈I ,n ∈ Z denote the generators of the loop algebra gl (˚ g )[ t, t − ], obeyingthe commutation relations (cid:2) S ab,n , S cd,m (cid:3) = δ cb S ad,n + m − δ ad S cb,n + m . (47)Let D denote the derivation element for the homogeneous gradation of this loopalgebra. By definition, it obeys [ D , S ab,n ] = n S ab,n . We have the homomorphism L ad : ˚ g [ t, t − ] → gl (˚ g )[ t, t − ] given by J a,n J a,n := X b,c ∈I f bac S bc,n , n ∈ Z . (48)and hence the homomorphism L ad : g → gl (˚ g )[ t, t − ] ⋊ C D , with k d D . There is an embedding ι : gl (˚ g )[ t, t − ] ⋊ C D ֒ → g Der O given by ι ( S bc,n ) := X m ∈ Z X b,m − n D c,m , ι ( D ) := X a ∈I ,m ∈ Z mX a,m D a,m . By means of this embedding, gl (˚ g )[ t, t − ] ⋊ C D acts (via the adjoint action) on g Der O . Lemma 16. This action stabilizes the Lie subalgebra Der O .Proof. Suppose P ( a,n ) ∈I× Z P a,n ( X ) D a,n ∈ Der O . Pick any K ≥ S bc,m of gl (˚ g )[ t, t − ] (it is clear that Der O is stable under D ). Since the { P a,n } havewidening gap, eventually P a,n has no factor X d,p with | p | > | n | − K − | m | . Therefore h S bc,m , P ( a,n ) ∈I× Z P a,n ( X ) D a,n i again has widening gap. (Note that for this argumentto work it is necessary that the gap is really widening , i.e. that the condition in (45)is that for every K eventually | m | < | n | − K .) (cid:3) Let us define D as the corresponding semi-direct product of Lie algebras, D := Der O ⋊ (cid:0) gl (˚ g )[ t, t − ] ⋊ C D (cid:1) . (49)3.2. Cartan involution. Now let τ : g Der O → g Der O be the involutive (i.e. τ = id)automorphism defined by τ ( X α,n ) = X − α, − n , τ ( X i,n ) = − X i, − n ,τ ( D α,n ) = D − α, − n , τ ( D i,n ) = − D i, − n , (50)for α ∈ ˚∆ \ { } , i ∈ ˚ I and n ∈ Z . Recall O ( n ) = C [ X a,n ] ( a,n ) ∈ A where A := { ( α, } α ∈ ˚∆ + ∪ I× Z ≥ indexes a basis of n . Let us introduce O ( n − ) := C [ X a,n ] ( a,n ) ∈− A ,where − A := { ( α, } α ∈ ˚∆ − ∪ I × Z ≤− . We define g Der O ( n − ) by obvious analogywith g Der O ( n ). Clearly, τ ( g Der O ( n )) = g Der O ( n − ) and τ ( g Der O ( n − )) = g Der O ( n ).Let σ : g → g be the Cartan involution of the affine Kac-Moody algebra g , defined by σ ( e i ) = f i , σ ( f i ) = e i , σ ( H ) = − H, (51) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 29 for i ∈ I and H ∈ h . One has σ ( J α,n ) = J − α, − n , σ ( J i,n ) = − J i, − n , for α ∈ ˚∆ \ { } , i ∈ ˚ I and n ∈ Z . We get a homomorphism of Lie algebras τ ◦ ρ ◦ σ : g → g Der O ( n − )and hence a homomorphism of Lie algebras ρ := ( ρ + τ ◦ ρ ◦ σ ) : g → g Der O ( n ) ⊕ g Der O ( n − ) ֒ → g Der O . (52)By construction, ρ has the equivariance property τ ◦ ρ = ρ ◦ σ. (53)Note that ι ( J a,n ) = ( ι ◦ L ad)( J a,n ) = X b,c ∈I f bac X m ∈ Z X b,m − n D c,m ∈ g Der O . Lemma 17. For all ( a, n ) ∈ I × Z , ρ ( J a,n ) − ι ( J a,n ) ∈ Der O . Proof. Recall the definition (38) of + J a,n . Let us give a name to the linear map f + : g → g Der O ( n ); J a,n → + J a,n . The key observation is that the difference( f + + τ ◦ f + ◦ σ )( J a,n ) − ι ( J a,n ) (54)is a finite sum of terms of the form X b,m D c,p , and thus an element of Der O ⊂ Der O .Together with Theorem 10, this implies the result. (cid:3) It follows from Lemma 17 that the image ρ ( g ) of g in g Der O lies in the embeddedcopy of D . In what follows we shall regard ρ as a homomorphism ρ : g → D (55)into the abstract copy of D we defined in (49).Observe that the difference ( f + + τ ◦ f + ◦ σ )( J a,n ) − ι ( J a,n ) in (54) genericallycontains terms which do not stabilize both O ( n ) and O ( n − ). (For example, termslike X a, D b, − or X a, − D b, .) For that reason, it is worth stressing the followingcrucial property (which is true by construction). Proposition 18. ρ ( g ) stabilizes O ( n ) and O ( n − ) in O . (cid:3) This is contrast to the obvious homomorphism g → g Der O sending J a,n ι ( J a,n )(and k d P n nX a,n D a,n ). So what we have shown is that it is possible to add,to each generator ι ( J a,n ), an infinite sum belonging to Der O of “correction terms”, insuch a way that the resulting action does stabilize O ( n ) and O ( n − ). It will be usefulto have a name for these “correction terms”. Let us write ρ ( J a,n ) − ι ( J a,n ) = X ( b,m ) ∈I× Z R b,ma,n ( X ) D b,m , (56) where the collection { R b,ma,n ( X ) } ( b,m ) ∈I× Z of polynomials has widening gap, for each( a, n ) ∈ I × Z . 4. Vertex algebras and main results Weyl algebra H . Let H denote the associative unital C -algebra obtained byquotienting the free associative unital C -algebra with generators β a,n [ N ] , γ a,n [ N ] , with N ∈ Z and ( a, n ) ∈ I × Z , by the two-sided ideal generated by the commutationrelations[ β a,n [ N ] , β b,m [ M ]] = 0 , [ β a,n [ N ] , γ b,m [ M ]] = δ b,ma,n δ N, − M , [ γ a,n [ N ] , γ b,m [ M ]] = 0 . As a vector space, H ∼ = C H − ⊗ H + , where H − ∼ = C [ γ a,n [ N ] , β a,n [ N − N ≤ a,n ) ∈I× Z (57)is the algebra of creation operators and H + ∼ = C [ γ a,n [ N ] , β a,n [ N − N> a,n ) ∈I× Z (58)is the algebra of annihilation operators . The fact that H + and H − are commutative(and that H is commutative modulo ) makes this an example of a system of freefields .4.2. Fock module M . Define M to be the induced H -module generated by a vector | i , the vacuum , annhilated by H + , i.e. β a,n [ M ] | i = 0 , M ∈ Z ≥ , γ a,n [ M ] | i = 0 , M ∈ Z ≥ (59)for all ( a, n ) ∈ I × Z , and on which | i = | i . Vectors in M are called states .There is an obvious Z × Q -gradation of H and of M in which β a,n [ N ] has grade( N, α ) and γ a,n [ N ] has grade ( N, − α ) whenever J a,n ∈ g α , and | i has grade (0 , depth gradation . Let M [ N ] denote the subspace ofdepth n , so that M = ∞ M n =0 M [ N ] . (60)Call the corresponding filtration, given by M [ ≤ m ] := m M n =0 M [ N ] , the depth filtration . Every v ∈ M belongs to M [ ≤ m ] for some m . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 31 The subspace M [ ≤ . Let us introduce the spaceΩ O := Hom O (Der O , O ) . It comes equipped with the derivative ∂ : O → Ω O defined by ( ∂f )( v ) = v ( f ). It isa free left O -module, with an O -basis consisting of basis vectors ∂X a,n , ( a, n ) ∈ A ,which obey ∂X a,n ( D b,m ) = δ a,nb,m : Ω O ∼ = O L ( a,n ) ∈ A O ∂X a,n . There is an action ofDer O on Ω O , given by v. ( f ∂g ) = ( v.f ) ∂g + f ∂ ( v.g ). This action extends to a well-defined action of g Der O . The space Ω O is graded by the root lattice Q , ∂ has grade0, and these structures respect the Q -grading.The subspace M [0] consists of states of the form P ( γ [0]) | i where P ( X ) ∈ O .Meanwhile the subspace M [1] consists of states of the form X ( a,n ) ∈I× Z P a,n ( γ [0]) β a,n [ − | i + X ( a,n ) ∈I× Z Q a,n ( γ [0]) γ a,n [ − | i for polynomials P a,n ( X ) , Q a,n ( X ) ∈ O , ( a, n ) ∈ I × Z . Thus, there are isomorphismsof vector spaces, M [0] ∼ = O , (61) M [1] ∼ = Ω O ⊕ Der O , (62)the latter given by identifying β a,n [ − | i with D a,n and γ a,n [ − | i with ∂X a,n . Remark . The subspace M [ ≤ M we are about to recall, is an example of a 1 -truncated vertex algebra .This in turn makes M [1] into a vertex O -algebroid . See [GMS04; Bre02; GMS01].4.4. Vertex algebra structure. For every N ∈ Z , there is a linear map M → End M ; A A ( N ) sending any given state A to its N th mode , A ( N ) ∈ End M , and these modes can bearranged in a formal series, the field Y ( A, x ) := X n ∈ Z A ( N ) x − N − ∈ Hom ( M , M (( x ))) . The state-field map Y ( · , x ) : M → Hom ( M , M (( x ))) obeys a collection of axioms thatmake M into a vertex algebra ; see e.g. [LL04; Kac98; FB04]. It is defined as follows.First, β a,n ( x ) := Y ( β a,n [ − | i , x ) = X N ∈ Z β a,n [ N ] x − N − , γ a,n ( x ) := Y ( γ a,n [0] | i , x ) = X N ∈ Z γ a,n [ N ] x − N . Next, let us write f ( N ) ( u ) := N ! ∂ N ∂u N f ( u ). Then Y ( β b,m [ − M ] | i , u ) = β ( M − b,m ( u ) and Y ( γ a,n [ − N ] | i , u ) = γ a,n ( N ) ( u ) and more generally if A = γ a ,n [ − N ] . . . γ a r ,n r [ − N r ] β b ,m [ − M ] . . . β b s ,m s [ − M s ] | i , (63) then A ( u ) := Y ( A, u ) = : γ a ,n ( N ) ( u ) . . . γ a r ,n r ( N r ) ( u ) β ( M − b ,m ( u ) . . . β ( M s − b s ,m s ( u ): . (64)Here : . . . : denotes the normal-ordered product of fields, which is defined in general asfollows. For any states A, B ∈ M ,: Y ( A, u ) Y ( B, v ): := X M< A ( M ) u − M − ! Y ( B, v ) + Y ( B, v ) X M ≥ A ( M ) u − M − ! , (65)and : Y ( A, u ) Y ( B, u ): is the specialization at u = v . For more than two fields, the nor-mal ordered product is understood to be right-associative, i.e. : Y ( A, u ) Y ( B, v ) Y ( C, w ): =: Y ( A, u ) (cid:0) : Y ( B, v ) Y ( C, w ): (cid:1) : and so on.For the βγ -system (and for systems of free fields more generally) there is a simplerdefinition of the normal ordered product for monomial states like A in (63). Givenany monomial m ∈ H , one defines : m : ∈ H to be the monomial with the same factorsas m but ordered so that all annihilation operators stand to the right of all creationoperators (see (57) and (58)). Then: γ a ,n ( N ) ( u ) . . . γ a r ,n r ( N r ) ( u r ) β ( M − b ,m ( v ) . . . β ( M s − b s ,m s ( v s ):is defined by normal-ordering the monomials in the series, term by term.The depth gradation, (60), makes M into a graded vertex algebra: the vacuum | i is in grade zero; for any states A ∈ M [ R ], B ∈ M [ S ] and any N ∈ Z , A ( N ) B ∈ M [ R + S − N − translation operator T ∈ End M of the vertex algebra M , defined by T A := A ( − | i . It acts on monomials in the generators of H as a derivation, accordingto [ T, β a,n [ − N ]] = − N β a,n [ N − , [ T, γ a,n [ − N ]] = − ( N − γ a,n [ N − , and by definition T | i = 0.4.5. The OPE. One has the commutator formula for modes of states: (cid:2) A ( M ) , B ( N ) (cid:3) = X K ≥ (cid:20) MK (cid:21)(cid:0) A ( K ) B (cid:1) ( M + N − K ) . (66)Here, for all M ∈ Z , (cid:2) MK (cid:3) := M ( M − ... ( M − K +1) K ! for K = 0 and (cid:2) M (cid:3) := 0. Fromthis and the definition of the normal ordered product, (65), one obtains the operatorproduct expansion (OPE) : Y ( A, u ) Y ( B, v ) = X N ≥ Y ( A ( N ) B, v )( u − v ) N +1 + : Y ( A, u ) Y ( B, v ):as an equality in Hom( M , M (( u ))(( v ))). Here ( u − v ) − N − is understood to be ex-panded in positive powers of v/u . Thus, computing the singular terms in the OPE isthe same things as computing the non-negative vertex algebra products. NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 33 The Wick formula. For the βγ -system (and for systems of free fields moregenerally) one has the usual Wick formula for OPEs of monomial states. We takethe statement from [FB04, § A ( u ) and B ( u ) be normal-ordered monomials in β a,n ( u ), γ a,n ( u ), ( a, n ) ∈ I × Z ,and their derivatives as in (64). A single pairing between A ( u ) and B ( v ) is a choice,for some fixed ( a, n ) ∈ I × Z , ofi) one factor β ( N ) a,n ( u ) from A ( u ) and one factor γ a,n ( M ) ( v ) from B ( v )– to such a pairing we associate the function ( − N (cid:2) N + MN (cid:3) u − v ) N + M +1 – orii) one factor γ a,n ( N ) ( u ) from A ( u ) and one factor β a,n ( M ) ( v ) from B ( v )– to such a pairing we associate the function ( − M +1 (cid:2) N + MN (cid:3) u − v ) N + M +1 .A pairing is a disjoint union of zero or more single pairings. To a pairing P we asso-ciate the function f P ( u, v ) obtained by taking the product the functions associatedto each constituent single pairing (or 1, for the empty pairing). Given a pairing P let ( A ( u ) B ( v )) P denote the product A ( u ) B ( v ) but with all factors belonging to thepairing removed (to leave 1, in the special case that there are no factors left). The contraction : A ( u ) B ( v ): P associated to a pairing P is by definition: A ( u ) B ( v ): P := f P ( u, v ) :( A ( u ) B ( v )) P : . Lemma 20 (Wick formula) . The product A ( u ) B ( v ) is equal to the sum of contrac-tions : A ( u ) B ( v ): P over all pairings P between the monomials A and B , counted withmultiplicity (and including the empty one). (cid:3) One can then compute the OPE by Taylor-expanding the fields β a,n ( u ) and γ a,n ( u )about u = v .4.7. Completion e M of M . Having in mind the completion g Der O of Der O intro-duced in Section 3 (and see Section 2.9), one sees that M [ ≤ 1] is not quite big enoughfor our purposes. Let us introduce a completion of e M of M in such a way as to pre-serve the depth filtration. To that end, we start by completing each filtered subspace M [ ≤ m ], as follows.Let H −≥ k denote the two-sided ideal in the commutative algebra H − of creationoperators, cf. (57), generated by { β a,n [ N ] : a ∈ I , | n | ≥ k, N ∈ Z } . Let I [ ≤ m ] k denote the subspace of M [ ≤ m ] given by I [ ≤ m ] k := M [ ≤ m ] ∩ (cid:16) H −≥ k | i (cid:17) . Thus, I [ ≤ m ] k is the subspace of M [ ≤ m ] spanned by monomials in the creation oper-ators that have some factor β a,n [ N ] with | n | ≥ k . We have I [ ≤ m ] ⊃ I [ ≤ m ] ⊃ I [ ≤ m ] ⊃ . . . and T ∞ i =0 I [ ≤ m ] i = { } and we define e M [ ≤ m ] := lim ←− k M [ ≤ m ] (cid:14) I [ ≤ m ] k . These completed subspaces e M [ ≤ m ] form a directed system, e M [ ≤ ⊂ e M [ ≤ ⊂ e M [ ≤ . . . ,and we define e M to be the direct limit e M := lim −→ m e M [ ≤ m ] . In other words, each element of e M is by definition an element of e M [ ≤ m ] for somesufficiently large m (depending on the element) with two elements of e M consideredequal if they are equal in e M [ ≤ m ] for some m .Explicitly, the sum X a ,...,a p ∈I n ,...,n p ∈ Z P a ,n ( γ ) . . . P a p ,n p ( γ ) β a ,n [ − N ] . . . β a p ,n p [ − N p ] | i , p ∈ Z ≥ , (67)belongs to e M if, for each i ∈ { , . . . , p } , N i ∈ Z ≥ and there is a bound on the depthof the polynomials P a i ,n i ( γ ) ∈ C [ γ b,m [ − M ]] b ∈I ,m ∈ Z ,M ∈ Z ≥ as ( a i , n i ) ranges over I × Z . Elements of e M are finite linear combinations of suchsums.The vertex algebra structure on M does not extend to a well-defined vertex algebrastructure on e M . For example in e M we have the state S := X ( a,n ) ∈I× Z γ a,n [0] β a,n [ − | i ∈ e M . If g were finite-dimensional, this would be a conformal vector. But here it is an infinitesum, and when we attempt to compute, for example, the action of the would-be firstmode of S , S (1) = X N ≥ X ( a,n ) ∈I× Z ( β a,n [ − N ] γ a,n [ N + 1] + γ a,n [ − N ] β a,n [ N + 1])on the state S , we encounter a double contraction yielding the ill-defined sum X ( a,n ) ∈I× Z β a,n [0] γ a,n [1] X ( b,m ) ∈I× Z γ b,m [0] β b,m [ − | i = X ( a,n ) ∈I× Z X ( b,m ) ∈I× Z δ b,ma,n δ a,nb,m = X ( a,n ) ∈I× Z . The vertex algebra M . Let M ⊂ e M denote the subspace of e M spanned bystates of the form (67) such that, for each i = 1 , . . . , p , the collection of polynomials { P a i ,n i } ( a i ,n i ) ∈I× Z has widening gap in same sense as in Section 3.1, i.e. for every K ≥ 1, we have P a i ,n i ( γ ) ∈ C h γ b,m [ − M ] : | m | < | n i | − K, b ∈ I , M ∈ Z ≥ i , (68)(for all a i ∈ I ) for all but finitely many n i ∈ Z . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 35 Evidently M ( M ( e M , and e M [1] ∼ = Ω O ⊕ g Der O M [1] ∼ = Ω O ⊕ Der O (and M [0] = M [0] = e M [0] ∼ = O ). Lemma 21. The vertex algebra structure on M extends uniquely to a well-definedvertex algebra structure on M .Proof. The non-trivial thing we have to check is the closure of the OPE, i.e. that theOPE of any two fields of states in M has coefficients that are fields of states in M .This follows from the Wick formula, Section 4.6. Consider a typical contraction: X ( a,n ) ∈I× Z : . . . P a,n ( γ )( z ) . . . β a,n ( z ): X ( b,m ) ∈I× Z : . . . Q b,m ( γ )( w ) . . . β b,m ( w ): . The resulting sum has, associated to the factor β b,m , the coefficient polynomial (wesuppress the mode numbers [ M ], which play no essential role here) X ( a,n ) ∈I× Z P a,n ∂Q b,m ∂ γ a,n . As in Corollary 12, this collection of polynomials again has widening gap, and thesame logic extends to longer chains of contractions. More generally, by Corollary 13the same is true in any situation in which the contractions form an acyclic directedgraph. (We think of a contraction of β a,n into P b,m ( γ ) as a directed edge ( a, n ) → ( b, m ).) And, in view of Corollary 14, any cycle in the directed graph of the contrac-tions merely gives rise to an overall factor belonging to C [ γ c,p [ − M ]] c ∈I ,p ∈ Z ,M ∈ Z ≥ .For example, in X ( a,n ) ∈I× Z : . . . P a,n ( γ )( z ) . . . β a,n ( z ): X ( b,m ) ∈I× Z : . . . Q b,m ( γ )( w ) . . . β b,m ( w ): , only finitely many n, m can give nonzero summands. (cid:3) Remark . As a vertex algebra, M is generated by the fields ( γ a,n ( z )) ( a,n ) ∈I× Z to-gether with all the fields X ( a,n ) ∈I× Z : P a,n ( γ ( z )) β a,n ( z ):as (cid:16) P a,n ( γ ( z )) ∈ C [ γ b,m ( N ) ( z ) : ( b, m ) ∈ I × Z , N ≥ (cid:17) ( a,n ) ∈I× Z runs over all collections of polynomials that have widening gap and bounded depth. The vertex algebras M and M . Let L denote the Lie algebra with generators { S ab,n [ N ] } a,b ∈I ,n ∈ Z ,N ∈ Z and { D [ N ] } N ∈ Z subject to the relations (cid:2) S ab,n [ N ] , S cd,m [ M ] (cid:3) = δ cb S ad,n + m [ N + M ] − δ ad S cb,n + m [ N + M ] . (cid:2) D [ N ] , S ab,n [ M ] (cid:3) = n S ab,n [ N + M ] . In other words, L is loop algebra of gl (˚ g )[ t, t − ] ⋊ C D . We can take the semi-directproduct L ⋉ H , where the action of L on H is given by (cid:2) S ab,n [ N ] , β c,m [ M ] (cid:3) = − δ ac β b,m + n [ N + M ] , [ D [ N ] , β c,m [ M ]] = − m β c,m [ N + M ] (cid:2) S ab,n [ N ] , γ c,m [ M ] (cid:3) = δ cb γ a,m − n [ N + M ] , [ D [ N ] , γ c,m [ M ]] = m γ c,m [ N + M ] . Let M denote the module over L ⋉ H induced from a vector | i obeying the conditions(59) together with S ab,n [ N ] | i = 0 , D [ N ] | i = 0 , N ∈ Z ≥ . Thus as a vector space M is the tensor product of the Fock module M over H andthe vacuum Verma module (at level 0) V gl (˚ g )[ t,t − ] ⋊C D , over L : M ∼ = C M ⊗ V gl (˚ g )[ t,t − ] ⋊C D , . This module M is a vertex algebra. The state-field map is given by the formulasin Section 4.4 together with S ab,n ( x ) := Y (cid:0) S ab,n [ − | i , x (cid:1) = X N ∈ Z S ab,n [ N ] x − N − , D ( x ) := Y ( D [ − | i , x ) = X N ∈ Z D [ N ] x − N − . (69)As a vertex algebra, M is generated (in the sense of the Strong ReconstructionTheorem [Fre+95],[Kac98, § § β a,n ( z ), γ a,n ( z ), S ab,n ( z ) and D ( z ), subject to the OPEs β a,n ( z ) γ b,m ( w ) = δ b,ma,n z − w + . . . S ab,n ( z ) γ c,m ( w ) = δ cb γ a,m − n ( w ) z − w + . . . S ab,n ( z ) β c,m ( w ) = − δ ac β b,m + n ( w ) z − w + . . . , D ( z ) γ c,m ( w ) = m γ a,m ( w ) z − w + . . . D ( z ) β c,m ( w ) = − m β b,m ( w ) z − w + . . . , S ab,n ( z ) S cd,m ( w ) = δ cb S ad,n + m ( w ) − δ ad S cb,n + m ( w ) z − w + . . . , D ( z ) S ab,n ( w ) = n S ab,n ( w ) z − w + . . . . (with the others being trivial). NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 37 We can enlarge the tensor factor M to M as in Section 4.8 above. Let M denotethe resulting vector space, M ∼ = C M ⊗ V gl (˚ g )[ t,t − ] ⋊C D , . Lemma 23. The vertex algebra structure on M extends uniquely to a well-definedvertex algebra structure on M . (cid:3) Proof. Given Lemma 21, the remaining thing to check is that the OPE of S ab,n ( z ) withthe field of any state in M has coefficients which are again the fields of well-definedstates in M . This is true by the same reasoning as in Lemma 16. (cid:3) We have the depth gradation on M , in which | i has grade 0 and γ a,n [ N ], β a,n [ N ], S ab,n [ N ] and D [ N ] contribute grade − N .We continue to identify M [0] = M [0] ∼ = O by identifying P ( X ) ∈ O with the state P ( γ [0]) | i in M [0], and to identify Ω O withthe subspace of M [1] consisting of states of the form X a,n ∈I× Z P a,n ( γ [0]) γ a,n [ − | i ∈ M [1] ⊂ M [1] , by identifying this state with P a,n ∈I× Z P a,n ( X ) ∂X a,n ∈ Ω O . Note that ∂ : O → Ω O then corresponds to the vertex algebra translation operator T .We also have the obvious injective linear map : D ֒ → M [1] (70)which maps the element X a,n ∈I× Z P a,n ( X ) D a,n + X a,b ∈I ,n ∈ Z p b,na S ab,n ∈ D to the state X a,n ∈I× Z P a,n ( γ [0]) β a,n [ − | i + X a,b ∈I ,n ∈ Z p b,na S ab,n [ − | i ∈ M [1](The first sum is possibly infinite but must obey the condition (45), cf. (68). Bydefinition of gl (˚ g )[ t, t − ] the second sum must have only finitely many nonzero sum-mands.)In this way, M [0] ∼ = O M [1] ∼ = Ω O ⊕ ( D ) . (71)Note that while we identify O and Ω O with their images in M [ ≤ for the embedding of D . Local Lie algebras. Given any vertex algebra V one has the Lie algebra L ( V )of formal modes of states in V . (See [FB04, § L ( V ) := V ⊗ C (( t )) (cid:14) Im( T ⊗ ⊗ ∂ t ) . (72)It is generated by formal modes A [ M ], A ∈ V , M ∈ Z , modulo the relations( T A )[ M ] = − M A [ M − The Lie bracket is given by the same commutator formula(66) obeyed by the modes A ( M ) (living in End( V )), i.e. (cid:2) A [ M ] , B [ N ] (cid:3) = X K ≥ (cid:20) MK (cid:21)(cid:0) A ( K ) B (cid:1) [ M + N − K ] . (73)Since this formula involves only the non-negative products (i.e., only the singularterms of the OPE), we get a Lie subalgebra L ( L ) associated to any subspace L ⊂ V closed under translation T and all the non-negative products. (Such a subspace L iscalled a vertex Lie subalgebra of V [Pri99; DLM02].) In fact, we don’t need to insiston closure under translation: let L be any subspace closed under the non-negativeproducts; then P ∞ n =0 T n L ⊂ V is also closed under all the non-negative products ,and under translation.Thus, given any vector subspace L ⊂ V , let L ( L ) := ∞ X n =0 T n L ! ⊗ C (( t )) (cid:14) Im( T ⊗ ⊗ ∂ t );then (73) defines the structure of a Lie algebra on L ( L ) whenever L closes underall the non-negative products.Such Lie algebras L ( L ) are called local Lie algebras.For any such Lie algebra L ( L ) we have (as one sees from the commutator formula)subalgebras L ≥ ( L ) and L ( L ) consisting of, respectively, the non-negative and zeromodes of states in L .4.11. The extension of L D by L ( O ⊕ Ω O ) . In our case we have the Lie algebra L ( M ) associated to the vertex algebra M . The following is clear on inspection. Lemma 24. (i) The subspace M [ ≤ is closed under all the non-negative products.(ii) The subspace O ⊕ Ω O is closed under all the non-negative products and moreoverthese non-negative products all vanish.(iii) O ⊕ Ω O ⊂ M [ ≤ is an ideal, in the sense that A ( M ) B ∈ O ⊕ Ω O for all A ∈O ⊕ Ω O , B ∈ M [ ≤ and M ≥ . (cid:3) As a result, we have the short exact sequence of Lie algebras0 → L ( O ⊕ Ω O ) → L ( M [ ≤ → L ( M [ ≤ (cid:14) L ( O ⊕ Ω O ) → Strictly speaking, it is spanned by linear combinations of the form P N f N A [ N ], f N ∈ C , A ∈ V , ofthese formal modes. Indeed, we have ( T A ) ( n ) B = − nA ( n − B (note the right-hand side is zero when n = 0) and A ( n ) T B = T ( A ( n ) B ) − ( T A ) ( n ) B . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 39 where L ( O ⊕ Ω O ) is commutative. Remark . We can describe the commutative Lie algebra L ( O ⊕ Ω O ) more explicitly:since ∂ = T : O → Ω O is injective and has kernel C | i , we have that L ( O ⊕ Ω O ) ∼ = L (Ω O ) ⊕ C ∼ = L Ω O ⊕ C where L Ω O := Ω O ⊗ C (( t )) is the loop algebra of Ω O , the latter regarded as acommutative Lie algebra, and where := | i [ − 1] is the only nonzero mode of thestate | i .Now let L D := D ⊗ C (( t )) denote the loop algebra of D . Lemma 26. There is an isomorphism L D ∼ = L ( M [ ≤ (cid:14) L ( O ⊕ Ω O ) of Lie algebras, so we have the exact sequence of Lie algebras → L ( O ⊕ Ω O ) → L ( M [ ≤ → L D → . (74) Proof. Checking the definitions, one first sees that at the level of vector spaces wehave L ( M [ ≤ ∼ = C L ( O ⊕ Ω O ) ⊕ L ( ( D ( g )))and L ( ( D ( g ))) ∼ = C L D . The fact that L D ∼ = L ( M [ ≤ (cid:14) L ( O ⊕ Ω O ) is also an isomorphism of Lie algebras isa consequence of the following observation. (cid:3) Lemma 27. For any X, Y ∈ D , we have ( X ) (0) ( Y ) ≡ ([ X, Y ]) mod Ω O , ( X ) (0) ( Y ) ≡ O . Proof. Let us compute the OPE of two states in ( D ). Throughout this proof we usesummation convention over repeated pairs of indices not only in I but also in Z . Let A = P a,n ( γ [0]) β a,n [ − | i + p b,na S ab,n [ − | i ,B = Q a,n ( γ [0]) β a,n [ − | i + q b,na S ab,n [ − | i . Then we find Y ( A, z ) Y ( B, w ) = : Y ( A, z ) Y ( B, z ): (75)+ 1 z − w : Y ( P a,n ( γ ) | i , z ) Y (cid:18) ∂Q b,m ∂ γ a,n [0] β b,m [ − | i , w (cid:19) :+ 1 z − w p b,na : Y (cid:18) ∂Q c,m ∂ γ b,p [0] γ a,p − n [0] β c,m [ − | i , w (cid:19) :+ 1 z − w p b,na : Y ( Q a,m ( γ ) β b,m + n [ − | i , w ): − z − w : Y (cid:18) ∂P a,n ∂ γ b,m [0] β a,n [ − | i , z (cid:19) Y (cid:16) Q b,m ( γ ) | i , w (cid:17) : − z − w q b,na : Y (cid:18) ∂P c,m ∂ γ b,p [0] γ a,p − n [0] β c,m [ − | i , z (cid:19) : − z − w q b,na : Y ( P a,m ( γ ) β b,m + n [ − | i , z ):+ 1 z − w p b,na q d,mb (cid:0) δ cb S ad,n + m ( w ) − δ ad S cb,n + m ( w ) (cid:1) − z − w ) : Y (cid:18) ∂P a,n ∂ γ b,m [0] | i , z (cid:19) Y (cid:18) ∂Q b,m ∂ γ a,n [0] | i , w (cid:19) : . All but the final line are modes of states in ( D ) and involve only single contractions,and we recognise the terms as correctly reproducing the Lie bracket in D . (cid:3) The final line in (75) involves a double contraction, and defines the extension of L D by L ( O ⊕ Ω O ) in Lemma 26. Namely, we see that[ A [ K ] , B [ L ]] = [ A, B ][ K + L ] + ω ( A [ K ] , B [ L ]) , with ω ( A [ K ] , B [ L ]):= − K (cid:18) ∂P a,n ∂ γ b,m [0] ∂Q b,m ∂ γ a,n [0] | i (cid:19) [ K + L − − (cid:18)(cid:20) T, ∂P a,n ∂ γ b,m [0] (cid:21) ∂Q b,m ∂ γ a,n [0] | i (cid:19) [ K + L ](76a)where we continue to employ summation convention over repeated indices. Let usstress that these implicit sums on b, m and a, n have only finitely many nonzero terms,by virtue of our definition of M ; cf. (68).The formula above defines ω on the subalgebra L (cid:0) gl (˚ g )[ t, t − ] (cid:1) ⊂ L D . When one ofthe arguments to ω is a mode D [ K ] of D , there are obviously no double contractions,so ω ( D [ K ] , · ) = 0 . (76b)This map ω defines a cocycle[ ω ] ∈ H ( L D , L ( O ⊕ Ω O )) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 41 in the usual Chevalley-Eilenberg cohomology of L D with coefficients in L ( O ⊕ Ω O ).(It is easy to see that this cocycle is non-trivial, as in the case of g of finite type in[Fre07, § Main result. The homomorphism ρ : g → D gives rise to a homomorphism ofthe corresponding loop algebras e ρ : L g → L D , (77)given by e ρ ( A [ N ]) := ρ ( A )[ N ] , A ∈ g , N ∈ Z . Pulling back the cocycle ω ∈ H ( L D , L ( O ⊕ Ω O )) from (76) by this homomorphism e ρ , we obtain a cocycle e ρ ∗ ( ω ) ∈ H ( L g , L ( O ⊕ Ω O )) (78)of L g with coefficients in L ( O ⊕ Ω O ). It defines an extension f L g of L g by L ( O ⊕ Ω O ),0 → L ( O ⊕ Ω O ) → f L g → L g → . We can now state the main result of this paper. Theorem 28. The cocycle e ρ ∗ ( ω ) is trivial, so this sequence splits.Specifically, there is a linear map φ : g → Ω O of Q -grade 0 and obeying φ ( h ) = 0 , φ ( n ∓ ) ⊂ Ω O ( n ± ) and φ ◦ σ = τ ◦ φ , such that the map ◦ ρ + φ : g → ( D ) ⊕ Ω O ∼ = M [1] gives rise to a homomorphismof Lie algebras, L g → L ( M [ ≤ A [ N ] (cid:0) ( ◦ ρ + φ )( A ) (cid:1) [ N ] , A ∈ g , N ∈ Z . Proof. The proof occupies Section 5 below. (cid:3) To state this result in a more concrete form, we note the map φ is given by J a,n X ( b,m ) ∈I× Z Q a,n ; b,m ( X ) ∂X b,m (and k d 0) for certain polynomials Q a,n ; b,m ( X ) ∈ O . We introduce also thegenerating series of the generators of L g , k ( z ) := X N k [ N ] z − N − , d ( z ) := X N d [ N ] z − N − , J a,n ( z ) := X N J a,n [ N ] z − N − , and let ( J a,n )( z ) = P N ∈ Z ( J a,n )[ N ] z − N − with J a,n as in (48). Recall the polyno-mials R b,ma,n ( X ) ∈ O from (56). Theorem 29. There is a homomorphism of Z × Q -graded Lie algebras L g → L ( M [ ≤ given by k ( z ) d ( z ) ( D )( z ) J a,n ( z ) ( J a,n )( z ) + X ( b,m ) ∈I× Z Y ( R b,ma,n ( γ [0]) β b,m [ − | i , z )+ X ( b,m ) ∈I× Z Y ( Q a,n ; b,m ( γ [0]) γ b,m [ − | i , z ) . Equivalently, there is a homomorphism of Z ≥ × Q -graded vertex algebras θ : V g , → M given by k [ − | i 7→ d [ − | i 7→ ( D )[ − | i J a,n [ − | i 7→ ( J a,n )[ − | i + X ( b,m ) ∈I× Z R b,ma,n ( γ [0]) β b,m [ − | i + X ( b,m ) ∈I× Z Q a,n ; b,m ( γ [0]) γ b,m [ − | i . Proof. The first part is a restatement of Theorem 28 in more concrete terms. For theequivalence of the statement about vertex algebras, see [Fre07, § (cid:3) Let us give the form of the homomorphism on the Chevalley-Serre generators of g .We write γ e i = γ α i , for i ∈ ˚ I and γ e = γ − δ + α , , cf. (20), and so on. Proposition 30. The homomorphism θ : V g , → M from Theorem 29 is given by e i [ − | i 7→ ( ρ ( e i )) + c i γ f i [ − | i h [ − | i 7→ ( ρ ( h )) f i [ − | i 7→ ( ρ ( f i )) + c i γ e i [ − | i for i ∈ I and h ∈ h , where c i := − X j ∈ Ij ≺ i a ij , i ∈ I. Proof. On Z ≥ × Q -grading grounds, the homomorphism must be of this form forsome values of coefficients c i . We compute these values in Appendix A. (cid:3) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 43 Action of L + g on M ( n ) . The vertex algebra M = M ( g ) was the vacuum Fockmodule of the βγ -system associated to the vector space g . In exactly the same way,one defines M ( n ± ), with M ( n ± ) ∼ = C C [ γ a,n [ N ]] N ≤ a,n ) ∈± A ⊗ C [ β a,n [ N − N ≤ a,n ) ∈± A as vector spaces. These M ( n ± ) are vertex subalgebras of M .We have the subalgebra L ≥ ( M ([ ≤ M [ ≤ A ∈ M [ ≤ 1] and B ∈ M , A ( N ) B ∈ M for all N ≥ 0. Therefore L ≥ ( M [ ≤ M , via A [ N ] v := A ( N ) v for A ∈ M [ ≤ N ≥ v ∈ M . The Lie algebra homomorphism L g → L ( M [ ≤ L + g → L ≥ ( M ([ ≤ L + g := g ⊗ C [[ t ]] . In this way, L + g acts on M . The following is analogous to Proposition 18. Proposition 31. This action of L + g stabilizes M ( n ) and M ( n − ) . That is, for all A ∈ g , N ≥ , and v ∈ M ( n ± ) , θ ( A [ − | i ) ( N ) v ∈ M ( n ± ) . Proof. Let N ≥ 0. Suppose without loss that A = J a,n . (The result is clear for A = d and trivial for A = k .)The conditions on φ given in Theorem 28 imply that φ ( J a,n ) ( N ) stabilizes thesubspace M ( n ± ), and annihilates M ( n ∓ ), whenever J a,n ∈ n ∓ . And φ ( h ) = 0. Hence φ ( J a,n ) ( N ) stabilizes both M ( n ± ).The same applies to all terms in ( ρ ( J a,n )) cubic or higher in the generators β , γ .(Recall that these terms are either in (Der O ( n )) or in (Der O ( n − )).)The term ( J a,n ) ( N ) in ( ρ ( J a,n )) ( N ) stabilizes M ( n ± ) if a ∈ ˚ I and n = 0, i.e.if J a,n ∈ h . Otherwise it does not, but by our construction there is then also asum of compensating quadratic terms in ( ρ ( J a,n )) ( N ) , of the form γ b,m ( M ) β c,p ( N − M ) with { ( b, m ) , ( c, p ) } 6⊂ ± A . The latter condition ensures the double contractionsbetween such quadratic terms and states in M ( n ± ) vanish. The single contractionsof such compensating quadratic terms ensure that M ( n ± ) is stabilized, just as inProposition 18 and the discussion following. (cid:3) Homomorphism to M ⊗ π . We have the loop algebra L h := h ⊗ C (( t )) ofthe Cartan subalgebra h ⊂ g . Let { b i } i =1 ,..., dim h = { H i } i ∈ ˚ I ⊔ { k , d } denote a copy ofour basis of h and let { b i } dim h i =1 ⊂ h be its dual basis with respect to the form κ (cid:0) ·|· (cid:1) : κ (cid:0) b i | b j (cid:1) = δ ji . Then L h has basis { b i [ N ] } i =1 ,dots, dim h ; N ∈ Z . Let π denote the L h -module π := U ( L h ) ⊗ U ( h ⊗ C [[ t ]]) C | i induced from the trivial one-dimensional h ⊗ C [[ t ]]-module C | i . There is a lin-ear isomorphism π ∼ = C [ b i,n ] i =1 ,..., dim h ; n< of of vector spaces, and of modules over U ( t − h [ t − ]) ∼ = C [ b i,n ] i =1 ,..., dim h ; n< . The space π is a commutative vertex algebra, the state-field map being given by b i ( x ) := Y ( b i [ − | i , x ) = X N ∈ Z b i [ N ] x − N − , cf. Section 4.4. (Like M , it is a system of free fields.) It has the depth gradation, inwhich b i [ N ] contributes grade − N , and it inherits the Q -gradation from h .We continue to write γ e i = γ α i , for i ∈ ˚ I and γ e = γ − δ + α , , etc.We have the tensor product of vertex algebras M ⊗ π , which is again Z ≥ × Q -graded. Theorem 32. There exists a Z ≥ ⊗ Q -graded homomorphism of vertex algebras w : V g , → M ⊗ π given by e i [ − | i 7→ ( ρ ( e i )) + c i γ f i [ − | i + (cid:10) b j , ˇ α i (cid:11) γ f i [0] b j [ − | i h [ − | i 7→ ( ρ ( h )) + (cid:10) b i , h (cid:11) b i [ − | i f i [ − | i 7→ ( ρ ( f i )) + c i γ e i [ − | i + (cid:10) b j , ˇ α i (cid:11) γ e i [0] b j [ − | i for i ∈ I and h ∈ h .Proof. The proof is given in Section 6. (cid:3) Equivalently, there is a Z × Q -graded homomorphism of Lie algebras L g → L ( M ⊗ π )given by e i ( z ) ( ρ ( e i ))( z ) + c i ∂ z γ f i ( z ) + (cid:10) b j , ˇ α i (cid:11) γ f i ( z ) b j ( z ) h ( z ) ( ρ ( h ))( z ) + (cid:10) b i , h (cid:11) b i ( z ) f i ( z ) ( ρ ( f i ))( z ) + c i ∂ z γ e i ( z ) + (cid:10) b j , ˇ α i (cid:11) γ e i ( z ) b j ( z )for i ∈ I and h ∈ h .4.15. On zero modes. We have the homomorphism of Lie algebras ρ : g → g Der O ( n )from Lemma 6, and the embedding i : g Der O ( n ) ֒ → e M ( n ) ⊂ e M given by X ( a,n ) ∈ A P a,n ( X ) D a,n X ( a,n ) ∈ A P a,n ( γ [0]) β a,n [ − | i , where P a,n ( X ) ∈ O ( n ) for each ( a, n ) ∈ A . Let φ : g → Ω O ( n ) be the linear mapdefined by φ ( x ) = ( x ∈ b + φ ( x ) x ∈ n − where φ : g → Ω O was the splitting map from Theorem 28. We continue to identifyΩ O ( n ) with a subspace of M ( n ) ⊂ e M ( n ) (with f ( X ) dX a,n f ( γ [0]) γ a,n [ − | i ). Weget a linear map ϑ = i ◦ ρ + φ : g ∼ = V g , [1] → e M ( n )[1] . NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 45 (The kernel is the centre C k ⊂ g ; if one wanted an embedding, one could introduce atensor factor π as in Section 4.14.)The would-be non-negative vertex algebra products ϑ ( x ) ( N ) ϑ ( y ), x, y ∈ g , N ≥ V g , − h ∨ → M ( n ) in finite types.Nonetheless, we do have the following. Theorem 33. The vertex algebra th product (0) : M ( n ) × M ( n ) → M ( n ) extends to awell-defined product (0) : ϑ ( g ) × ϑ ( g ) → ϑ ( g ) on the image ϑ ( g ) ⊂ e M ( n )[1] of g in e M ( n ) .Moreover, we have ϑ ( x ) (0) ϑ ( y ) = ϑ ([ x, y ]) for all x, y ∈ g .Proof. Let x, y ∈ g . We first want to show that ϑ ( x ) (0) ϑ ( y ) is well-defined.Calculating as in the proof of Lemma 27, one sees that for any two X, Y ∈ Der O ( n ),we have i ( X ) (0) i ( Y ) = i ([ X, Y ]) + d ( X, Y ) (79)where we define a ( C -)bilinear map d ( · , · ) : Der O ( n ) × Der O ( n ) → Ω O ( n ) by d X ( a,n ) ∈ A P a,n ( X ) D a,n , X ( b,m ) ∈ A Q b,m ( X ) D b,m := − X ( a,n ) , ( b,m ) ∈ A (cid:0) ∂D b,m P a,n ( X ) (cid:1)(cid:0) D a,n Q b,m ( X ) (cid:1) . (80)Now, d ( · , · ) does not extend to a well-defined bilinear form on g Der O ( n ) × g Der O ( n ):for example if v = P n ∈ Z ≥ X a,n D a,n for some fixed a ∈ I , then − d (cid:0) X c, v, v (cid:1) = (cid:16) P m ∈ Z ≥ (cid:17) ∂X c, is ill-defined. But we can define a Lie subalgebra large enoughto contain the image of g and small enough that d ( · , · ) remains well-defined, asfollows. First let us define a Lie subalgebra D ⊂ g Der O ( n ), by stipulating that P ( a,n ) ∈ A P a,n ( X ) D a,n belongs to D if, for some K ∈ Z ≥ , P a,n ( X ) ∈ M ( b,m ) ∈ A n + K ≥ m ≥ n − K C X b,m for each ( a, n ) ∈ A . One checks that Der O ( n ) ⋊ D is a Lie subalgebra of g Der O ( n ).According to Theorem 10, the image ρ ( x ) of x belongs to this subalgebra: ρ ( x ) ∈ Der O ( n ) ⋊ D . Lemma 34. d ( · , · ) extends to a well-defined C -bilinear map d ( · , · ) : (cid:0) Der O ( n ) ⋊ D (cid:1) × (cid:0) Der O ( n ) ⋊ D (cid:1) → Ω O ( n ) , with the property that d ( D , · ) = 0 .Proof of Lemma 34. d ( · , · ) vanishes, summand by summand, whenever its first ar-gument is linear in the generators { X a,n } ( a,n ) ∈ A , since ∂ ( D a,n X b,m ) = 0 for all( a, n ) , ( b, m ) ∈ A . So by linearity it is enough to consider the case when the firstargument, P ( a,n ) ∈ A P a,n ( X ) D a,n belongs to Der O ( n ), i.e. has widening gap. Thesecond argument has bounded grade, so there is some M such that D a,n Q b,m ( X ) = 0for all n, m such that n − m ≥ M . By the assumption of widening gap, there is some N (depending on M ) such that, for all n > N , D b,m P a,n ( x ) = 0 whenever n − m < M .So at most the first N terms in the sum on n can be nonzero: X ( a,n ) , ( b,m ) ∈ A (cid:0) ∂D b,m P a,n ( X ) (cid:1)(cid:0) D a,n Q b,m ( X ) (cid:1) = X ( a,n ) , ( b,m ) ∈ A n ≤ N (cid:0) ∂D b,m P a,n ( X ) (cid:1)(cid:0) D a,n Q b,m ( X ) (cid:1) , and then for each n the sum on m is also finite, since P a,n ( X ) is a polynomial. (cid:3) At this stage, we have shown that (79) holds for all X, Y ∈ Der O ( n ) ⋊ D . Inparticular, it holds for the images ρ ( x ) , ρ ( y ) of x, y ∈ g : i ( ρ ( x )) (0) i ( ρ ( y )) = i ( ρ ([ x, y ])) + d ( ρ ( x ) , ρ ( y )) . It follows that ϑ ( x ) (0) ϑ ( y ) is well-defined. (There are no possible double contractions,and hence no possible divergences, of the products between the subspaces Ω O ( n ) and i ( g Der O ( n )) of e M ( n ).)The “moreover” part is then essentially a corollary of Theorem 28. Consider thelinear map π : M [1] → e M [1] which acts as the identity on M [1] ⊂ M [1] and sends S ab,n [ − | i → P m ∈ Z γ a,m [0] β b,m + n [ − | i and D → P ( a,m ) ∈I× Z m γ a,m [0] β a,m [ − | i .By construction, for any x ∈ g , π ( θ ( x [ − | i )) = ϑ ( x ) + τ ( ϑ ( σ ( x ))), where ϑ ( x ) ∈ e M ( n )[1] and τ ( ϑ ( σ ( x ))) ∈ e M ( n − )[1]. (See (54) and the discussion following.) Thus π ( θ ( x [ − | i )) (0) π ( θ ( y [ − | i )) = ϑ ( x ) (0) ϑ ( y ) + τ ( ϑ ( σ ( x ))) (0) τ ( ϑ ( σ ( y ))) (81)for all x, y ∈ g . At the same time, for any X, Y ∈ M [1], we check that π ( X ) (0) π ( Y )is well-defined and equal to π ( X (0) Y ). Thus π (cid:0) θ ( x [ − | i ) (cid:1) (0) π (cid:0) θ ( y [ − | i ) (cid:1) = π (cid:0) θ ( x [ − | i ) (0) θ ( y [ − | i ) (cid:1) = π ( θ ([ x, y ][ − | i ))= ϑ ([ x, y ]) + τ ( ϑ ( σ ([ x, y ]))) . (82)On comparing (81) and (82), and projecting onto the summand e M ( n )[1] of e M ( n )[1] ⊕ e M ( n − )[1], we have the result. (cid:3) Corollary 35. There is a well-defined Lie algebra L ( ϑ ( g )) of the formal 0-modes ofstates in the image of ϑ , and the map g → L ( ϑ ( g )); x ϑ ( x )[0] is a homomorphism of Lie algebras. (cid:3) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 47 Proof. This follows from the theorem since, when the Lie bracket formula (73) forformal modes is specialized to the Lie bracket of formal zero modes, only the 0thproduct contributes: [ A [0] , B [0]] = ( A (0) B )[0]. (cid:3) Proof of Theorem 28 In (78) we obtained a cocycle e ρ ∗ ( ω ) ∈ H ( L g , L ( O ⊕ Ω O )). In fact, we can berather more precise: recall ρ ( k ) = 0, ρ ( d ) = D , and ω ( D [ N ] , · ) = 0. So we haveactually defined a L ( O ⊕ Ω O )-valued 2-cocycle on the double-loop algebra LL ˚ g := L (˚ g [ t, t − ]) := ˚ g [ t, t − ] ⊗ C (( s )) . Our goal is to show that this cocycle is trivial.We shall follow closely the strategy of proof due to Feigin and Frenkel [FF90a],and specifically the treatment in [Fre07, § O ⊕ Ω O ⊂ M is contained in the larger subspace M := C [ γ a,n [ − N ]] a ∈I ,n ∈ Z ,N ∈ Z ≥ | i ⊂ M . This subspace is a commutative vertex algebra. It is also an ideal for the action of M [ ≤ 1] in the same sense as in Lemma 24: A ( M ) B ∈ M for all A ∈ M , B ∈ M [ ≤ M ≥ 0. It follows that e ρ ∗ ( ω ) ∈ H ( LL ˚ g , L ( M )) , and it is convenient to show our cocycle is zero in the latter space.To do so, we first show that the cocycle e ρ ∗ ( ω ) actually belongs to the local sub-complex of this CE complex. Let us define this local complex.5.1. bc -system. Let Cl denote the Clifford algebra with generators b a,n [ N ] , c a,n [ N ] , with N ∈ Z and ( a, n ) ∈ I × Z , and anticommutation relations[ b a,n [ N ] , b b,m [ M ]] + = 0 , [ b a,n [ N ] , c b,m [ M ]] + = δ b,ma,n δ N, − M , [ c a,n [ N ] , c b,m [ M ]] + = 0 , where we write [ X, Y ] + := XY + Y X for the anticommutator.Define Λ to be the induced Cl -module generated by a vector | i such that b a,n [ M ] | i = 0 , M ∈ Z ≥ , c a,n [ M ] | i = 0 , M ∈ Z ≥ . (83)for all ( a, n ) ∈ I × Z . It is Z × Q -graded just as is M , cf. Section 4.4. Cl is a superalgebra (with all generators b , c of odd degree) and its module Λ is avector superspace. Λ is moreover a vertex superalgebra (for the definition of whichsee e.g. [Kac98; FB04]). The state-field map Y ( · , x ) : Λ → Hom (Λ , Λ(( x ))) is definedas follows. First, b a,n ( x ) := Y ( b a,n [ − | i , x ) = X N ∈ Z b a,n [ N ] x − N − , c a,n ( x ) := Y ( c a,n [0] | i , x ) = X N ∈ Z c a,n [ N ] x − N , and then in general if A = c a ,n [ − N ] . . . c a r ,n r [ − N r ] b b ,m [ − M ] . . . b b s ,m s [ − M s ] | i , (84)then A ( u ) := Y ( A, u ) = : c a ,n ( N ) ( u ) . . . c a r ,n r ( N r ) ( u ) b ( M − b ,m ( u ) . . . b ( M s − b s ,m s ( u ): . (85)The normal-ordering here is defined just as in Section 4.4 only with the addition ofsigns to allow for the superspace Z / Z -grading: we have: Y ( A, u ) Y ( B, v ): := X M< A ( M ) u − M − ! Y ( B, v )+( − p ( A ) p ( B ) Y ( B, v ) X M ≥ A ( M ) u − M − ! . (86)whenever A and B are in grades p ( A ) and p ( B ) respectively.We have the (super)translation operator T ∈ End Λ of the vertex superalgebraΛ, defined by T A := A ( − | i . It acts on monomials in the generators of Cl as asuperderivation, according to[ T, b a,n [ − N ]] + = − N b a,n [ N − , [ T, c a,n [ − N ]] + = − ( N − c a,n [ N − , and we have T | i = 0.For each r ≥ 0, let Λ r denote the subspace of Λ spanned by states of the form c a ,n [ − N ] . . . c a r ,n r [ − N r ] | i , with a , . . . , a r ∈ I , n , . . . , n r ∈ Z , N , . . . , N r ∈ Z ≥ . The sum, Λ := L ∞ r =0 Λ r is asupercommutative vertex superalgebra.5.2. Chevalley-Eilenberg cochains for L + L ˚ g with coefficients in M . Let L + L ˚ g denote the Lie algebra L + L ˚ g := ˚ g [ t, t − ] ⊗ C [[ s ]] . It is a topological Lie algebra, with the the linear topology coming from the Z ≥ -grading (i.e. the linear topology in which ˚ g [ t, t − ] ⊗ s N C [[ s ]], N ∈ Z ≥ , are a base ofthe open neighbourhoods of 0).The space M is a module over L + L ˚ g . Indeed, let us write J a,n := ( ρ ( J a,n )) ∈ M ,that is J a,n = X ( b,m ) ∈I× Z R b,ma,n ( γ [0]) β b,m [ − | i + X ( b,c ) ∈I f bac S bc,n [ − | i , where we recall the definition (56) of the polynomials R b,ma,n and the definition (70) ofthe injective map . From Lemma 27 and the fact that ρ : g → D is a homomorphismof Lie algebras, it follows that J a,n (0) J b,m ≡ f abc J c,n + m mod Ω O , J a,n (1) J b,m ≡ O . (87)The mode J a,n ( N ) restricts to a linear map M → M for all non-negative N , as inLemma 24(iii). This defines an action of L + L ˚ g because, in view the commutatorformula (66) and (87), we have[ J a,n ( N ) , J b,m ( M ) ] = f abc J c,n + m ( N + M ) mod L ( M ) , N, M ≥ , NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 49 (and equality modulo L ( M ) is enough, cf. Lemma 24(ii)).As an L + L ˚ g -module, M is smooth, which is to say that for all v ∈ M and all a ∈ g , a [ N ] v = 0 for all sufficiently large N . In other words the action is continuous,when we endow M with the discrete topology.We have the r th exterior power V r L + L ˚ g of the topological vector space L + L ˚ g ,with its natural topology. The space of r -cochains of the Chevalley-Eilenberg com-plex of L + L ˚ g with coefficients in the module M is the space C r ( L + L ˚ g , M ) :=Hom cont ( V r L + L ˚ g , M ) of continuous linear maps λ : ^ r L + L ˚ g → M . Since M has the discrete topology, continuity means that for each such λ there mustbe some N such that λ kills everything in grades ≥ N . ThusHom cont (cid:16)^ r L + L ˚ g , M (cid:17) = M N ≥ Hom (cid:16)(cid:16)^ r L + L ˚ g (cid:17) N , M (cid:17) . (88)5.2.1. Case of finite-dimensional g . Let us digress to recall what happens when g isof finite dimension. In that case the subspaces ( V r L + L ˚ g ) N appearing in (88) are alsoall finite-dimensional, and so M N ≥ Hom (cid:16)(cid:16)^ r L + L ˚ g (cid:17) N , M (cid:17) ∼ = M ⊗ M N ≥ (cid:16)(cid:16)^ r L + L ˚ g (cid:17) N (cid:17) ∗ ∼ = M ⊗ Λ r , where in the second step we identify the restricted dual, L N ≥ (( V r L + L ˚ g ) N ) ∗ , withthe space Λ r spanned by states of the form c a [ − N ] . . . c a r [ − N r ] | i , by means of thebilinear pairing given by( c a [ − N ] . . . c a r [ − N r ] | i , J b [ K ] ∧ · · · ∧ J b r [ K r ]) ( sign( σ ) a i = b σ ( i ) and N i = K σ ( i ) for each i , for some σ ∈ S r J a of g , where a runs over a finite set of indices.)In this way, one has a linear isomorphism C • ( L + L ˚ g , M ) ∼ = M ⊗ Λ .5.2.2. Case of infinite-dimensional g . Now we return to our case, in which g hascountably infinite dimension. The tensor product M ⊗ Λ now corresponds only to asubspace of C • ( L + L ˚ g , M ), and one which will not be large enough for our purposes.It is convenient simply to define , for each r , M e ⊗ Λ r := M N ≥ Hom (cid:16)(cid:16)^ r L + L ˚ g (cid:17) N , M (cid:17) = C r ( L + L ˚ g , M ) . An element Φ ∈ M e ⊗ Λ r can be regarded as a possibly infinite sum of states ofthe form c a ,n [ − N ] . . . c a r ,n r [ − N r ] v ∈ M ⊗ Λ r , (89) with P ri =1 N i ≤ N for some bound N depending on Φ, and subject to the conditionthat for any λ ∈ V r L + L ˚ g , the pairing (Φ , λ ), defined as follows, yields a well-defined(i.e. finite) element of M . We first define the pairing between a state in M ⊗ Λ r ofthe form (89) and a vector in V r L + L ˚ g of the form J b ,k [ K ] ∧ · · · ∧ J b r ,k r [ K r ] (90)by declaring that( c a ,n [ − N ] . . . c a r ,n r [ − N r ] m, J b ,k [ K ] ∧ · · · ∧ J b r ,k r [ K r ]) ( sign( σ ) m a i = b σ ( i ) , n i = k σ ( i ) and N i = K σ ( i ) for each i , for some σ ∈ S r Remark . Neither L + L ˚ g nor V r L + L ˚ g are complete. For example, pick an a ∈ I andlet s N = P Nn =1 J a,n [ n ] for N = 1 , , , . . . . This sequence is Cauchy but not convergent(because the infinite sum P ∞ n =1 J a,n [ n ] does not belong to L + L ˚ g = g ⊗ C [[ t ]]).By not completing these spaces, we preserve a useful property of “compact sup-port”: for any λ ∈ V r L + L ˚ g there exists some k such that λ can be written as apossibly infinite linear combination of terms of the form (90) with | k i | < k for each i = 1 , . . . , r .5.3. The complex C • ( L + L ˚ g , M ) and the state Q . Thus, we have the spaces of theChevalley-Eilenberg complex C • ( L + L ˚ g , M ) of L + L ˚ g with coefficients in the module M . The differential d : C r ( L + L ˚ g , M ) → C r +1 ( L + L ˚ g , M ) is given by the usualformula, ( df )( x , . . . , x r +1 ) := r +1 X p =1 ( − p +1 x p .f ( x , . . . , x p − , x p +1 , . . . , x r +1 )+ X ≤ p Here is the first such property. The zero mode Q (0) unambiguously defines a linearmap M e ⊗ Λ → M e ⊗ Λ . (It involves only single contractions.) One easily seesthat it coincides with the CE differential: d Φ = Q (0) Φ . The local complex C • loc ( L g , L ( M )) . Recall the definition of the local Lie alge-bra L ( L ) from Section 4.10. One sees from (73) that the formal zero modes generatea Lie subalgebra, which we shall denote by L ( L ) ⊂ L ( L ) . (92)We can apply this in particular to the subspaces M ⊗ Λ r of the supercommutativevertex superalgebra M ⊗ Λ , to obtain the vector superspaces (in fact, supercommu-tative Lie superalgebras) L ( M ⊗ Λ r ). We see that these spaces are given by L ( M ⊗ Λ r ) = X k ≥ T k ( M ⊗ Λ r ) (cid:14) (Im T + C | i )= ( M ⊗ Λ r ) (cid:14) (Im T + C | i ) . (Note that we have to quotient by the subspace C | i = ker T too, not just Im T ,because the zero mode | i [0] = | i ⊗ t = ∂ t ( | i ⊗ t ) = ( T + ∂ t )( | i ⊗ t ) ≡ L ( M ⊗ Λ r ).)Now, when g has finite dimension, the spaces of the local complex are, by definition,precisely these L ( M ⊗ Λ r ). In our case, in which g has countably infinite dimension,we must use the larger spaces M e ⊗ Λ r from Section 5.2.2. By analogy with theabove, let us define C r loc ( LL ˚ g , L ( M )) := (cid:0) M e ⊗ Λ r (cid:1)(cid:14) (Im T + C | i ) (93)for each r . Here we use the fact that the definition of the translation operator extendsin a well-defined way to M e ⊗ Λ r .Let us write R for the projection map R : C r ( L + L ˚ g , M ) → C r loc ( LL ˚ g , L ( M )); Φ R Φ := Φ[0]. Let us attempt to define a differential d : C r loc ( LL ˚ g , L ( M )) → C r +1loc ( LL ˚ g , L ( M ))by setting d (Φ[0]) := ( d Φ)[0] = ( Q (0) Φ)[0] . The fact that this is a consistent definition is a consequence of the following lemma. Lemma 37. We have the commutative diagram C r loc ( LL ˚ g , L ( M )) C r +1loc ( LL ˚ g , L ( M )) C r ( L + L ˚ g , M ) C r +1 ( L + L ˚ g , M ) C r ( L + L ˚ g , M ) C r +1 ( L + L ˚ g , M ) . d R d R T d T (94) Proof. What has to be checked is that the lower square is commutative. This isseen by direct calculation: as in Section 5.3, there is a well-defined notion of thezero mode Q (0) of Q acting on M e ⊗ Λ , and we check that T d Φ = T ( Q (0) Φ) =( T Q ) (0) Φ + Q (0) T Φ = Q (0) T Φ = dT Φ. (cid:3) Thus, we obtain a complex, ( C • loc ( LL ˚ g , L ( M )) , d ). This is the local complex . Asthe notation suggests (and as we now check) it forms a subcomplex of the usualChevalley-Eilenberg complex C • ( LL ˚ g , L ( M )) of LL ˚ g with coefficients in L ( M ).Consider first a state Ψ ∈ M ⊗ Λ r ⊂ M e ⊗ Λ r of the formΨ = c a ,n [ − N ] . . . c a r ,n r [ − N r ] v. We can apply the state-field map to this state, and in particular we can take the zeromode Ψ (0) ∈ End( M ⊗ Λ). This zero mode Ψ (0) is a (generically infinite) sum of terms c a ,n [ M ] . . . c a r ,n r [ M r ] v [ M ] (95)(Here v [ M ] ∈ L ( M ) is the M th mode of the state v ∈ M . )We have the exterior algebra V r LL ˚ g . The pairing from Section 5.2.2 goes over tothis setting, and using it we can certainly interpret each term (95) as an r -cochain in C r ( LL ˚ g , L ( M ) := Hom cont ( ^ r LL ˚ g , L ( M )) . Moreover, the infinite sum Ψ[0] is again a well-defined r -cochain, i.e. it is continuous,when L ( M ) gets its natural (i.e. t -adic) linear topology.These statements are exactly as in the case in which g has finite dimension. Theonly new aspect of the present case is that a general state Φ ∈ M e ⊗ Λ r may itselfbe an infinite sum of such states Ψ ∈ M ⊗ Λ r , subject to the conditions we gave inSection 5.2.2. But, for any given µ ∈ V r LL ˚ g , we can arrange that for only finitelymany of these summands Ψ does Ψ[0] have nonzero pairing with µ . (This is clearfrom the notion of “compact support” from Remark 36.) In this way, we can indeedinterpret Φ[0] ∈ C r loc ( LL ˚ g , L ( M )) as an element of C r ( LL ˚ g , L ( M )).Finally, one checks that the derivative on C • loc ( LL ˚ g , L ( M )) coincides with the usualChevalley-Eilenberg derivative.5.5. The cocycle ω [0] . As the relevant example for us, consider the state ω ∈ M e ⊗ Λ given by ω := − c a,n [ − c b,m [0] ∂R c,pa,n ∂ γ d,q [0] ∂R d,qb,m ∂ γ c,p [0] | i − c a,n [0] c b,m [0] (cid:20) T, ∂R c,pa,n ∂ γ d,q [0] (cid:21) ∂R d,qb,m ∂ γ c,p [0] | i (96)where the polynomials R are as we defined them in (56).(Here and below we use summation convention, for brevity.) Here and in what follows we are equivocating between formal modes X [ M ] ∈ L ( M ⊗ Λ) and modes X ( M ) ∈ End( M ⊗ Λ) of states in X ∈ M ⊗ Λ. There is no loss in this because the vertex superalgebra M ⊗ Λ of free fields has the property that the Lie algebra homomorphism L ( M ⊗ Λ) → End M ⊗ Λ isinjective. NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 53 The formal zero mode ω [0] is realized in End( M e ⊗ Λ ) as the following infinitesum ω (0) = − Z c a,n ′ ( x ) c b,m ( x ) Y ∂R c,pa,n ∂ γ d,q [0] ∂R d,qb,m ∂ γ c,p [0] | i , x ! dx − Z c a,n ( x ) c b,m ( x ) Y (cid:20) T, ∂R c,pa,n ∂ γ d,q [0] (cid:21) ∂R d,qb,m ∂ γ c,p [0] | i , x ! dx = − X K,L ∈ Z K c a,n [ − K ] c b,m [ − L ] ∂R c,pa,n ∂ γ d,q [0] ∂R d,qb,m ∂ γ c,p [0] | i ! [ K + L − − X K,L ∈ Z c a,n [ − K ] c b,m [ − L ] (cid:20) T, ∂R c,pa,n ∂ γ d,q [0] (cid:21) ∂R d,qb,m ∂ γ c,p [0] | i ! [ K + L ] . At the same time, from (76) we have e ρ ∗ ( ω )( J a,n [ K ] , J b,m [ L ])= − K ∂R c,pa,n ∂ γ d,q [0] ∂R d,qb,m ∂ γ c,p [0] | i ! [ K + L − − (cid:20) T, ∂R c,pa,n ∂ γ d,q [0] (cid:21) ∂R d,qb,m ∂ γ c,p [0] | i ! [ K + L ] . We see that ω [0] is identified with our cocycle e ρ ∗ ( ω ), and so our cocycle belongs tothe local complex.5.6. Submodules M ( n ) and M ( n − ) . To proceed, we need more information aboutthe structure of M as an L + L ˚ g -module. Recall that O ( n ) = C [ X a,n ] ( a,n ) ∈ A , O ( n − ) = C [ X a,n ] ( a,n ) ∈− A and that our action of g on O stabilizes both of these, as in Proposition 18. It followsthat if we now define M ( n ) := C [ γ a,n [ − N ]] ( a,n ) ∈ A ,N ≥ | i , M ( n − ) := C [ γ a,n [ − N ]] ( a,n ) ∈− A ,N ≥ | i , then our action of L + L ˚ g on M (given by J a,n [ N ] v = J a,n ( N ) v , as in Section 5.2)stabilizes these subspaces (in fact, commutative vertex subalgebras) of M . (Cf.Proposition 31.)As modules over L + L ˚ g , these subspaces turn out to be isomorphic to contragredientVerma modules, as we now describe.5.7. Contragredient Verma modules. The contragredient Verma module M ∗ λ over g of highest weight λ ∈ h ∗ is by definition the coinduced left U ( g )-module M ∗ λ = Coind gb − C v λ := Hom res U ( b − ) ( U ( g ) , C v λ ) , where C v λ denotes the one-dimensional U ( b − )-module defined by n − .v λ = 0 and h.v λ = λ ( h ) v λ for h ∈ h . Here Hom res means the following: we have the isomorphismof vector spacesHom U ( b − ) ( U ( g ) , C v λ ) ∼ = Hom U ( b − ) ( U ( b − ) ⊗ U ( n ) , C v λ ) ∼ = Hom C ( U ( n ) , C ) = U ( n ) ∗ , and Hom res means we allow only maps that, under this isomorphism, belong to the restricted dual U ( n ) ∨ := L α ∈ Q ( U ( n ) α ) ∗ ⊂ U ( n ) ∗ of the Q -graded vector space U ( n ).The isomorphism above is also one of left U ( n ) modules. So, as left U ( n )-modules, M ∗ λ ∼ = U ( n ) ∨ . (97)We also have the contragredient Verma modules “in the opposite category O ”, i.e.the twists of the modules above by the Cartan involution σ of (51).Define M ∗ ,σλ to be the coinduced left U ( g )-module M ∗ ,σλ = Coind gb + C v − λ := Hom res U ( b + ) ( U ( g ) , C v − λ ) . where C v − λ denotes the one-dimensional U ( b + )-module defined by n .v − λ = 0 and h.v − λ = λ ( h ) v − λ for h ∈ h .As vector spaces, and as modules over U ( n − ), we have M ∗ ,σλ ∼ = U ( n − ) ∨ . These definitions go over to the half loop algebra L + g in an obvious way: C v λ becomes a module over L + b − if we declare that ( b − ⊗ t C [[ t ]]) .v λ = 0 and then we getthe left U ( L + g )-module Hom res U ( L + b − ) ( U ( L + L ˚ g ) , C v λ ), and likewise its twist by σ . Proposition 38. (i) There are isomorphisms of g -modules O ( n ) ∼ = Hom res U ( b − ) ( U ( g ) , C v ) , O ( n − ) ∼ = Hom res U ( b + ) ( U ( g ) , C v ) . (ii) There are isomorphisms of L + g -modules M ( n ) ∼ = Hom res U ( L + b − ) ( U ( L + g ) , C v ) M ( n − ) ∼ = Hom res U ( L + b + ) ( U ( L + g ) , C v ) . Proof. The proof is the same as in the case of g of finite type in [Fre07, § § O ( n ) ∼ = U ( n ) ∨ as n -modules. To do that we consider thepairing U ( n ) × O ( n ) C ; ( x, P ) 7→ h x, P i := x.P | ∈ n . It respects the Q -gradationsof U ( n ) and O ( n ), in the sense that U ( n ) α pairs as zero with O ( n ) β unless α + β = 0.Consider the restriction of the pairing to U ( n ) α × O ( n ) − α for some α ∈ Q > . Recallour ordered basis B + = { J a,n } ( a,n ) ∈ A of n from (23) and (19). We have the PBWbasis of U ( n ) α consisting of ordered monomials these basis elements, and we have thebasis of O ( n ) − α consisting of monomials in the X a,n , ( a, n ) ∈ A . Both these bases NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 55 have the lexicographical ordering coming from the ordering of A . The action of J a,n ,( a, n ) ∈ A , on O ( n ) is by a differential operator of the form D a,n + X ( b,m ) ∈ A wgt( J b,m ) − wgt( J a,n ) ∈ Q > P b,ma,n ( X ) D b,m (98)From this one sees that the matrix of the restricted pairing with respect to the twoordered bases above is diagonal with non-zero entries. Thus the restriction of thepairing to U ( n ) α × O ( n ) − α is non-degenerate, for each α ∈ Q > . This shows that O ( n ) ∼ = U ( n ) ∨ as a vector space. But the pairing is also manifestly n -invariant: h xe i , P i = h x, e i P i . This shows that O ( n ) ∼ = U ( n ) ∨ as left modules over U ( n ), wherethe left U ( n )-module structure on U ( n ) ∨ is the canonical one, coming from the rightaction of U ( n ) on itself by right multiplication.Thus, given (97), we have O ( n ) ∼ = M ∗ as n -modules. Now we show it is an isomor-phism of g -modules. The coinduced module M ∗ λ has the following universal property.Suppose M is a g -module and N ⊂ M a b − -submodule of M such that the quotient M/N is isomorphic to C v λ as a b − -module. Then there is a homomorphism of g -modules M → M ∗ λ sending v v ∗ λ , where v ∈ M is such that v + N spans M/N ,and where v ∗ λ ∈ M ∗ λ is a non-zero vector of weight λ . In our case, as a b − -module, O ( n ) has the submodule N = L α ∈ Q > O ( n ) − α (or equivalently, the ideal in O ( n )generated by ( X a,n ) ( a,n ) ∈ A ). The quotient O ( n ) /N is a b − -module of dimension one,spanned by the class 1 + N of the vector 1 ∈ O ( n ). This vector 1 + N has weightzero and is annihilated by n − (since n − . ∈ N ). Hence there exists a homomorphismof g -modules φ : O ( n ) → M ∗ sending 1 to a non-zero vector v ∗ ∈ M ∗ of weight zero.Now, for any P ∈ O ( n ) there exists x ∈ U ( n ) such that x.P = 1: indeed, take the lastnonzero monomial m of P with respect to the lexicographical ordering and considerthe corresponding PBW basis element m ∗ of U ( n ). We see that m ∗ .m is a nonzeromultiple of 1. Thus x.φ ( P ) = φ ( x.P ) = φ (1) = v ∗ is nonzero and hence φ ( P ) isalso nonzero. That is, φ : O ( n ) → M ∗ is injective. But we know O ( n ) ∼ = M ∗ as an n -module, as above, so in fact φ must be a bijection. This completes the proof that O ( n ) ∼ = M ∗ ≡ Hom res U ( b − ) ( U ( g ) , C v ) as g -modules.The argument for O ( n − ) ∼ = Hom res U ( b + ) ( U ( g ) , C v ) is the same, just twisted by theCartan involution σ so that O ( n ) and O ( n − ), and U ( n ) and U ( n − ), are interchanged.(Compare (55).)For part (ii), the argument is again essentially the same, with the Q -gradationabove replaced by the Q × Z ≥ gradation. One shows first that M ( n ) ∼ = U ( L + n ) ∨ as L + n -modules, and then uses that fact to show that the canonical homomorphism of L + g -modules M ( n ) → Hom res U ( L + b − ) ( U ( L + g ) , C v ) is an isomorphism. (cid:3) Therefore there is an isomorphism of modules over g ′ / C k ∼ = L ˚ g := ˚ g [ t, t − ], O ( n ) ∼ = Hom res U ( b ′− / C k ) ( U ( g ′ / C k ) , C v )(where b ′− := g ′ ∩ b − ) and an isomorphism of modules over L + ( g ′ / C k ) ∼ = L + L ˚ g , M ( n ) ∼ = Hom res U ( L + ( b ′− / C k )) ( U ( L + ( g ′ / C k )) , C v ) . Given an r -cochain V r L + L ˚ g → M ( n ), we may restrict it to V r L + ˚ h and thencompose the resulting map V r L + ˚ h → M ( n ) with the canonical projection of L + L ˚ g -modules M ( n ) → C v . This defines a map of complexes µ : C • ( L + L ˚ g , M ( n )) → C • ( L + ˚ h , C v ) . Lemma 39. This map µ is a quasi-isomorphism, i.e. it induces an isomorphism ofthe cohomologies, H • ( L + L ˚ g , M ( n )) ∼ = H • ( L + ˚ h , C v ) . Proof. The proof, using the Serre-Hochschild spectral sequence (see [Fuk86, § (cid:3) The following is [Fre07, Lemma 5.6.7]. Lemma 40. If the restriction of a cocycle γ ∈ C r loc ( LL ˚ g , L ( M ( n ))) to V r L ˚ h is zero,then γ represents the zero cohomology class, [ γ ] = [0] , in H r loc ( LL ˚ g , L ( M ( n ))) .Proof. We can suppose r ≥ γ ∈ C r loc ( LL ˚ g , L ( M ( n ))). We have γ = X [0] for some cochain X ∈ C r ( L + L ˚ g , M ( n )). The closure of γ implies dX is in the image of T : dX = T Y ,say, for some Y ∈ C r +1 ( L + L ˚ g , M ( n )). We have T dY = − dT Y = − ddX = 0 andsince T has kernel 0 (for all r ≥ dY = 0.Let γ denote the restriction of γ to V r L ˚ h . We have γ = X [0], where X denotes therestriction of X to V r L + ˚ h . If γ is zero then X is in the image of T . Therefore so toois µ ( X ). So we have µ ( X ) = T h , say, for some h ∈ C r ( L + ˚ h , C v ). Now, µ is a map ofcomplexes, so dT h = dµ ( X ) = µ ( dX ) = µ ( T Y ). It is clear that T commutes with µ .Thus − T dh = T µ ( Y ) and hence, again since the kernel of T is trivial, dh = − µ ( Y ).That is, µ ( Y ) is exact. But µ is a quasi-isomorphism as in Lemma 39. So Y mustalso be exact: Y = dB , say, for some B ∈ C r ( L + L ˚ g , M ( n )).Let X ′ = X + T B . We see that γ = X ′ [0] and X ′ is a cocycle: dX ′ = dX − T dB = dX − T Y = 0.(At this point, effectively we have shown we were at liberty to assume our X – nowcalled X ′ – was not only a cochain, but a cocycle. We now repeat many steps fromabove, but armed with that extra fact.)We have µ ( X ′ ) = T h ′ (where h ′ = h + µ ( B )). So − T dh ′ = dT h ′ = dµ ( X ′ ) = µ ( dX ′ ) = 0, and hence dh ′ = 0. So h ′ is a cocycle in C r ( L + ˚ h , C v ) = (cid:16)V r L + ˚ h (cid:17) ∨ .Thus, again since µ is a quasi-isomorphism, we have h ′ = µ ( B ′ ) for some cocycle B ′ ∈ C r ( L + L ˚ g , M ( n )). Finally, we see that µ ( X ′ ) = T µ ( B ′ ) = µ ( T B ′ ). Hencethe cocycles X ′ and T B ′ in C r ( L + L ˚ g , M ( n )) represent the same cohomology classin H r ( L + L ˚ g , M ( n )). Therefore the cocycles X ′ [0] and ( T B ′ )[0] in C r loc ( L + L ˚ g , M ( n ))represent the same cohomology class in H r loc ( L + g , M ( n )). (Indeed X ′ − T B ′ = dC forsome C ∈ C r ( L + L ˚ g , M ( n )) implies X ′ [0] − ( T B ′ )[0] = ( dC )[0] = d ( C [0]).) But γ = X ′ [0] and 0 = ( T B ′ )[0], so we have shown γ is cohomologous to zero, as required. (cid:3) This is the key lemma. However, to use it, we have to get around one final obsta-cle: our cochain ω [0] lives not in C ( LL ˚ g , L ( M ( n ))) but only in the larger space C ( LL ˚ g , L ( M )). NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 57 Cohomology equivariant with respect to τ . Recall the involutive automor-phism τ defined in Section 3. Let us also denote by τ the involutive automorphismof Λ ⊗ M defined by τ | i = | i , τ ( γ α,n )[ N ] = γ − α, − n [ N ] , τ ( γ i,n )[ N ] = − γ i, − n [ N ] ,τ ( β α,n )[ N ] = β − α, − n [ N ] , τ ( β i,n )[ N ] = − β i, − n [ N ] ,τ ( c α,n )[ N ] = c − α, − n [ N ] , τ ( c i,n )[ N ] = − c i, − n [ N ] ,τ ( b α,n )[ N ] = b − α, − n [ N ] , τ ( b i,n )[ N ] = − b i, − n [ N ] ,τ ( S αβ,n [ N ]) = S − α − β, − n [ N ] , τ ( S αi,n [ N ]) = − S − αi, − n [ N ] ,τ ( S iβ,n [ N ]) = − S i − β, − n [ N ] , τ ( S ij,n [ N ]) = S ij, − n [ N ] , for α, β ∈ ˚∆ \ { } , i, j ∈ ˚ I , n ∈ Z , and N ∈ Z , and τ ( D [ N ]) = − D [ N ].Let C τ, • loc ( LL ˚ g , L ( M )) = L ((Λ • ⊗ M ) τ ) denote the subspace consisting of zeromodes of states Φ ∈ Λ ⊗ M such that τ Φ = 0. Lemma 41. τ Q = Q and hence C τ, • loc ( LL ˚ g , L ( M )) is a subcomplex of C • loc ( LL ˚ g , L ( M )) .Proof. The term f abc c a,m [0] c b,n [0] b c,n + m [ − | i in Q is τ -invariant because σ is anautomorphism of ˚ g [ t, t − ]. (More explicitly, this term is equal to X α ∈ ˚∆ + X i ∈ ˚ I X n,m ∈ Z f α, − αi c α,m [0] c − α,n [0] b i,n + m [ − | i + X α ∈ ˚∆ + X i ∈ ˚ I X n,m ∈ Z (cid:0) f αiα c α,m [0] c i,n [0] b α,n + m [ − | i + f − α,i − α c − α,m [0] c i,n [0] b − α,n + m [ − | i (cid:1) + 12 X i,j,k ∈ ˚ I X n,m ∈ Z f ijk c i,m [0] c j,n [0] b k,n + m [ − | i and each line of this expression is τ -invariant.) The other term in Q , P ( a,n ) ∈I× Z c a,n [0] J a,n ,is τ -invariant because, in view of (53), τ J a,n = τ ( ρ ( J a,n )) = ( τ ( ρ ( J a,n ))) = ( ρ ( σJ a,n ))and thus τ J α,n = J − α, − n , τ J i,n = − J i, − n for α ∈ ˚∆ \ { } , i ∈ ˚ I and n ∈ Z . (cid:3) Lemma 42. The element ω ∈ Λ ⊗ M obeys τ ω = 0 . Proof. Indeed, we defined ω as in (96) but one sees (in view of (75)) that, equivalently, ω = c a,n [0] c b,m [0] ( J a,n (0) J b,m − f abc J c,n + m ) + c a,n [ − c b,m [0] J a,n (1) J b,m (summation convention). The fact that τ ω = 0 follows, using the statements aboveand the fact that τ is an automorphism for all the non-negative products. (cid:3) Thus our cocycle ω [0] belongs to the subcomplex C τ, • loc ( LL ˚ g , L ( M )). More is true.We have the subspace M ( n ) + M ( n − ) of M . It is closed (trivially) under all thenon-negative products. Therefore L ( M ) has the Lie subalgebra L ( M ( n ) + M ( n − )). Lemma 43. The cocycle ω [0] has coefficients in this subalgebra, i.e ω [0] ∈ C τ, ( LL ˚ g , L ( M ( n ) + M ( n − ))) . Proof. We use summation convention. By definition, (96), ω = − c a,n [ − c b,m [0] ∂R c,pa,n ∂ γ d,q [0] ∂R d,qb,m ∂ γ c,p [0] | i − c a,n [0] c b,m [0] (cid:20) T, ∂R c,pa,n ∂ γ d,q [0] (cid:21) ∂R d,qb,m ∂ γ c,p [0] | i where R b,ma,n ( X ) are the polynomials from (56). On recalling (44) and (54), we seethat for every ( a, n ) ∈ I × Z , we have that R b,ma,n ( X ) = A b,ma,n ( X ) + B b,ma,n ( X )where A b,ma,n ( X ) ∈ C [ X c,p ] ( c,p ) ∈± A , B b,ma,n ( X ) ∈ M ( c,p ) ∈∓ A C X c,p , for all ( b, m ) ∈ ± A .Now we shall argue that ω = − c a,n [ − c b,m [0] ∂A c,pa,n ∂ γ d,q [0] ∂A d,qb,m ∂ γ c,p [0] | i + ∂B c,pa,n ∂ γ d,q [0] ∂B d,qb,m ∂ γ c,p [0] | i ! − c a,n [0] c b,m [0] (cid:20) T, ∂A c,pa,n ∂ γ d,q [0] (cid:21) ∂A d,qb,m ∂ γ c,p [0] | i . (99)Indeed, we see all A - B cross terms are zero just by inspecting the index contractions.The remaining term is − c a,n [0] c b,m [0] h T, ∂B c,pa,n ∂ γ d,q [0] i ∂B d,qb,m ∂ γ c,p [0] | i but this is zero by thelinearity of the B b,ma,n and the fact that [ T, 1] = 0. So we have the equality (99). Theterm ∂B c,pa,n ∂ γ d,q [0] ∂B d,qb,m ∂ γ c,p [0] | i is proportional to | i , again by the linearity of B b,ma,n . And bydefinition of the A b,ma,n , the A - A terms all belong to Λ ⊗ ( M ( n ) + M ( n − )). So wehave established that ω ∈ Λ ⊗ ( M ( n ) + M ( n − ))and hence the result. (cid:3) Lemma 44. There is an isomorphism of complexes C τ, • loc ( LL ˚ g , L ( M ( n ) + M ( n − ))) ∼ = C • loc ( LL ˚ g , L ( M ( n ))) Proof. There is certainly a linear isomorphismΛ ⊗ M ( n ) ∼ −→ Λ ⊗ ( M ( n ) + M ( n − )) v (1 + τ ) v, and hence a linear isomorphism L (Λ ⊗ M ( n )) ∼ = L (Λ ⊗ ( M ( n ) + M ( n − ))). Wehave (1 + τ )( Q (0) v ) = Q (0) v + ( τ Q ) (0) τ v = Q (0) v + Q (0) τ v = Q (0) (1 + τ ) v , so thisisomorphism commutes with the differential. (cid:3) NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 59 Finally we can complete the proof of Theorem 28.We have shown that our cocycle e ρ ∗ ( ω ) = ω [0] belongs to C τ, ( LL ˚ g , L ( M ( n ) + M ( n − ))) and therefore corresponds to a cocycle ζ ∈ C ( LL ˚ g , L ( M ( n ))). By thekey lemma, Lemma 40, such a cocycle is cohomologous to zero if its restriction to L ˚ h vanishes. The restriction of ζ to L ˚ h is zero if the restriction of e ρ ∗ ( ω ) to L ˚ h is zero.And the restriction of e ρ ∗ ( ω ) to L ˚ h is indeed zero because, for all i ∈ ˚ I , ρ ( J i, ) = J i, (and hence R b,ni, ( X ) = 0).Thus there exists a 1-cochain ξ ∈ C ( LL ˚ g , L ( M ( n ))) such that ζ = dξ . It maybe written in the form ξ = Ξ[0] , with Ξ := c a,n [0] X ( b,m ) ∈ A Q a,n ; b,m ( γ [0]) γ b,m [ − | i for some polynomials Q a,n ; b,m ( X ) ∈ O ( n ), ( b, m ) ∈ A , such that Ξ has Q -grade 0.In this way we obtain a 1-cochain, (1 + τ ) ξ ∈ C τ, ( LL ˚ g , L ( M ( n ) + M ( n − )), suchthat e ρ ∗ ( ω ) = d (1 + τ ) ξ , as required.6. Proof of Theorem 32 In Section 2.11 we studied the infinitesimal right action of e g on the right coset space U ( C ǫ ) := exp( ǫ b − ) / exp( f ǫ ). In the same way we may consider the right action of e g on the right coset space exp( ǫ n − ) / exp( f ǫ ). We get the following analog of Lemma 5and Lemma 6. Lemma 45. Let A ∈ e g . Then there exist polynomials n P b,mA ( X ) ∈ O ( n ) o ( b,m ) ∈ A and { p iA ( X ) ∈ O ( n ) } i ∈ I (depending linearly on A ) such that e ǫ n − e P dim h i =1 y i b i −→ Y ( b,m ) ∈ A e x b,m J b,m e ǫA = e ǫ n − e P dim h i =1 ( y i + ǫp iA ( x ) ) b i −→ Y ( b,m ) ∈ A e (cid:16) x b,m + ǫP b,mA ( x ) (cid:17) J b,m for every element e P dim h i =1 y i b i −→ Q ( b,m ) ∈ A e x b,m J b,m of the group B := H ⋉ U .Hence, the linear map e g → g Der O ( n ) ⋉ dim h M i =1 O ( n ) ∂ Y i given by A dim h X i =1 p iA ( X ) ∂ Y i + X ( b,m ) ∈ A P b,mA ( X ) D b,m is a homomorphism of Lie algebras. It respects the Q -gradation (where we assign Q -grade zero to the generators ∂ Y i ). (cid:3) Explicitly, this homomorphism sends e i ρ ( e i ) , h ρ ( h ) + dim h X j =1 (cid:10) b j , h (cid:11) ∂ Y j , f i ρ ( f i ) + dim h X j =1 (cid:10) b j , ˇ α i (cid:11) X e i ∂ Y j , where X e i := X i are as in (101). Let X f i := τ X e i . It follows that there is ahomomorphism of Lie algebras (here, recall (55)) ρ ′ : g → D ⋉ dim h M i =1 O ∂ Y i defined by e i ρ ( e i ) + dim h X j =1 (cid:10) b j , ˇ α i (cid:11) X f i ∂ Y j ,h ρ ( h ) + dim h X j =1 (cid:10) b j , h (cid:11) ∂ Y j ,f i ρ ( f i ) + dim h X j =1 (cid:10) b j , ˇ α i (cid:11) X e i ∂ Y j , for i ∈ I and h ∈ h . This yields a homomorphism of the loop algebras, just as in (77), L g → L D ⋉ dim h M i =1 O ∂ Y i ! . Now we note that( M ⊗ π )[0] = C O and ( M ⊗ π )[1] = C Ω O ⊕ D ⋉ dim h M i =1 O ∂ Y i ! where we continue to identify O and Ω O with subspaces of M = M ⊗ C | i as before,and we extend the definition of the injective linear map , (70), by setting dim h X i =1 p i ( X ) ∂ Y i ! = dim h X i =1 p i ( γ [0]) b i [ − | i . As in Lemma 26 and Lemma 27, we then find that there is an isomorphism of Liealgebras L D ⋉ dim h M i =1 O ∂ Y i ! ∼ = L (( M ⊗ π )[ ≤ (cid:14) L ( O ⊕ Ω O ) . We get an exact sequence of Lie algebras0 → L ( O ⊕ Ω O ) → L (( M ⊗ π )[ ≤ → L D ⋉ dim h M i =1 O ∂ Y i ! → NALOG OF FEIGIN-FRENKEL HOMOMORPHISM FOR DOUBLE LOOP ALGEBRAS 61 As in the proof of Lemma 27, the cocycle defining this extension is given by doublecontraction terms in the OPE. Following [Fre07], the key observation is then thatthere are no possible double contractions between the new terms we have added(which belong to the subspace O ⊗ π ⊂ M ⊗ π ) and the existing terms (which eachhave at most one factor of β or S ). It follows that the statement of Theorem 28 stillholds (with the same lifting map φ ) when ρ is replaced by the map ρ ′ above. Appendix A. The coefficients c i In this section, for i ∈ I , let us write X i := X a,n for the unique ( a, n ) such that J a,n = e i , i.e. X i := ( X α i , , i ∈ ˚ I = I \ { } X − δ + α , , i = 0 (101)and similarly X [ i,j ] := X a,n for the unique ( a, n ) such that J a,n ∝ [ e i , e j ]. Define D i , D [ i,j ] likewise. Recall the homomorphism ρ : e g → g Der O ( n ) from Lemma 6. Lemma 46. Let i ∈ I . The terms in ρ ( f i ) of the form X i X a,n D a,n for some ( a, n ) ∈ A are − X i X i D i + X j ≺ i a ij X i X [ i,j ] D [ i,j ] − X j ≺ i a ij X i X j D j . Proof. This follows from a direct calculation, of the sort in the proof of Lemma 5 andTheorem 10. Let us give the outline. We have ρ ( f i ) = X ( a,n ) ∈ A P a,nf i ( X ) D a,n . (102)By inspection, one sees that if P a,nf i ( X ) is to have both X a,n and X i as factors, thenit must be that [ f i , J a,n ] is proportional to a basis vector that precedes e i in our basis.(We have to pick up the dependence on X a,n as exp( ǫf i ) moves leftwards through theproduct, and then pick up the dependence on X i as some term is pushed throughexp (cid:0) x i e i (cid:1) .) This is a strong constraint: we must have either(1) J a,n = e j for some j ∈ I , or(2) J a,n ∝ [ e i , e j ] for some j ∈ I such that e j ≺ e i .Let us compute the coefficients of the resulting terms. Consider case (2): suppose e j ≺ e i and J a,n = c [ e i , e j ] for some nonzero c ∈ C , so that [ J a,n , f i ] = c [[ e i , e j ] , f i ] = c [ ˇ α i , e j ] = ca ij e j . We haveexp( x a,n J a,n ) exp( ǫf i )= exp( ǫf i ) (cid:16) exp( ǫx a,n [ J a,n , f i ]) exp (cid:0) ǫ [ J a,n , [ J a,n , f i ]] (cid:1) . . . (cid:17) exp( x a,n J a,n )= exp( ǫf i ) exp( ǫx a,n ca ij e j ) exp( x a,n J a,n ) . . . . Here exp( ǫx a,n ca ij e j ) still needs to be pushed left through the factor exp (cid:0) x i e i (cid:1) whichappears further left in the product. We getexp (cid:0) x i e i (cid:1) exp( ǫx a,n ca ij e j ) = exp( ǫx a,n ca ij e j ) exp (cid:0) ǫx i x a,n a ij c [ e i , e j ] (cid:1) exp (cid:0) x i e i (cid:1) . . . . We obtain the terms P j ≺ i a ij X i X [ i,j ] D [ i,j ] in ρ ( f i ).Now consider terms J a,n = e j for some j . When i ≺ j we just getexp (cid:0) x j e j (cid:1) exp( ǫf i ) = exp( ǫf i ) exp (cid:0) x j e j (cid:1) since [ e j , f i ] = 0. Eventually we reach the factor exp (cid:0) x i e i (cid:1) . We continue to shuffleterms, gettingexp (cid:0) x i e i (cid:1) exp( ǫf i ) = exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) exp (cid:18) ǫ 12 ( − x i x i e i ) (cid:19) exp (cid:0) x i e i (cid:1) = exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) exp (cid:0)(cid:0) x i − ǫx i x i (cid:1) e i (cid:1) and then finally we have to move exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) further left through factorsexp (cid:0) x j e j (cid:1) with j ≺ i :exp (cid:0) x j e j (cid:1) exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) = exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) exp (cid:0) − ǫx j x i a ij e j (cid:1) exp (cid:0) x j e j (cid:1) = exp( ǫf i ) exp (cid:0) ǫx i ˇ α i (cid:1) exp (cid:0)(cid:0) x j − ǫx j x i a ij (cid:1) e j (cid:1) From these last two expressions we read off the terms − X i X i D i and − P j ≺ i a ij X i X j D j in ρ ( f i ). (cid:3) For all h ∈ h , we have ρ ( h ) = − X ( a,n ) ∈ A h wgt( a, n ) , h i X a,n D a,n (103)and hence ρ ( h )( X i ) = − h α i , h i X i (as it certainly should, on Q -grading grounds). Proposition 47. For all h ∈ h and for each i ∈ I , we have ( ρ ( h )) (1) ( ρ ( e i )) = 0 , ( ρ ( h )) (1) ( ρ ( f i )) = − c i ρ ( h )( X i ) , with c i as in Proposition 30.Proof. Recall the Wick lemma from Section 4.6. First products, like ( ρ ( h )) (1) ( ρ ( f i )),involve a double contraction. In view of (102) and (103), and then making use of thelemma above, we find ( ρ ( h )) (1) ( ρ ( f i )) = X ( a,n ) ∈ A h wgt( a, n ) , h i (cid:16) D a,n P a,nf i ( X ) (cid:17) = − h α i , h i X i + X j ≺ i a ij h α i + α j , h i X i − X j ≺ i a ij h α j , h i X i = h α i , h i − X j ≺ i a ij X i = − c i ρ ( h )( X i ) . (cid:3) Therefore ( ρ ( h )) (1) ( ( ρ ( f i )) + c i γ e i [ − | i ) = 0for each i ∈ I . This shows that φ ( f i ) = c i γ e i [ − | i and, hence, φ ( e i ) = c i γ f i [ − | i . 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