aa r X i v : . [ m a t h . QA ] D ec SUPEROPERS ON SUPERCURVES
ANTON M. ZEITLIN
Abstract.
In this note, we introduce the generalization of op-ers (superopers) for a certain class of superalgebras, which havepure odd simple root system. We study in detail
SP L -superopersand in particular derive the corresponding Bethe ansatz equations,which describe the spectrum of osp (2 |
1) Gaudin model. Introduction
Opers are necessary ingredients in the study of the geometric Lang-lands correspondence (see e.g. [12]). They also play important role inmany aspects of mathematical physics. For example, opers are very im-portant in the theory of integrable systems, and recently they becamea necessary component even in the modern Quantum Field Theoryapproaches to the knot theory (see e.g. [24]).Originally, opers were studied locally in the seminal paper of Drinfeldand Sokolov [8] as gauge equivalence classes of certain differential oper-ators with values in some simple Lie algebra, which are the L-operatorsof the generalized Korteweg-de Vries (KdV) integrable models. Later,Beilinson and Drinfeld generalized this local object making it coordi-nate independent [2]. Namely, a G -oper on a smooth curve Σ, where G is a simple algebraic group of the adjoint type with the Lie algebra g , is a triple ( F , F B , ∇ ), where F is a G -bundle over Σ, F B is its B -reduction with respect to Borel subgroup B , and ∇ is a flat connection,which behaves in a certain way with respect to F B . For example, inthe case of P GL -oper, this condition just means that the reduction F B is nowhere preserved by this connection. Moreover, it appears, fol-lowing the results of Drinfeld and Sokolov, that the space of G -opers isequivalent to a certain space of scalar pseudodifferential operators. Inthe P GL case, the resulting space of scalar operators is just a familyof Sturm-Liouville operators and the connection transformation prop-erties allows to consider them on all Σ as projective connections.A really interesting story starts when we allow opers to have reg-ular singularities. It turns out that the opers on the projective linecan be described via the Bethe ansatz equations for the Gaudin model corresponding to the Langlands dual Lie algebra [11], [13]. An im-portant object on the way to understand this relation is the so-calledMiura oper, which was introduced by E. Frenkel [13]. A M iura oper is an oper with one extra constraint: the connection preserves another B -reduction of F , which we call F ′ B . The space of the Miura opers,associated to a given G -oper with trivial monodromy, is isomorphic tothe flag manifold G/B . If the reduction F ′ B corresponds to the pointin a big cell of G/B , then such a Miura oper is called generic . It wasshown by E. Frenkel that any Miura oper is generic on the punctureddisc and that there is an isomorphism between the space of genericMiura opers on the open neighborhood with certain H -bundle connec-tions ( H = B/ [ B, B ]) [13]. The map from H -connections to G -opersis just a generalization of the standard Miura transformation in thetheory of KdV integrable models.By means of the above relation with the H -connections, it was provedfor P GL -oper in [11] and then generalized to the higher rank in [9],[13] that the eigenvalues of the Gaudin model for a Langlands dualLie algebra g L can be described by the G -opers on C P with givenregular singularities and trivial monodromy. Namely, the consistencyconditions for the H -connections underlying such opers coincide withthe Bethe ansatz equations for the Gaudin Model.In this article, we are trying to generalize some of the above notionsand results on the level of superalgebras. We define an analogue of theoper in the case of supergroups which allow the pure fermionic familyof simple roots on a super Riemann surface, following some local con-siderations of [14], [7], [17]. We call such objects superopers , and insome sense they turn out to be “square roots” of standard opers. Un-fortunately for all other superalgebras, the resulting formalism allowsonly locally defined objects (on a formal superdisc). We study in detailthe simplest nontrivial case of superoper, related to the group SP L (see e.g. [5]), related to superprojective transformations, and explic-itly establish the relation between osp (2 |
1) Gaudin model studied in[16] and the
SP L -oper on super Riemann sphere with given regularsingularities.In section 2 we explain the relation between super projective struc-tures on super Riemann surface and the supersymmetric version of theSturm-Liouville operator. Then we relate it to the flat connection on SP L -bundle which will give us the first example of superoper.In Section 3 we use this experience to generalize the notion of super-oper to the case of higher rank simple supergroups. However, only thesupergroups which permit a pure fermionic system of simple roots allowus to construct a globally defined object on a super Riemann surface. UPEROPERS ON SUPERCURVES 3
We define Miura superopers and superopers with regular singularitiesin section 4. There we study the consistency conditions for the super-opers on the superconformal sphere and derive the corresponding Betheequations. We compare the results with the osp (2 |
1) Gaudin model andfind that the Bethe ansatz equations coincide with the “body” part ofthe consistency condition for corresponing
SP L Miura superopers.Some remarks and open questions are given in section 5.
Acknowledgments.
I am very grateful to I. Penkov for useful dis-cussions and to D. Leites for pointing out important references. I amindebted to E. Frenkel and E. Vishnyakova for comments on the man-uscript.2.
Superprojective structures, super Sturm-Liouvilleoperator and osp (2 | superoper We remind that a supercurve of dimension (1 |
1) (see e.g. [4])over some Grassman algebra S is a pair ( X, O X ), where X is a topolog-ical space and O X is a sheaf of supercommutative S -algebras over X such that ( X, O red X ) is an algebraic curve (where O red X is obtained from O X by quoting out nilpotents) and for some open sets U α ⊂ X andsome linearly independent elements { θ α } we have O U α = O red U α ⊗ S [ θ α ].These open sets U α serve as coordinate neighborhoods for supercurveswith coordinates ( z α , θ α ). The coordinate transformations on the over-laps U α ∪ U β are given by the following formulas z α = F αβ ( z β , θ β ), θ α = Ψ αβ ( z β , θ β ), where F αβ , Ψ αβ are even and odd functions corre-spondingly. A super Riemann surface Σ over some Grassmann algebra S (for more details see e.g. [25]) is a supercurve of dimension 1 | S , with one more extra structure: there is a subbundle D of T Σ ofdimension 0 |
1, such that for any nonzero section D of D on an opensubset U of Σ, D is nowhere proportional to D , i.e. we have the exactsequence: 0 → D → T Σ → D → . (1)One can pick the holomorphic local coordinates in such a way that thisodd vector field will have the form f ( z, θ ) D θ , where f ( z, θ ) is a nonvanishing function and: D θ = ∂ θ + θ∂ z , D θ = ∂ z . (2) ANTON M. ZEITLIN
Such coordinates are called superconf ormal . The transformation be-tween two superconformal coordinate systems ( z, θ ), ( z ′ , θ ′ ) is deter-mined by the condition that D should be preserved, i.e.: D θ = ( D θ θ ′ ) D θ ′ , (3)so that the constraint on the transformation coming from the localchange of coordinates is D θ z ′ − θ ′ D θ θ ′ = 0. An important nontrivialexample of a super Riemann surface is the Riemann super sphere SC ∗ :there are two charts ( z, θ ), ( z, θ ′ ) so that z ′ = − z , θ ′ = θz . (4)There is a group of superconformal transformations, usually denotedas SP L which acts transitively on SC ∗ as follows: z → az + bcz + d + θ γz + δ ( cz + d ) ,θ → γz + δcz + d + θ δγcz + d , (5)where a, b, c, d are even, ad − bc = 1, and γ, δ are odd. The Lie algebraof this group is isomorphic to osp (2 | D n the superconf ormal f ields ofdimension − n/
2. In particular, taking the dual of the exact sequence 1,we find that a bundle of superconformal fields of dimension 1 (i.e. D − )is a subbundle in T ∗ Σ. Considering the superconformal coordinatesystem, a nonzero section of this bundle is generated by η = dz − θdθ ,which is orthogonal to D θ under standard pairing.At last, we introduce one more notation. For any element A whichbelongs to some free module over S [ θ ], where θ is a local odd coor-dinate, we denote the body of this element (i.e. A is stripped of thedependence on the odd variables) as ¯ A . Let us at first define what a superprojective connection is. Weconsider the following differential operator, defined locally with coor-dinates ( z, θ ): D θ − ω ( z, θ ) . (6)The following proposition holds. UPEROPERS ON SUPERCURVES 5
Proposition 2.1. [21]
Formula (6) defines the operator L , such that L : D − → D (7) iff the transformation of ω on the overlap of two coordinate charts ( z, θ ) , ( z ′ , θ ′ ) is given by the following expression: ω ( z, θ ) = ω ( z ′ , θ ′ )( D θ θ ′ ) + { θ ′ ; z, θ } (8) where { θ ′ ; z, θ } = ∂ z θ ′ D θ θ − ∂ z θD θ θ ′ ( D θ θ ′ ) (9) is a supersymmetric generalization of Schwarzian derivative. One can show that the only coordinate transformations for which thesuper Schwarzian derivative vanishes, are the fractional linear transfor-mations (5).Let us consider the covering of Σ by open subsets, so that the tran-sition functions are given by (5). Two such coverings are consideredequivalent if their union has the same property of transition functions.The corresponding equivalence classes are called superprojective struc-tures .It appears that like in the pure even case, there is a bijection be-tween super projective connections and super projective structures.For a given super projective structure one can define a superprojectiveconnection by assigning operator D θ in every coordinate chart. FromProposition 2.1 we find that the resulting object is defined globally onΣ. On the other hand, given a super projective connection on Σ, onecan consider the following linear problem:( D θ − ω ( z, θ )) ψ ( z, θ ) = 0 . (10)From the results of [1] we know that this equation has 3 independentsolutions: two even x ( z, θ ), y ( z, θ ) and one odd ξ ( z, θ ). Defining C = y/x , α = ξ/x , we find that ω ( z, θ ) is expressed via super Schwarzianderivative, i.e. w ( z, θ ) = { α ; θ, z } and the consistency conditions on C and α are such that C can be represented in terms of α in the followingway: C = cA + γAα + δα, (11)where A is such a function that ( z, θ ) → ( A, α ) is a superconformaltransformation. In a different basis (
A, α ) will be transformed via
SP L (5) and hence ( A, α ) form natural coordinates for a projective structureon Σ. Therefore we have the following proposition.
ANTON M. ZEITLIN
Proposition 2.2
There is a bijection between the set of superprojectivestructures and the set of superprojective connections on Σ . Let us consider a vector bundle V over the super Riemannsurface with the fiber C m | nS . Let E (Σ , V ) be the space of sections on V over Σ and let E (Σ , V ) be the space of 1-form valued sections. Asusual, the connection is a differential operator d A ( f s ) = df ⊗ s + ( − | f | f d A s, (12)where f is a smooth even/odd function on Σ and s ∈ E (Σ , V ). Locally,in the chart ( z, θ ) the connection has the following form: d A = d + A = d + ( ηA z + dθA θ ) + (¯ ηA ¯ z + d ¯ θA ¯ θ ) =( ∂ + ηA z + dθA θ ) + ( ¯ ∂ + ¯ ηA ¯ z + d ¯ θA ¯ θ ) =( ηD Az + dθD Aθ ) + (¯ ηD A ¯ z + d ¯ θD A ¯ θ ) . (13)We note that we used here the fact that d = ∂ + ¯ ∂ and ∂ = η∂ z + dθD θ .The expression for the curvature is: F = d A = dθdθF θθ + ηdθF zθ + d ¯ θd ¯ θF ¯ θ ¯ θ + ¯ ηd ¯ θF ¯ z ¯ θ + η ¯ ηF z ¯ z + ηd ¯ θF z ¯ θ + ¯ ηdθF ¯ zθ + dθd ¯ θF θ ¯ θ , (14)where F θθ = − D Aθ + D Az , F zθ = [ D Az , D Aθ ], F z, ¯ z = [ D Az , D A ¯ z ], F z ¯ θ =[ D Az , D A ¯ θ ], F θ ¯ θ = − [ D Aθ , D A ¯ θ ], etc.It appears that if the connection d A offers partial flatness, which im-plies F θθ = F zθ = F ¯ θ ¯ θ = F ¯ z ¯ θ = 0, then there is a superholomorphicstructure on V (i.e. transition functions of the bundle can be madesuperholomorphic) [22]. We are interested in the flat superholomor-phic connections. In this case, since F θθ = 0, the connection is fullydetermined by the D Aθ locally. In other words it is determined by thefollowing odd differential operator, which from now on will denote ∇ and call long superderivative : ∇ = D θ + A θ ( z, θ ) , (15)which gives a map: D →
EndV so that the transformation propertiesfor A θ are: A θ → gA θ g − − D θ gg − , where g is a superholomorphicfunction providing change of trivialization. SP L -opers. In this subsection, we give a description of the firstnontrivial superoper. Suppose we have a superprojective structure onΣ. Naturally we have a structure of a flat
SP L -bundle F over Σ, UPEROPERS ON SUPERCURVES 7 since on on the overlaps there is a constant map to
SP L . Let usstudy the corresponding flat connection on Σ. Since SP L is a groupof superconformal automorphisms of SC ∗ , one can form an associatedbundle SC ∗F = F ×
SP L SC ∗ . This bundle has a global section which isjust given by the superprojective coordinate functions ( z, θ ) on Σ. Wenote that it has nonvanishing (super)derivative at all points.One can view SC ∗ as a flag supermanifold. Namely, consider thegroup SP L acting in C | = span ( e , ξ, e ), where we put the odd vec-tor in the middle. Then e is stabilized by the Borel subgroup of uppertriangular matrices. Therefore, one can identify SC ∗ with SP L /B .Since we have a nozero section of SC ∗F , we have a B -subbundle F B ofa G -bundle, where G stands for SP L . Hence, a superprojective struc-trure gives the flat SP L -bundle F with a reduction F B . However,there is one more piece of data we can use: it is the condition that the(super)derivative of the section of SC ∗F is nowhere vanishing. It meansthat the flat connection on F does not preserve the B -reduction any-where. Let us figure out which conditions does it put on the connectionif we choose a trivialization of F induced from the F B trivialization.As we discussed above, the connection is determined by the followingodd differential operator: ∇ = D θ + α ( z, θ ) b ( z, θ ) β ( z, θ ) − a ( z, θ ) 0 b ( z, θ ) γ ( z, θ ) a ( z, θ ) − α ( z, θ ) , , (16)so that the matrix is in the defining representation of the Lie superal-gebra of SP L , namely osp (2 | SC ∗ . Since we havethe condition that both of them are nonvanishing, and identifying tan-gent space with osp (2 | / b (where b is the Borel subalgebra), we obtainthat a is nonvanishing. It is possible to make γ = 0, by redefining ∇ byadding µ ( ∇ ) with appropriate odd function µ , which just correspondsto the choice of superconformal coordinates on SC ∗ . We call such atriple ( F , F B , ∇ ) a superoper . We notice that taking the square ofthe odd operator ∇ , reducing such even operator from Σ to the under-lying curve Σ and getting rid of all the odd variables, we obtain theoper connection for the P GL -bundle. Thus superopers can be thoughtabout as “square roots” of opers.Using B -valued gauge transformations one can bring ∇ θ to the canon-ical form: ∇ = D θ + ω ( z, θ ) − . (17) ANTON M. ZEITLIN
Therefore on a superdisc with coordinate ( z, θ ) the space of
SP L superopers can be identified with the space of differential operators D θ − ω ( z, θ ). We will see in the next section that the coordinate trans-formations of ω are the same as in Proposition 2.1.Therefore we see that there is a full analogy with the bosonic case,where the space of P GL -superopers was identified with the set of pro-jective connections or equivalently with the set of projective structures.Let us summarize the results of this section in the following theorem. Theorem 2.3.
There are bijections between the following three sets ona super Riemann surface Σ :i) Superprojective structuresii) Superprojective connectionsiii) SP L -opers. Superopers for higher rank superalgebras
In this section we generalizethe results of the previous section to higher rank. Suppose G is a sim-ple algebraic supergroup [3] of adjoint type over Grassmann algebra S , B is its Borel subgroup, N = [ B, B ], so that for the correspond-ing Lie superalgebras we have n ⊂ b ⊂ g . Note that g = S ⊗ g red ,where g red is a simple Lie superalgebra over C . As usual, H = B/N with the Lie algebra h and there is a decomposition: g = n − ⊕ h ⊕ n + .The corresponding generators of simple roots will be denoted as usual: e , . . . , e l ; f , . . . , f l . We are interested in the superalgebras, which havea pure fermionic system of simple roots, namely psl ( n | n ), sl ( n + 1 | n ), sl ( n | n + 1), osp (2 n ± | n ), osp (2 n | n ), osp (2 n + 2 | n ) with n ≥ D (2 , α ) with α = 0 , ±
1. Moreover, a necessary ingredient forour construction is the presence of the embeddining of superprinci-pal osp (1 |
2) subalgebra [10], [6], namely that for χ − = P i f i andˇ ρ = P i ˇ ω i , where ˇ ω i are fundamental coweights, there is such χ thatmakes a triple ( χ , χ − , ˇ ρ ) an osp (1 |
2) superalgebra. Almost all seriesof superalgebras from the list above allow such an embedding, however, psl ( n | n ) does not and we do not consider these series in this article.As in the standard bosonic case we define an open orbit O ⊂ [ n , n ] ⊥ / b consisting of vectors, stabilized by N and such that all the negative rootcomponents of these vectors with respect to the adjoint action of H arenon-zero.Let us consider a principal G -bundle F over X , which can be a su-per Riemann surface Σ or a formal superdisc SD x = Spec S [ θ ][[ z ]], or UPEROPERS ON SUPERCURVES 9 a punctured superdisc D Sx × = Spec S [ θ ](( z )) (see e.g. [18] or [15] forthe definitions of the spectra of supercommutative rings), and its re-duction F B to the Borel subgroup B . We assume that it has a flatconnection determined by a long superderivative ∇ (see (15)). Ac-cording to the example, considered in section 2 we do not want ∇ topreserve F B . However, in the higher rank case this is not enough, sowe have to specify extra conditions. Namely, suppose ∇ ′ is anotherlong superderivative, which preserves F B . Then we require that thedifference ∇ ′ − ∇ has a structure of superconformal field of dimension1 / g F B . We can project it onto( g / b ) F B ⊗ D − . Let us denote the resulting ( g / b ) F B -valued supercon-formal field as ∇ / F B . Now we are ready to define the superoper, whichis a natural generalization of the oper.A G − superoper on X is the triple ( F , F B , ∇ ), where F is a principle G -bundle, F B is its B -reduction and ∇ is a long superderivative on F ,such that ∇ / F B takes values in O F B .Locally this means that in the coordinates ( z, θ ) and with respect tothe trivialization of F B , the structure of the long superderivative is: D z,θ + l X i =1 a i ( z, θ ) f i + µ ( z, θ ) , (18)where each a i ( z, θ ) is an even nonzero function (meaning that thesefunctions have nonzero body and are invertible) and µ ( z, θ ) is an odd b -valued function. Therefore locally on the open subset U , where wechose coordinates ( z, θ ), the space of G -superopers on U , which willbe denoted as s Op G ( U ), can be characterized the space of all odd op-erators of type (18) modulo gauge transformations from B ( R ) group,where R are either analytic or algebraic functions on U . Let usnotice that one can use the H -action to make the operator (18) lookas follows: D θ + l X i =1 f i + µ ( z, θ ) , (19)where µ ∈ b ( R ). Therefore the space s Op G ( U ) can be considered asthe quotient of the space of operators of the form (19) (denoted as g sOp G ( U )) by the action of N ( R ). As in the pure bosonic case, ˇ ρ givesa principal gradation (for those classes of superalgebras we consider),i.e. we have a direct sum decomposition b = ⊕ i ≥ b i . Moreover, letus remind that we denoted χ − = P li =1 f i and there exists a unique element χ of degree 1 in b , such that χ ± , ˇ ρ generate osp (1 |
2) super-algebra. Let ˜ χ k ( k = 1 , . . . , l ) (which can be either odd or even), sothat ˜ chi = χ , be the basis of the space of the ad ( χ ) invariants. Wenote, that the decompositions of g with respect to the adjoint actionof such osp (1 |
2) triple were studied in [10]. Based on that, we have thefollowing Lemma which is proved in a similar way as in [8] (see alsoLemma 4.2.2 of [12]).
Lemma 3.1.
The gauge action of N ( R ) on ^ s Op G ( U ) is free and eachgauge equivalence class contains a unique operator of the form (19)with µ ( θ, z ) = l X i =1 g i ( z, θ ) ˜ χ i , (20) where g i has opposite parity to χ i . Now let us discuss the transformation properties of operators ^ s Op G ( U ).Assume we have a superconformal coordinate change ( z, θ ) = ( f ( w, ξ ) , α ( w, ξ )).Then according to the transformations of the long derivative we have ∇ =(21) D ξ + ( D ξ α )( w, ξ ) χ − + ( D ξ α )( w, ξ )( µ ( f (( w, ξ ) , α ( w, ξ )) . Considering 1-parameter subgroup C × S | → H which corresponds to ˇ ρ ,applying adjoint transformation with ˇ ρ ( D ξ α ) we obtain: D ξ +(22) χ − + ( D ξ α )( w, ξ ) Ad ˇ ρ ( D ξ α ) · µ ( f (( w, ξ ) , α ( w, ξ )) − ∂ w α ( w, ξ ) D ξ α ( w, ξ ) ˇ ρ. This gives us the gluing formula for superopers on any super Riemannsurface Σ.Consider the H -bundle D − ˇ ρ on Σ, which is determined by the prop-erty that the line bundle D − ˇ ρ × C λ is D −h ˇ ρ,λ i , where λ is from the latticeof characters and C λ is the corresponding 1-dimensional representation.The coordinate transformation formulas for superoper connectionimmediately lead to another characterization of this bundle via F B -reduction. The following statement is the supersymmetric version ofLemma 4.2.1 of [12]. Lemma 3.2.
The H-bundle F H = F B × B H = F B /N is isomorphic to D − ˇ ρ . UPEROPERS ON SUPERCURVES 11
Now one can derive the transformation properties for the canonicalrepresentatives of opers from Lemma 3.1, which will provide the trans-formation formulas for g , . . . , g n . In order to do that, one needs toapply to the operator (19) the gauge transformation of the formexp (cid:16) κχ −
12 ( Dκ )[ χ , χ ] (cid:17) ˇ ρ ( D ξ α ) , (23)where κ = ∂ w α ( w,ξ ) D ξ α . Then we have that˜ g ( w, ξ ) = g ( w, ξ )( D ξ α ) , ˜ g ( w, ξ ) = g ( w, ξ )( D ξ α ) + { α ; w, ξ } , ˜ g j ( w, ξ ) = g j ( w, ξ )( D ξ α ) d j +1 , j > . (24)Therefore (23) are transition functions for F B and F bundles. Remark.
Note that the g -term is absent in the osp (1 | sl (2 | ∼ = osp (2 | s Op G (Σ) ∼ = sP roj (Σ) × ⊕ lj =1 ,j =2 Γ(Σ , D − d j − ) , (25)where sP roj (Σ) stands for superprojective connections on Σ.In the previous section we indicated that in the osp (1 |
2) one canintroduce the oper related to a superoper, by considering ∇ , thenstripping it from the θ and S dependence, we obtain that the resulting ∇ has all the needed properties of sl (2) oper on the curve X which isa base manifold for Σ.A similar construction is possible in the higher rank case. Let G be the reductive group, which is a base manifold for G . Due to thestructure of the coordinate transformations we derived above, we findout that indeed ∇ = ∇ defines an oper on X . We refer to this objectas G - oper , associated with the G -superoper, which we will denote astriple ( F , ∇ , F B ), where F , F B denote the appropriate pure evenreductions of the principal bundles.4. Superopers with regular singularities, Miurasuperopers and Bethe ansatz equations . Consider a point x onon the superc Riemann surface Σ and the formal superdisc SD x aroundthat point with the coordinates ( z, θ ). We define a G -superoper with regular singularity on SD x as an operator of the form D θ + X a i ( z, θ ) f i + (cid:16) µ ( z ) + θz µ ( z ) (cid:17) , (26)modulo the B ( K x )-transformations ( K x = C [ θ ](( z ))), where a i ( z, θ ) ∈O x are nowhere vanishing and invertible, µ i ( z, θ ) ∈ b ( K x ) ( i = 0 , µ i ( z, θ ), i.e. µ i ∈ b red ( O red x ). As before, onecan eliminate a i -dependence via H -transformations, therefore we cantalk about N ( K x ) equivalence class of operators of the type (26) with a i = 1. Let us denote by sOp RSG ( SD x ) the space of superopers withregular singularity. Clearly, we have the embedding: sOp RSG ( SD x ) ⊂ sOp G ( SD × x ).The G -oper, corresponding to G -superoper (26) is the oper withregular singularity. It has the following form: ∂ + χ − + [ χ − , µ ] + ( µ ) + 1 z ( µ ) , (27)which can be transformed to the standard form via the gauge transfor-mation by means of ρ ( z ): ∂ z + 1 z (cid:16) χ − − ˇ ρ Ad ˇ ρ ( z ) · ¯ µ (cid:17) + v ( z ) , (28)where v ( z ) is regular.Denoting − ˇ λ the projection of µ on h , we find that the residue ofthis differential operator is equal to χ − − λ − ˇ ρ , however since this isan oper, only the corresponding class in h /W is well defined, and wedenote it as ( − λ − ˇ ρ ) W , i.e. this oper belongs to Op RSG ( D x ) ˇ λ , see e.g.[11].Let us refer to the space of superopers with regular singularity suchthat ¯ µ (0) = ˇ λ , as s Op RSG ( D x ) λ .If we consider the representation V of G one can talk about a systemof differential equations ∇ · φ V ( z, θ ) and their monodromy like in thepure even case.Let ˇ λ be the dominant integral coweight and let us introduce thefollowing class of operators: ∇ = D θ + (cid:16) X a i ( z, θ ) f i + µ ( z, θ ) (cid:17) , (29)where a i = z h α i , ˇ λ i ( r i ( θ ) + z ( . . . )), so that the body of r i is nonzero, and µ ( z, θ ) ∈ b ( O x ). We call the quotient of the space of operators aboveby the action of B ( O x ) as s Op G ( SD x ) λ .The following Lemma is an analogue of Lemma 2.4. of [11]. UPEROPERS ON SUPERCURVES 13
Lemma 4.1.
There is an injective map i : s Op G ( SD x ) ˇ λ → sOp ( SD × x ) ,so that Im i ⊂ s Op RSG ( SD x ) ˇ λ . The image of i is a subset in the set ofthose elements of s Op RSG ( SD x ) ˇ λ , such that the resulting oper has a triv-ial monodromy around x. Remark.
Notice that the superopers corresponding to s Op G ( SD x ) ˇ λ belong to Op G ( D x ) ˇ λ . However, here ˇ λ is the integral dominant weightfor Lie superalgebra. If we consider λ to be an integral dominant weightfor the underlying Lie algebra, the monodromy for the correspondingsuperoper would not be necessarily trivial: the expression will includethe half-integer powers of z and the monodromy will correspond to thereflection: θ → − θ . Miura superoper is defined in completeanalogy with the pure even case. Namely,
Miura G-superoper is aquadruple ( F , ∇ , F B , F ′ B ) where the triple ( F , ∇ , F B ) is a G -superoperand F ′ B is another B-reduction preserved by ∇ . Let us denote the spaceof such superopers as s MOp G (Σ).Such B -reductions of F are completely determined by the B-reductionof the fiber F x at any point x on Σ and a set of all such reductionsis given by ( G/B ) F x = F x × G G/B = (
G/B ) F ′ x . Then if superoper ξ has the regular singularity and a trivial monodromy, then there is anisomorphism between the space of Miura opers for such ξ and ( G/B ) F ′ x .The structure of the flag manifold G/B is usually quite complicated[20],[19], however we just need the structure determined by its ”body”,i.e. G/ B . For the pure even flag variety G/ B , we have the standardSchubert cell decomposition, where cells S w = Bw w B are labeledby the Weyl group elements w ∈ W and w is the longest element ofthe Weyl group (from now on when we say Weyl group, we mean onlythe Weyl group corresponding to pure even Weyl reflections of the G root system).Let us denote S w the preimage of P : G/B → G/ B . We assumethat the preimage of a big cell Bw B allows factorization Bw B . TheB-reduction F ′ B defines a point in G/B . We say that B-reductions F B,x and F ′ B,x are in relative position w if F B,x belongs to F ′ x × B S w . When w =1, we say that F x , F ′ x are in generic position. A Miura superoper iscalled generic at a given point x ∈ Σ if the B -reductions F B,x , F ′ B,x . aregeneric. Notice that if a Miura superoper is generic at x , it is genericin the neighborhood of x. We denote the space of Miura superopers onU as s MOp G ( U ) gen . It is clear that the reduction of Miura superoperto ( F , ∇ , F B , F ′ B ) gives a Miura oper. Therefore the following Proposition holds, which follows directlyfrom the reduction to the pure even case, although one can also goalong the lines of the proof of Lemma 2.6. and Lemma 2.7 of [11].
Proposition 4.2. i) The restriction of the Miura superoper to thepunctured disk is generic.ii) For a generic Miura superoper ( F , ∇ , F B , F ′ B ) the H -bundle F ′ H isisomorphic to w ∗ ( F h )As in the even case we can define an H -connection associated toMiura superoper on F H ∼ = D − ˇ ρ , which is determined by ˜ ∇ = D θ + u ( z, θ ), where u is h -valued function. Under the change of coordinates( z, θ ) = ( f ( w, ξ ) , α ( w, ξ )), the long superderivative transforms as fol-lows: D ξ + D ξ α · u ( f ( w, ξ ) , θ ( w, ξ )) − ∂ w α ( w, ξ ) D ξ α ) ˇ ρ. (30)Let us call the resulting morphism s MOp G ( U ) gen to the space Conn U ofthe described above flat H -connections on U as a . Now suppose we aregiven a long superderivative ˜ ∇ on H -bundle D − ˇ ρ , one can construct ageneric superoper as follows. Let us set F = D − ˇ ρ × H G , F B = D − ˇ ρ × H B . Then, defining F ′ B as D − ˇ ρ × H w B and the long superderivative on F as ∇ = χ − + ˜ ∇ , we see that the constructed quadruple ( F , ∇ , F B , F ′ B )is a generic Miura oper.Therefore, we obtained the following statement which is analogue ofProposition 2.8 of [11]. Proposition 4.3.
The morphism a : s MOp G ( U ) gen → Conn U is anisomorphism of algebraic supervarieties. Similarly one can define the space of Miura G -superopers of coweightˇ λ on SD x via the same definition applied to s Op G ( SD x ) ˇ λ . Again,we have isomorphism s MOp G ( SD x ) λ ∼ = s Op G ( SD x ) λ × ( G/B ) F ′ x . Wedefine relative positions as in the case of standard Miura superopers (ˇ λ = 0) and let s MOp G ( SD x ) ˇ λ,gen denote the variety of generic Miuraopers of weight λ .Finally, there is an analogue of Proposition 4.3 in this case. Let Conn
RSSD x , ˇ λ denote the set of of long derivatives on the H-bundle D − ˇ ρ with regular singularity and residue − ˇ λ , namely the long derivatives ofthe form: ˜ ∇ = D θ + θz ˇ λ + u ( z, θ ) , (31) UPEROPERS ON SUPERCURVES 15 where u ( z, θ ) ∈ h [[ z, θ ]]. Then as before, one can construct a connec-tion ∇ = ˜ ∇ + χ − and making the gauge transformation with ˇ λ ( z )we obtain the connection from s Op G ( SD x ) ˇ λ . Therefore, there is anisomorphism between Conn
RSSD x ,λ and s MOp G ( SD x ) gen, ˇ λ . SC ∗ . First,let us consider a Miura superoper of coweight ˆ λ on the disc SD x .Assume, it is not generic, but F ′ B,x has the relative position w with F B,x at x . Let us denote the space of all such Miura superopers by s MOp G ( SD x ) ˇ λ,w .From previous subsection we know that each such Miura superopercorresponds to some H -connection on D − ˇ ρ over SD × x . Using the re-sults from the pure even case, one can show that the corresponding H -connection has the form D θ + θz ˇ ν + f ( z, θ ) , (32)where ˇ ν − ˇ ρ = w ( − ˇ λ − ˇ ρ ), w defines the relative position at x , f ( u, θ ) is such that the body of its superderivative is regular in z , i.e. D θ f ( z, θ ) ∈ h red [[ z ]]. Let us call the space of such connections by Conn
RSSD x , ˇ λ,w .Therefore, we can construct a map b RSλ,w : Conn
RSSD x , ˇ λ,w → s Op RSG ( SD x )similarly to the previous subsection, by constructing the triple ( F , ∇ , F B )via identification F = D − ˇ ρ × H G , F B = D − ˇ ρ × H B and ∇ = ˜ ∇ + χ − ,where ˜ ∇ ∈ Conn
RSSD x , ˇ λ,w . We denote by Conn regSD x , ˇ λ,w the preimage of s Op G ( SD x ) ˇ λ,w under this morphism, therefore we have the map: b λ,w : Conn regSD x , ˇ λ,w → s MOp
RSG ( SD x ) ˇ λ , so that in the quadruple ( F , ∇ , F B , F ′ B )first three terms are as above and F ′ B = D − ˇ ρ × H w B . If we denote s MOp
RSG ( SD x ) ˇ λ,w those Miura superopers of coweight ˇ λ which have therelative position w at x , then the following Proposition is true, basedon the results from the pure even case (see Proposition 2.9 of [11]). Proposition 4.4.
For each w ∈ W , b ˇ λ,w is an isomorphism of super-varieties Conn regSD x , ˇ λ,w and s MOp G ( SD x ) ˇ λ,w Let us now consider the case of ˇ λ = 0 and assume that the relativeposition is given by s α i , where α i is a simple black root. In local coordi-nates, the corresponding H -connection will be given by the differentialoperator: ˜ ∇ = D θ + θ z ˇ α i + u ( z, θ ) , (33) where u ( z, θ ) ∈ h [ θ ](( z )) and u ( z, θ ) = u ( z ) + θu ( z ) and u ( z ) ∈ h red [[ z ]]. Then applying the gauge transformationexp (cid:16) − θ z e i + 14 z e i (cid:17) (34)to the Miura superoper ˜ ∇ + χ − , we obtain that the resulting element of s Op G ( SD x ) ˇ λ,s αi gives the element Op G D x ˇ λ,s αi if h ˇ α i , u (0) i = 0. If weconsider the associate bundle corresponding to the 3-dimensional rep-resentation of the osp (2 |
1) triple { e i , f i , ˇ α i } , writing explicitly all thesolutions we find that this condition is also a necessary one. Namely,the following Proposition holds. Proposition 4.5.
A superoper corresponding to the H-connectiongiven by (33) corresponds to Op G ( D x ) ˇ λ,s αi if and only if h ˇ α i , u (0) i =0 . Now we are ready to study superopers with regular singularities overthe super Riemann surface SC ∗ . Let us consider Z = ( z , θ ) , . . . , Z N =( z N , θ N ) on SC ∗ . Also, let ˇ λ , . . . , ˇ λ N , ˇ λ ∞ be the set of dominantcoweights of g . Let us consider the H-connections on SC ∗ with reg-ular singularities at the points Z , . . . , Z N , ( ∞ ,
0) and a finite num-ber of other points W = ( w , ξ ) , . . . , W n = ( w m , ξ m ) such that theresidues of the corresponding even H -connection at z i , w j , ∞ are equalto − y i (ˇ λ + ˇ ρ )+ ˇ ρ , − y ′ j ( ˇ ρ )+ ˇ ρ , − y i (ˇ λ ∞ + ˇ ρ )+ ˇ ρ , where y i , y ′ j , y ∞ ∈ W .In other words, we are considering the H-connections determined by thedifferential operator of the following type: D θ − (cid:16) N X i =1 θ − θ i z − z i + θθ i ( y i (ˇ λ + ˇ ρ − ˇ ρ m X j =1 θ − ξ j z − w j + θθ j ( y ′ j ( ˇ ρ − ˇ ρ (cid:17) + nilp(35)on SC ∗ \∞ , where nilp stands for elements f ( z, θ ) from h [ θ ](( z )) suchthat f ( z, θ ) = D θ f ( z, θ ) = 0. Let us study its behaviour at infinity.Any connection D θ + α ( θ, u ) on D − ˇ ρ has the following expansion withrespect to the coordinates ( u, η ) = ( − z , θz ): D η + u − α ( − u − , − ηu − ) + u − η ˇ ρ. (36) UPEROPERS ON SUPERCURVES 17
Therefore, considering ηu -coefficient in the expansion, we obtain thefollowing constraint: N X i =1 ( y i (ˇ λ + ˇ ρ − ˇ ρ m X i =1 ( y ′ i ( ˇ ρ − ˇ ρ y ′∞ ( − w (ˇ λ ∞ ) + ˇ ρ − ˇ ρ , (37)where y ′∞ w = y ∞ . This expression is expected from the considerationof the pure even case [11].Let us denote the set of the considered above H -connections by Conn ( SC ∗ ) RS ( Z i ) , ( ∞ , λ i , ˇ λ ∞ .Now one can associate to any such connection a G -oper on SC ∗ with regular singularities at the points ( Z i ), ( W j ), ( ∞ ,
0) by setting,in familar way, F = D − ˇ ρ × H G , F B = D − ˇ ρ × H B .Let us denote the set of superopers with regular singularities at Z . . . Z N , ( ∞ , Z i or ( ∞ ,
0) belongs to the space s Op G ( SD Z i ) ˇ λ or s Op G ( SD ( ∞ , ) ˇ λ ∞ ,by s Op G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ .Then let Conn ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ ⊂ Conn ( SC ∗ ) RS ( Z i ) , ( ∞ , λ i , ˇ λ ∞ be those H -connections with regular singularities, which are associated to s Op G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ under the above correspondence. Thereforewe have the map(38) Conn ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ → s Op G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ . We can construct a Miura superoper associated with the image of thismap, namely F ′ B = D − ˇ ρ × H w B . Therefore, this map can be lifted to b ( Z i ) , ( ∞ , λ i , ˇ λ ∞ : Conn ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ → s MOp G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ . (39)Similarly to the pure even case, one can argue that this map is an iso-morphism. Notice that for a given superoper τ ∈ s Op G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ (because of the absence of nontrivial monodromy), the space s MOp G ( SC ∗ ) τ of the corresponding Miura superopers is isomorphic to G/B .Similarly to the argument in the pure even case, we obtain the fol-lowing theorem, which is an analogue of Theorem 3.1 of [11].
Theorem 4.6.
The set of all connections
Conn ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ , whichcorrespond to a given oper τ ∈ s Op G ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ , is isomorphicto the set of points of the flag variety G/B. SP L -superopers and super Bethe ansatz equations. Inthis section we return back to the simplest nontrivial example of thesuperoper, related to supergroup
SP L . In the previous section weobtained that for a fixed superoper τ one can trivialize F by using thefiber at ( ∞ , G/B - bundleand the map: φ τ : SC ∗ → G/B , so that ( ∞ ,
0) maps into the pointorbit of
G/B . Also, in the case G = SP L , G/B ∼ = SC ∗ .Similar to the pure even case, let us call the superoper τ non − degenerate if i) φ τ ( Z i ) is in generic position with B , for any i =1 , . . . , N , ii) The relative position of φ τ ( x ) and B is either generic or cor-responds to a reflection for all x ∈ SC ∗ \ ( ∞ , P GL opers arenon-degenerate for the generic choice of z i , and those are the opers cor-responding to SP L -superopers, then any τ ∈ s Op SP L (2) ( SC ∗ ) ( Z i ) , ( ∞ , λ i , ˇ λ ∞ for the generic choice of Z i is non-degenerate. Also, let us consider theunique Miura superoper structure for τ , such that F B, ( ∞ , and F ′ B, ( ∞ , coincide, i.e. correspond to the point orbit in G/B .The corresponding H -connections will have the following form:˜ ∇ = D θ − N X i =1 θ − θ i z − z i − θθ i ˇ λ i + m X j =1 θ − ξ j z − w j − θξ j ˇ α , (40)where ˇ λ i = l i ˇ ω , so that l i ∈ Z + . Imposing the constraint from Propo-sition 4.5, we obtain that the following equations should hold for thecorresponding oper to be monodromy free: N X i =1 l i w j − z i − m X s =1 w j − w s = 0(41)Also, let us recall that the coweights ˇ λ i should also satisfy (37), whichin our case simpifies to: N X i =1 l i − m = l ∞ (42)Note, that the corresponding P GL -oper coweights, i.e. 2 l i are even:superopers associated with the odd weights will have a monodromywhich will correspond to a reflection in θ variable, as it was explainedabove. The equations (41) are exactly the Bethe ansatz equations for osp (2 |
1) Gaudin model studied in [16].
UPEROPERS ON SUPERCURVES 19 Some remarks
In this article, we studied superopers for superalgebras with pureodd simple root system. However, one can define a similar object forother types of superalgebras, just in such case it can be only locallydefined (i.e. on a superdisc). The analogue of the expression (18) willbe: ∇ = D z,θ + X e a e ( z, θ ) f e + X o θa o ( z, θ ) f o + µ ( z, θ ) , (43)where the summation is over even and odd roots correspondingly and a e,f ( z, θ ) are the even functions of z, θ with nonzero body. The resultingconnection cannot be defined globally on the super Riemann surface,however the operator ∇ θ =0 can give rise to a connection for a G-bundle over a smooth curve underlying the super Riemann surface,while ∇ will give an oper for the underlying semisimple supergroup.This construction gives a generalization of opers in the case of anysimple superalgebra.In this paper we briefly considered an important relation betweenthe spectrum of the Gaudin model and superopers on SC ∗ , which infact could give an example of geometric Langlands correspondence inthe case of superalgebras. For SP L -superopers and Gaudin modelfor osp (2 |
1) the spectrum was determined in fact by the underlying
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UPEROPERS ON SUPERCURVES 21
Department of Mathematics,Columbia University,2990 Broadway, New York,NY 10027, USA.Max Planck Institute for Mathematics,Vivatsgasse 7, Bonn, 53111, GermanyIPME RAS, V.O. Bolshoj pr., 61, 199178, St. [email protected]://math.columbia.edu/ ∼∼