aa r X i v : . [ m a t h . AG ] J u l TOROIDAL q -OPERS PETER KOROTEEV AND ANTON M. ZEITLIN
Abstract.
We define and study the space of q -opers associated with Bethe equations forintegrable models of XXZ type with quantum toroidal algebra symmetry. Our constructionis suggested by the study of the enumerative geometry of cyclic quiver varieties, in par-ticular, the ADHM moduli spaces. We define ( GL ( ∞ ) , q )-opers with regular singularitiesand then, by imposing various analytic conditions on singularities, arrive at the desiredBethe equations for toroidal q -opers. Contents
1. Introduction: Geometric facets of Bethe equations 12. ( SL ( r + 1) , q )-opers 113. Z-twisted Miura ( SL ( r + 1) , q )-opers 144. Miura-Pl¨ucker ( SL ( r + 1) , q )-opers 165. Z-twisted Miura ( SL ( r + 1) , q )-opers and QQ -systems 196. q -Opers via Quantum Wronskians 277. GL ( ∞ ) and the Fermionic Fock Space 358. ( GL ( ∞ ) , q )-opers 389. Z -twisted Miura ( GL ( ∞ ) , q )-opers and QQ -systems 4110. Toroidal q-opers 46References 511. Introduction: Geometric facets of Bethe equations
Integrable Models and Bethe Ansatz.
The study of one-dimensional quantumintegrable models fueled modern mathematics with a variety of interesting ideas, in partic-ular, in discovery of the quantum groups and related structures. A particularly useful toolin the study of integrable models is the so-called algebraic Bethe ansatz method (see, e.g.,[KBI, R]) having its roots in the original papers of Hans Bethe from the 1930s.Let us briefly describe here the modern mathematical perspective on how algebraic Betheansatz works for integrable models of specific type, namely the spin chains. Let g be asimple Lie algebra and ˆ g k =0 = g [ t ± ] be the corresponding loop algebra (affine algebra withvanishing central charge k = 0). The finite-dimensional modules { V i } of g give rise to theso-called evaluation modules { V i ( a i ) } , where a i stands for the value of the loop parameter t .These modules form a tensor category. Passing from g [ t ± ] to the corresponding quantumaffine algebra U ~ (ˆ g ) or the Yangian Y ~ ( g ) one obtains a deformation of such tensor category, Date : July 24, 2020. known as braided tensor category [CP]. This object features a new (braiding) intertwiningoperator R V i ( a i ) ,V j ( a j ) : V i ( a i ) ⊗ V j ( a j ) −→ V j ( a j ) ⊗ V i ( a i ) , satisfying famous Yang-Baxter equation.To describe the integrable model, we choose a specific object in such braided tensorcategory H = V i ( a i ) ⊗ · · · ⊗ V i n ( a i n ) , which we refer to as physical space , the vectors in this space are called states . For a givenmodule W ( u ) called auxiliary module with parameter u known as spectral parameter , wedefine the transfer matrix T W ( u ) = Tr W ( u ) h ( Z ⊗ P R W ( u ) , H i . Here the twist Z is given by Z = Q ri =1 z ˇ α i i ∈ e h , where h is the Cartan subalgebra in g , { ˇ α i } i =1 ,...,r are the simple coroots of g , and P is a permutation operator. The monodromymatrix M ZW ( u ) = ( Z ⊗ P R W ( u ) , H is an operator in W ( u ) ⊗ H . Notice, that the transfermatrix T W ( u ) is an operator acting in the physical space H . The Yang-Baxter equationimplies that transfer matrices, corresponding to various choices of W ( u ) form a commutativealgebra, known as Bethe algebra . The commutativity of Bethe algebra implies integrability and the expansion coefficients of the transfer matrix yield (nonlocal)
Hamiltonians of theXXX or XXZ spin chain depending on whether we deal with the Yangian or the quantumaffine algebra. From now on, we will primarily focus on quantum affine algebra and theXXZ model, although most of the construction below applies to the Yangian and the XXXmodels as well.The classic example of the XXZ Heisenberg magnet corresponds to the quantum algebra U ~ ( ˆ sl (2)) in which the physical space H is constructed from V i ( a i ) = C ( a i ) – the standardtwo-dimensional evaluation modules of U ~ ( ˆ sl (2)).The solution of the integrable model implies finding the eigenvalues and eigenvectorsof simultaneously diagonalized Hamiltonians, i.e. elements of the Bethe algebra. Oneway to accomplish the task is to follow the old-fashioned procedure from the 1980s knownas algebraic Bethe ansatz . It implies that the eigenvalues of the transfer-matrices (uponrescaling) are symmetric functions of the roots of the system of algebraic equations, knownas Bethe ansatz equations. Although this approach is straightforward and effective, wewill explore other modern techniques, which provide insights into representation-theoreticaspects of the problem.1.2. Modern Approach to Bethe Ansatz.
Quantum Knizhnik-Zamolodchikov equations.
The intertwining operators for the quan-tum affine algebra U ~ (ˆ g ) and thus the matrix elements of their products, known as conformalblocks satisfy certain difference equations known as quantum Kniznik-Zamolodchikov (qKZ)equations (also known as Frenkel-Reshetikhin equations) [FR].Explicitly, qKZ equations can be written as follows: difference equationsΨ( a i , . . . , qa i k , . . . , a i n , { z i } ) = H ( q ) i k Ψ( a i , . . . , , a i n , { z i } ) , (1.1)where the solutions Ψ take values in H and operators H qi are expressed in terms of productsof R-matrices. The analytic properties of the solutions of qKZ equations will be discussed OROIDAL q -OPERS 3 later in subsection 1.3.2. There is also a commuting system of equations in { z i } -variablesfor Ψ, known as dynamical equations see e.g. [TV1, TV2].The solution to the qKZ equation is given by an integral expression, so that the integrandhas the following asymptotic behavior in the limit q → e Y ( { ai } , { zi } ,xi )log( q ) ( φ ( { a i } , { z i } , x i ) + O ( η )) , (1.2)where { x i } are the variables of integration. In the limit q → e Sη (Ψ + O ( η )), where S = Y | σ i , where σ i are the solutions of theequations ∂ x i Y = 0 which need to be solved with respect to the variables { x i } . Theseequations coincide with the Bethe equations, and Ψ is the eigenvector for operators H (1) i ,known as the nonlocal Hamiltonians of the corresponding XXZ model: they emerge as coef-ficients from the expansion of the transfer matrices with respect to the spectral parameter,viz. H (1) i Ψ = e p i Ψ , where p i = ∂ a i S. QQ -systems and Baxter operators. When we earlier discussed the transfer matrices T W ( u ) we considered W ( u ) to be a finite-dimensional module of U ~ (ˆ g ). We notice thatthe universal R-matrix , which produces particular braiding operators R V i ( a i ) ,V j ( a j ) belongsto the completion of the tensor product U ~ (ˆ b + ) ⊗ U ~ (ˆ b − ), where U ~ (ˆ b ± ) are the Borelsubalgebras of U ~ (ˆ g ). Therefore there is no obstruction in taking auxiliary representations W ( u ) to be representations of U ~ (ˆ b + ).The purpose of that is as follows. There exist prefundamental representations of U ~ (ˆ b + )which are infinite-dimensional. If one extends the braided tensor category of finite-dimensionalmodules by such representations, it appears that prefundamental representations generatethe entire category.The corresponding transfer matrices turn out to be well-defined and moreover, the eigen-values of the transfer matrices are polynomials of the spectral parameter, generating elemen-tary symmetric functions of the solutions of Bethe equations. Such transfer matrices wereoriginally introduced by Baxter and thus are known as Baxter operators ad hoc via theireigenvalues. Their representation-theoretic meaning was realized much later, in the papersof Frenkel and Hernandez [FH1], [FH2], following earlier ideas of Bazhanov, Lukyanov andZamolodchikov [BLZ] and Hernandez and Jimbo [HJ].There are two series of prefundamental representations { V i + ( u ) } i =1 ,...,r , { V i − ( u ) } i =1 ,...,r and the associated Baxter operators { Q i ± ( u ) } i =1 ,...,r . They obey the following key relation e ξ i Q i − ( u ) Q i + ( ~ u ) − ξ i Q i − ( ~ u ) Q i + ( u ) = Λ i ( u ) Y j = i " − a ij Y k =1 Q j + ( ~ b kij u ) i = 1 , . . . , r, b kij ∈ Z Here polynomials Λ i ( u ) are known as Drinfeld polynomials, characterizing the representa-tion H of U ~ (ˆ g ) and ξ i , e ξ i are some monomials of { z i } .This system of equations, known as the QQ -system , considered as equations on { Q i ± ( u ) } i =1 ,...,r and subject to some nondegeneracy conditions, are equivalent to the Bethe ansatz equations.We note, that similar construction and the analogue of the QQ -system should also existfor Yangians with some progress being made in [BFL + ]. P. KOROTEEV AND A.M. ZEITLIN
Geometric Interpretations.
Quantum K-theory of Nakajima Varieties.
The relation between enumerative alge-braic geometry and integrability was known for some time. Starting from the pioneeringworks of Witten and Dubrovin, it flourished in the works of A. Givental and his schoolin the 90s. Recently, the progress in the understanding of supersymmetric gauge theorymerged with the developments in the geometric representation theory. In particular, thestudy of the so-called symplectic resolutions from the representation-theoretic point of viewgave a new life to this fruitful relationship in works of A. Okounkov and his collaborators[BMO],[O], [MO]. It was observed that some integrable systems based on quantum groups,specifically XXX and XXZ models, naturally emerge from enumerative geometry for a largeclass of algebraic varieties, known as
Nakajima quiver varieties [N1],[G].Let us recall this connection in the simplest nontrivial examples of such varieties, namelythe cotangent bundles over Grassmannians T ∗ Gr k,n . The standard objects in the enumer-ative geometry are the appropriate deformations of the cup product and the tensor productin the equivariant cohomology and K-theory correspondingly, where the deformation ischaracterized by the series in K¨ahler parameters , with coefficients being produced by curvecounting.The physics results of Nekrasov and Shatashvili [NS] lead to the following conjectureabout the equivariant quantum K-theory K T ( T ∗ Gr k,n ): the quantum multiplication by thegenerating function for the exterior algebra of tautological bundle coincides with the Baxter Q -operator for the Heisenberg XXZ-spin chain. Also, since tautological bundles generatethe entire quantum K-theory, one can describe the equivariant quantum K-theory ring asthe ring of symmetric functions of Bethe roots.The proof of that conjecture was given in [PSZ]. It uses the theory of quasimaps toNakajima varieties as the ‘curve counting’ which is different from the older approach toquantum K-theory due to A. Givental. To relate the quantum equivariant K -theory withspin chains, it is not enough to consider the operators of quantum multiplication by classical K -theory classes: in fact, both the multiplication in the equivariant K -theory and thetautological classes should be deformed simultaneously: in our case of T ∗ Gr k,n , by justone K¨ahler parameter z . One can define elements ˆ V τ ∈ K T ( T ∗ Gr k,n )[[ z ]] which we callquantum tautological bundles. In the classical limit z →
0, these elements coincide withthe corresponding classical bundles V τ , which is a certain tensorial polynomial of standardtautological bundles, corresponding to the symmetric polynomial τ in k variables in thestandard K -theory. The localized equivariant quantum K-theory K locT ( T ∗ Gr k,n ) can beidentified with appropriate weight subspace in the space H of the XXZ Heisenberg magnet,so that considering the union of such spaces for all k , one obtains the entire space of states H .To prove this conjecture one needs to define and compute vertex functions , which arequasimap analogues of Givental’s I-functions . These are certain Euler characteristics, whichcount quasimaps and determine the quantum K-theory classes. Such vertex functions satisfythe quantum difference equations (QDE) which coincide with qKZ and dynamical equations[OS], and will be discussed in subsection 1.2.1. To understand the action of the operators ofquantum multiplication by the quantum tautological bundles, one has to study the q → OROIDAL q -OPERS 5 Later these results have been proven for larger classes of Nakajima varieties, e.g., partialflag varieties, see, e.g., [KPSZ].Notice that this approach gives geometric interpretation to qKZ and dynamical equations.Moreover, each of the Q-operators on its own has a geometric meaning. The Q i + -operatorscorrespond to the exterior powers of tautological bundles. Their Q i − counterparts corre-spond to the exterior powers of tautological bundles of Nakajima varieties with a differentchoice of stability parameters. In the case of flag varieties, such a change in the stabilityparameters is provided by the action of Weyl reflection.However, the QQ -system relations themselves do not arise naturally, since in particular, Q i ± operators do not act in the same space. In the next section, we will discuss anothergeometric viewpoint on Bethe ansatz, specifically related to the geometric interpretation ofthe QQ -system.1.3.2. Quantum q-Langlands Correspondence.
To oversee the geometric interpretation of QQ -systems, we take several steps back in time and deformation-wise. Earlier in the intro-duction we described the construction of the XXZ spin chain. There is a certain scaling limitof the XXZ model which is called the Gaudin model. This limit can be understood quasi-classically as ~ → q = ~ k + h ∨ , where h ∨ is a dual Coxeter element. Thenthe qKZ equation turns into the differential equation, known as Knizhnik-Zamolodchikov(KZ) equation: ( k + h ∨ ) ∂ a i Ψ = H i Ψ(1.3)The solution Ψ belongs to the classical limit of H , viz. the tensor product of some evaluationrepresentations of Lie algebra ˆ g with evaluation parameters a i : H cl = V ( a ) ⊗ · · · ⊗ V n ( a n ).The mutually commuting Gaudin Hamiltonians H i , have easy to read expressions H i = n X j =1 ,i = j t αi ⊗ t αj a i − a j + Z i , where t α form an orthonormal basis in g with respect to the Kiling form, Z belongs toCartan subalgebra of g and indices i, j indicate on which of the representations V i theseelements act.One can see that in the limit k → − ~ ∨ , known as critical level limit, this equationturns into an eigenvalue problem for Gaudin Hamiltonians. It is possible to interpret thesolutions of the KZ equations in a particular analyticity region in evaluation parameters | a | > | a | > · · · > | a n | , as the equations for the intertwiners of ˆ g with central charge k .Moreover, the Gaudin Hamiltonians were shown in [FFR] to be part of a bigger structure,namely the center Z ( U (ˆ g )) of U (ˆ g ) at the critical level k = − h ∨ .There is a natural Poisson structure on Z ( U (ˆ g )), arising from standard commutators awayfrom the critical level. The famous theorem of Feigin and Frenkel [FF] provides isomor-phism of this Poisson algebra and the classical limit of W -algebra W ( L g ), associated to theLanglands dual Lie algebra L g , also known as Gelfand-Dickey algebra of pseudodifferentialoperators in the case g = sl n .Later this statement was reformulated [F1] in terms of special connections for principal L ˜ G -bundles, known as oper connections , on the punctured disk, where L ˜ G is an adjointLie group associated to L g . The reformulated Feigin-Frenkel theorem implies that there isan isomorphism between Z ( U (ˆ g )) and the space of functions on L ˜ G -oper connections on P. KOROTEEV AND A.M. ZEITLIN a punctured disk. The path from such connections to Gelfand-Dickey pseudo-differentialoperators is given by a well-known construction, known as
Drinfeld-Sokolov reduction [DS].Let us return to the eigenvalue problem for Gaudin Hamiltonians, arising from criticallevel limit of (1.3). E. Frenkel’s theorem [F2] gives a geometric description of the spectrumin terms of opers. Explicitly, Frenkel’s theorem states that there is a one-to-one correspon-dence between the space of
Miura oper connections with regular singularities with trivialmonodromies around them on P in case when Z = 0. The word ‘Miura’ there means thatthere is an extra condition on such oper connections: they have to preserve the reductionof L ˜ G -bundle to Borel subgroup. Later, this theorem was generalized for Z = 0 by addingirregular singularity at ∞ ∈ P [FFTL].The constraints on such connections could be expressed in terms of Wronskian-typerelations, which are particularly manifest in the case of SL ( N ). That suggests, that the QQ -system, which is a deformation of the Wronskian relation, should arise from an appropriate ~ -deformation of Miura opers. Below we shall provide further motivation and hints in thisdirection.The pseudodifferential operators, corresponding to such Miura oper connections withregular singularities through Drinfeld-Sokolov reduction, describe the constraints on theconformal blocks/intertwiners of W ( L g )-algebra in the limit when central charge c → ∞ .The most famous such constraint is known as the Belavin-Polyakov-Zamolodchikov (BPZ)equation (essentially the Sturm-Liouville problem with singular potential) for conformalblocks of Virasoro algebra, which is the case when g = sl (2).Naturally, that led to the quantum Langlands correspondence linking conformal blocksof W ( L g )-algebras and ˆ g -conformal blocks away from the critical level. Recently, a q-deformation of this correspondence was proposed in [AFO]. The proof of the latter, given in[AFO], provided in the case of simply-laced g is based on the enumerative geometry approachwhich we touched briefly in 1.3.1. The key to that is to further deform this correspondence,namely, identify conformal blocks for the quantum affine algebra U ~ (ˆ g ) and the deformedW-algebra W q,t ( L g ), which is the 2-parametric deformation of Gelfand-Dickey algebra [FR].The conformal blocks for U ~ (ˆ g ), as we discussed in subsection 1.2.1, satisfy the qKZequation. We remind, that as in the classical case, they correspond to the solution of qKZ,analytic in the region | a | > | a | > · · · > | a n | . However, the solutions of qKZ which areprovided by enumerative geometry, i.e. vertex functions, are analytic in { z i } -variables.Also, it turns out that they are the ones, producing the conformal blocks of W q,t ( L g )-algebra. The transition between two families of solutions is crucial for establishing theexact correspondence between such conformal blocks. We refer to [AFO] for the details.1.3.3. Miura ~ -opers. A natural question is to understand the difference analogues of BPZ-type equations which serve as constraints for the conformal blocks of W q,t ( L g ). As we havediscussed, the differential BPZ equations on the critical level correspond to the classicalobjects, namely L ˜ G -oper connections with regular singularities on P . Let L G be the simply-conected group with Lie algebra L g . There is a natural classical object, the ~ -differenceconnection, locally a meromorphic L G -valued function A ( z ) on Zariski open set of P , whichtransform upon trivialization change A ( z ) −→ g ( ~ z ) A ( z ) g − ( z ).In [FKSZ], following the constructions in [KSZ] done for SL ( N ), we developed the ~ -difference analogue of opers as such ~ -difference connections for any simply connectedsemisimple Lie group L G with a fixed Borel subgroup L B − . Locally, these ~ -connectionshave the form A ( z ) = n ′ ( z ) Q ri =1 s i φ ˇ α i i ( z ) n ( z ). Here n ( z ) , n ′ ( z ) ∈ G ( z ), φ i ( z ) ∈ C ( z ), s i are OROIDAL q -OPERS 7 the lifts of the fundamental Weyl reflections to L G . In other words A ( z ) ∈ B − ( z ) cB − ( z ),where c = Q ri =1 s i is a Coxeter element.Moreover, we defined such ( L G, ~ )-opers and their Miura versions with regular singu-larities, which amounts to the connections of this type which preserve the opposite Borelsubgroup of B + and taking φ i ( z ) = Λ i ( z ) ∈ C [ z ]. We proved several structural theoremsabout them.One of the major statements we make in [FKSZ] is devoted to the explicit relation of theseobjects to the QQ -systems and Bethe ansatz. To do that, we work with two versions of whatwe call Z-twisted condition for Miura opers. The simplest Z -twisted condition implies thatthe ( L G, ~ )-oper connection can be ~ -gauge equivalent to semisimple element Z ∈ H ⊂ L G ,where H is the Cartan subgroup. That means A ( z ) = g ( ~ z ) Zg − ( z ). This condition is adifference version of zero monodromy condition and double pole irregular singularity at ∞ point of P .The relaxed version of this Z -twisted condition is as follows. Given the principal L G -bundle, one can construct an associated bundle for any fundamental representation V ω i forthe fundamental weight ω i . It turns out, one can associate a ( GL (2) , ~ )-oper to any suchpair ( L G, ~ )-oper and V ω i : this is done by restricting the Miura ( L G, ~ )-oper to the two-dimensional subspace, spanned by two top weights in V ω i . This is possible, since Miura( L G, ~ )-oper preserves the reduction to positive Borel subgroup L B + ⊂ L G .We say that the resulting Miura oper is Z -twisted Miura-Pl¨ucker ( L G, ~ )-oper if for everysuch ( GL (2) , ~ ) oper is ~ -gauge equivalent to the restriction of Z to the correspondingtwo-dimensional space.In [FKSZ] we showed that Z -twisted Miura-Pl¨ucker ( L G, ~ )-opers with mild non-degene-racy conditions are in one-to-one correspondence with certain QQ -systems and that doesnot depend on the order in the Coxeter element. In simply-laced case such QQ -systems areequivalent to standard Bethe ansatz equations. The non-simply laced case is more involved(see the discussion in [FKSZ] and the upcoming paper [FHR].While it immediately follows that any Z -twisted Miura oper is indeed Z -twisted Miura-Pl¨ucker one, the opposite statement, however, is highly nontrivial. In [FKSZ] we introducea chain of ~ -gauge transformations, which we refer to as ~ -B¨acklund transformations, whichon the level of QQ -systems amounts to the Q i + ( z ) → Q i − ( z ), Z → s i ( Z ), where s i iselementary Weyl reflection. However, at every step, in order to progress further, we have toimpose the nondegeneracy condition on the QQ -system and the associated Miura oper. Wehave shown that if one can proceed with this transformations to Z -twisted Miura-Pl¨uckeroper, corresponding to the w ( Z ), where w is the longest Weyl group element, then such Z -twisted Miura-Pl¨ucker Miura oper is Z -twisted. We call such Miura-Pl¨ucker opers andthe associated QQ -system w -generic.We also discuss the explicit version of ~ -version of Drinfeld-Sokolov reduction, followingthe ideas of [SS]. The scalar difference equations emerging this way from Z-twisted Miura( L G, ~ )-opers and the correspondence with the difference equations which the conformalblocks for W q,t ( L g )-algebras remains an interesting open problem.1.4. Our goals in this paper.
Two approaches to ( SL ( r + 1) , ~ ) -opers. In this paper, we are investigating severalproblems. The first one is devoted to the correspondence between the results of [KSZ],where we work with SL ( r + 1) case only, and a more general approach of [FKSZ]. In [KSZ]we used a definition of (Miura) ~ -oper which is very specific to SL ( r + 1). It can be P. KOROTEEV AND A.M. ZEITLIN deduced from the ‘universal’ definition of ( SL ( r + 1) , ~ )-oper as an ~ -connection for theprincipal SL ( r + 1)-bundle, which we discussed in the previous section, with the standardorder of reflections in the corresponding Coxeter element (following the order in the Dynkindiagram), so that in the defining representation it is represented as the matrix with zeroesabove its superdiagonal.Considering the associated bundle, corresponding to the defining representation, one canreinterpret the oper condition in the following way. Namely, it is the condition on the operaction in the complete flag of subbundles of this associated bundle, which reflects its matrixstructure described above. In particular, that implies that on a Zariski dense subset in P the total space of the flag can be recreated by the consecutive action of the ( SL ( r + 1) , ~ )-oper connection on the section of the line bundle. The Miura condition can be reformulatedas the constraint, that the connection preserves a different complete flag of subbundles.Such definition lead to another approach to the derivation of the QQ -systems from Z -twisted Miura ( SL ( r + 1) , ~ )-opers with regular singularities. This is done using ~ -deformedWronskian matrices. Their matrix elements are components of the nontrivial section ofthe line bundle in the trivialization when oper connection is represented by the regularsemisimple twist element Z and describes the relative position of two flags of subbundles.It turns out, that the points where these flags are in a non-generic position correspond toBethe roots and QQ -systems, as we have demonstrated in [KSZ]. Here we show that theextension of the QQ -system by ~ -B¨acklund transformations is provided by various minors inthis ~ -Wronskian matrix. More importantly, we explicitly construct the element g ( z ), suchthat connection A ( z ) = g ( z ) Zg − ( z ). This element can be represented both in abstractLie-theoretic form as well explicitly in the matrix notation, which uses polynomials of theextended QQ -system. As a consequence, we obtain that the w -generic condition, whichwas needed in general for Z -twisted Miura-Pl¨ucker ( L G, ~ )-oper to be just Z -twisted, is notneeded for L G = SL ( N ).1.4.2. Completion to ( GL ( ∞ ) , ~ ) -opers. Following the calculations of ( SL ( r + 1) , ~ )-opersit is not hard to extend this construction to SL ( ∞ ) – the group of infinite-dimensionalmatrices with unit determinant with a finite amount of nonzero off-diagonal entries andfinite amount of non-unit elements on the diagonal. However, for any Miura ( SL ( ∞ ) , ~ )-oper, the corresponding QQ -system will always be finite. Let us explain how to constructa Miura oper, which corresponds to the ‘complete’ QQ -system associated with the Dynkindiagram of A ∞ . We note, that SL ( ∞ ) has a well-defined set of fundamental representationsbased on semi-infinite wedge spaces, which has an interpretation in terms of Dirac sea , andthe generators of Lie algebra sl ( ∞ ) are represented via quadratic expressions of the fermionicoperators of exterior and interior multiplication, thereby generating Clifford algebra.One can complete the corresponding Lie algebra sl ( ∞ ) by allowing infinite sums of gener-ators. The resulting Lie algebra, endowed by central extension equal to 1, has fundamentalrepresentations realized in the same spaces as sl ( ∞ ). This is an important construction,that plays a central role in the celebrated boson-fermion correspondence [F, KRR]To address related Miura opers, we take a certain completion of SL ( ∞ ), which will besufficient to put an infinite number of terms in the QQ -system. Namely, we constructthe group corresponding to the completion of the upper Borel subgroup in the Bruhatdecomposition of SL ( ∞ ). The resulting object, denoted by GL ( ∞ ), is the group of theinfinite matrices with an infinite number of elements above the diagonal and an infinitenumber of nonunital elements on the diagonal, while the number of elements below the OROIDAL q -OPERS 9 diagonal remains finite. It has the same set of fundamental representations realized in thesame set Dirac sea spaces as described above.The resulting Miura ( GL ( ∞ ) , ~ )-opers satisfy similar properties as the SL ( r + 1) ones.One can define Z -twisted and Z -twisted Miura-P¨ucker opers and explicitly construct the op-erator from completed upper Borel subalgebra, diagonalizing the corresponding connectionmatrix. As before, it is constructed from the elements of the extended QQ -system.As an application of this construction, we can built the main novel objects of the currentpaper, namely toroidal opers .1.4.3. Toroidal opers and the q-Langlands correspondence for toroidal algebras.
There is anatural family of automorphisms of sl ( ∞ ) algebra, corresponding to the Dynkin diagramtranslations through n vertices. On the group-theoretic level, such transformations arerealized via the n -th power of the ‘completed’ Coxeter element c (infinite matrix withthe only nonzero elements being units on the superdiagonal). Imposing the condition c n A ( z ) c − n = A ( pz ) for Z -twisted ( GL ( ∞ ) , ~ ), where p is a new parameter, we obtainthat the resulting constrained infinite QQ -system generate Bethe equations for the toroidalalgebras bb gl ( n ).While the corresponding QQ -system for toroidal algebras has yet to emerge from theperspective of prefundamental representations and Grothendieck ring, the Bethe equationsfor toroidal algebra bb gl ( n ) in representation-theoretic setting emerged through the shortcut,namely Baxter T Q -relation [FJMM], the relation between the Q-operator and the transfer-matrix.However, a more natural approach to generate Bethe equations for toroidal algebrasemanates from enumerative geometry. In section 1.3.1, we discussed elementary examples ofquiver varieties, namely T ∗ Gr k,n and, in general, cotangent bundles to partial flag varieties.The corresponding quantum K-theory ring reproduces the Bethe algebra for the XXZ modelrelated to b sl ( n ). Another set of varieties, which have been extensively studied, are theframed cyclic quiver varieties, which are related to bb gl ( n ) toroidal algebras, where n is thenumber of vertices. In the simplest situation of one vertex, such variety is identified withthe space of ADHM instantons [SV]. One can find more details on algebraic properties ofquantum toroidal algebras and their geometric realization in the recent reviews [N2], [N3].According to general construction, the q → z -analytic solutions of theresulting qKZ equations reproduce the Bethe equations, which serve as constraints for thequantum K-theory ring. These are exactly the equations we reproduce from toroidal opers.Given that t , t are the standard deformation parameters of U t ,t (cid:0) bb gl ( n ) (cid:1) we obtain thefollowing exchange of parameters:(1.4) ( ~ , p ) ↔ (( t t ) − , t ) , which serve as the first example of the analogue q-Langlands correspondence for toroidalalgebras.1.4.4. String Theory Motivation.
In string theory literature it is common to study limitswhen the number of objects, like branes becomes infinite. The most relevant example to thispaper is topological holography program which was initiated by Gopakumar and Vafa [GV].According to loc. cit. a topological phase transition can be regarded as an interpolationbetween two desingularizations of the conifold geometry – deformed conifold T ∗ S andresolved conifold O ( − ⊕ P . The M-theory description of the former phase, in the presence of certain defects and fluxthrough one of the complimentary complex directions, after dimensional reduction, leads toa three-dimensional quiver gauge theory on S × C q . Massive spectrum of such 3d theoriesis described by the equivariant quantum K-theory of the corresponding quiver varietieswhich we discussed earlier. Parameter ~ from above plays the role of a N = 4 R-symmetryequivariant parameter.The latter, resolved phase yields five dimensional gauge theory. The moduli space ofinstantons in this 5d theory is given by the ADHM quiver which will later in this paper bediscussed in connection with toroidal q -opers.The topological phase transition from the deformed phase to the resolved phase occurswhen the number of branes which wrap S cycle in the deformed geometry, and determinesthe number of gauge groups in the 3d theory becomes infinite. In addition, a certainquantization condition between the Omega-background parameters of the 3d gauge theoryand other mass parameters of the problem must be satisfied. Namely, if s i and s i +1 arecomplexified gauge field vacuum expectation values of vector superfields of the i th and( i + 1)st gauge groups respectively, then the condition reads s i +1 s i = p n where n is aninteger. On the resolved side of the transition parameter p becomes equivariant parameterof the K-theory of the ADHM moduli space. Same parameters already appeared in (1.4).Representation-theoretic aspects of the Gopakumar-Vafa transition in connection withquantum geometry of quiver varieties of A-type were discussed in [K]. This paper providesan alternative description of the same physics in terms of bona fide classical objects – q -opers. By combining our results with those of the first author in [K] we can establishthe quantum/classical duality between quantum XXZ spin chain of b A -type, whose Betheequations coincide with relations in quantum equivariant K-theory of the ADHM quivervariety, and the so-called 1-toroidal q -opers. This correspondence can be regarded as thelarge rank limit of the quantum/classical duality which was discussed in both physics [GK,KS] and mathematics [KSZ] literature.We also note that oper-related structures in type-A as well as their super analoguesappeared in recent physics literature on integrability in the AdS/CFT correspondence [KLV,CLV] as well as some earlier work [KLWZ].1.4.5. Structure of the paper.
In Section 2 we give two equivalent definitions of ( SL ( r +1) , q )-opers and their Miura versions as q -connections, which were introduced in [FKSZ] and [KSZ]correspondingly. The first definition uses the Lie-theoretic approach and the second one isusing complete flags of subbundles.In Sections 3,4, and 5 we elaborate on the Lie-theoretic definition and remind basicconstructions of [FKSZ]. Section 3 is devoted to Z -twisted ( SL ( r + 1) , q )-opers, which areq-gauge equivalent to a diagonal matrix. In Section 4 a more mild version of Z -twistedcondition is introduced, which is related to associated bundles leading to the notion of Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q )-opers. We also discuss nondegeneracy conditionsfor these objects. Section 5 addresses the one-to-one correspondence between Z -twistedMiura-Pl¨ucker opers and the nondegenerate solutions of the QQ-systems (and thus Betheansatz equations) as well as their extension. We also prove that Z -twisted Miura-Pl¨ucker( SL ( r + 1) , q )-opers are Z -twisted and relate extended QQ-system to quantum B¨acklundtransformations, introduced in [FKSZ]. OROIDAL q -OPERS 11 In Section 6 we use the second definition of Miura ( SL ( r + 1) , q )-opers and show how Z -twisted condition as well as quantum B¨acklund transformations can be reformulated interms of q-Wronskian matrices, extending the results of [KSZ].In Section 7 we describe the fermionic realization of Z -twisted Miura ( SL ( r + 1) , q )-opersusing the realization of the fundamental representations in the fermionic Fock space. Wethen use it as a motivation to write an infinite rank formula. To do that, in Section 7 weintroduce the group GL ( ∞ ) and its representations in the fermionic Dirac sea, i.e. semi-infinite wedge space and then in Sections 8 and 9 we extend the finite-dimensional notionsof ( SL ( r + 1) , q )-oper theory from earlier Sections to the case of GL ( ∞ ). In particular, weshow the relation between the corresponding infinite generalization of the QQ -system and Z -twisted Miura ( GL ( ∞ ) , q )-opers.Finally, Section 10 is devoted to the main target of the paper, the toroidal opers. Theseare nondegenerate Z -twisted Miura opers with certain periodicity conditions. The maingoal of Section 10 is to show that they are in one-to-one correspondence with the non-degenerate solutions of the QQ -system for toroidal algebras. We also discuss the relationto the enumerative geometry of ADHM spaces and generalizations to framed cyclic quivervarieties. Acknowledgments.
We thank E. Frenkel for his advices. P.K. is partially supported byAMS Simons grant. A.M.Z. is partially supported by Simons Collaboration Grant, AwardID: 578501. 2. ( SL ( r + 1) , q ) -opers Group-theoretic data and notations.
Consider SL ( r + 1) be the simple algebraicgroup of invertible ( r + 1) × ( r + 1) matrices over C . We fix a Borel subgroup B − withunipotent radical N − = [ B − , B − ] of lower triangular matrices and strictly lower triangularmatrices correspondingly. The maximal torus is the corresponding set of diagonal matrices H ⊂ B − . Let B + be the opposite Borel subgroup containing H . Let { α , . . . , α r } be theset of positive simple roots for the pair H ⊂ B + . Let { ˇ α , . . . , ˇ α r } be the correspondingcoroots. Then the elements of the Cartan matrix of the Lie algebra sl ( r + 1) of G are givenby a ij = h α j , ˇ α i i . The Lie algebra sl ( r + 1) has Chevalley generators { e i , f i , ˇ α i } i =1 ,...,r , sothat b − = Lie( B − ) is generated by the f i ’s and the ˇ α i ’s and b + = Lie( B + ) is generatedby the e i ’s and the ˇ α i ’s. In the defining representation ˇ α i ≡ E ii − E i +1 ,i +1 , e i ≡ E i,i +1 , f i ≡ E i − ,i , where E ij stand for the matrix with the only nonzero element 1 at ij-th place.The fundamental weights ω , . . . ω r are defined by the condition h ω i , ˇ α j i = δ ij .Let W SL ( r +1) = N ( H ) /H ≡ S r +1 be the Weyl group of SL ( r + 1). Let w i ∈ W SL ( r +1) ,( i = 1 , . . . , r ) denote the simple reflection corresponding to α i . We also denote by w be the longest element of W , so that B + = w ( B − ). Recall that a Coxeter element of W is a product of all simple reflections in a particular order. It is known that the setof all Coxeter elements forms a single conjugacy class in W G . We will fix once and forall (unless specified otherwise) a particular ordering of the simple roots, according to thenatural ordering provide by Dynkin diagram. Let c = w r w r − . . . w be the Coxeter elementassociated to this ordering. In what follows (unless specified otherwise) all products over i ∈ { , . . . , r } will be taken in this order; thus, for example, we write c = Q i w i . We alsofix representatives s i ∈ N ( H ) of w i . In particular, s = Q i s i will be a representative of c in N ( H ). In the following we will denote the deformation parameter q instead of ~ for conveniencepurposes.2.2. ( SL ( r + 1) , q ) -opers: two definitions. Let’s consider the automorphism M q : P −→ P sending z qz , where q ∈ C × is not a root of unity.Given a principal SL ( r + 1)-bundle F SL ( r +1) over P (in Zariski topology), let F qSL ( r +1) denote its pullback under the map M q : P −→ P sending z qz . A meromorphic( SL ( r + 1) , q )- connection on a principal SL ( r + 1)-bundle F SL ( r +1) on P is a section A ofHom O U ( F SL ( r +1) , F qSL ( r +1) ), where U is a Zariski open dense subset of P . We can alwayschoose U so that the restriction F SL ( r +1) | U of F SL ( r +1) to U is isomorphic to the trivial SL ( r + 1)-bundle. Choosing such an isomorphism, i.e. a trivialization of F SL ( r +1) | U , wealso obtain a trivialization of F SL ( r +1) | M − q ( U ) . Using these trivializations, the restrictionof A to the Zariski open dense subset U ∩ M − q ( U ) can be written as section of the trivial SL ( r + 1)-bundle on U ∩ M − q ( U ), and hence as an element A ( z ) of SL ( r + 1)( z ), wherewe set K ( z ) = K ( C ( z )). Changing the trivialization of F SL ( r +1) | U via g ( z ) ∈ SL ( r + 1)( z )changes A ( z ) by the following q - gauge transformation :(2.1) A ( z ) g ( qz ) A ( z ) g ( z ) − . This shows that the set of equivalence classes of pairs ( F SL ( r +1) , A ) as above is in bijectionwith the quotient of SL ( r + 1)( z ) by the q -gauge transformations (2.1). Equivalently, onecould consider the associated to F SL ( r +1) the vector bundle E of rank r + 1 over P anddefine ( SL ( r + 1) , q )-connection as a section of Hom O U ( E, E q ), which is invertible and hasdeterminant 1.Following [FKSZ] we define a ( SL ( r + 1) , q )-oper as follows. Definition 2.1.
A meromorphic ( SL ( r + 1) , q )- oper (or simply a q - oper ) on P is a triple( F SL ( r +1) , A, F B − ), where A is a meromorphic ( SL ( r + 1) , q )-connection on a SL ( r + 1)-bundle F SL ( r +1) on P and F B − is the reduction of F SL ( r +1) to B − satisfying the followingcondition: there exists a Zariski open dense subset U ⊂ P together with a trivialization ı B − of F B − , such that the restriction of the connection A : F SL ( r +1) −→ F qSL ( r +1) to U ∩ M − q ( U ), written as an element of G ( z ) using the trivializations of F SL ( r +1) and F qSL ( r +1) on U ∩ M − q ( U ) induced by ı B − takes values in the Bruhat cell B − ( C [ U ∩ M − q ( U )]) cB − ( C [ U ∩ M − q ( U )]).Thus locally, any q -oper connection A can be written (using a particular trivialization ı B − ) in the form A ( z ) = n ′ ( z ) Y i ( φ i ( z ) ˇ α i s i ) n ( z )(2.2)where φ i ( z ) ∈ C ( z ) and n ( z ) , n ′ ( z ) ∈ N − ( z ) are such that their zeros and poles are outsidethe subset U ∩ M − q ( U ) of P .However, we used another definition in [KSZ], namely: Definition 2.2.
A meromorphic ( GL ( r + 1) , q )- oper on P is a triple ( A, E, L • ), where E is a vector bundle of rank r + 1 and L • is the corresponding complete flag of the vectorbundles, L r +1 ⊂ ... ⊂ L i +1 ⊂ L i ⊂ L i − ⊂ ... ⊂ L = E, OROIDAL q -OPERS 13 where L r +1 is a line bundle, so that A ∈ Hom O U ( E, E q ) satisfies the following conditions:i) A · L i ⊂ L i − ,ii) There exists a Zariski open dense subset U ⊂ P , such that the restriction of the connec-tion A ∈ Hom ( L • , L q • ) to U ∩ M − q ( U ), which belongs to GL ( r +1) and satisfies the conditionthat the induced operator ¯ A : L i / L i +1 −→ L i − / L i is an isomorphism on U ∩ M − q ( U ).An ( SL ( r + 1) , q )- oper is a ( GL ( r + 1) , q )-oper with the condition that det ( A ) = 1 on U ∩ M − q ( U ).One can derive the second definition from the first by considering the associated bundle E = ( F SL ( r +1) × V ω ) /SL ( r + 1), where V ω in the defining representation of G . Notice thatthe second definition implies local formula (2.2) in the defining representation and thus byfaithfulness the first definition follows from the second.2.3. Miura ( SL ( r + 1) , q ) -opers. Miura condition for the the q -opers corresponds to theintroduction of an additional datum: reduction of the underlying SL ( r + 1)-bundle to theBorel subgroup B + (opposite to B − ) that is preserved by the oper q -connection. Definition 2.3. A Miura ( SL ( r + 1) , q ) -oper on P is a quadruple ( F SL ( r +1) , A, F B − , F B + ),where ( F SL ( r +1) , A, F B − ) is a meromorphic ( SL ( r + 1) , q )-oper on P and F B + is a reductionof the SL ( r + 1)-bundle F SL ( r +1) to B + that is preserved by the q -connection A .An equivalent definition using flag of subbundles can be obtained by using the explicitidentification of G/B + with the flag variety. Definition 2.4. A Miura ( SL ( r + 1) , q ) -oper on P is a quadruple ( E, A, L • , ˆ L • ), where( E, A, L • ) is a meromorphic ( SL ( r + 1) , q )-oper on P and ˆ L • = { L i } is another full flag ofsubbundles in E that is preserved by the q -connection A .Forgetting F B + , we associate a ( SL ( r + 1) , q )-oper to a given Miura ( SL ( r + 1) , q )-oper.We will refer to it as the ( SL ( r + 1) , q )-oper underlying this Miura ( SL ( r + 1) , q )-oper.From a point of view of local consideration, let U be a Zariski open dense subset on P as in Definition 2.1. Choosing a trivialization ı B − of F SL ( r +1) on U ∩ M − q ( U ), we can writethe q -connection A in the form (2.2). On the other hand, using the B + -reduction F B + , wecan choose another trivialization of F SL ( r +1) on U ∩ M − q ( U ) such that the q -connection A acquires the form e A ( z ) ∈ B + ( z ). Hence there exists SL ( r + 1)( z ) ∈ G ( z ) such that(2.3) g ( zq ) n ′ ( z ) Y i ( φ i ( z ) ˇ α i s i ) n ( z ) g ( z ) − = e A ( z ) ∈ B + ( z ) . Suppose we are given a principal SL ( r + 1)-bundle F SL ( r +1) on any smooth complexmanifold X equipped with reductions F B − and F B + to B − and B + , respectively. Then weassign to any point x ∈ X an element of the Weyl group S r +1 . Namely, the fiber F G,x of F SL ( r +1) at x is a G -torsor with reductions F B − ,x and F B + ,x to B − and B + , respectively.Choose any trivialization of F SL ( r +1) ,x , i.e. an isomorphism of G -torsors F SL ( r +1) ,x ≃ G .Under this isomorphism, F B − ,x gets identified with aB − ⊂ SL ( r + 1) and F B + ,x with bB + .Then a − b is a well-defined element of the double quotient B − \ G/B + , which is in bijectionwith W SL ( r +1) . Hence we obtain a well-defined element of W SL ( r +1) = S r +1 .We will say that F B − and F B + have a generic relative position at x ∈ X if the elementof W G assigned to them at x is equal to 1 (this means that the corresponding element a − b belongs to the open dense Bruhat cell B − · B + ⊂ G ). Using Bruhat decomposition: SL ( r + 1)( z ) = F w ∈ W SL ( r +1) B + ( z ) wN − ( z ), we claim that g ( z ) from (2.3) lies in the w = 1 cell, namely: g ( z ) ∈ B + ( z ) N − ( z ) . Using the notion of relative position, we can reformulate this local statement as thefollowing theorem, which was proven in [FKSZ]:
Theorem 2.5.
For any Miura ( SL ( r + 1) , q ) -oper on P , there exists an open dense subset V ⊂ P such that the reductions F B − and F B + are in generic relative position for all x ∈ V . Returning back to the local expression (2.3) we now wish to characterize the explicitrepresentatives for ˜ A ( z ). Theorem 2.6.
Every element of the set N − ( z ) Q i φ i ( z ) ˇ α i s i N − ( z ) ∩ B + ( z ) can be writtenin the form (2.4) Y i g ˇ α i i e ti ( z ) φi ( z ) gi e i , g i ∈ C ( z ) , where each t i ( z ) ∈ C ( z ) is determined by the lifting s i . This fact was proven in higher generality in [FKSZ]. Note that in the case of SL ( r + 1)for a given order of s i this follows directly from the matrix realization.From now on we consider the liftings s i of simple reflections w i ∈ W in such a way that t i = 1 for ( i = 1 , . . . , r ).3. Z-twisted Miura ( SL ( r + 1) , q ) -opers Z-twisted (Miura) opers.
In this paper we consider a class of (Miura) q -opers thatare gauge equivalent to a constant element of SL ( r + 1) (as ( SL ( r + 1) , q )-connections).Moreover, we assume that such element Z be the regular element of the maximal torus H .One can express it as follows(3.1) Z = r Y i =1 ζ ˇ α i i , ζ i ∈ C × . Definition 3.1. A Z -twisted ( SL ( r + 1) , q ) -oper on P is a ( SL ( r + 1) , q )-oper that isequivalent to the constant element Z ∈ H ⊂ H ( z ) under the q -gauge action of SL ( r + 1)( z ),i.e. if A ( z ) is the meromorphic oper q -connection (with respect to a particular trivializationof the underlying bundle), there exists g ( z ) ∈ G ( z ) such that A ( z ) = g ( qz ) Zg ( z ) − . (3.2)A Z -twisted Miura ( SL ( r + 1) , q ) -oper is a Miura ( SL ( r + 1) , q )-oper on P that is equivalentto the constant element Z ∈ H ⊂ H ( z ) under the q -gauge action of B + ( z ), i.e. A ( z ) = v ( qz ) Zv ( z ) − , v ( z ) ∈ B + ( z ) . (3.3)It follows from Definition 3.1 that any Z -twisted ( SL ( r + 1) , q )-oper is also Z ′ -twisted forany Z ′ in the S r +1 -orbit of Z . But if we endow it with the structure of a Z -twisted Miura( SL ( r + 1) , q )-oper (by adding a B + -reduction F B + preserved by the oper q -connection),then we fix a specific element in this S r +1 -orbit.Thus we have the following Proposition, which allows to characterize Z -twisted Miuraq-opers associated to Z -twisted q-opers. OROIDAL q -OPERS 15 Proposition 3.2.
Let Z ∈ H be regular. For any Z -twisted ( SL ( r +1) , q ) -oper ( F SL ( r +1) , A, F B − ) and any choice of B + -reduction F B + of F SL ( r +1) preserved by the oper q -connection A , theresulting Miura ( SL ( r + 1) , q ) -oper is Z ′ -twisted for a particular Z ′ ∈ S r +1 · Z . The set of A -invariant B + -reductions F B + on the ( SL ( r + 1) , q ) -oper is in one-to-one correspondencewith the elements of W . Cartan connections.
Consider a Miura ( SL ( r + 1) , q )-oper. By Corollary 3.4, theunderlying ( SL ( r + 1) , q )-connection can be written in the form (3.9). Since it preservesthe B + -bundle F B + underlying this Miura ( SL ( r + 1) , q )-oper (see Definition 2.3), it maybe viewed as a meromorphic ( B + , q )-connection on P . Taking the quotient of F B + by N + = [ B + , B + ] and using the fact that B/N + ≃ H , we obtain an H -bundle F B + /N + andthe corresponding ( H, q )-connection, which we denote by A H ( z ). According to formula(3.9), it is given by the formula(3.4) A H ( z ) = Y i g i ( z ) ˇ α i . We call A H ( z ) the associated Cartan q –connection of the Miura q -oper A ( z ).Now, if our Miura q -oper is Z -twisted (see Definition 3.1), then we also have A ( z ) = v ( qz ) Zv ( z ) − , where v ( z ) ∈ B + ( z ). Since v ( z ) can be written as(3.5) v ( z ) = Y i y i ( z ) ˇ α i n ( z ) , n ( z ) ∈ N + ( z ) , y i ( z ) ∈ C ( z ) × , the Cartan q -connection A H ( z ) has the form(3.6) A H ( z ) = Y i y i ( qz ) ˇ α i Z Y i y i ( z ) − ˇ α i and hence we will refer to A H ( z ) as Z - twisted Cartan q -connection . This formula shows that A H ( z ) is completely determined by Z and the rational functions y i ( z ). Indeed, comparingthis equation with (3.4) gives(3.7) g i ( z ) = ζ i y i ( qz ) y i ( z ) . We note that A H ( z ) determines the y i ( z )’s uniquely up to scalar.3.3. (Miura) q-opers with regular singularities. Let { Λ i ( z ) } i =1 ,...,N − be a collectionof non-constant polynomials. Definition 3.3.
A ( SL ( r +1) , q )- oper with regular singularities determined by { Λ i ( z ) } i =1 ,...,r is a q -oper on P whose q -connection (2.2) may be written in the form(3.8) A ( z ) = n ′ ( z ) Y i (Λ i ( z ) ˇ α i s i ) n ( z ) , n ( z ) , n ′ ( z ) ∈ N − ( z ) . A Miura ( SL ( r +1) , q ) -oper with regular singularities determined by polynomials { Λ i ( z ) } i =1 ,...,r is a Miura ( SL ( r + 1) , q )-oper such that the underlying q -oper has regular singularities de-termined by { Λ i ( z ) } i =1 ,...,r .The following theorem follows from Theorem 2.6 and gives an explicit parameterizationof generic elements from the space of Miura opers. Theorem 3.4.
For every Miura ( SL ( r + 1) , q ) -oper with regular singularities determinedby the polynomials { Λ i ( z ) } i =1 ,...,r , the underlying q -connection can be written in the form (3.9) A ( z ) = Y i g i ( z ) ˇ α i e Λ i ( z ) gi ( z ) e i , g i ( z ) ∈ C ( z ) × . Miura-Pl¨ucker ( SL ( r + 1) , q ) -opers In this section, we will talk about the notion of nondegeneracy and we will relax theZ-twisted condition slightly. We will associate to the given ( SL ( r + 1) , q )-Miura oper acollection of ( GL (2) , q )-opers and require that all of them are Z-twisted with some nonde-generacy conditions. This will lead to the notion of Z -twisted Miura-Pl¨ucker q-opers. In thenext section we will show that for SL ( r + 1) this relaxed Z-twisted condition is equivalentto the original Z -twisted condition.4.1. The associated Miura (GL(2) , q ) -opers and Miura-Pl¨ucker condition. Let V i be the irreducible representation of SL ( r + 1) with the highest weight ω i . Notice, that theone-dimensional and two-dimensional subspaces L i and W i of V i spanned by the weightvectors ν ω i (the highest weight vector), and ν ω i , f i · ν ω i are a B + -invariant subspaces of V i .Now let ( F SL ( r +1) , A, F B − , F B + ) be a Miura ( SL ( r + 1) , q )-oper with regular singularitiesdetermined by polynomials { Λ i ( z ) } i =1 ,...,r (see Definition 3.3). Recall that F B + is a B + -reduction of a G -bundle F SL ( r +1) on P preserved by the ( SL ( r + 1) , q )-connection A .Therefore for each i = 1 , . . . , r , the vector bundle V i = F B + × B + V i = F SL ( r +1) × SL ( r +1) V i associated to V i contains a rank two subbundle W i = F B + × B + W i associated to W i ⊂ V i , and W i in turn contains a line subbundleˆ L i = F B + × B + L i associated to L i ⊂ W i .Denote by φ i ( A ) the q -connection on the vector bundle V i corresponding to the aboveMiura q -oper connection A . Since, by definition A preserves F B + , we obtain that φ i ( A )preserves the subbundles L i and W i of V i and thus produces ( GL (2) , q )-oper on W i . Letus denote such q-oper by A i .If we trivialize F B + on a Zariski open subset of P so that A ( z ) has the form (3.9) withrespect to this trivialization (see Corollary 3.4). This trivializes the bundles V i , W i , and L i ,so that the q -connection A i ( z ) becomes a 2 × C ( z ). Moreover, W i decomposes into direct sum of two subbundles, ˆ L i , preserved by B + and L i with respectto which it satisfies the (GL(2),q)-oper condition. We can unify all that in the followingProposition. Proposition 4.1.
The quadruple ( A i , W i , L i , ˆ L i ) forms a ( GL (2) , q ) Miura oper, so thatexplicitly: (4.1) A i ( z ) = g i ( z ) Λ i ( z ) Q j>i g j ( z ) − a ji g − i ( z ) Q j = i g j ( z ) − a ji , OROIDAL q -OPERS 17 where we use the ordering of the simple roots determined by the Coxeter element c . Now we impose the condition (3.6) on the corresponding A H connection, namely g i = ζ i y i ( qz ) y i ( z ) . Let G i ∼ = SL(2) be the subgroup of SL ( r + 1) corresponding to the sl (2)-triple spanned by { e i , f i , ˇ α i } , which preserves W i . Performing the gauge transformation via diagonal matrixfor 4.1, we can represent the resulting connection as follows: e A i ( z ) = u ( qz ) A i ( z ) u − ( z ) = (cid:18) Q j = i ζ − a ji j (cid:19) A i ( z )= (cid:18) Q j = i ζ − a ji j (cid:19) g ˇ α i i ( z ) e ρi ( z ) gi ( z ) e i . (4.2)where, where(4.3) ρ i ( z ) = Λ i ( z ) Y j>i ( ζ j y j ( qz )) − a ji Y j
1. It should benoted that it has regular singularities in the sense of Definition 3.3 if and only if ρ i ( z ) is apolynomial. For example, this holds for all i if all y j ( z ) , j = 1 , . . . , N −
1, are polynomials,an observation we will use below.Now we are ready to relax Z -twisted condition as follows. Definition 4.2. A Z - twisted Miura-Pl¨ucker ( SL ( r + 1) , q ) -oper is a meromorphic Miura( SL ( r + 1) , q )-oper on P with the underlying q -connection A ( z ), such that there exists v ( z ) ∈ B + ( z ) such that for all i = 1 , . . . , r , the Miura (GL(2) , q )-opers A i ( z ) associated to A ( z ) by formula (4.1) can be written in the form:(4.4) A i ( z ) = v ( zq ) Zv ( z ) − | W i = v i ( zq ) Z i v i ( z ) − where v i ( z ) = v ( z ) | W i and Z i = Z | W i .Note, that it follows from the above definition that the ( H, q )-connection A H ( z ) asso-ciated to a Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q )-oper can be written in the same form(3.6) as the ( H, q )-connection associated to a Z -twisted Miura ( SL ( r + 1) , q )-oper.However, while it is true that every Z -twisted Miura ( SL ( r + 1) , q )-oper is automaticallya Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q )-oper, but the converse is not necessarily true if N = 2.4.2. Nondegeneracy conditions.
In what follows, we will say that v, w ∈ C × are q -distinct if q Z v ∩ q Z w = ∅ .In this subsection we introduce two nondegeneracy conditions for Z -twisted Miura-Pl¨ucker q -opers. The first of them, called the H -nondegeneracy condition, is applicableto arbitrary Miura q -opers with regular singularities. Recall from Corollary 3.4 that theunderlying q -connection can be represented in the form (3.9). Definition 4.3.
A Miura ( SL ( r + 1) , q )-oper A ( z ) of the form (3.9) is called H - nondegene-rate if the corresponding ( H, q )-connection A H ( z ) can be written in the form (3.6), wherefor all i, j, k with i = j and a ik = 0 , a jk = 0, the zeros and poles of y i ( z ) and y j ( z ) are q -distinct from each other and from the zeros of Λ k ( z ). Next, we define the second nondegeneracy condition. This condition applies to Z -twistedMiura-Pl¨ucker ( SL ( r + 1) , q )-opers. Firstly, we start from ( SL (2) , q )-opers.Consider a Miura (SL(2) , q )-oper given by formula (3.9), which reads in this case: A ( z ) = g ( z ) ˇ α exp (cid:18) Λ( z ) g ( z ) e (cid:19) so that the corresponding Cartan q -connection A H ( z ) is A H ( z ) = g ( z ) ˇ α , where y ( z ) is arational function. Let us assume that A ( z ) is H -nondegenerate (see Definition 4.3). Thismeans that the zeros of Λ( z ) are q -distinct from the zeros and poles of y ( z ).If we apply to A ( z ) a q -gauge transformation by an element of h ( z ) ˇ α ∈ H [ z ], we obtaina new q -oper connection(4.5) e A ( z ) = e g ( z ) ˇ α exp e Λ( z ) e g ( z ) e ! , where e g ( z ) = g ( z ) h ( zq ) h ( z ) − , e Λ( z ) = Λ( z ) h ( zq ) h ( z ). It also has regular singularities, butfor a different polynomial e Λ( z ), and e A ( z ) may no longer be H -nondegenerate. However,it turns out there is an essentially unique gauge transformation from H [ z ] for which theresulting e A ( z ) is H -nondegenerate e A H ( z ) and e y ( z ) is a polynomial. This choice allows usto fix the polynomial Λ( z ) determining the regular singularities of our ( SL (2) , q )-oper. Lemma 4.4. [FKSZ](1)
There is an (SL(2) , q ) -oper e A ( z ) in the H [ z ] -gauge class of A ( z ) for which e A H ( z ) = e g ( z ) ˇ α is nondegenerate and the rational function e y ( z ) is a polynomial. This operis unique up to a scalar a ∈ C × that leaves e g ( z ) unchanged but multiplies e y ( z ) and e Λ( z ) by a and a , respectively. (2) This (SL(2) , q ) -oper e A ( z ) may also be characterized by the property that e Λ( z ) hasmaximal degree subject to the constraint that it is H -nondegenerate. This motivates the following definition.
Definition 4.5. A Z -twisted Miura (SL(2) , q )-oper is called nondegenerate if it is H -nondegenerate and the rational function y ( z ) appearing in formula (3.6) is a polynomial.We now turn to the general case. Recall Definition 4.2 of Z -twisted Miura-Pl¨ucker ( SL ( r +1) , q )-opers. Definition 4.6.
Suppose that r >
1. A Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q )-oper A ( z )is called nondegenerate if its associated Cartan q -connection A H ( z ) is nondegenerate andeach associated Z i -twisted Miura (SL(2) , q )-oper A i ( z ) is nondegenerate.It turns out that this simply means that in addition to A H ( z ) being nondegenerate, each y i ( z ) from formula (3.6) is a polynomial. Here we provide the complete proof, since we willneed it for infinite-dimensional case. Proposition 4.7.
Suppose that r > , and let A ( z ) be a Z -twisted Miura-Pl¨ucker ( SL ( r +1) , q ) -oper. The following statements are equivalent: (1) A ( z ) is nondegenerate. (2) The Cartan q -connection A H ( z ) is nondegenerate, and each A i ( z ) has regular sin-gularities, i.e. ρ i ( z ) given by formula (4.3) is in C [ z ] . OROIDAL q -OPERS 19 (3) Each y i ( z ) from formula (3.6) is a polynomial, and for all i, j, k with i = j and a ik = 0 , a jk = 0 , the zeros of y i ( z ) and y j ( z ) are q -distinct from each other andfrom the zeros of Λ k ( z ) .Proof. To prove that (2) implies (3), we need only show that if each ρ i ( z ) given by formula(4.3) is in C [ z ], then the y i ( z )’s are polynomials. Suppose y i ( z ) is not a polynomial, andchoose j = i such that a ij = 0. Then − a ij > y i ( z ) or y i ( qz )appears in the denominator of ρ j ( z ). Moreover, since the poles of y i ( z ) are q -distinct fromthe zeros of Λ j ( z ) and the other y k ( z )’s, the poles of y i ( z ) or y i ( qz ) would give rise to polesof ρ j ( z ). But then A j ( z ) would not have regular singularities.Next, assume (3). Then A H ( z ) is nondegenerate by Definition 4.3. Since all the y i ( z )’sare polynomials, the same if true for the ρ i ( z )’s. (Here we are using the fact that the off-diagonal elements of the Cartan matrix, a ij with i = j , are less than or equal to 0.) Since ρ i ( z ) is a product of polynomials whose roots are q -distinct from the roots of y i ( z ), we seethat the Cartan q -connection associated to A i ( z ) is nondegenerate.Finally, (2) is a trivial consequence of (1). (cid:3) If we apply a q -gauge transformation by an element h ( z ) ∈ H [ z ] to A ( z ), we get a new Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q )-oper. However, the following proposition shows thatit is only nondegenerate if h ( z ) ∈ H . As a consequence, the Λ k ’s of a nondegenerate q -operare determined up to scalar multiples. Proposition 4.8. [FKSZ] If A ( z ) is a nondegenerate Z -twisted Miura-Pl¨ucker ( SL ( r + 1) , q ) -oper and h ( z ) ∈ H [ z ] , then h ( qz ) A ( z ) h ( z ) − is nondegenerate if and only if h ( z ) is a constant element of H . Z-twisted Miura ( SL ( r + 1) , q ) -opers and QQ -systems QQ -systems and Miura-Pl¨ucker ( SL ( r + 1) , q ) -opers. One of the main results ofprevious section was the explicit structure of the non-degenerate Miura-Pl¨ucker ( SL ( r +1) , q )-oper with regular singularities defined by { Λ i ( z ) } i =(1 ,...,r ) and associated with regularelement Z = Q i ζ ˇ α i i . Following Proposition 4.7 the local expression, namely A ( z ) can beexpressed as follows:(5.1) A ( z ) = Y i g i ( z ) ˇ α i e Λ i ( z ) gi ( z ) e i , g i ( z ) = ζ i Q + i ( qz ) Q + i ( z ) . where Q + i ( z ) are monic polynomials (here we changed the notation y i ( z ) ≡ Q + i ( z )). Fromnow on, we will assume that Z satisfies the following property:(5.2) r Y i =1 ζ a ij i = ζ j ζ j +1 ζ j − / ∈ q Z , ∀ j = 1 , . . . , r , where a ij are matrix elements of the Cartan matrix for sl r +1 . Since Q ri =1 ζ a ij i = 1 is aspecial case of (5.2), this implies that Z is regular semisimple .5.2. The SL ( r + 1) QQ -system. In [FKSZ] the following statement was proven (we spe-cialize that result to the case of SL ( r + 1)): Theorem 5.1.
There is a one-to-one correspondence between the set of nondegenerate Z -twisted Miura-Pl¨ucker ( SL ( r +1) , q ) -opers and the set of nondegenerate polynomial solutionsof the QQ -system (5.3) ξ i Q + i ( qz ) Q − i ( z ) − ξ i +1 Q + i ( z ) Q − i ( qz ) = Λ i ( z ) Q + i − ( z ) Q + i +1 ( qz ) , i = 1 , . . . , r subject to the boundary conditions Q ± ( z ) = Q ± r +1 ( z ) = 1 and ξ = ξ r +2 = 1 so that ξ = ζ , ξ = ζ ζ , . . . ξ r = ζ r ζ r − , ξ r +1 = 1 ζ r . Note, that ξ i is the i th element on the diagonal of Z from (3.1).We will say that a polynomial solution { Q + i ( z ) , Q − i ( z ) } i =1 ,...,r of (5.3) is nondegenerate ifthe following conditions are satisfied: relation (5.2) holds; for i = j the zeros of Q + i ( z ) and Q − j ( z ) are q -distinct from each other and from the zeros of Λ k ( z ) for | i − k | = 1 , | j − k | = 1.For the convenience we will rewrite (5.3) as follows:(5.4) ξ i φ i ( z ) − ξ i +1 φ i ( qz ) = ρ i ( z ) , where(5.5) φ i ( z ) = Q − i ( z ) Q + i ( z ) , ρ i ( z ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) Q + i ( z ) Q + i ( qz ) . Extended QQ -system and Z -twisted ( SL ( r +1) , q ) -opers. As it was demonstratedin [FKSZ] for a simply-connected simple Lie group G the set of nondegenerate Z -twistedMiura-Pl¨ucker q-opers includes as a subset the set of Z -twisted Miura ( G, q )-opers. Theopposite inclusion was possible provided that Z -twisted Miura-Pl¨ucker q-opers are in addi-tion w -generic (see Theorem 7.10). We will discuss this notion in detail later, in subsection5.6.In this section we shall demonstrate that when G is a special linear group then we do notneed this extra condition and that the corresponding Z -twisted Miura-Pl¨ucker ( SL ( r +1) , q )-oper will be Z -twisted Miura q-oper, namely there exists v ( z ) ∈ B + ( z ), such that theq-connection A ( z ) reduces to an element of the form (3.1), or, equivalently(5.6) v ( qz ) − A ( z ) = Zv ( z ) − . Moreover, we will construct explicit expression for v ( z ).The following statement is a generalization of the result of [MV] to Z -twisted q-opers. Theorem 5.2.
Let A ( z ) be as in (5.1) and Z as in (3.1) . Suppose Q − i,i +1 ,...,j ( z ) ( i, j ∈ Z , i < j ) are polynomials, satisfying equations: ξ i φ i ( z ) − ξ i +1 φ i ( qz ) = ρ i ( z ) , i = 1 , . . . , rξ i φ i,i +1 ( z ) − ξ i +2 φ i,i +1 ( qz ) = ρ i +1 ( z ) φ i ( qz ) , i = 1 , . . . , r − . . . . . . (5.7) ξ i φ i,...,r − i ( z ) − ξ r + i − φ i,...,r − i ( qz ) = ρ r − ( z ) φ i,...,r − i ( qz ) , i = 1 , ξ φ ,...,r ( z ) − ξ r +1 φ ,...,r ( qz ) = ρ r ( z ) φ ,...,r − ( qz ) , where for all j > i (5.8) φ i,...,j ( z ) = Q − i,...,j ( z ) Q + j ( z ) . OROIDAL q -OPERS 21 Then there exist v ( z ) ∈ B + ( z ) such that (5.6) holds and is given by (5.9) v ( z ) = r Y i =1 Q + i ( z ) ˇ α i · r Y i =1 V i ( z ) , where (5.10) V i ( z ) = r Y j = i exp ( − φ i,...,j ( z ) e i,...,j ) , e i,...,j = [ . . . [[ e i , e i +1 ] , e i +2 ] . . . e j ] . We shall prove a more general statement in Section 9 about ( GL ( ∞ ) , q ) opers which willcontain Theorem 5.2 as a corollary. Here, to illustrate how the theorem works, we willregard some low rank examples.Notice that although the expression for v ( z ) in (5.9) is rather complicated, the inverse v ( z ) − can be succinctly presented as(5.11) v ( z ) − = Q +1 ( z ) Q − ( z ) Q +2 ( z ) Q − ( z ) Q +3 ( z ) . . . Q − ,...,r − ( z ) Q + r ( z ) Q − ,...,r ( z )0 Q +1 ( z ) Q +2 ( z ) Q − ( z ) Q +3 ( z ) . . . Q − ,...,r − ( z ) Q + r ( z ) Q − ,...,r ( z )0 0 Q +2 ( z ) Q +3 ( z ) . . . Q − ,...,r − ( z ) Q + r ( z ) Q − ,...,r ( z )... ... ... . . . ... ...0 . . . . . . . . . Q + r − ( z ) Q + r ( z ) Q − r ( z )0 . . . . . . . . . Q + r ( z ) . Before we continue the following statement will be needed.
Lemma 5.3.
The following relations hold for any u, v ∈ C and i, j = 1 , . . . , ru ˇ α i e ve j = exp ( u a ji v e i ) u ˇ α i . (5.12) In general, if [ X, Y ] = sY we have (5.13) u X e vY = exp( u s vY ) u X . Using this Lemma we can rewrite the q-connection (5.1) such that the roots of SL ( r + 1)are placed in the decreasing order. Lemma 5.4.
Let (5.14) ρ i ( z ) = Λ i ( z ) Q i − ( qz ) Q i +1 ( z ) Q i ( qz ) Q i ( z ) . Then the ( SL ( r + 1) , q ) -oper reads (5.15) A ( z ) = Y i = r Q + i ( qz ) ˇ α i · Y i = r e ζiζi +1 ρ i ( z ) e i · Y i = r ζ ˇ αi Q + i ( z ) − ˇ α i , or as a matrix (5.16) A ( z ) = g ( z ) Λ ( z ) 0 0 . . . g ( z ) g ( z ) Λ ( z ) 0 . . . g ( z ) g ( z ) Λ ( z ) . . . ... ... · · · . . . . . . ... ...... ... · · · . . . . . . Λ r − ( z ) 00 0 0 . . . . . . g r ( z ) g r − ( z ) Λ r ( z )0 0 0 . . . . . . g r ( z ) At this point the above choice of the order of simple roots may seem unsubstantiated,however, it will be justified in later sections, where we will consider ( GL ( ∞ ) , q )-opers.5.4. Examples.
Miura ( SL (2) , q ) -oper. The twist element Z = ζ ˇ α = diag( ζ, ζ − ) = diag( ξ , ξ ) The q -connection (5.15) reads(5.17) A ( z ) = Q + ( qz ) ˇ α · e ζρ ( z ) e · ζ ˇ α Q + ( z ) − ˇ α = (cid:18) g ( z ) Λ( z )0 g ( z ) − (cid:19) We look for the gauge transformation in the form(5.18) v ( z ) = Q + ( z ) ˇ α e − φ ( z ) e , where φ ( z ) = Q − ( z ) Q + ( z ) . The left hand side of (5.6) reads v ( qz ) − A ( z ) = e φ ( qz ) e e ζρ ( z ) e · ζ ˇ α Q + ( z ) − ˇ α (5.19)where ρ ( z ) = Λ( z ) Q + ( z ) Q + ( qz ) . Meanwhile, the right hand side of (5.6) equals Zv ( z ) − = ζ ˇ α e − φ ( z ) e Q + ( z ) − ˇ α = e − ζ φ ( z ) e ζ ˇ α Q + ( z ) − ˇ α , (5.20)where we used Lemma 5.3 in the last step. Comparing the above two expressions yields thedesired QQ -system equation(5.21) ζφ ( z ) − ζ − φ ( qz ) = ρ ( z ) , or equivalently as(5.22) ξ φ ( z ) − ξ φ ( qz ) = ρ ( z ) . Miura ( SL (3) , q ) -oper. Consider Z = ζ ˇ α ζ ˇ α = diag( ζ , ζ ζ , ζ ) = diag( ξ , ξ , ξ ). Theq-connection is given by A ( z ) = Q +1 ( qz ) ˇ α Q +2 ( qz ) ˇ α · e ζ ρ ( z ) e e ζ ρ z ) ζ e · ζ ˇ α ζ ˇ α Q +1 ( z ) − ˇ α Q +2 ( z ) − ˇ α = g ( z ) Λ ( z ) 00 g ( z ) g ( z ) Λ ( z )0 0 g ( z ) (5.23)while the gauge transformation reads(5.24) v ( z ) = Q +1 ( z ) ˇ α Q +2 ( z ) ˇ α e − φ ( z ) e e − φ ( z )[ e ,e ] e − φ ( z ) e OROIDAL q -OPERS 23 Thus the left hand side of (5.6) becomes v ( qz ) − A ( z ) = e φ ( qz ) e e φ ( qz )[ e ,e ] e φ ( qz ) e · e ζ ρ ( z ) e e ζ ρ z ) ζ e · ζ ˇ α ζ ˇ α Q +1 ( z ) − ˇ α Q +2 ( z ) − ˇ α = e ( φ ( qz )+ ζ ρ ) e e ( φ ( qz )+ ζ ρ ( z ) φ ( qz ))[ e ,e ] e ( φ ( qz )+ ζ ζ ρ ( z )) e · ζ ˇ α ζ ˇ α Q +1 ( z ) − ˇ α Q +2 ( z ) − ˇ α (5.25)Meanwhile the right hand side of (5.6) equals(5.26) Zv ( z ) − = ζ ˇ α ζ ˇ α · e φ ( qz ) e e φ ( qz )[ e ,e ] e φ ( qz ) e Q +1 ( z ) − ˇ α Q +2 ( z ) − ˇ α Now we need to move all Cartan elements from the front of the above expression to its rearusing Lemma 5.3 Zv ( z ) − = e ζ φ z ) ζ e e ζ ζ φ ( z )[ e ,e ] e ζ φ z ) ζ e · ζ ˇ α ζ ˇ α Q +1 ( z ) − ˇ α Q +2 ( z ) − ˇ α (5.27)By comparing (5.25) with (5.27) we get ζ φ ( z ) − ζ ζ φ ( qz ) = ρ ( z ) ,ζ ζ φ ( z ) − ζ φ ( qz ) = ρ ( z ) ,ζ φ ( z ) − ζ φ ( qz ) = ρ ( z ) φ ( qz ) , (5.28)or equivalently, ξ φ ( z ) − ξ φ ( qz ) = ρ ( z ) ,ξ φ ( z ) − ξ φ ( qz ) = ρ ( z ) ,ξ φ ( z ) − ξ φ ( qz ) = ρ ( z ) φ ( qz ) . (5.29)5.5. The Extended QQ -System and Bethe ansatz. The first line of (5.7) is the SL ( r +1) QQ -system (5.4). In the rest of the equations we introduced new functions (5.8). Noticethat ρ i +1 ( z ) φ i ( qz ) = Λ i +1 ( z ) Q − i ( qz ) Q + i +2 ( z ) Q + i +1 ( z ) Q + i +1 ( qz ) =: ρ i,i +1 ( z ) , where ρ i,i +1 ( z ) is ρ i +1 ( z ) with Q + i ( z ) replaced by Q − i ( z ). In terms of this new notation wecan rewrite (5.7) as follows ξ i φ i ( z ) − ξ i +1 φ i ( qz ) = ρ i ( z ) , i = 1 , . . . , rξ i φ i,i +1 ( z ) − ξ i +2 φ i,i +1 ( qz ) = ρ i,i +1 ( z ) , i = 1 , . . . , r − . . . . . . (5.30) ξ i φ i,...,r − i ( z ) − ξ r + i − φ i,...,r − i ( qz ) = ρ i,...,r − i ( z ) , i = 1 , ξ φ ,...,r ( z ) − ξ r +1 φ ,...,r ( qz ) = ρ ,...,r ( z ) , For the future reference let us rewrite the above equations in terms of the Q-polynomials: ξ i Q + i ( qz ) Q − i ( z ) − ξ i +1 Q + i ( z ) Q − i ( qz ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) ,ξ i Q + i +1 ( qz ) Q − i,i +1 ( z ) − ξ i +2 Q + i +1 ( z ) Q − i,i +1 ( qz ) = Λ i +1 ( z ) Q − i ( qz ) Q + i +2 ( z ) ,. . . . . . (5.31) ξ i Q + r − i ( qz ) Q − i,...,r − i ( z ) − ξ r − i Q + r − i ( z ) Q − i,...,r − i ( qz ) = Λ r − i ( z ) Q − i,...,r − i ( qz ) Q + r + i ( z ) ,ξ Q + r ( qz ) Q − ,...,r ( z ) − ξ r +1 Q + r ( z ) Q − ,...,r ( qz ) = Λ r ( z ) Q − ,...,r − ( qz ) . We shall refer to all equations of (5.31) as the extended QQ -system for SL ( r +1). We call itssolution nondegenerate , if the resulting solution of the original QQ -system is nondegenerate.Let us now show that starting from the solution of the nondegenerate QQ -system, weobtain solutions to the extended QQ -system as well. To do that we need the result (whichis true for other simply laced groups) of [FKSZ]: Theorem 5.5.
The solutions of the nondegenerate SL ( r + 1) QQ -system are in one-to-onecorrespondence to the solutions of the Bethe Ansatz equations for sl ( r + 1) XXZ spin chain: (5.32) Q + i ( qw ik ) Q + i ( q − w ki ) ξ i ξ i +1 = − Λ i ( w ik ) Q + i +1 ( qw ik ) Q + i − ( w ik )Λ i ( q − w ik ) Q + i +1 ( w ik ) Q + i − ( q − w ik ) , where i = 1 , . . . , r ; k = 1 , . . . , m i . We will extend the statement of this Theorem as follows.
Theorem 5.6.
There is a one-to-one correspondence between the set of nondegeneratesolutions of the extended QQ -system (5.31) , the set of nondegenerate solutions of the QQ -system (5.3) , and the set of solutions of Bethe Ansatz equations (5.32) .Proof. Consider the first line of (5.31) which is also presented in (5.3). If the QQ -systemis nondegenerate, then, according to the previous theorem, there is a bijection between itspolynomial nondegenerate solutions and Bethe equations. We will show now recursively thatgiven the nondegenerate solution of the QQ -system one can construct elements Q i,i +1 ,...,j satisfying the equations of the extended QQ -system. Let us immediately consider thedegenerate case, when Q − i ( z ) and Q + i +1 ( z ) have common roots: without loss of generality,let us now assume that Q − i ( z ) and Q + i +1 ( z ) have just one common root u . Now we show thatwe can construct the solution to the second equation in (5.31), namely Q + i,i +1 . Introducingthe notation(5.33) Q + i +1 ( z ) = ( z − u ) ˇ Q + i +1 ( z ) , we see from nondegeneracy condition QQ -system, namely the fact that Q + i ( z ) and Q + i +1 ( z )have q-distinct roots, we have:(5.34) Q − i ( z ) = ( z − u ) (cid:0) q − z − u (cid:1) e Q − i ( z ) . Now consider the following equation from the second line from (5.30) ξ i φ i,i +1 ( z ) − ξ i +1 φ i,i +1 ( qz ) = Λ i +1 ( z ) Q − i ( qz ) Q + i +2 ( z ) Q + i +1 ( z ) Q + i +1 ( qz ) . Substituting (5.33) and (5.34) we get(5.35) ξ i Q − i,i +1 ( z )( z − u ) ˇ Q + i +1 ( z ) − ξ i +2 Q − i,i +1 ( qz )( qz − u ) ˇ Q + i +1 ( qz ) = Λ i +1 ( z ) e Q − i ( qz ) Q + i +2 ( z )ˇ Q + i +1 ( z ) ˇ Q + i +1 ( qz ) . OROIDAL q -OPERS 25 From the residue decomposition of both sides of the above equation we conclude that u must be the root of polynomial Q − i,i +1 ( z ) Q − i,i +1 ( z ) = ( z − u ) ˇ Q − i,i +1 ( z ) . Thus, one can represent the resulting system as follows: ρ i,i +1 ( z ) = h i ( z ) + m i +1 X k =1 ,k = s b k z − w i +1 k + m i +1 X k =1 ,k = s c k qz − w i +1 k , (5.36) φ i,i +1 ( z ) = e φ i ( z ) + m i +1 X k =1 ,k = s d k z − w i +1 k , (5.37)where w i +1 s = u , h i ( z ) and e φ i ( z ) are polynomials. By matching the polar and polynomialparts of (5.35) we can readily find coefficients d k and polynomials e φ i ( z ) and hence Q + i,i +1 ( z ).The only constraint we need to satisfy is the one on b k , c k , namely b k ξ i +2 + c k ξ i = 0, where k = s . These equations are explicitly given by:(5.38) Q + i +1 ( qw i +1 k ) Q + i +1 ( q − w i +1 k ) ξ i ξ i +2 = − Λ i +1 ( w i +1 k ) Q + i +2 ( qw i +1 k ) Q − i ( w i +1 k )Λ i +1 ( q − w i +1 k ) Q + i +2 ( w i +1 k ) Q − i ( q − w i +1 k ) , where k = s. At the same time the i th equation can be rewritten as: ξ i Q + i ( qz ) Q + i ( z ) − ξ i +1 Q − i ( qz ) Q − i ( z ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) Q + i ( z ) Q − i ( z )which leads to ξ i Q + i ( qw i +1 k ) Q + i ( w i +1 k ) = ξ i +1 Q − i ( qw i +1 k ) Q − i ( w i +1 k ) , where w i +1 k are the roots of Q + i +1 ( z ) for k = s .Thus the equations (5.38) are equivalent to the Bethe equations emerging from the QQ -system:(5.39) Q + i +1 ( qw i +1 k ) Q + i +1 ( q − w i +1 k ) ξ i +1 ξ i +2 = − Λ i +1 ( w i +1 k ) Q + i +2 ( qw i +1 k ) Q + i ( w i +1 k )Λ i +1 ( q − w i +1 k ) Q + i +2 ( w i +1 k ) Q + i ( q − w i +1 k )Therefore, we found that the equation (5.35) follows from the XXZ Bethe equations.The above step can be iterated if Q − i,...,j polynomials have coincident roots with Q + j .Therefore we have shown that any QQ -system of the form (5.30) with such degeneracies isequivalent to a nondegenerate QQ -system. (cid:3) In the next section we shall present a different proof of the above theorem, exploring thedefinition of Miura ( SL ( r + 1 , q )-opers involving flags of subbundles.5.6. The extended QQ -System, Bethe equations and B¨acklund Transformations. We would like to understand the representation theoretic meaning of the extended QQ -system a bit better. In fact, motivated by Lemma 7.3 of [FKSZ] we can demonstratethat, starting from the original QQ -system (the first line of (5.31) or (5.3)), upon certainassumptions, one can recover all the remaining equations of the entire extended QQ -systemby B¨acklund transformations.
In [FKSZ] B¨acklund transformations were introduced for Miura q-opers (Proposition 7.1)and were associated to the i -th simple reflection from the Weyl group: Proposition 5.7.
Consider the q -gauge transformation of the q -connection given by (5.1) A A ( i ) = e µ i ( qz ) f i A ( z ) e − µ i ( z ) f i , where µ i ( z ) = Q + i − ( z ) Q + i +1 ( z ) Q i + ( z ) Q i − ( z ) . (5.40) Then A ( i ) ( z ) can be obtained from A ( z ) by substituting in formula (5.1) Q j + ( z ) Q j + ( z ) , j = i, (5.41) Q i + ( z ) Q i − ( z ) , Z −→ s i ( Z ) (cid:16) ζ i ζ i − ζ i +1 ζ i (cid:17) . (5.42)It is possible that after the transformation the resulting operator gives rise to the non-degenerate QQ -system. Denoting the the QQ -system after the B¨acklund transformation as { e Q ± i } i =1 ,...,r , we obtain: { e Q + j } j =1 ,...,r = { Q +1 , . . . , Q + i − , Q − i , Q + i +1 . . . , Q r + } ;(5.43) { e Q − j } j =1 ,...,r = n Q − , . . . , Q ∗− i − , − Q + i , Q − i,i +1 . . . , Q − r o { e ζ j } j =1 ,...,r = (cid:26) ζ , . . . , ζ i − , ζ i − ζ i +1 ζ i , . . . , ζ r (cid:27) The last line can be also rewritten in terms of ξ variables as follows: { e ξ j } j =1 ,...,r = { ξ , . . . , ξ i − , ξ i +1 , ξ i , ξ i +2 , . . . , ξ r +1 } Here we note, that the notation Q − i,i +1 was used for Q − i +1 , since the equation this newpolynomial satisfies, is the second one from the extended QQ -system. At the same time,the new polynomial Q ∗− i − ( z ) does not belong to what we called the extended QQ -system.As an example, if we apply the 1st B¨acklund transformation Q +1 Q − , Q −
7→ − Q +1 , ξ ξ , ξ ξ , Q − Q − , , to the QQ system for SL (3): ξ Q +1 ( qz ) Q − ( z ) − ξ Q +1 ( z ) Q − ( qz ) = Λ ( z ) Q +2 ( z ) ,ξ Q +2 ( qz ) Q − ( z ) − ξ Q +2 ( z ) Q − ( qz ) = Λ ( z ) Q +1 ( qz )(5.44)the first equation above will not change, however, the second will become(5.45) ξ Q +2 ( qz ) Q − , ( z ) − ξ Q +2 ( z ) Q − , ( qz ) = Λ ( z ) Q − ( qz ) , which completes its extended QQ -system (5.29).In general, one can talk about the successive B¨acklund transformations associated withthe Weyl group element w . If such transformations are possible, namely if after each of theelementary B¨acklund transformation one arrives to the nondegenerate q-oper (i.e. nonde-generate solution of the QQ -system), such oper is called w -generic in [FKSZ]. As one can see,the equations of the extended QQ -system emerge as a part of the all QQ -system equationsobtained after every B¨acklund transformation if the Weyl group element is constructed bysuccessive reflections along the order in the Dynkin diagram: w = s i s i +1 . . . s j − s j .We will refer to the collection of QQ -system equations, obtained via B¨acklund transfor-mations for all Weyl group elements w as the full QQ -system .One of the applications of the B¨acklund transformations which was proven in [FKSZ] isthat Z -twisted Miura-Pl¨ucker ( G, q )-oper is Z -twisted Miura ( G, q )-oper if it is w -generic,where w is the longest root. OROIDAL q -OPERS 27 Here we show that a stronger result holds for ( SL ( r + 1) , q )-opers. Combining Theorems5.2 and 5.6, we obtain the following theorem, which is the central result of this section. Theorem 5.8.
The nondegenerate Z-twisted Miura-Pl¨ucker ( SL ( r + 1) , q ) -opers are Z-twisted Miura ( SL ( r + 1) , q ) -opers. They are in one-to-one correspondence with the nonde-generate solutions of the QQ system and thus sl ( r + 1) XXZ Bethe equations. q -Opers via Quantum Wronskians Sections of line bundles and q-Wronskians.
In this section we will make useof an alternative definition of Miura ( SL ( r + 1) , q )-opers (see Definition 2.4) to describeZ-twisted Miura q-opers with regular singularities, following [KSZ]. Namely, we have acomplete flag of subbundles L • such that q-connection A maps L i into L qi − and the inducedmaps ¯ A i : L i / L i +1 −→ L qi − / L qi are isomorphisms for i = 1 , . . . , r on U ∩ M − q ( U ), where U is the Zariski open dense subset. Explicitly, considering the determinants(6.1) (cid:16) i − Y j =0 ( A ( q i − − j z ) (cid:17) s ( z ) ∧ · · · ∧ A ( q i − z ) s ( q i − z ) ∧ s ( q i − z ) (cid:12)(cid:12)(cid:12)(cid:12) Λ i L qi − r − i +2 for i = 1 , . . . , r + 1, where s is a local section of L r +1 , we claim that ( E, A, L • ) is an( SL ( r + 1) , q )-oper if and only if at every point of U ∩ M − q ( U ), there exists local section forwhich each such determinant is nonzero (see [KSZ]). When we encounter the case of regularsingularities (see Section 3.3), each ¯ A i is an isomorphism except at zeroes of Λ i and thuswe require the determinants to vanish at zeroes of the following polynomial W k ( s ): W k ( s ) = P ( z ) · P ( q z ) · · · P k ( q k − z ) , P i ( z ) = Λ r Λ r − · · · Λ r − i +1 ( z ) . (6.2)Now we discuss the Z -twisted Miura condition. Recall from Section 2.3 that Miuracondition implies that there exist a flag ˆ L • which is preserved by the q-connection A . The Z -twisted condition implies that in the gauge when A is given by fixed semisimple diagonalelement Z ∈ H such flag is formed by the standard basis e , . . . , e r +1 .The relative position between two flags is generic on U ∩ M − q ( U ). The regular singularitycondition implies that quantum Wronskians , namely determinants(6.3) D k ( s ) = e ∧ · · · ∧ e r +1 − k ∧ Z k − s ( z ) ∧ Z k − s ( qz ) ∧ · · · ∧ Zs ( q k − ) ∧ s ( q k − z )have a subset of zeroes, which coincide with those of W k ( s ). To be more explicit, for k = 1 , . . . , r + 1, we have nonzero constants α k and polynomials(6.4) V k ( z ) = r k Y a =1 ( z − v k,a ) , for which(6.5)det . . . ξ k − s ( z ) · · · ξ s ( q k − z ) s ( q k − z )... . . . ... ... ... . . . ...0 . . . ξ k − k s r +1 − k ( z ) . . . ξ k s r +1 − k ( q k − z ) s k ( q k − z )0 . . . ξ k − k +1 s r +1 − k +1 ( z ) . . . ξ r +1 − k +1 s k +1 ( q k − z ) s k +1 ( q k − z )... . . . ... ... ... . . . ...0 . . . ξ k − r +1 s r +1 ( z ) . . . ξ r +1 s r +1 ( q k − z ) s r +1 ( q k − z ) = α k W k V k ; Since D r +1 ( s ) = W r +1 ( s ), we have V r +1 = 1. We also set V = 1; this is consistent withthe fact that (6.3) also makes sense for k = 0, giving D = e ∧ · · · ∧ e r +1 .We can also rewrite (6.5) as(6.6) det i,j h ξ k − jr +1 − k + i s r +1 − k + i ( q j − z ) i = α k W k V k , where i, j = 1 , . . . , k .Note that the above determinants have slightly different form those of [KSZ] – twistparameters ξ i entered in different powers. This is due to a different order of the simpleroots in the definition of the q-oper. Theorem 6.1 ([KSZ]) . Polynomials { V k ( z ) } k =1 ,...,r give the solution to the QQ -system 5.3so that Q + j ( z ) = V j ( z ) under the nondegeneracy condition that for all i, j, k with i = j and a ik = 0 , a jk = 0 , the zeros of V i ( z ) and V j ( z ) are q -distinct from each other and from thezeros of Λ k ( z ) .Remark . Technically, we used stronger conditions in [KSZ], namely that zeroes of { Λ i } i =1 ...r and { V j ( z ) } j =1 ...r have to be q-disjoint to satisfy QQ -system equations, butone can relax it easily and even more than it is done in the statement above.In the next subsection we will show that the extended QQ -system can be obtained fromvarious minors in q-Wronskian matrices. This theorem allows to relate the section s ( z ),generating the line bundle L r +1 with the elements of the extended QQ -system using thetransformation (5.6). Proposition 6.3.
Let v ( z ) be the gauge transformation from (5.6) and s ( z ) be the sectiongenerating L r +1 in the definition of the ( SL ( r + 1) , q ) -oper. Then the components of s ( z ) in the gauge when q-oper connection is equal to Z is given by: (6.7) s r +1 ( z ) = Q + r ( z ) , s r ( z ) = Q − r ( z ) , s k ( z ) = Q − k,...,r ( z ) , for k = 1 , . . . , r − .Proof. The Proposition follows from the direct application of (5.11) Starting from (5.11) theProposition follows after acting with v ( z ) − on the basis vector e r +1 = (0 , , . . . , , (cid:3) In the next subsection we will show that the extended QQ -system can be obtained fromvarious minors in q-Wronskian matrices.6.2. Wronskians and extended QQ -systems. First, we will rewrite the extended QQ -system in a more convenient way to relate it to the minors in the q-Wronskian matrix.Namely, we multiply Q -terms by certain polynomials to get rid of the Λ-polynomials in theright hand side. This is done in the following Lemma. Lemma 6.4.
The system of equations (5.7) is equivalent to the following set of equations ξ i D + i ( qz ) D − i ( z ) − ξ i +1 D + i ( z ) D − i ( qz ) = ( ξ i − ξ i +1 ) D + i − ( qz ) D + i +1 ( z ) ,ξ i D + i +1 ( qz ) D − i,i +1 ( z ) − ξ i +2 D + i +1 ( z ) D − i,i +1 ( qz ) = ( ξ i − ξ i +2 ) D − i ( qz ) D + i +2 ( z ) ,. . . . . . (6.8) ξ i D + r + i − ( qz ) D − i,...,r − i ( z ) − ξ r + i − , D + r + i − ( z ) D − i,...,r − i ( qz ) = ( ξ i − ξ r + i − ) D − i,...,r − i ( qz ) D + r + i − ( z ) ,ξ D + r ( qz ) D − ,...,r ( z ) − ξ r +1 D + r ( z ) D − ,...,r ( qz ) = ( ξ − ξ r +1 ) D − ,...,r − ( qz ) . OROIDAL q -OPERS 29 where index i ranges between the same values as in the corresponding equations in (5.7) ,for the polynomials (6.9) D + k = Q + k F k , D − k = Q − k F k η k , D − l,...,k = Q − l,...,k F k η l,...,k . where F i ( z ) = W r − i ( q r − i z ) , η l,...,i = i − l Y a =0 ( ξ l − ξ l + a +1 ) . For the future we shall refer to (6.8) as the extended
D D -system for SL ( r + 1) and to itsfirst line specifically as merely the D D -system.
Proof.
The proof is the direct extension of the proof of Lemma 4.2 in [KSZ] to other equa-tions in (5.31). Since all equations are treated analogously, let us consider the second set of(6.8) which we can write as(6.10) ξ i − D + i ( qz ) D − i − ,i ( z ) − ξ i +1 D + i ( z ) D − i − ,i ( qz ) = ( ξ i − − ξ i +1 ) D − i − ( qz ) D + i +1 ( z ) . After replacing D + i = Q + i F i , D − i = Q − i F i η i , D − i − ,i = Q − i − ,i F i η i − ,i and assigning η i − = ξ i − − ξ i , η i − ,i = ( ξ i − − ξ i )( ξ i − − ξ i +1 ) , we can see that (6.10) is equivalent to the second equation of (5.31) provided that thefollowing difference equation is satisfied: F i − ( qz ) F i +1 ( z ) F i ( qz ) F i ( z ) · η i − η i − ,i ( ξ i − − ξ i +1 ) = Λ i ( z ) . The validity of this relation follows from the above formulae and form the definitions (6.2) F i − ( qz ) F i +1 ( z ) F i ( qz ) F i ( z ) = W r − i +1 ( q r − i +1 z ) W r − i ( q r − i z ) W r − i − ( q r − i z ) W r − i ( q r − i +1 z ) = P r − i +1 ( q r − i z ) P r − i ( q r − i z ) = Λ i ( z ) . (cid:3) As we shall see below, one can express the solutions of the QQ - and D D -systems in termsof section s ( z ) of subbundle L r +1 . Following the discussion of [KSZ] (Section 4) we considerthe following matrices:(6.11) M i ,...,i j = ξ j − i s i ( z ) · · · ξ i s i ( q j − z ) s i ( q j − z )... . . . ... ... ξ j − i j s i j ( z ) · · · ξ i j s i j ( q j − z ) s i j ( q j − z ) , V i ,...,i j = ξ j − i · · · ξ i ξ j − i j · · · ξ i j , where s i are polynomials and V i ,...,i j is the Vandermonde-like matrix whose determinant is(6.12) det V i ,...,i j = Y i Given polynomials D + i , D − i for i = 1 , . . . , r satisfying the first line of (6.8) , there exist unique polynomials s , . . . , s r +1 such that (6.13) D + i ( z ) = det M r +2 − i,...,r +1 ( z )det V r +2 − i,...,r +1 and D − i ( z ) = det M r +1 − i,r +3 − i,...,r +1 ( z )det V r +1 − i,r +3 − i,...,r +1 , where matrix M is given in (6.11) .Proof. The proof is based on the determinant Desnanot-Jacobi identity which holds for any l × l matrix M . In this proof we shall use this identity in the following form(6.14) M M l − M l M = M , ,l M , where M ab is the determinant of matrix M with row a and column b removed (respectively M a,cb,d is the determinant of matrix M with rows a and c and column b and d removed). Notethat the Desnanot-Jacobi identity holds for any pairs of indices { a, c } and { b, d } as long as a = c and b = d .In [KSZ] it was shown, using periodic properties of matrix M that the first line of (6.8)can be identified with (6.14) if M i = M r +1 − i,...,r +1 ( z ) is the determinant of the bottom-right i × i submatrix of the ( r + 1) × ( r + 1) matrix M ,...,r +1 ( z ). It is easy to see, for instance,that ( M i ) ( z ) = M r +2 − i,...,r +1 ( qz ) , ( M i ) ( z ) = M r +1 − i,r +3 − i,...,r +1 ( qz ) , ( M i ) i ( z ) = ξ r +2 − i r +1 Y a = r +3 − i ξ i ! · M r +2 − i,...,r +1 ( z ) , ( M i ) i ( z ) = ξ r +1 − i r +1 Y a = r +3 − i ξ i ! · M r +1 − i,r +3 − i,...,r +1 ( z ) , ( M i ) , ,i ( z ) = r +1 Y a = r +3 − i ξ i ! · M r +3 − i,...,r +1 ( qz ) . We can substitute the above five relations into (6.14) and then divide its both sides by V r +2 − i,...,r +1 V r +1 − i,r +3 − i,...,r +1 . The first D D -relation will follow after observing that V r +3 − i,...,r +1 V r +1 − i,...,r +1 = ( ζ r +1 − i − ζ r +2 − i ) V r +2 − i,...,r +1 V r +1 − i,r +3 − i,...,r +1 . (cid:3) In the above proof we have derived an alternative presentation of D ± i polynomials andtheir q -shifted counterparts in terms of minors M i (6.15) D + i ( z ) = ( M i ) i ( z )( V i ) i , D − i ( z ) = ( M i ) i ( z )( V i ) i , D + i ( qz ) = ( M i ) ( z )( V i ) , D − i ( qz ) = ( M i ) ( z )( V i ) , where V i = V r +1 − i,...,r +1 ( z ) is the determinant of the bottom-right i × i submatrix of the( r + 1) × ( r + 1) matrix V ,...,r +1 ( z ). This way all polynomials which appear in the D D -system can be universally presented as ratios of (unshifted) minors of two sets of matrices { M i } and { V i } for i = 1 , . . . , r .Thus i -th equation of the D D -system represents a Densanot-Jacobi determinant identityfor matrix M i of the form (6.11). In the following subsection we shall demonstrate that allequations of the extended D D -system can also be thought of as determinant identities for OROIDAL q -OPERS 31 matrices which are obtained from M i s by permutation of rows and columns. The latter isprovided by B¨acklund transformations.6.3. B¨acklund transformations and the extended D D -System. We have shown thatthe extended QQ -system is equivalent to the extended D D -system in Lemma 6.4. Let usfocus on D D -system, namely the equations, corresponding to the first line in (6.8). Wealready mentioned that all the equations from (5.31) can be obtained from the QQ -systemby applying B¨acklund transformations. The same works for the D D -system. The i -thB¨acklund transformation replaces the data { D + j , D − j } j =1 ,...,r , { ξ j } j =1 ,...,r +1 with the following { e D + j } j =1 ,...,r = { D +1 , . . . , D + i − , D − i , D + i +1 . . . , D + r } ;(6.16) { e D − j } j =1 ,...,r = n D − , . . . , D ∗ − i − , D + i , D − i − ,i , D − i +2 . . . , D − r o { e ξ j } j =1 ,...,r = { ξ , . . . , ξ i − , ξ i +1 , ξ i , . . . , ξ r +1 } Notice that in the QQ -system this rule works as Q − i 7→ − Q + i . In the D D -system the signdisappears due to the presence of the multiplicative factor η i between Q − i and D functions.We also note, that polynomials D ∗ − i − do not belong to the extended D D -system, ratherthey will be a part of the full D D -system. By applying B¨acklund transformations fur-ther we can readily find all polynomials from the full D D -system, which is in one-to-onecorrespondence with the full QQ -system we discussed in section 5.We can now find a similar presentation for other polynomials D − i,...,j in terms of ratios ofdeterminants by combining the ideas above and Proposition 6.3. In particular, we need tounderstand how B¨acklund transformations act on the matrices (6.11).Let us start with the ( r + 1) × ( r + 1)-matrices from (6.11) M = M ,...,r +1 and V = V ,...,r +1 . During the i th B¨acklund transformation (5.43) the following functions in theextended D D -system get interchanged: D + i ( z ) ↔ D − i ( z ), D − i +1 ( z ) ↔ D − i,i +1 ( z ) and ξ i ↔ ξ i +1 which, using the identification (6.7), amounts to acting by permutations r i on the setof indices as { , . . . , i, i + 1 , . . . , r + 1 } 7→ { , . . . , i + 1 , i, . . . , r + 1 } . Therefore we can define a new tuple of matrices r i ( M )( z ) = M ,...,i +1 ,i,...,r +1 and r i ( V )( z ) = V ,...,i +1 ,i,...,r +1 as well as their submatrices r i ( M j )( z ) and r i ( V j )( z ), which are obtained byexcising the corresponding ( r − j + 1) × ( r − j + 1) bottom-right blocks. Then by Proposition6.5 we must have(6.17) D + i ( z ) = det r i ( M ) r +2 − i,...,r +1 ( z )det r i ( V ) r +2 − i,...,r +1 , D − i,i +1 ( z ) = det r i ( M ) r +2 − i,r +4 − i,...,r +1 ( z )det r i ( V ) r +2 − i,r +4 − i,...,r +1 , or, equivalently,(6.18) D − i ( z ) = r i ( M i ) i ( z ) r i ( M i ) i , D − i,i +1 ( z ) = r i ( M i +1 ) i − ( z ) r i ( V i +1 ) i − . Notice that the second equality of (6.17) can be written in terms of the original matrix M (6.19) D − i,i +1 = det M i,i +3 ...,r +1 det V i,i +3 ...,r +1 . In order to determine similar expressions for other D − i,...,i + k one needs to act by otherelements of the Weyl group W = S r +1 . Essentially, the B¨acklund transformation, beingassociated with the elementary Weyl reflection interchanges two rows in the q-Wronskianmatrix. This brings us to the following statement, which an be verified by a direct calcula-tion along the lines of Proposition 6.5 for each set of equations of the extended D D -system. Proposition 6.6. The polynomials D − i,...i + k from (6.8) read (6.20) D − i,...,i + k = det M i,i + k +2 ...,r +1 det V i,i + k +2 ...,r +1 , i = 1 , . . . , r , k = 0 , . . . , r − i . or equivalently, (6.21) D − i,...,i + k = r i + k − ( . . . r i +1 ( r i ( M i + k )) . . . ) i +1 r +1 r i + k − ( . . . r i +1 ( r i ( V i + k )) . . . ) i +1 r +1 , Although we do not discuss polynomials D ∗ − i − which belong to the full D D -system, ratherthan to the extended D D -system, we can nevertheless provide a formula for these polyno-mials. Proposition 6.7. The polynomials D ∗ − i − from (6.8) read (6.22) D ∗ − i − = r i − ( M i − ) i +1 ( z ) r i − ( V i − ) i +1 . Proof. The ( i − D D -system after applying the i th B¨acklund trans-formation reads ξ i − D + i − ( qz ) D ∗ − i − ( z ) − ξ i +1 D + i − ( z ) D ∗ − i − ( qz ) = ( ξ i − − ξ i +1 ) D + i − ( qz ) D − i ( z ) . Given the description of D polynomials in terms of minors (6.15) it can be shown thatthe above equation is equivalent to the Jacobi determinant identity of the form (6.14) formatrix M i − . (cid:3) This statement implies that the solutions of the full D D - and thus the full QQ -systemare well-defined, if the original QQ -system is nondegenerate. Also, notice that all the otherequations in the QQ -system correspond to all possible Miura ( SL ( r + 1) , q )-opers for a given( SL ( r + 1) , q )-oper. Thus, the following theorem is true, which generalizes Theorem 5.8. Theorem 6.8. i)The solution of the nondegenerate ( SL ( r + 1)) QQ -system can be extendedto the solution of the the full QQ -system.ii) This full QQ -system is comprised of ( r + 1)! QQ -systems, with B¨acklund transforma-tions acting transitively between them.iii) Each such QQ -system determine one of the ( r + 1)! Z -twisted Miura ( SL ( r + 1) , q ) -opers, corresponding to a unique Z-twisted ( SL ( r + 1) , q ) -oper. We can combine Lemma 6.4 with Propositions 6.5 and 6.6 to get the following theoremwhich will be used in later sections to study infinite-dimensional q -opers. OROIDAL q -OPERS 33 Theorem 6.9. The polynomials which appear in the extended QQ -system (5.30) are givenby Q + i ( z ) = 1 F i ( z ) · ( M i ) i ( z )( V i ) i , Q − i ( z ) = 1 F i ( z ) η i · ( M i ) i ( z )( V i ) i ,Q − i,...,i + k ( z ) = 1 F i ( z ) η i,...,k · s i + k − ( . . . s i +1 ( s i ( M i + k )) . . . ) i ( z ) s i + k − ( . . . s i +1 ( s i ( V i + k )) . . . ) i . (6.23) or, equivalently, Q + i ( z ) = 1 F i ( z ) · det M r +2 − i,...,r +1 ( z )det V r +2 − i,...,r +1 ,Q − i ( z ) = 1 F i ( z ) η i · det M r +1 − i,r +3 − i,...,r +1 ( z )det V r +1 − i,r +3 − i,...,r +1 , (6.24) Q − i,...,i + k ( z ) = 1 F i ( z ) η i,...,k · det M i,i + k +2 ...,r +1 ( z )det V i,i + k +2 ...,r +1 . Note also the following expressions for shifted Q -functions which will be used later.(6.25) Q + i ( qz ) = 1 F i ( z ) · ( M i ) ( z )( V i ) , Q − i ( qz ) = 1 F i ( z ) η i · ( M i ) ( z )( V i ) . Example: Miura ( SL (3) , q ) -oper. Define matrices M ( z ) = ξ s ( z ) ξ s ( qz ) s ( q z ) ξ s ( z ) ξ s ( qz ) s ( q z ) ξ s ( z ) ξ s ( qz ) s ( q z ) , V = ξ ξ ξ ξ ξ ξ ,M ( z ) = ( M ) ( z ) = (cid:18) ξ s ( z ) s ( qz ) ξ s ( z ) s ( qz ) (cid:19) , V = ( V ) = (cid:18) ξ ξ (cid:19) , and the matrices which are obtained from the above by the Weyl action r ( M )( z ) = ξ s ( z ) ξ s ( qz ) s ( q z ) ξ s ( z ) ξ s ( qz ) s ( q z ) ξ s ( z ) ξ s ( qz ) s ( q z ) , r ( V ) = ξ ξ ξ ξ ξ ξ ,M , ( z ) = r ( M ) ( z ) = (cid:18) ξ s ( z ) s ( qz ) ξ s ( z ) s ( qz ) (cid:19) , V , = r ( V ) = (cid:18) ξ ξ (cid:19) . Then the D D -system reads ξ D +1 ( qz ) D − ( z ) − ξ D +1 ( z ) D − ( qz ) = ( ξ − ξ ) D +2 ( qz ) W ( z ) ,ξ D +2 ( qz ) D − ( z ) − ξ D +2 ( z ) D − ( qz ) = ( ξ − ξ ) D +1 ( z ) ,ξ D +2 ( qz ) D − , ( z ) − ξ D +2 ( z ) D − , ( qz ) = ( ξ − ξ ) D − ( z ) , (6.26)with the following D +1 ( z ) = ( M ) ( z ) V , D − ( z ) = ( M ) ( z ) V , W ( z ) = det M ( z )det V , as well as D +2 ( z ) = ( M ) ( z )( V ) = s ( z ) , D − ( z ) = ( M ) ( z )( V ) = s ( z ) , D − , ( z ) = ( M , ) ( z )( V , ) = s ( z ) , where ( M i ) ab ( z ) is the determinant of matrix M i ( z ) with row a and column b removed. Theshifted D -functions read D +1 ( qz ) = ( M ) ( z )( V ) , D − ( qz ) = ( M ) ( z )( V ) , D +2 ( qz ) = ( M ) ( z )( V ) , D − ( qz ) = ( M ) ( z )( V ) , D − , ( qz ) = ( M , ) ( z )( V , ) , Then the solutions of the SL (3) QQ -system read Q +1 ( z ) = 1 F ( z )( ξ − ξ ) · ( M ) ( z ) V , Q − ( z ) = 1 F ( z )( ξ − ξ ) · ( M ) ( z ) V , as well as Q +2 ( z ) = 1 F ( z )( ξ − ξ ) · ( M ) ( z )( V ) = s ( z ) F ( z )( ξ − ξ ) ,Q − ( z ) = 1 F ( z )( ξ − ξ ) · ( M ) ( z )( V ) = s ( z ) F ( z )( ξ − ξ )(6.27) Q − , ( z ) = 1 F ( z )( ξ − ξ ) · ( M , ) ( z )( V , ) = s ( z ) F ( z )( ξ − ξ ) , Explicit Formula for ( SL ( r + 1) , q ) -Oper via minors. We can now collect all theresults of this section in order to present the Miura ( SL ( r + 1) , q )-oper (5.16) in terms oftrivialization of subbundle L r +1 . Consider functions g i ( z ) which appear on the diagonal g i ( z ) = ζ i Q + i ( qz ) Q + i ( z ) = ζ i F i ( z ) F i ( qz ) · ( M i ) ( M i ) i ( V i ) i ( V i ) = ζ i · i Y a =1 ξ r +2 − a ! · i Y b =1 Λ r − i + b ( q − b z )Λ r − i + b ( z ) · ( M i ) ( M i ) i . (6.28)Then the diagonal entry of (5.16) becomes the following meromorphic function g i +1 g i ( z ) = ξ i +1 H ( r ) i ( z, q ) · G ( r ) i ( z, q ) , (6.29)where(6.30) H ( r ) i ( z, q ) = i Y b =1 Λ r − i + b ( q − b z )Λ r − i + b ( q − b z ) , G ( r ) i ( z, q ) = ( M i +1 ) ( M i ) · ( M i ) i ( M i +1 ) i +1 . Relation to Berenstein-Fomin-Zelevinsky work on generalized minors. Wedevoted this section to the description of Miura ( SL ( r + 1) , q )-opers via various minors ofthe q-Wronskian matrix. That matrix is produced by the components of the section ofthe line bundle and the components of the constant regular element Z , representing theq-connection in the given trivialization. One may wonder if such construction exists in thegeneral case, for simply connected simple group G , namely whether there exists an analogueof the q-Wronskian. Of course, in that case, we do not have a line bundle, since the definitionof ( G, q ) in terms of the flag of bundles is SL -specific. Nevertheless, there is a notion ofgeneralized minors [BZ, BFZ, FZ]. These are the functions on G , defined on the dense setcorresponding to the dense Bruhat cell N − HN + . For any g = n + hn − , the so-called principalminors [ g ] ω i are defined as the value of the multiplicative characters [ · ] ω i : H −→ C ∗ on h , OROIDAL q -OPERS 35 namely [ h ] ω i , corresponding to the fundamental weight ω i for i = 1 , . . . , r . Other generalizedminors are obtained by the action of the Weyl group elements on the left and the right of g and then applying the appropriate lifts of Weyl group elements u, v on the right and theleft and then applying [ · ] ω i , thus producing generalized minors ∆ uω i ,vω i . In the case of SL ( r + 1), the nondegeneracy conditions imply that the full q-Wronskian matrix belongs tothe dense Bruhat cell (i.e. it has Gauss decomposition) and the action of the Weyl groupelements correspond to the permutations of rows and columns.One of the fundamental relations between generalized minors is as follows [FZ]. Let, u, v ∈ W , such that for i ∈ { , . . . , r } , ℓ ( us i ) = ℓ ( u ) + 1, ℓ ( vs i ) = ℓ ( v ) + 1. Then∆ uω i ,vω i ∆ us i ω i ,vs i ω i − ∆ us i ω i ,vω i ∆ uω i ,vs i ω i = Y j = i ∆ − a ji uω j ,vω j , When applied to the q-Wronskian matrix in SL ( r + 1) case, these equations reproducethe D D -system. In the case of general G the left and right-hand sides of this relation isvery similar to the analogue of D D -system for general G (see [FKSZ]). Thus it is reason-able to assume the existence of the analogue of the Wronskian matrix as an element in n − ( z ) h ( z ) n + ( z ) ∈ N − ( z ) H ( z ) N + ( z ) ⊂ G ( z ). We will discuss this in future work.Note that one important feature of generalized minors is that relations between themgive cluster algebra structure for double Bruhat cells, so that our B¨acklund transformationsdescend from mutations for the cluster algebra elements.We believe that these cluster structures stand behind known cluster structures relevantfor Grothendieck rings of quantum affine algebras.7. GL ( ∞ ) and the Fermionic Fock Space This section contains the material on infinite-dimensional generalizations of GL ( N ) andtheir representations which will be later needed. The reader may consult with [KRR] formore details.7.1. ( SL ( r + 1) , q ) -Miura opers and the fermionic Fock space. First, we note thatgiven a defining representation V ω of SL ( r + 1), one can construct all other fundamentalrepresentations V ω i by considering wedge powers ≃ Λ i ( V ω ). If ν , . . . , ν r +1 are the standardbasis vectors in V ω , so that ν is the highest weight, then the highest weight vectors in V ω i are ν i ∧ ν i − · · · ∧ ν . Introducing operators ψ i of exterior multiplication on ν i and ψ ∗ i of interior multiplicationby ν i , we find that they satisfy Clifford algebra relation(7.1) ψ ∗ i ψ j + ψ j ψ ∗ i = δ ij . Using those operators we can realize the Chevalley generators as follows:ˇ α i = ψ i ψ ∗ i − ψ i +1 ψ ∗ i +1 , e i = ψ i ψ ∗ i +1 , f i = ψ i +1 ψ ∗ i , (7.2)such that [ e i , f i ] = ˇ α i . We arrive at the following Proposition. Proposition 7.1. In any fundamental representation the q-connection, corresponding to ( SL ( r + 1) , q ) -Miura oper (5.1) reads as follows: A ( z ) = Y i = r g ˇ α i i e Λ i ( z ) gi ( z ) e i = g − ψ r +1 ψ ∗ r +1 r Y i = r e Λ i ( z ) ψ i ψ ∗ i +1 · h g i g i − i ψ i ψ ∗ i , (7.3) where g = 1 . Our goal in the following will be to make sense of the completion of the above formulain the infinite-dimensional Fock space.7.2. SL ( ∞ ) and its Completions. Let GL ( ∞ ) is a set of infinite-dimensional matriceswhich can be characterized as follows: GL ( ∞ ) = { A = ( a ij ) i,j ∈ Z | A is invertible and all but finite number of a ij − δ ij are 0 } (7.4)The SL ( ∞ ) is the subgroup of GL ( ∞ ) of unimodular matrices. The Lie algebra gl ( ∞ ) of GL ( ∞ ) is given by gl ( ∞ ) = { A = ( a ij ) i,j ∈ Z | a ij = 0 for all but finite number } (7.5)and sl ( ∞ ) is the subalgebra of traceless matrices. The Lie algebra sl ( ∞ ) is the explicitrealization of the simple Kac-Moody algebra a ∞ , which one associates to the infinite Dynkindiagram A ∞ . However, there exist a bigger algebra, known as ¯ a ∞ , which consists of elementsof the form: x = X i ∈ Z c i ˇ α i + X α η α e α (7.6)where e α is an element of Cartan-Weyl basis corresponding to the root α with the heightht( α ) of sl ( ∞ ), so that the set S x = { k ∈ Z (cid:12)(cid:12) ∃ α, η α = 0 , ht( α ) = k } is finite. This algebra has two nontrivial central elements c = P i i ˇ α i and c = P i ˇ α i . Theexplicit realization of this algebra is given by the central extension of the algebra gl ∞ : gl ∞ = { A = ( a ij ) i,j ∈ Z | a ij = 0 for | i − j | ≫ } . (7.7)Namely, there exists a homomorphism from ¯ a ∞ to gl ∞ ⊕ C c where c is mapped to theidentity matrix and c = c correspond to the central extension c . Indeed, one can modifyrelations on the fundamental generators of a ∞ , namely[ e , f ] c = ˇ α + c (7.8)leaving all other relations between Chevalley generators intact. This leads to a nontrivialcentral extension for gl ∞ , although for any gl ( n ) subalgebra, this central extension is trivial.In order to describe these algebraic structures it is convenient to use matrix notation.Let us denote by E ij the matrix whose ( i, j ) entry is 1 and the rest are equal to zero. Thesematrices obey the following commutation relations[ E ij , E mn ] = δ jm E in − δ ni E mj . One can then representˇ α i = E ii − E i +1 ,i +1 , e i = E i,i +1 , f i = E i,i − . If we define a i = X k ∈ Z E k,k + i , i = 0and a = X k> E k,k − X k ≤ E k,k . OROIDAL q -OPERS 37 we have the following Heisenberg subalgebra[ a n , a m ] = n c δ n, − m , However, we will be interested in a smaller subalgebra ¯ a ′∞ ⊂ ¯ a ∞ , so that for every x ∈ ¯ a ′∞ in the form (7.6) only finite number of coefficients λ α = 0 for negative α . The correspondingsubalgebra gl ′∞ ⊂ gl ∞ is formed by matrices (7 . 7) with only finite number of elements belowthe main diagonal. It is easy to exponentiate this Lie algebra and we denote the resultingLie group as GL ( ∞ ): GL ( ∞ ) = { A = ( a ij ) i,j ∈ Z | a ij = 0; k for i > j for all but finite number; a ii = 0 ∀ i ∈ Z } . (7.9)Given the upper Borel part b + of gl ∞ , generated by ˇ α i , e i , one can construct an upperBorel subgroup B + by exponentiating elements of b + , which we denote as B + , namely B + = { A = ( a ij ) i,j ∈ Z | a ij = 0 for i > j, a ii = 0 ∀ i ∈ Z } (7.10)Combining it with B − , the Borel subgroup of SL ( ∞ ): B − =(7.11) { A = ( a ij ) i,j ∈ Z | a ij = 0 for i < j,a ii = 0 for all i, a ii = 1 for all but finite number , (7.12) a ij = 0 for i > j for all but finite number , det( A ) = 1 } Then one can write Bruhat decomposition GL ( ∞ ) = ⊔ ¯ w B − ¯ wB + , where ¯ w is a Weylgroup element inherited from a Weyl group element of SL ( k + 1) subgroup for some finite k . Now we can construct the appropriate generalization of the q-connection (3.9)(7.13) A ( z ) = −∞ Y i =+ ∞ g i ( z ) ˇ α i e Λ i ( z ) gi ( z ) e i , g i ( z ) ∈ C ( z ) × , Λ i ( z ) ∈ C [ z ] . which belongs to B + ( z ) ⊂ GL ( ∞ )( z ). Remark . In principle, it could be possible to consider further completions of GL ( ∞ ) andmake full use of the central extension, however, since we are interested in Miura q-opers,we only need to complete one of the Borels to arrive to the formula above.7.3. Infinite wedge space representations for GL ( ∞ ) . Here we will explain the con-struction of the fundamental representations of ¯ a ′∞ with central charge 1, which will serveas fundamental representations for GL ( ∞ ) as well.Let V = ⊕ j ∈ Z C ν j be the infinite-dimensional space where ν j are basis elements. Thereis a natural action of sl ( ∞ ) on V as infinite-dimensional matrices. Consider the followingexpression: Ψ m = ν m ∧ ν m − ∧ ν m − ∧ . . . (7.14)We will call it the highest weight vector in the vector space F m . The other basis vectors in F m have the form Ψ = ν i m ∧ ν i m − ∧ ν i m − ∧ . . . , (7.15)where i m > i m − > i m − > . . . and i k = k for k ≪ 0. Action of sl ( ∞ ) algebra on F m isdefined in the following way. We identify e i , f i , ˇ α i with matrix generators E i,i +1 , E i +1 ,i , E ii − E i +1 ,i +1 respectively. Then we define the action of any element X of sl ( ∞ ) on F m isgiven by the following formula: X Ψ = Xν i m ∧ ν i m − ∧ ν i m − ∧ · · · + ν i m ∧ Xν i m − ∧ ν i m − ∧ · · · + ν i m ∧ ν i m − ∧ Xν i m − ∧ . . . Remark . A famous representation of ¯ a ∞ with central charge c = 1 is achieved in thefolowing way. One has to modify the action of ˇ α , via a shift ˇ α −→ ˇ α − 1, namely:ˇ α Ψ =ˇ α ν i m ∧ ν i m − ∧ ν i m − ∧ · · · + ν i m ∧ ˇ α ν i m − ∧ ν i m − ∧ · · · + ν i m ∧ ν i m − ∧ ˇ α ν i m − ∧ . . . − v i m ∧ ν i m − ∧ ν i m − ∧ . . . Notice that ¯ n + Ψ m = 0 , where ¯ n + = [¯ b + , ¯ b + ] and ˇ α k Ψ m = δ k,m Ψ m . Thus { F m } canbe interpreted as fundamental representations of ¯ a ′∞ and fundamental representations of GL ( ∞ ) as well. The group action is given by the formula: g · Ψ = gν i m ∧ gν i m − ∧ gν i m − ∧ . . . (7.16)Using the formalism of the Clifford algebra (7.1), we have again formulas (7.2) for thegenerators ˇ α , e i , f i , where now i ∈ Z . This allows us to write the expression for the elementof B + ( z ) from (7.13) acting on F m as A ( z ) = Y i =+ ∞ e Λ i ( z ) ψ i ψ ∗ i +1 h g i g i − i ψ i ψ ∗ i · −∞ Y i =0 e Λ i ( z ) ψ i ψ ∗ i +1 h g i g i − i − ψ ∗ i ψ i . (7.17) 8. ( GL ( ∞ ) , q ) -opers Definitions of ( GL ( ∞ ) , q ) -opers and the canonical form of ( GL ( ∞ ) , q ) -Miuraopers. Given a principal GL ( ∞ )-bundle F GL ( ∞ ) over P , let F qGL ( ∞ ) denote its pullbackunder the map M q : P −→ P sending z qz . A meromorphic ( GL ( ∞ ) , q )- connection ona principal GL ( ∞ )-bundle F GL ( ∞ ) on P is a section A of Hom O U ( F GL ( ∞ ) , F qGL ( ∞ ) ), where U is an open dense subset of P in the standard topology. Notice, that now the number ofzeroes and poles which we have to exclude from P could be infinite. We assume that theonly two accumulations points points possible are 0 , ∞ . We can always choose U so thatthe restriction F GL ( ∞ ) | U of F GL ( ∞ ) to U is isomorphic to the trivial GL ( ∞ )-bundle. Therestriction of A to the Zariski open dense subset U ∩ M − q ( U ) can be written as section ofthe trivial GL ( ∞ )-bundle on U ∩ M − q ( U ), and hence as an element A ( z ) of GL ( ∞ )( z ). Definition 8.1. A meromorphic ( GL ( ∞ ) , q )- oper on P is a triple ( F GL ( ∞ ) , A, F B − ), where A is a meromorphic ( GL ( ∞ ) , q )-connection on a GL ( ∞ )-bundle F GL ( ∞ ) on P and F B − is the reduction of F GL ( ∞ ) to B − satisfying the following condition: there exists an opendense subset U ⊂ P together with a trivialization ı B − of F B − , such that the restriction ofthe connection A : F GL ( ∞ ) −→ F qGL ( ∞ ) to U ∩ M − q ( U ), written as an element of GL ( ∞ )( z )using the trivializations of F GL ( ∞ ) and F qGL ( ∞ ) on U ∩ M − q ( U ) induced by ı B − , takes valuesin the infinte product of Bruhat cells Q −∞ i =+ ∞ B − ( C [ U ∩ M − q ( U )]) s i B − ( C [ U ∩ M − q ( U )]),where the ordering in the product follows the infinite version of the one in SL ( r + 1). OROIDAL q -OPERS 39 Therefore any q -oper connection A can be written in the form A ( z ) = −∞ Y i =+ ∞ h n ′ i ( z )( φ i ( z ) ˇ α i s i ) n i ( z ) i (8.1)where φ i ( z ) ∈ C ( z ) and n i ( z ) , n ′ i ( z ) ∈ ¯ N − = [ ¯ B − , ¯ B − ]( z ) are such that their zeros and polesare outside the subset U ∩ M − q ( U ) of P . As we stated before, we require that the onlyaccumulation points of zeroes and poles of φ i ( z ), n i ( z ) , n ′ i ( z ) are 0 , ∞ .We can give an alternative definition of ( GL ( ∞ ) , q )-oper connection using associatedbundles as well. Definition 8.2. A meromorphic ( GL ( ∞ ) , q )- oper on P is a triple ( A, E, L • ), where E is anambient vector bundle with the fiber being infinite-dimensional vector space with countablebasis and L • is the corresponding complete flag of the vector bundles, ... ⊂ L i +1 ⊂ L i ⊂ L i − ⊂ ... ⊂ E, i.e. with the fibers for L i being semi-infinite spaces so that A ∈ Hom O U ( E, E q ) satisfies thefollowing conditions:i) A · L i ⊂ L i − ii) There exists an open dense subset U ⊂ P , such that the restriction of the con-nection A ∈ Hom ( L • , L q • ) to U ∩ M − q ( U ), which belongs to GL ( ∞ ) and satisfiesthe condition that the induced operator ¯ A : L i / L i +1 −→ L i − / L i is an isomorphism.Let us give two equivalent definitions of the ( GL ( ∞ ) , q )-Miura oper, which is the sameas in finite-dimensional case. Definition 8.3. i) A Miura ( GL ( ∞ ) , q ) -oper on P is a quadruple ( F GL ( ∞ ) , A, F B − , F B + ),where ( F GL ( ∞ ) , A, F B − ) is a meromorphic ( GL ( ∞ ) , q )-oper on P and F B + is a re-duction of the GL ( ∞ )-bundle F GL ( ∞ ) to B + that is preserved by the q -connection A .ii) A Miura ( GL ( ∞ ) , q ) -oper on P is a quadruple ( E, A, L • , ˆ L • ), where ( E, A, L • ) is ameromorphic GL ( ∞ )-oper on P and ˆ L • = { L i } , is another full flag of subbundlesin E that is preserved by the q -connection A .As in SL ( r + 1) case, we can define relative position (see Section 2.3) between F B + , F B − because of the Bruhat decomposition of G . We will say that F B − and F B + have a genericrelative position at x ∈ X if the element of W G assigned to them at x is equal to 1 (thismeans that the corresponding element a − b belongs to the open dense Bruhat cell B − · B + ⊂ GL ( ∞ )).We immediately have the following result, which is a generalization of a finite-dimensionalcase Theorem 8.4. For any Miura ( GL ( ∞ ) , q ) -oper on P , there exists an open dense subset V ⊂ P such that the reductions F B − and F B + are in generic relative position for all x ∈ V .Proof. Notice, that according to the local expression for the q-oper connection (2.2) andthe condition that it belongs to GL ( ∞ )( z ) means that there is a finite number of elements below the diagonal. This means that for some k, l we have A ( z ) = " k Y i =+ ∞ g ˇ α i i ( z ) e φi ( z ) eigi ( z ) n ′ ( z ) l +1 Y j = k − ( φ j ( z ) ˇ α j s j ) n ( z ) " −∞ Y i = l g ˇ α i i ( z ) e φi ( z ) eigi ( z ) , (8.2)where n ( z ) , n ′ ( z ) ∈ N − ( z ) belong to the SL ( k + l ) subgroup with H generated by { ˇ α j } j = l +1 j = k − .The expression in the middle, namely A ′ ( z ) = n ′ ( z ) Q j ( φ j ( z ) ˇ α j s j ) n ( z ) ∈ SL ( k + l )( z ) isthe local expression for Miura SL ( k + l ) q-oper for which generic property follows fromthe finite-dimensional case (see Theorem 2.5) and thus we have generic relative position forMiura ( GL ( ∞ ) , q )-oper. (cid:3) That leads to the following Corollary. Corollary 8.5. For any Miura ( GL ( ∞ ) , q ) -oper on P , there exists a trivialization of theunderlying GL ( ∞ ) -bundle F GL ( ∞ ) on an open dense subset of P for which the oper q -connection has the form (8.3) −∞ Y i =+ ∞ g ˇ α i i e λitigi e i , g i ∈ C ( z ) × , where each t i ∈ C ( z ) is determined by the lifting s i and the order in the product is canonical. As in the finite-dimensional case, we fix t i ≡ Z-twisted Miura q-opers. Now we are ready to define the notion of ( GL ( ∞ ) , q )-oper and ( GL ( ∞ ) , q )-Miura oper, which are straightforward definitions of their SL ( r + 1)-counterparts. As in SL ( r + 1) case, let Z be the regular element of the maximal torus H = B + / [ B + , B + ]. One can express it as follows:(8.4) Z = −∞ Y i =+ ∞ ζ ˇ α i i , ζ i ∈ C × . Definition 8.6. A Z -twisted ( GL ( ∞ ) , q ) -oper on P is a ( GL ( ∞ ) , q )-oper that is equivalentto the constant element Z ∈ H ⊂ H ( z ) under the q -gauge action of GL ( ∞ )( z ), i.e. if A ( z )is the meromorphic oper q -connection (with respect to a particular trivialization of theunderlying bundle), there exists g ( z ) ∈ GL ( ∞ )( z ) such that A ( z ) = g ( qz ) Zg ( z ) − . (8.5)A Z -twisted Miura ( GL ( ∞ ) , q ) -oper is a Miura ( GL ( ∞ ) , q )-oper on P that is equivalent tothe constant element Z ∈ H ⊂ H ( z ) under the q -gauge action of B + ( z ), i.e. A ( z ) = v ( qz ) Zv ( z ) − , v ( z ) ∈ B + ( z ) . (8.6)Naturally, we have the proposition addressing characterization of Z -twisted Miura q-opersassociated to Z -twisted q-opers. Proposition 8.7. Let Z ∈ H be regular. For any Z -twisted ( GL ( ∞ ) , q ) -oper ( F GL ( ∞ ) , A, F B − ) and any choice of B + -reduction F B + of F GL ( ∞ ) preserved by the oper q -connection A , theresulting Miura ( GL ( ∞ ) , q ) -oper is Z ′ -twisted for a particular Z ′ ∈ S ∞ · Z . The set of A -invariant B + -reductions F B + on the ( GL ( ∞ ) , q ) -oper is in one-to-one correspondence withthe elements of W = S ∞ . OROIDAL q -OPERS 41 Given a Miura ( GL ( ∞ ) , q )-oper. By Corollary 3.4, the underlying ( G, q )-connection canbe written in the form (3.9). As in SL ( r + 1) case we obtain an ¯ H -bundle F B + / ¯ N + , where¯ N + = [ B + , B + ]. The corresponding ( ¯ H , q )-connection A ¯ H ( z ) according to (8.3) is given by:(8.7) A H ( z ) = Y i g i ( z ) ˇ α i . We call A ¯ H ( z ) the associated Cartan q –connection of the Miura q -oper A ( z ).The same can be done in the infinite-dimensional case. If our Miura q -oper is Z -twisted(see Definition 8.6), then we also have A ( z ) = v ( qz ) Zv ( z ) − , where v ( z ) ∈ B + ( z ). Notice,that v ( z ) can be written as(8.8) v ( z ) = Y i y i ( z ) ˇ α i n ( z ) , n ( z ) ∈ ¯ N + ( z ) , y i ( z ) ∈ C ( z ) × , We refer to the associated to Cartan q -connection A ¯ H ( z ) as Z - t wisted, so that the explicitrealization is given by the following formula:(8.9) A H ( z ) = Y i " ζ i y i ( qz ) y i ( z ) ˇ α i and we note, that A H ( z ) determines the y i ( z )’s uniquely up to a scalar.9. Z -twisted Miura ( GL ( ∞ ) , q ) -opers and QQ -systems Definition and explicit realization. Let { Λ i ( z ) } i ∈ Z be a collection of non-constantpolynomials with accumulation points of roots at 0 or ∞ only. Definition 9.1. i) An ( GL ( ∞ ) , q )- oper with regular singularities determined by { Λ i ( z ) } i ∈ Z is a q -oper on P whose q -connection may be written in the form(9.1) A ( z ) = " k Y i =+ ∞ g ˇ α i i ( z ) e Λ i ( z ) tieigi ( z ) n ′ ( z ) l +1 Y j = k − ( φ j ( z ) ˇ α j s j ) n ( z ) " −∞ Y i = l g ˇ α i i ( z ) e Λ i ( z ) tieigi ( z ) , for some k, l ∈ Z , where n ( z ) , n ′ ( z ) ∈ N − ( z ) and belong to SL ( k + l ) subgroup with H generated by { ˇ α j } j = l +1 j = k − .ii) A Miura ( GL ( ∞ ) , q ) -oper with regular singularities determined by polynomials { Λ i ( z ) } i =1 ,...,r is a Miura ( GL ( ∞ ) , q )-oper such that the underlying q -oper has regular singularities deter-mined by { Λ i ( z ) } i =1 ,...,r .As in the SL ( r + 1) case, from now on we set t i ( z ) = 1, i ∈ Z . Then we have an analogueof the Theorem 3.4. Corollary 9.2. For every Miura ( GL ( ∞ ) , q ) -oper with regular singularities determined bythe polynomials { Λ i ( z ) } i ∈ Z , the underlying q -connection can be written in the form (9.2) A ( z ) = −∞ Y i =+ ∞ g i ( z ) ˇ α i e Λ i ( z ) gi ( z ) e i , g i ( z ) ∈ C ( z ) × . Fermionic realization. Let F i be the irreducible representation of GL ( ∞ ) with thehighest weight ω i which we discussed in Section 7. Notice, that the one-dimensional andtwo-dimensional subspaces L i and W i of F i spanned by the weight vectors Ψ i , and f i · Ψ i are a B + -invariant subspaces of V i .Now let ( F GL ( ∞ ) , A, F B − , F B + ) be a Miura ( GL ( ∞ ) , q )-oper with regular singularities de-termined by polynomials { Λ i ( z ) } i ∈ Z (see Definition 3.3). Recall that F B + is a B + -reductionof a i ∈ Z -bundle F ( GL ( ∞ ) on P , preserved by the ( GL ( ∞ ) , q )-connection A . Therefore foreach i ∈ Z , the vector bundle F i = F B + × B + V i = F G × G F i . Thus we have the following Proposition. Proposition 9.3. For every Miura ( GL ( ∞ ) , q ) -oper with regular singularities determinedby the polynomials { Λ j ( z ) } j ∈ Z , the underlying q -connection φ i ( A ) in the associated bundle F i for any i ∈ Z can be written in the form: (9.3) φ i ( A )( z ) = Y j =+ ∞ e Λ j ( z ) ψ j ψ ∗ j +1 h g j ( z ) g j − ( z ) i ψ j ψ ∗ j · −∞ Y j =0 e Λ j ( z ) ψ j ψ ∗ j +1 h g j ( z ) g j − ( z ) i − ψ ∗ j ψ j . Miura-Pl¨ucker ( GL ( ∞ ) , q ) -opers. For all i ∈ Z , the infinite rank bundle F i containsa rank two subbundle W i = F B + × B + W i associated to W i ⊂ F i , and W i in turn contains a line subbundleˆ L i = F B + × B + L i associated to L i ⊂ W i .Note, that φ i ( A ) preserves subbundles L i and W i of F i and thus produces ( GL (2) , q )-operon W i . We denote such q-oper by A i as in subsection 4.1.Notice that W i decomposes into direct sum of two subbundles, ˆ L i , preserved by B + and L i with respect to which it satisfies the ( GL (2) , q )-oper condition. We can unify all that inthe following Proposition. Proposition 9.4. The quadruple ( A i , W i , L i , ˆ L i ) for any i ∈ Z forms a ( GL (2) , q ) Miuraoper, so that explicitly: (9.4) A i ( z ) = g i ( z ) Λ i ( z ) g i − ( z )0 g − i ( z ) g i +1 ( z ) g i − ( z ) , i = 0 , A ( z ) = ( z ) g − ( z )0 g − ( z ) g ( z ) g − ( z ) , where we use the ordering of the simple roots determined by the Coxeter element c . We can see that the expression for A ( z ) looks slightly differently than the rest of A i ( z )in (9.4). However, if we multiply A ( z ) by the diagonal matrix proportional to the identitydiag( g ( z ) , g ( z )) then it will be of the same form as the rest of the matrices. This is dueto the central extension in ¯ a ∞ algebra and shift of Chevalley generator ˇ α .Now we impose the Z -twisted condition on the corresponding A H connection, namely g i = ζ i y i ( qz ) y i ( z ) . OROIDAL q -OPERS 43 Let G i ∼ = SL(2) be the subgroup of GL ( ∞ ) corresponding to the sl (2)-triple spanned by { e i , f i , ˇ α i } , which preserves W i , using a diagonal gauge transformation as in subsection 4.1,we associate to A i connection a ( G i , q )-oper with the explicit form: A i ( z ) = g ˇ α i i ( z ) e βi ( z ) gi ( z ) e i , where β i ( z ) = Λ i ( z ) ζ i − y i +1 ( z ) y i − ( qz ) . (9.5)Note that the diagonal transformation for A ( z ) looks a bit different than for other A i ( z )because of the aforementioned shift.Now we are ready to define Miura-Pl¨ucker ( GL ( ∞ ) , q )-opers. Definition 9.5. A Z - twisted Miura-Pl¨ucker ( GL ( ∞ ) , q ) -oper is a meromorphic Miura ( GL ( ∞ ) , q )-oper on P with the underlying q -connection A ( z ), such that there exists v ( z ) ∈ B + ( z ) suchthat for all i ∈ Z , the Miura (GL(2) , q )-opers A i ( z ) associated to A ( z ) by formula (9.4) canbe written in the form:(9.6) A i ( z ) = v ( zq ) Zv ( z ) − | W i = v i ( zq ) Z i v i ( z ) − where v i ( z ) = v ( z ) | W i and Z i = Z | W i .9.4. Nondegeneracy conditions. In this subsection we will generalize two nondegeneracyconditions we had in subsection 4.2 for Z -twisted Miura-Pl¨ucker ( GL ( ∞ ) , q )-opers.The first nondegeneracy condition deals with the associated H connection. Definition 9.6. A Miura ( GL ( ∞ ) , q )-oper A ( z ) of the form (3.9) is called H - nondegenerate if the corresponding ( ¯ H, q )-connection A ¯ H ( z ) can be written in the form (8.9), where zeroesand poles y i ( z ) and y i ± ( z ) are q -distinct from each other and from the zeros of Λ k ( z ).The second nondegeneracy condition addresses the associated ( G i , q )-opers. Definition 9.7. A Z -twisted Miura-Pl¨ucker ( GL ( ∞ ) , q )-oper A ( z ) is called nondegenerate if its associated Cartan q -connection A H ( z ) is nondegenerate and each associated Z i -twistedMiura (SL(2) , q )-oper A i ( z ) is nondegenerate.This we arrive to the analogue of the Proposition 4.7, which is proven in exactly the sameway. Proposition 9.8. Let A ( z ) be a Z -twisted Miura-Pl¨ucker ( G, q ) -oper. The following state-ments are equivalent: (1) A ( z ) is nondegenerate. (2) The Cartan q -connection A H ( z ) is nondegenerate, and each A i ( z ) has regular sin-gularities, i.e. ρ i ( z ) given by formula (4.3) is in C [ z ] . (3) Each y i ( z ) from formula (3.6) is a polynomial, and for all i ∈ Z the zeros of y i ( z ) and y i ± ( z ) are q -distinct from each other and from the zeros of Λ k ( z ) . Z-twisted Miura-Pl¨ucker ( GL ( ∞ ) , q ) -oper is Z-twisted. From the previous Sec-tion we see that the the q-connection of the nondegenerate Miura-Pl¨ucker ( GL ( ∞ ) , q )-operwith regular singularities defined by polynomials { Λ i ( z ) } i =1 ,...,r reads as follows:(9.7) A ( z ) = −∞ Y i =+ ∞ g i ( z ) ˇ α i e Λ i ( z ) gi ( z ) e i , g i ( z ) = ζ i Q + i ( qz ) Q + i ( z ) . Let us assume as in the SL ( r +1) case (see (5.2)) that ξ i is q-distinct from ξ i +1 . In particularthis means Z is regular semisimple. First we define the QQ -system for GL ( ∞ ) as infinite generalization of (5.3):(9.8) ξ i +1 Q + i ( qz ) Q − i ( z ) − ξ i Q + i ( z ) Q − i ( qz ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) , i ∈ Z . We say that a polynomial solution { Q + i ( z ) , Q − i ( z ) of (9.8) is nondegenerate if for i = j thezeros of Q + i ( z ) and Q − j ( z ) are q -distinct from each other and from the zeros of Λ k ( z ) for | i − k | = 1 , | j − k | = 1.The following Theorem is the the direct analogue of Theorem (5.1): Theorem 9.9. There is a one-to-one correspondence between the set of nondegenerate Z -twisted Miura-Pl¨ucker ( GL ( ∞ ) , q ) -opers and the set of nondegenerate polynomial solutionsof the GL ( ∞ ) QQ -system (9.8) . It can proved the same way as in [FKSZ] with the use of the Proposition 9.4. Thefollowing theorem serves as an infinite-dimensional generalization of Theorem 5.2. Theorem 9.10. Let A ( z ) be as in (9.7) and Z = Q i ζ ˇ α i i . Then the q -gauge transformation v ( z ) which diagonalizes q-connection A ( z ) = v ( qz ) Zv ( z ) − reads (9.9) v ( z ) = + ∞ Y i = −∞ Q + i ( z ) ˇ α i · + ∞ Y i = −∞ V i ( z ) , where (9.10) V i ( z ) = exp − X j>i φ i,...,j ( z ) e i,...,j , in which e i,...,j = [ . . . [[ e i , e i +1 ] , e i +2 ] . . . e j ] and functions φ i,...,j ( z ) satisfy the following rela-tions ξ i +1 φ i ( z ) − ξ i φ i ( qz ) = ρ i ( z ) ,ξ i +2 φ i,i +1 ( z ) − ξ i φ i,i +1 ( qz ) = ρ i +1 ( z ) φ i ( z ) ,. . . . . . (9.11) ξ i + j +1 φ i,...,i + j ( z ) − ξ i φ i,...,i + j ( qz ) = ρ i + j ( z ) φ i,...,i + j − ( z ) ,. . . . . . where i ∈ Z , j ∈ Z + and we use the same notations as in Section 5. The set of equations (9.11) is called extended QQ -system for GL ( ∞ ) which can also bepresented as ξ i +1 Q + i ( qz ) Q − i ( z ) − ξ i Q + i ( z ) Q − i ( qz ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) ,ξ i +2 Q + i +1 ( qz ) Q − i,i +1 ( z ) − ξ i Q + i +1 ( z ) Q − i,i +1 ( qz ) = Λ i +1 ( z ) Q − i ( qz ) Q + i +1 ( z ) ,. . . . . . (9.12) ξ i + j +1 Q + i + j ( qz ) Q − i,...,i + j ( z ) − ξ i Q + i + j ( z ) Q − i,...,i + j ( qz ) = Λ i + j ( z ) Q − i,...,i + j − ( qz ) Q + i + j +1 ( z ) ,. . . . . . OROIDAL q -OPERS 45 Proof. Let us first rewrite the diagonalization condition as(9.13) v ( qz ) − A ( z ) = Zv ( z ) − as it will be easier to compute the left and right hand sides of the above equation and thencompare them. We can make a statement similar to Lemma 5.4 and write the ( GL ( ∞ ) , q )-oper as(9.14) A ( z ) = −∞ Y i =+ ∞ Q + i ( qz ) ˇ α i · −∞ Y i =+ ∞ e ζiζi +1 ρ i ( z ) e i · −∞ Y i =+ ∞ ζ ˇ αi Q + i ( z ) − ˇ α i . Then the left hand side of (9.13) reads(9.15) v ( qz ) − A ( z ) = −∞ Y i =+ ∞ exp X j>i φ i,...,j ( qz ) e i,...,j · −∞ Y i =+ ∞ e ζiζi +1 ρ i ( z ) e i · −∞ Y i =+ ∞ ζ ˇ α i i Q + i ( z ) − ˇ α i . We now need to move i th element from the middle product above to the left until they com-bine with the corresponding e − φ i ( qz ) e i terms. This way e ζiζi +1 ρ i ( z ) e i will need to be carriedover to V i +1 ( qz ) − inside the first product. Each term e ζiζi +1 ρ i ( z ) e i will have nontrivial com-mutators with exponentials containing e i +1 – the others will vanish due to Serre relations.Thus (9.15) reads v ( qz ) − A ( z ) = . . . . . . · exp (cid:18) ζ ζ ρ ( z ) + φ ( qz ) (cid:19) e · · · · · exp (cid:18) ζ i ζ i +1 ρ r ( z ) φ ,...,i − ( qz ) + φ ,...,i ( qz ) (cid:19) e ,...,i · . . . (9.16) · exp (cid:18) ζ ζ ρ ( z ) + φ ( qz ) (cid:19) e · · · · · exp (cid:18) ζ j ζ j +1 ρ j ( z ) φ ,...,j − ( qz ) + φ ,...,j ( qz ) (cid:19) e ,...,j · . . . · . . . . . . · −∞ Y i =+ ∞ ζ ˇ α i i Q + i ( z ) − ˇ α i . This expression needs to be compared against the right hand side of (9.13) which is givenby(9.17) Zv ( z ) − = Y i ζ ˇ α i · −∞ Y i =+ ∞ exp X j>i φ i,...,j ( qz ) e i,...,j · −∞ Y i =+ ∞ Q + i ( z ) − ˇ α i In order to make the comparison manifest one needs to move the Cartan terms from theend to the front using the second equation from Lemma 5.3 Zv ( z ) − = . . . . . . · exp (cid:18) ζ ζ ζ φ ( z ) e (cid:19) · · · · · exp (cid:18) ζ ζ j ζ φ ,...,j ( z ) e ,...,j (cid:19) . . . (9.18) · exp (cid:18) ζ ζ ζ φ ( z ) e (cid:19) · · · · · exp (cid:18) ζ ζ l ζ φ ,...,l ( z ) e ,...,l (cid:19) · . . . · . . . . . . · −∞ Y i =+ ∞ ζ ˇ α i i Q + i ( z ) − ˇ α i . Here we used the following fact about nested commutators in Chevalley basis:(9.19) [ ˇ α a , e i,...,j ] = e i,...,j , a = i, or a = j , − e i,...,j , a = i − , or a = j + 1 , , otherwise . Comparing (9.16) with (9.18) leads to (9.11). (cid:3) Corollary 9.11. Theorem 5.2 follows.Proof. In the proof of Theorem 9.10 one needs to replace all infinite products with productsranging between 1 and r and put ζ = ζ r +1 = 1. (cid:3) Analogously to Theorem 5.6 we can make the following statement. Theorem 9.12. (1) Every nondegenerate solution of the QQ -system for GL ( ∞ ) (9.8) is also a nondegenerate solution of the extended QQ -system for GL ( ∞ ) (9.12) . (2) There is a one-to-one correspondence between the set of nondegenerate solutions ofthe the QQ -system for GL ( ∞ ) the set of solutions of Bethe Ansatz equations for GL ( ∞ ) : (9.20) Q + i ( qw ik ) Q + i ( q − w ki ) ξ i ξ i +1 = − Λ i ( w ik ) Q + i +1 ( qw ik ) Q + i − ( w ik )Λ i ( q − w ik ) Q + i +1 ( w ik ) Q + i − ( q − w ik ) , where i ∈ Z ; k = 1 , . . . , m i . Toroidal q-opers Quantum toroidal algebras, Bethe equations and QQ -System of c A type. The quantum toroidal algebra U t ,t (cid:0) bb gl (1) (cid:1) attracted a lot of attention in the recent years.On one hand it has explicit geometric realization: there is a natural action of this algebraon equivariant K-theory of ADHM instanton spaces [SV],[N2],[OS] which corresponds tosimplest framed quiver varieties with one loop and one vertex.One can describe such moduli spaces M N of rank- N torsion-free sheaves F on P withframing at infinity (also known as the moduli space of U ( N ) instantons on R ). Theframing condition forces the first Chern class to vanish, however, the second Chern class Sometimes in the literature different notation U q ,q ,q ( bb gl (1)) is used, where q = ( t t ) − , q = t , q q q = 1. OROIDAL q -OPERS 47 can range over the non-positive integers c ( F ) = k . The moduli space can be representedas a disjoint sum M N = ⊔ k M k,N . Each M k,N can be described as the moduli spaces ofstable representations of the ADHM quiver below, where W is a trivial bundle of rank N and V is a bundle of rank k . For N = 1 this quiver variety describes Hilbert scheme of k points on C . We refer for the details and the equivalence of various descriptions of ADHMmoduli spaces to [N1]. WV Let us denote G = A × ( C × ) , where A be the framing torus, i.e. maximal torus of GL ( N ) and the second factor ( C × ) is the torus acting on C ⊂ P . We denote equivariantparameters corresponding to A and G/A as a , . . . , a N and t , t correspondingly.The G -equivariant K-theory of M k,N is generated by the equivariant vector bundle V ofrank k over M k,N as in the case of the cotangent bundles to Grassmannians discussed in theintroduction. The space of the localized K G ( M k,N ) is a module for spanned by the fixedpoints of U t ,t ( bb gl (1)).This module has the structure of the analogue of XXZ-module for the toroidal algebra.Namely, the physical space H = F (a ) ⊗ · · · ⊗ F (a N )is the product of Fock space representations of the toroidal algebra { F (a i ) } , where theparameters { a i } ,which have the meaning of evaluation parameters, correspond to the zeromode value of the infinite-dimensional Heisenberg algebra. We refer to [OS] for more details.As we described in the Introduction, the quantum equivariant K-theory based on quasimapsis described by difference equations, which coincide with quantum Knizhnik-Zamolodchikovequations and the related dynamical equations. The solutions to these difference equationscan be computed as certain Euler characteristics on the moduli spaces of quasimaps. It isgiven by certain integral formula with the asymptotics given by the Yang-Yang function Y ,which can be described as follows.Let ℓ ( x ) be a multi-valued function, which can be written in terms of dilogarithm (see[GK]), such that exp 2 π ∂ℓ ( x ) ∂x = 2 sinh πx . The Yang-Yang function for ADHM space M k,N is given by [AO]: Y ADHM ( σ , α , ǫ , ǫ ) = k X a =1 N X m =1 ℓ ( σ a − α m ) + ℓ ( − σ a + α m − ǫ − ǫ )+ k X a = b ℓ ( σ a − σ b + ǫ ) + ℓ ( σ a − σ b + ǫ ) + ℓ ( σ a − σ b − ǫ − ǫ ) − κ k X a =1 σ a . (10.1) where s b = e πσ b , a b = e πα b , t = e πǫ , t = e πǫ , κ = e π κ , κ = ( t t ) − N z , so that z is a K¨ahler parameter of M k,N .Then Bethe equations in the case of M k,N can be computed as critical points for Y ADHM : Lemma 10.1. The equations (10.2) exp 2 π ∂Y ADHM ∂σ a = 1 , a = 1 , . . . , k . are equivalent to the following Bethe equations (10.3) N Y l =1 s a − a l t t s a − a l · k Y b =1 b = a s a − t s b t s a − s b s a − t s b t s a − s b s a − t t s b t t s a − s b = z , a = 1 , . . . , k . Recall that equations (10.3) describe relations in quantum equivariant K-theory of M k,N .Generalizing the results of [PSZ] with the help of [S], one can prove the following. Proposition 10.2. The eigenvalues of quantum multiplication operators by bundles Λ l V , ≤ l ≤ k in localized quantum G -equivarinat K-theory of M N,k are given by elementary sym-metric polynomials e l ( s , . . . s k ) of Bethe roots which satisfy the following Bethe equations (10.3) . Thus quantum equivariant K-theory ring of M N,k can be described by the symmetricfunctions of variables s , . . . , s k subject to Bethe equations. We refer to that as Bethealgebra of the XXZ model for quantum toroidal algebra.On the other hand, Feigin, Jimbo, Miwa and Mukhin [F, FJMM] studied such XXZmodel explicitly and derived such Bethe equations for the corresponding transfer matrices.Another important issue, featured in [FJMM] is the explicit construction of the Q -operator.We remind that this is the operator in the Bethe algebra whose eigenvalues form a generatingfunction of the elementary symmetric functions of Bethe roots, i.e. it is a generating functionof operators from Proposition (10.2).Recently Frenkel and Hernandez [FH2] wrote down the QQ -system leading to Betheansatz equations for quantum toroidal gl algebra. It reads as follows:(10.4) ξ Q + (cid:0) ( t t ) − z (cid:1) Q − ( z ) − Q + ( z ) Q − (cid:0) ( t t ) − z (cid:1) = L ( z ) Q + ( t − z ) Q + ( t − z ) , where we altered the authors’ notation slightly and introduced ‘framing polynomial’ L ( z ).We will refer to this functional equation as ˆ A QQ -system. The above QQ -system equationscan also be rewritten as(10.5) ξ φ ( z ) − φ (( t t ) − z ) = ρ ( z ) , where(10.6) φ ( z ) = Q − ( z ) Q + (( t t ) − z ) , ρ ( z ) = L ( z ) Q + ( t − z ) Q + ( t − z ) Q + ( z ) Q + (( t t ) − z ) . We call the solutions for such system nondegnerate if Q + ( z ), Q − ( z ), L are t , t -distinct and ξ = 1.That leads to the following Lemma. Lemma 10.3. There is a one-to-one correspondence between the set of nondegenerate so-lutions of (10.4) and (10.3) . OROIDAL q -OPERS 49 Proof. Since Q ( u ) = k Y a =1 ( z − s a ) , L ( u ) = N Y i =1 ( z − a i )we can first evaluate (10.21) at u = s a , then shift variable u by t t and evaluate thisequation again at z = s a . This leads us to the following L ( s a ) L ( t t s a ) · Q ( t − s a ) Q ( t s a ) Q ( t − s a ) Q ( t s a ) Q ( t t s a ) Q (( t t ) − s a ) = − ξ . This shows the implication in one direction – from the QQ -system to Bethe equations.The opposite statement can be proved analogously to Theorem 5.6. We leave it to thereader. (cid:3) There exists a generalization of this construction to higher rank quantum toroidal algebra U t ,t ( bb gl ( N )) for cyclic quiver varieties with N vertices. It is easy to write the Yang-Yangfunction in this case as well as Bethe equations (see [AO] for universal treatment). It is alsoeasy to present the analogue of the QQ -system (see below). However, the representation-theoretic approach along the lines of [FJMM] has not yet been developed, i.e. constructionof the Q -operator as a transfer matrix for special auxiliary representation of U t ,t ( bb gl ( N )).10.2. Miura 1-toroidal q-opers. We can now define toroidal opers. Let us consider theautomorphism of the Dynkin diagram of a ∞ , which correspond to the shift by one vertex.This automorphism can be realized by the transformation corresponding to the conjugationvia the infinite Coxeter element Q −∞ i =+ ∞ s i . In the matrix notation such infinite Coxeterelement can be realized via V = P i ∈ Z E i,i − . Definition 10.4. Let p, ξ ∈ C × . We refer to a Z-twisted Miura ( GL ( ∞ ) , q )-oper (3.8)satisfying(10.7) V A ( z ) V − = ξA ( pz ) , as the Z -twisted 1-toroidal Miura q-oper . We call it nondegnerate if it is nondegenerate asZ-twisted Miura ( GL ( ∞ ) , q )-oper.The above definition (10.7) translates to the following conditions on polynomials whichappear in the QQ-system(10.8) g i +1 ( z ) g i ( z ) = ξ g i ( pz ) g i − ( pz ) , Λ i +1 ( z ) = ξ Λ i ( pz ) , The first equation above becomes (recall that ξ i = ζ i ζ i − )(10.9) ξ i +1 Q + i +1 ( qz ) Q + i ( z ) Q + i +1 ( z ) Q + i ( qz ) = ξ ξ i Q + i ( qz ) Q + i − ( pz ) Q + i ( z ) Q + i − ( pqz ) , which can be satisfied provided that for(10.10) ξ i = ξ i , Q + i ( z ) = Q + ( p i z ) , Λ i ( z ) = ξ i Λ( p i z ) . Let us now study how the above periodic conditions affect the QQ -system for GL ( ∞ ).The QQ -equations (9.8) can be rewritten in the following way(10.11) ξ i +1 φ ( p i z ) − ξ i φ ( p i qz ) = ξ i Λ( p i z ) Q + ( p i − z ) Q + ( p i +1 qz ) Q + ( p i z ) Q + ( p i qz ) , where we defined φ i ( z ) is replaced by φ ( p i z ) and thusly(10.12) Q − i ( z ) = Q − ( p i z ) . By shifting the variable z p − i z , we get(10.13) ξ φ ( z ) − φ ( qz ) = Λ( z ) Q + ( p − z ) Q + ( pqz ) Q + ( z ) Q + ( qz ) , Equivalently we can impose ρ i ( z ) = ρ ( p i z ).Notice that the above equations coincide with (10.5) and (10.6); therefore we recover the b A QQ -system provided that(10.14) L ( z ) = Λ( z ) , p = t , q = ( t t ) − . This brings us to the following Theorem 10.5. The space of nondegenerate Z -twisted Miura 1-toroidal q-opers with regularsingularities at a , . . . , a N is isomorphic to the space of solutions of the nondegenerate b A QQ -system (10.6) or equivalently to (10.15) ξ Q + ( qz ) Q − ( z ) − Q + ( z ) Q − ( qz ) = Λ( z ) Q + ( p − z ) Q + ( pqz ) , where Λ( z ) = Q Nj =1 ( z − a j ) . The full set of equations for the extended QQ -system for GL ( ∞ ) (9.12) reads Q + ( qz ) Q − ( z ) − ξ Q + ( z ) Q − ( qz ) = Λ( z ) Q + ( p − qz ) Q + ( pz ) , Q + ( qz ) Q − (1) ( z ) − ξ Q + ( z ) Q − (1) ( qz ) = ξ Λ( z ) Q − ( p − qz ) Q + ( pz ) ,. . . . . . (10.16) Q + ( qz ) Q − ( j ) ( z ) − ξ j +1 Q + ( z ) Q − (1) ( qz ) = ξ j Λ( z ) Q − ( j − ( p − qz ) Q + ( pz ) ,. . . . . . where(10.17) Q − ( j ) ( p i z ) = Q − i − j +1 ,i − j +2 ,...,i ( z ) . The gauge transformation which brings toroidal q-oper to the diagonal form can bedirectly generalized from (5.11) using (10.10), (10.12) and (10.17):(10.18) v ( z ) − = . . . ... ... . . . . . . ... . . .0 Q + ( p i − z ) Q + ( p i z ) Q − ( p i − z ) Q + ( p i +1 z ) . . . . . . Q − ( j ) ( p i − z ) Q + ( p i + j z ) ...0 0 Q + ( p i z ) Q + ( p i +1 z ) . . . . . . Q − ( j − ( p i z ) Q + ( p i + j z ) ...... ... ... . . . ... ... ...0 . . . . . . . . . Q + ( p i + j − z ) Q + ( p i + j − z ) Q − ( p i + j − z ) Q + ( p i + j z ) ...0 . . . . . . . . . . . . Q + ( p i + j − z ) Q + ( p i + j z ) ...0 . . . . . . . . . . . . . OROIDAL q -OPERS 51 Miura N -toroidal q-Opers. Here we briefly show, how the above construction canbe immediately generalized to higher rank. Namely, one has to generalize the periodicityconditions we Definition 10.6. Let p, ξ ∈ C × . The Z -twisted N-toroidal Miura q-oper is a ( GL ( ∞ ) , q )-oper (3.8) satisfying(10.19) V N A ( z ) V − N = ξ N A ( p N z ) , where V N = V N . We call it nondegnerate if it is nondegenerate as Z-twisted Miura( GL ( ∞ ) , q )-oper.If we impose (10.19) on the q-connection we get the following family of equations for i ≥ j :(10.20) g i + N ( z ) g j + N ( z ) = ξ N g i ( p N z ) g j ( p N z ) , Λ i + N ( z ) = ξ N Λ i ( p N z ) , which imposes N -periodicity on all functions Q ± i + N ( z ) = Q ± i ( p N z ) , ξ i + N = ξ N ξ i . for all i .Thus we arrive to the generalization of the Theorem 10.5: Theorem 10.7. The nondegenerate Z -twisted Miura N-toroidal q -opers with regular sin-gularities given by Λ i ( u ) = Q N j =1 ( z − a ( i ) j ) is in one-to-one correspondence with the nonde-generate solutions of the following b A N − QQ -system: ξ Q +1 ( qz ) Q − ( z ) − ξ Q +1 ( z ) Q − ( qz ) = Λ ( z ) Q + N ( qz ) Q +2 ( z ) , (10.21) ξ i Q + i ( qz ) Q − i ( z ) − ξ i +1 Q + i ( z ) Q − i ( qz ) = Λ i ( z ) Q + i − ( qz ) Q + i +1 ( z ) , i = 2 , . . . , N − ξ N Q + N ( qz ) Q − N ( z ) − ξ Q + N ( z ) Q − N ( qz ) = Λ N ( z ) Q + N − ( qz ) Q +1 ( z ) . with the nondegeneracy conditions induced from original GL ( ∞ ) QQ -system. References [AO] M. Aganagic and A. Okounkov, Quasimap counts and Bethe eigenfunctions , Moscow Math. J. (2017), no. 4, 565–600, 1704.08746.[AFO] M. Aganagic, E. Frenkel, and A. Okounkov, Quantum q -Langlands Correspondence , Trans. MoscowMath. Soc. (2018), 1–83, 1701.03146.[BFL + ] V. Bazhanov, R. Frassek, T. Lukowski, C. Meneghelli, and M. 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