Abstract
A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for \({{\,\mathrm{\mathrm {SL}}\,}}(N)\). We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars–Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties.
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1 Introduction
1.1 Opers and the Gaudin model
One formulation of the geometric Langlands correspondence is the existence of an isomorphism between spaces of conformal blocks for the classical W-algebra associated to a simple complex Lie algebra \(\mathfrak {g}\) and the dual affine Kac-Moody algebra \({}^L{\hat{\mathfrak {g}}}\) at the critical level. Since both these algebras admit deformations, it is natural to conjecture the existence of deformed versions of the Langlands correspondence, and indeed, this has been the subject of considerable recent interest [AFO, GF, P]. In this paper, we describe a q-Langlands correspondence which is a deformation of an important example of geometric Langlands, the classical correspondence between the spectra of the Gaudin model and opers on the projective line with regular singularities and trivial monodromy.
Let G be a simple complex algebraic group of adjoint type, and let \({}^L\mathfrak {g}\) be the Lie algebra of the Langlands dual group \({}^L G\). Fix a collection of distinct points \(z_1,\dots ,z_n\) in \(\mathbb {C}\). The Gaudin Hamiltonians are certain mutually commuting elements of the algebra \(U({}^L \mathfrak {g})^{\otimes n}\). They are contained in a commutative subalgebra \(\mathcal {Z}_{(z_i)}({}^L \mathfrak {g})\) called the Gaudin algebra. The simultaneous eigenvalues of the actions of the Gaudin Hamiltonians on N-fold tensor products of \({}^L\mathfrak {g}\)-modules is given by the (maximal) spectrum of this algebra, namely, the set of algebra homomorphisms \(\mathcal {Z}_{(z_i)}({}^L \mathfrak {g})\longrightarrow \mathbb {C}\).Footnote 1
Feigin, Frenkel, and Reshetikhin found a geometric interpretation of this spectrum in terms of flat G-bundles on \(\mathbb {P}^1\) with extra structure [FFR1, F2, F1]. Let B be a Borel subgroup of G. A G-oper on a smooth curve X is a triple \((\mathcal {F},\nabla ,\mathcal {F}_B)\), where \((\mathcal {F},\nabla )\) is a flat G-bundle on X and \(\mathcal {F}_B\) is a reduction of \(\mathcal {F}\) satisfying a certain transversality condition with respect to \(\nabla \). As an example, for \({{\,\mathrm{\mathrm {PGL}}\,}}(2)\)-opers, this condition is that \(\mathcal {F}_B\) is nowhere preserved by \(\nabla \). The space of G-opers can be realized more concretely as a certain space of differential operators. For example, a \({{\,\mathrm{\mathrm {PGL}}\,}}(2)\)-oper can be identified with projective connections: second-order operators \(\partial _z^2-f(z)\) mapping sections of \(K^{-1/2}\) to sections of \(K^{3/2}\), where K is the canonical bundle. It turns out that the spectrum of \(\mathcal {Z}_{(z_i)}({}^L\mathfrak {g})\) may be identified with the set of G-opers on \(\mathbb {P}^1\) with regular singularities at \(z_1,\dots ,z_n\) and \(\infty \), which therefore has the structure of an algebraic variety.
We now consider the action of the Gaudin algebra on the tensor product of irreducible finite-dimensional modules \(V_{\varvec{\lambda }}=V_{\lambda _1}\otimes \dots \otimes V_{\lambda _n}\), where \(\varvec{\lambda }\) is an n-tuple of dominant integral weights. The Bethe ansatz is a method of constructing such simultaneous eigenvectors. One starts with the unique (up to scalar) vector \(|0\rangle \in V_{\varvec{\lambda }}\) of highest weight \(\sum \lambda _i\); it is a simultaneous eigenvector. Given a set of distinct complex numbers \(w_1,\dots ,w_m\) labeled by simple roots \(\alpha _{k_j}\), one applies a certain order m lowering operator with poles at the \(w_j\)’s to \(|0\rangle \). If this vector is nonzero and \(\sum \lambda _i -\sum \alpha _{k_j}\) is dominant, it is an eigenvector of the Gaudin Hamiltonians if and only if certain equations called the Bethe ansatz equations are satisfied (see (2.13)). Frenkel has shown that the corresponding point in the spectrum of the Gaudin algebra is a G-oper with regular singularities at the \(z_i\)’s and \(\infty \) and and with trivial monodromy [F1].
In fact, it is possible to give a geometric description of all solutions of the Bethe equations (i.e., without assuming \(\sum \lambda _i -\sum \alpha _{k_j}\) is dominant) in terms of an enhanced version of opers. A Miura G-oper on \(\mathbb {P}^1\) is a G-oper together with an additional reduction \(\mathcal {F}'_B\) which is preserved by \(\nabla \). The set of Miura opers with the same underlying oper is parametrized by the flag manifold G/B. Frenkel has shown that there is a one-to-one correspondence between the set of solutions to the \({}^L \mathfrak {g}\) Bethe ansatz and “nondegenerate” Miura G-opers with regular singularities and trivial monodromy [F2]. To see how this works, let \(H\subset B\) be a maximal torus. The initial data of the Bethe ansatz gives rise to the explicit flat H-bundle (a Cartan connection)
There is a map from Cartan connections to Miura opers given by the Miura transformation; this is just a generalization of the standard Miura transformation in the theory of KdV integrable models. It turns out that the Bethe equations are precisely the conditions necessary for the corresponding Miura oper to be regular at the \(w_j\)’s.
In the global geometric Langlands correspondence for \(\mathbb {P}^1\), the objects on the Galois side are flat G-bundles (with singularities) on \(\mathbb {P}^1\) while on the automorphic side, one considers D-modules on enhanced versions of the moduli space of \({}^L G\)-bundles over \(\mathbb {P}^1\). The correspondence between opers and spectra of the Gaudin model provides an example of geometric Langlands. Indeed, the eigenvector equations for the Gaudin Hamiltonians for fixed eigenvalues determines a D-module on the moduli space of \({}^L G\)-bundles with parabolic structures at \(z_1,\dots ,z_n\) and \(\infty \) while the oper gives the flat G-bundle.
1.2 q-opers and the q-Langlands correspondence
Recall that the geometric Langlands correspondence may be viewed as an identification of conformal blocks for the classical \(\mathcal {W}\)-algebra associated to \(\mathfrak {g}\) and conformal blocks for the affine Kac-Moody algebra \({}^L{\hat{\mathfrak {g}}}\) at the critical level. Both these algebras admit deformations. For example, one may pass from \({}^L {\hat{\mathfrak {g}}}\) to the associated quantum affine algebra while at the same time moving away from the critical level. This led Aganagic, Frenkel, and Okounkov to formulate a two-parameter deformation of geometric Langlands called the quantum q-Langlands correspondence [AFO]. This is an identification of certain conformal blocks of a quantum affine algebra with those of a deformed \(\mathcal {W}\)-algebra, working over the infinite cylinder. They prove this correspondence in the simply-laced case; their proof is based on a study of the equivariant K-theory of Nakajima quiver varieties whose quiver is the Dynkin diagram of \(\mathfrak {g}\).
In this paper, we take another more geometric approach, involving q-connections, a difference equation version of flat G-bundles. Our goal is to establish a q-Langlands correspondence between q-opers with regular singularities and the spectra of the XXZ spin chain model. Here, we only consider this correspondence in type A, where we can describe q-opers explicitly as a vector bundle endowed with a difference operator, together with a complete flag of subbundles that is well-behaved with respect to the operator.Footnote 2
Fix a nonzero complex number q which is not a root of unity. We are interested in (multiplicative) difference equations of the form \(s(qz)=A(z)s(z)\); here A(z) is an \(N\times N\) invertible matrix whose entries are rational functions. To express this more geometrically, we start with a trivializable rank n vector bundle E on \(\mathbb {P}^1\), and let \(E^q\) denote the pullback of E via the map \(z\mapsto qz\). A \(({{\,\mathrm{\mathrm {GL}}\,}}(N),q)\)-connection on \(\mathbb {P}^1\) is an invertible operator A taking sections of E to sections of \(E^q\). If the matrices A(z) have determinant one in some trivialization, (E, A) is called an \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-connection. Just as in the classical setting, an \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-oper is a triple \((E,A,E_B)\), where \(E_B\) is a reduction to a Borel subgroup satisfying a certain transversality condition with respect to A. We also define a Miura q-oper to be a q-oper with an additional reduction \(E'_B\) preserved by A. We remark that these definitions make sense when \(\mathbb {P}^1\) is replaced by the formal punctured disk. In this setting, a concept equivalent to \(({{\,\mathrm{\mathrm {GL}}\,}}(N),q)\)-connections was introduced by Baranovsky and Ginzburg [BG] while the notion of a formal q-oper is inherent in the work of Frenkel, Reshetikhin, Semenov-Tian-Shansky, and Sevostyanov on Drinfeld-Sokolov reduction for difference operators [FRSTS, SS].
We now explain how q-opers can be viewed as the Galois side of a q-Langlands correspondence. The XXZ spin chain model is an integrable model whose dynamical symmetry algebra is the quantum affine algebra \(U_q({\hat{\mathfrak {g}}})\) [R1]. Under an appropriate limiting process, it degenerates to the Gaudin model. The model depends on certain twist parameters which can be described by a diagonal matrix Z. We will always assume that Z has distinct eigenvalues. Eigenvectors of the Hamiltonians in the XXZ model can again be found using the Bethe ansatz, and the spectra can be expressed in terms of Bethe equations (see (3.6), (4.10) below).
It turns out that these equations also arise from appropriate q-opers. We consider q-opers with regular singularities on \(\mathbb {P}^1{\setminus }\{0,\infty \}\). We further assume that the q-oper is Z-twisted, where Z is the diagonal matrix appearing in the Bethe equations; this simply means that the underlying q-connection is q-gauge equivalent to the q-connection with matrix Z. (This may be viewed as the quantum analogue of the opers with a double pole singularity at \(\infty \) considered by Feigin, Frenkel, Rybnikov, and Toledano-Laredo in their work on an inhomogeneous version of the Gaudin model [FFTL, FFR2].) Given a Z-twisted q-oper with regular singularities, we examine a certain associated Miura q-oper. The assumption that this Miura q-oper is “nondegenerate” imposes certain conditions on the zeros of quantum Wronskians arising from the q-oper, and these conditions lead to the XXZ Bethe equations. Thus, in type A, we obtain the desired q-Langlands correspondence. It should be noted that in contrast to the results of [AFO], our results do not depend on geometric data related to the quantum K-theory of Nakajima quiver varieties. In particular, there are no restrictions on the dominant weights that can appear in our correspondence.
Our approach has some similarities with the earlier work of Mukhin and Varchenko on discrete opers and the spectra of the XXX model [MV2]. Here, they considered additive difference equations, i.e., equations of the form \(f(z+h)=A(z)f(z)\) where A is a G-valued function and \(h\in \mathbb {C}^*\) is a fixed parameter. They defined a discrete oper to be the linear difference operator \(f(z)\mapsto f(z+h)-A(z)f(z)\) if A(z) had a suitable form. They also introduced a notion of discrete Miura oper and showed that they correspond to solutions of the XXX Bethe ansatz equations. Unlike our q-opers, these discrete opers do not seem to be related to the difference equation version of Drinfeld-Sokolov reduction considered in [FRSTS].
Since the XXZ model may be viewed as a deformation of the Gaudin model, one would expect that we should recover the Gaudin Bethe equations under an appropriate limit. In fact, by taking this limit in two steps, one can say more. First, a suitable limit takes one to a twisted version of the XXX spin chain, giving rise to a correspondence between the solutions of the Bethe equations for this model and a twisted analogue of the discrete opers of [MV2]. A further limit brings one back to the inhomogeneous Gaudin model and opers with irregular singularity considered in [FFTL, FFR2].
1.3 QQ-systems, XXZ models, and Baxter operators
As we have seen, Bethe ansatz equations arise from q-opers by considering quantum Wronksians and their properties. More precisely, the algebraic structure of the set of quantum Wronksians governs both twisted Z-opers and solutions of the XXZ Bethe equations and leads to the correspondence between the two. This algebraic structure is a special case of a more general phenomenon, which appears in the representation theory of quantum affine algebras. Hernandez and Jimbo have introduced a category \(\mathcal {O}\) of highest weight representations of a Borel subalgebra \(U_q(\hat{\mathfrak {b}}_+)\) of the quantum affine algebra \(U_q(\hat{\mathfrak {g}})\) [HJ]. This category includes all finite-dimensional representations of \(U_q(\hat{\mathfrak {g}})\), but also contains infinite-dimensional representations on which the Borel action does not extend to the entire quantum affine algebra. Within this category, they have constructed certain “prefundamental modules” which generate the Grothendieck ring of \(\mathcal {O}\). The relations among these generators give rise to so-called QQ-systems (or \(Q\tilde{Q}\)-systems), which generalize the the quantum Wronskian relations for \(\hat{\mathfrak {sl_2}}\) obtained in [BLZ].
These QQ-systems, in their manifestation as relations in the Grothendieck ring, have arisen in various circumstances [KLWZ, MV1, KLV, B]. However, they also have an explicit realization in terms of certain polynomial equations. As this is the form in which QQ-systems appear in the context of q-opers, we briefly explain how this works.
Upon fixing a finite-dimensional representation \(\mathcal {H}\) of \(U_q(\hat{\mathfrak {g}})\) (known as the physical space), one obtains a particular XXZ integrable model associated to the given quantum group. The spectrum of this integrable model is described by the eigenvalues of the transfer matrices, which are constructed in the following way. The R-matrix, associated to \(U_q(\hat{\mathfrak {g}})\) belongs to \(U_q(\hat{\mathfrak {b}}_+)\otimes U_q(\hat{\mathfrak {b}}_-)\), where \(\mathfrak {b}_\pm \) are opposite Borel subalgebras. For any \(\mathcal {V}\in \mathcal {O}\), the R-matrix acts on \(\mathcal {V}\otimes \mathcal {H}\), and one can now take the weighted trace of the R-matrix with respect to the first factor. This trace is an operator acting on \(\mathcal {H}\). In the particular cases where \(\mathcal {V}\) is a finite-dimensional representation of \(U_q(\hat{\mathfrak {g}})\) or a prefundamental module, the resulting operators are called transfer matrices and Baxter Q-operators respectively. Upon certain canonical rescalings of the transfer matrices and Q-operators, the eigenvalues of these operators become polynomials, and the relations in the Grothendieck ring turn into polynomial equations [FH1, FH2], thereby proving conjectures on transfer matrices found in [FR].
We observe that the QQ-system which describes twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-opers is the one associated to \(\hat{\mathfrak {sl}}_n\) in [FH2].
1.4 Quantum/classical duality and applications to enumerative geometry
Quantum/classical duality is a relationship between a quantum and a classical integrable system. Well-known examples are the relationship in type A between the Gaudin model and the rational Calogero-Moser system and between the XXX spin-chain and the rational Ruijsenaars–Schneider model. Both of these can be viewed as limits of the duality between the XXZ spin-chain and the trigonometric Ruijsenaars–Schneider model [HR1, HR2, MTV].Footnote 3 This duality is given by a transformation relating two sets of generators for the quantum K-theory ring of cotangent bundles of full flag varieties [KPSZ]. One set of generators is obtained from the XXZ Bethe equations. One considers certain Bethe equations where the dominant weights all come from the defining representation and then takes symmetric functions on the corresponding Bethe roots. The other generators are functions on a certain Lagrangian subvariety in the phase space for the tRS model.
This correspondence has a direct interpretation in terms of twisted q-opers; indeed, it may be viewed as a special case of the q-Langlands correspondence. As we discussed in the previous section, Bethe equations arise from nondegenerate twisted q-opers. The Bethe roots are precisely those zeros of the quantum Wronskians associated to the q-oper which are not singularities of the underlying q-connection. On the other hand, there is an embedding of the tRS model into the space of twisted q-opers. More precisely, a q-oper structure on a given q-connection (E, A) is determined uniquely by a full flag \(\mathcal {L}_\bullet \) of vector subbundles which behave in a specified way with respect to A. A section s generating the line bundle \(\mathcal {L}_1\) over \(\mathbb {P}^1{\setminus }\infty \) may be viewed as an N-tuple of monic polynomials \((s_1,\dots ,s_N)\). If these polynomials are all linear, then their constant terms are precisely the momenta in the phase space of the tRS model. Quantum/classical duality is then equivalent to the statement that the Bethe roots and the constant terms of these monic linear polynomial both give coordinates for an appropriate spaces of twisted q-opers.
If the monic polynomials \(s_i\) are no longer linear, it is still the case that the Bethe roots and the coefficients of these polynomials are equivalent sets of coordinates for a space of twisted q-opers. It is more complicated to interpret this statement as a duality between the XXZ spin-chain and a classical multiparticle integrable system. However, we do get an application to the quantum K-theory of the cotangent bundles of partial flag varieties. This K-theory ring is again generated by symmetric functions in appropriate Bethe roots. In [RTV], Rimanyi, Tarasov, and Varchenko gave another conjectural set of generators for this ring. We show that these generators are precisely those obtained from the coordinates for the set of twisted q-opers coming from the coefficients of the polynomials \(s_i\), thereby proving this conjecture.
We remark that other applications of the XXZ Bethe ansatz equations to geometry have appeared in the recent physics literature [NS1, NS2, GK, NPS].
1.5 Structure of the paper
In Sect. 2, we recall the relationship between monodromy-free \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-opers with regular singularities on the projective line and Gaudin models [F1, F2]. We follow an approach hinted at in [GW], describing opers in terms of vector bundles instead of principal bundles and obtaining the Bethe equations from Wronskian relations. We also discuss the correspondence between an inhomogeneous version of the Gaudin model and opers with an irregular singularity at infinity.
Next, in Sect. 3, we consider a q-deformation of opers in the case of \({{\,\mathrm{\mathrm {SL}}\,}}(2)\). We adapt the techniques of the previous section to give a correspondence between twisted q-opers and the Bethe ansatz equations for the XXZ spin chain for \(\mathfrak {sl}_2\). In Sect. 4, we generalize these constructions to \({{\,\mathrm{\mathrm {SL}}\,}}(N)\) and again prove a correspondence between q-opers and the XXZ spin chain model. We then discuss the case of \({{\,\mathrm{\mathrm {SL}}\,}}(3)\) in detail in Sect. 5.
In Sect. 6, we consider classical limits of our results. We show that an appropriate limit leads to a correspondence between a twisted analogue of the discrete opers considered in [MV2] and the spectra of a version of the XXX spin chain. By taking a further limit, we recover the relationship between opers with an irregular singularity and the inhomogeneous Gaudin model [FFTL, FFR2].
Finally, Sect. 7 is devoted to some geometric implications of the results of this paper. The quantum K-theory ring of the cotangent bundle to the variety of partial flags is known to be described via the Bethe ansatz equations [KPSZ]. We find a new set of generators defined in terms of canonical coordinates on an appropriate set of q-opers. These generators turn out to be the same as the conjectural generators given in [RTV].
2 \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-Opers with Trivial Monodromy and Regular Singularities
2.1 \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-opers and Bethe equations
In this section, we describe a simple reformulation of the results of [F1, F2] due to Gaiotto and Witten [GW].
Definition 2.1
A \({{\,\mathrm{\mathrm {GL}}\,}}(2)\)-oper on \(\mathbb {P}^1\) is a triple \((E,\nabla ,\mathcal {L})\), where E is a rank 2 vector bundle on \(\mathbb {P}^1\), \(\nabla :E\longrightarrow E\otimes K\) is a connection (here K is the canonical bundle), and \(\mathcal {L}\) is a line subbundle such that the induced map \({\bar{\nabla }}:\mathcal {L}\longrightarrow E/\mathcal {L}\otimes K\) is an isomorphism. The triple is called an \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-oper if the structure group of the flat \({{\,\mathrm{\mathrm {GL}}\,}}(2)\)-bundle may be reduced to \({{\,\mathrm{\mathrm {SL}}\,}}(2)\).
We always assume that the vector bundle E is trivializable.
The oper condition may be checked explicitly in terms of a determinant condition on local sections. Indeed, \({\bar{\nabla }}\) is an isomorphism in a neighborhood of a given point z if for some (or for any) local section s of \(\mathcal {L}\) with \(s(z)\ne 0\),
Here, \(\nabla _z=\iota _{\frac{d}{dz}}\circ \nabla \), where \(\iota _{\frac{d}{dz}}\) is the inner derivation by the vector field \(\frac{d}{dz}\).
In this section, we will be interested in \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-opers with regular singularities. An \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-oper with regular singularities of weights \(k_1,\dots ,k_L,k_\infty \) at the points \(z_1,\dots ,z_L,\infty \) is a triple \((E,\nabla ,\mathcal {L})\) as above where \({\bar{\nabla }}\) is an isomorphism everywhere except at each \(z_i\) (resp. \(\infty \)), where it has a zero of order \(k_i\) (resp. \(k_\infty \)). Concretely, near the point \(z_i\), we have
We will always assume that our opers have trivial monodromy, i.e., that the monodromy of the connection around each \(z_i\) is trivial. This means that after an appropriate gauge change, we can assume that the connection is trivial. (Recall that changing the trivialization of E by g(z) induces gauge change in the connection; explicitly, the new connection is \(g(z)\nabla g(z)^{-1}-d(g(z))g(z)^{-1}\).) In terms of this trivialization of E over \(\mathbb {P}^1{\setminus }\infty \), the line bundle \(\mathcal {L}\) is generated over this affine space by the section
where \(q_{\pm }(z)\) are polynomials without common roots. The condition (2.1) leads to the following equation on the Wronskian:
where \(\rho (z)\) is a polynomial whose zeros are determined by (2.1). After multiplying s by a constant, we may take \(\rho (z)=\prod ^L_{i=1}(z-z_i)^{k_i}\). By applying a constant gauge transformation in \({{\,\mathrm{\mathrm {SL}}\,}}(2,\mathbb {C})\), we may further normalize s so that \(\deg (q_{-})<\deg (q_{+})\) and \(q_{-}(z)=\prod ^{l_-}_{i=1}(z-w_i)\) has leading coefficient 1. (More precisely, transforming by \(\left( {\begin{matrix} 0 &{} 1\\ -1 &{} 0 \end{matrix}}\right) \) if necessary allows us to assume that \(\deg (q_{-})\le \deg (q_{+})\); if the degrees are equal, transforming by an elementary matrix brings us to the case \(\deg (q_{-})<\deg (q_{+})\). The final reduction uses a diagonal gauge change.)
We now make the further assumption that our oper is nondegenerate, meaning that none of the \(z_i\)’s are roots of \(q_{-}\). It is now an immediate consequence of (2.3) that each root of \(q_{-}\) has multiplicity 1.
Let \(k=\sum ^L_{i=1} k_i\) denote \(\deg (\rho )\). An easy calculation using the fact that \(\deg (q_-)<\deg (q_+)\) gives \(\deg (q_-)+\deg (q_+)=k+1\); this implies that \(\deg (q_{-})=l_-\le k/2\). We now rewrite (2.3) in the equivalent form
Since the residue at each \(w_i\) of the left-hand side of this equation is 0, computing the residues of the right-hand side leads to the conditions
These are the Bethe ansatz equations for the \(\mathfrak {sl}_2\)-Gaudin model at level \(k-2l_{-}\ge 0\); they determine the spectrum of this model.
A local section for \(\mathcal {L}\) at \(\infty \) is given by
where \(l_{+}=\deg (q_{+})\). If we set \(k_{\infty }=k-2l_{-}=l_{+}-l_{-}-1\), we obtain
Thus, we have proved the following theorem.
Theorem 2.2
There is a one-to-one correspondence between the spectrum of the Gaudin model, described by the Bethe equations for dominant weights, and the space of nondegenerate \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-opers with trivial monodromy and regular singularities at the points \(z_1, \dots , z_L, \infty \) with weights \(k_1, \dots , k_L, k_{\infty }\).
2.2 Miura opers and the Miura transformation
The previous theorem raises the natural question of whether one can give a geometric interpretation to solutions of the Bethe equations without assuming that the level \(k-2l_-\) is nonnegative. Miura opers provide such an description. A Miura oper is an oper \((E,\nabla ,\mathcal {L})\) together with an additional line bundle \({\hat{\mathcal {L}}}\) preserved by \(\nabla \). There may be a finite set of points where \(\mathcal {L}\) and \({\hat{\mathcal {L}}}\) do not span E. It turns out that one can associate to any oper with regular singularities a family of Miura opers parameterized by the flag variety [F1].
Given a Miura oper, we may choose a trivialization of E so that the line bundle \({\hat{\mathcal {L}}}\) is generated by the section \({\hat{s}}=(1,0)\). We retain our notation for the section \(s=\left( {\begin{matrix}q_{+}\\ q_{-} \end{matrix}} \right) \) generating \(\mathcal {L}\), but here, we do not impose any restrictions on \(\deg (q_{-})\).
Theorem 2.2 can be generalized to give the following theorem, which is proved in a similar way.
Theorem 2.3
There is a one-to-one correspondence between the set of solutions of the Bethe Ansatz equations (2.5) and the set of nondegenerate \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-Miura opers with trivial monodromy and regular singularities at the points \(z_1, \dots , z_L, \infty \) with weights at the finite points given by \(k_1, \dots , k_L\).
We now give a different formulation of \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-opers which shows how the eigenvalues of the Gaudin Hamiltonian can be seen directly from the oper. We will do this by applying several \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-gauge transformations to our trivial connection to reduce it to a canonical form. We start with a gauge change by \(g(z)=\left( {\begin{matrix} q_-(z) &{} -q_+(z)\\ 0 &{} q_-^{-1}(z) \end{matrix}}\right) \); note that \(g(z)s(z)=\left( {\begin{matrix}0\\ 1 \end{matrix}}\right) \). The new connection matrix is
Next, the diagonal transformation \(\left( {\begin{matrix} \rho (z)^{-1/2}&{} 0\\ 0 &{} \rho (z)^{1/2} \end{matrix}}\right) \) brings us to the Cartan connection
where
Finally, we apply the Miura transformation: gauge change by the lower triangular matrix \(\left( {\begin{matrix} 1&{} 0\\ u(z) &{} 1 \end{matrix}}\right) \) gives the connection matrix
An explicit computation using the Bethe equations (2.5) gives
where
This shows that t(z) does not have any singularities at \(z= w_i\); moreover, since the \(c_m\) are the eigenvalues of the Gaudin Hamiltonians, it depends only on this spectrum. In particular, the Gaudin eigenvalues can be read off explicitly as the negative of the residue of the nonconstant entry of the connection matrices B(z). Note that a horizontal section \(f=\left( {\begin{matrix}f_1\\ f_2 \end{matrix}}\right) \) to the connection in this gauge is determined by a solution to the linear differential equation
The differential operator \(\partial _z^2-t(z)\) can be viewed as a projective connection.
2.3 Generalization to \({{\,\mathrm{\mathrm {SL}}\,}}(N)\): a brief summary
We now give a brief description of the interpretation of the spectrum of the \(\mathfrak {sl}_N\)-Gaudin model in terms of \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-opers.
Definition 2.4
A \({{\,\mathrm{\mathrm {GL}}\,}}(N)\)-oper on \(\mathbb {P}^1\) is a triple \((E,\nabla ,\mathcal {L}_\bullet )\), where E is a rank n vector bundle on \(\mathbb {P}^1\), \(\nabla :E\longrightarrow E\otimes K\) is a connection, and \(\mathcal {L}_\bullet \) is a complete flag of subbundles such that \(\nabla \) maps \(\mathcal {L}_i\) into \(\mathcal {L}_{i+1}\otimes K\) and the induced maps \({\bar{\nabla }}_i:\mathcal {L}_i/\mathcal {L}_{i-1}\longrightarrow \mathcal {L}_{i+1}/\mathcal {L}_i\otimes K\) are isomorphisms for \(i=1,\dots ,N-1\). The triple is called an \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-oper if the structure group of the flat \({{\,\mathrm{\mathrm {GL}}\,}}(N)\)-bundle may be reduced to \({{\,\mathrm{\mathrm {SL}}\,}}(N)\).
As in the \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-case, one can interpret the fact that the \({\bar{\nabla }}_i\)’s are isomorphisms in terms of the nonvanishing of certain determinants involving local sections of \(\mathcal {L}_1\). Given a local section s of \(\mathcal {L}_1\), for \(i=1,\dots ,N\), let
Then \((E,\nabla ,\mathcal {L}_\bullet )\) is an oper if and only if for each z, there exists a local section of \(\mathcal {L}_1\) for which \(\mathcal {W}_i(s)(z)\ne 0\) for all i. Note that \(\mathcal {W}_1(s)\ne 0\) simply means that s locally generates \(\mathcal {L}_1\).
We again will need to relax the isomorphism condition in the above definition to allow the oper to have regular singularities. Recall that the weight lattice for \({{\,\mathrm{\mathrm {SL}}\,}}(N)\) is the free abelian group on the fundamental weights \(\omega _1,\dots ,\omega _{N-1}\). Moreover, a weight is dominant if it is a nonnegative linear combination of the \(\omega _i\)’s.
Fix a collection of points \(z_1,\dots ,z_L\) and corresponding dominant integral weights \(\lambda _1,\dots ,\lambda _L\). Write \(\lambda _m=\sum l_m^i\omega _i\). We say that \((E,\nabla ,\mathcal {L}_\bullet )\) is an \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-oper with regular singularities of weights \(\lambda _1,\dots ,\lambda _L\) at \(z_1,\dots ,z_L\) if \((E,\nabla )\) is a flat \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-bundle, and each of the \({\bar{\nabla }}_i\)’s is an isomorphism except possibly at \(z_m\), where it has a zero of order \(l_m^{i}\), and \(\infty \). The conditions at the singularities may be expressed equivalently in terms of a nonvanishing local section. For each j with \(1\le j \le N-1\), set \(\Lambda _{j} = \prod ^{L}_{m=1}(z-z_m)^{l_m^{j}}\) and \(\ell _m^{j}=\sum _{k=1}^{j} l_m^{k}\). Then, for \(2\le i\le N\),
As we saw for \({{\,\mathrm{\mathrm {SL}}\,}}(2)\), to get the Bethe equations for nondominant weights, we need to introduce Miura opers. Again, a Miura oper is a quadruple \((E,\nabla ,\mathcal {L}_\bullet ,{\hat{\mathcal {L}}}_\bullet )\) where \((E,\nabla ,\mathcal {L}_\bullet )\) is an oper with regular singularities and \({\hat{\mathcal {L}}}_\bullet \) is a complete flag of subbundles preserved by \(\nabla \). Given a Miura oper, choose a trivialization of E on \(\mathbb {P}^1{\setminus }\infty \) such that \({\hat{\mathcal {L}}}_\bullet \) is the standard flag, i.e., the flag generated by the ordered basis \(e_1,\dots , e_N\). If s is a section generating \(\mathcal {L}_1\) on this affine line, consider the following determinants for \(i=1,\dots ,N\):
Each of these is a polynomial multiple of the volume form. Note that \(\mathcal {D}_N(s)(z)=\mathcal {W}_N(s)(z)\); in particular, \(\mathcal {D}_N(s)(z)\ne 0\) away from the \(z_m\)’s. We will call a Miura oper nondegenerate if the orders of the zero of \(\mathcal {D}_i(s)\) and \(\mathcal {W}_i(s)\) at each \(z_m\) are the same and moreover, if \(\mathcal {D}_i(s)\) and \(\mathcal {D}_k(s)\) for \(i\ne k\) both vanish at a point z, then \(z=z_m\) for some m.
These conditions may be expressed in a more Lie-theoretic form. Let B be the upper triangular Borel subgroup of \({{\,\mathrm{\mathrm {SL}}\,}}(N)\). Under the usual identification of \({{\,\mathrm{\mathrm {SL}}\,}}(N)/B\) as the variety of complete flags, B corresponds to the standard flag \(\mathcal {E}\). If \(\mathcal {F}\) is another flag, we say that \((\mathcal {E},\mathcal {F})\) have relative position w (with w an element of the Weyl group \(S_N\)) if \(\mathcal {F}=g\cdot \mathcal {E}\) for some g in the double coset BwB. If the relative position is \(w_0\), where \(w_0\) is the longest element given by the permutation \(i\mapsto N+1-i\) for all i, we say that the flags are in general position.
Given an ordered basis \(f=(f_1,\dots ,f_N)\) for \(\mathbb {C}^N\), let \(Q_k(f)=e_1\wedge \dots \wedge e_{N-k}\wedge f_1\wedge \dots \wedge f_k\). It is immediate that the zeros of the function \(k\mapsto Q_k(f)\) depend only on the flag determined by f. (Of course, \(Q_N(f)\) is always nonzero, since f is a basis.) Let \(\sigma _k=(k\ k+1)\in S_N\).
Lemma 2.5
Let \(\mathcal {F}\) be a flag determined by the ordered basis \(f=(f_1,\dots ,f_N)\).
-
1.
The pair \((\mathcal {E},\mathcal {F})\) are in general position if and only if \(Q_j(f)\ne 0\) for all j.
-
2.
The pair \((\mathcal {E},\mathcal {F})\) have relative position \(w_0 \sigma _k\) if and only if \(Q_k(f)= 0\) and \(Q_j(f)\ne 0\) for all \(j\ne k\).
Proof
In both cases, the forward implication is an easy direct calculation. For example, if \((\mathcal {E},\mathcal {F})\) are in general position, then \(\mathcal {F}=bw_0\mathcal {E}\) for some \(b\in B\), hence is determined by the ordered basis \(f_i=bw_0e_i=be_{N+1-i}\). Since \(Q_j(f)\ne 0\) is equivalent to the fact that the projection of \({{\,\mathrm{span}\,}}(f_1,\dots ,f_j)\) onto \({{\,\mathrm{span}\,}}(e_{N+1-j},\dots ,e_N)\) is an isomorphism, it is now immediate that for \((\mathcal {E},\mathcal {F})\) in general position, \(Q_j(f)\ne 0\) for all j.
For the converse, first assume that \(Q_j(f)\ne 0\) for all j. One shows inductively that the basis f can be modified to give a new ordered basis \({\hat{f}}\) for \(\mathcal {F}\) for which the matrix \(b=\left( {\begin{matrix} {\hat{f}}_N&{\hat{f}}_{N-1}\dots&{\hat{f}}_1 \end{matrix}}\right) \in B\). Thus, \(\mathcal {F}=b w_0\mathcal {E}\).
Now, assume that \(Q_k(f)=0\), but the other \(Q_j(f)\)’s are nonzero. The same argument as above shows that without loss of generality, we may assume that for \(j=1,\dots ,k-1\), \(f_j\) is a column vector with lowest nonzero component in the \(N-j\) place. We may further assume that all other \(f_i\)’s have bottom \(k-1\) components zero. Since \(Q_k(f)=0\), \((f_k)_{N-k}=0\). However, \(Q_{k+1}(f)\ne 0\) now gives \((f_{k+1})_{N-k}\ne 0\) and \((f_k)_{N-k-1}\ne 0\). It is now clear that the flag \(\mathcal {F}\) is determined by an ordered basis \({\hat{f}}\) for which \(b=\left( {\begin{matrix} {\hat{f}}_N&\dots {\hat{f}}_{k}&{\hat{f}}_{k+1}\dots&{\hat{f}}_1 \end{matrix}}\right) \in B\). This means that \(\mathcal {F}= b w_0 w_k\mathcal {E}\). \(\quad \square \)
Returning to our Miura oper, recall that \(s(z),\nabla _z s(z),\dots ,\nabla _z^{N-1} s(z)\) is an ordered basis for the flag \(\mathcal {L}(z)\) as long as z is not a singular point. If we denote this basis by \(\varvec{s}(z)\), we see that \(\mathcal {D}_i(s)(z)=Q_i(\varvec{s}(z))\). The lemma now shows that the fact that the \(\mathcal {D}_i(s)\)’s have no roots in common outside of regular singularities is equivalent to the statement that the relative position of \(({\hat{\mathcal {L}}}_\bullet (z),\mathcal {L}_\bullet (z))\) is either \(w_0\) or \(w_0 \sigma _k\) for some k. Furthermore, \(s(z),(z-z_m)^{-l_m^1}\nabla _z s(z),\dots ,(z-z_m)^{-l_m^{N-1}}\nabla _z^{N-1}s(z)\) is an ordered basis for \(\mathcal {L}_\bullet \) at \(z_m\). Hence, \(\mathcal {D}_i(s)(z)\) and \(\mathcal {W}_i(s)(z)\) having zeros of the same order at \(z_m\) is equivalent to the fact that the flags \({\hat{\mathcal {L}}}_\bullet (z_m)\) and \(\mathcal {L}_\bullet (z_m)\) are in general position.
The determinant conditions for the zeros of \(\mathcal {D}_k(s)\) lead to Bethe equations in the same way as before [F1]:
where the \(w_j\)’s are distinct points corresponding to zeros of the determinants \(\mathcal {D_k}(s)\).
We can now state the \({{\,\mathrm{\mathrm {SL}}\,}}(N)\) analogue of Theorem 2.3. Here, \(\lambda _\infty \) is a dominant weight determined by the \(\lambda _i\)’s and the \(\alpha _{i_j}\)’s.
Theorem 2.6
There is a one-to-one correspondence between the set of solutions to the Bethe ansatz equations (2.13) and the set of nondegenerate \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-Miura opers with trivial monodromy and regular singularities at the points \(z_1, \dots , z_L,\infty \) with weights \(\lambda _1, \dots , \lambda _L,\lambda _\infty \).
2.4 Irregular singularities
In this section, we recall the relationship between opers with irregular as well as regular singularities and an inhomogeneous version of the Gaudin model introduced in [FFTL, FFR2]. Here, we will only consider the simplest case of a double pole irregularity at \(\infty \). We also restrict the discussion to \({{\,\mathrm{\mathrm {SL}}\,}}(2)\).
Let \((E,\nabla ,\mathcal {L})\) be an \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-oper with regular singularities on \(\mathbb {P}^1{\setminus }\infty \) whose underlying connection is gauge equivalent to \(d+\varvec{a}\,dz\), where \(\varvec{a}={{\,\mathrm{diag}\,}}(a,-a)\) with \(a\ne 0\). Changing variables to 1/z, we see that this connection has a double pole at \(\infty \). It is no longer possible to trivialize the connection algebraically, but it can be trivialized using the exponential transformation \(h(z)=e^{\varvec{a}z}\). If we let \(\left( {\begin{matrix} q_{+}(z)\\ q_{-}(z) \end{matrix}}\right) \) be a section generating the line bundle \(\mathcal {L}\) (so \(q_{+}(z)\) and \(q_{-}(z)\) are polynomials with no common zeros), then in the trivial gauge, this section becomes
Note that we cannot assume that \(\deg (q_{-})<\deg (q_{+})\), since the necessary constant gauge changes do not preserve \(d+\varvec{a}\,dz\). However, we can assume that \(q_{-}\) is monic: \(q_{-}(z)=\prod ^{l_{-}}_{i=1}(z-w_i)\).
The condition \(s(z)\wedge \nabla _z s(z)=\rho (z)\) gives a “twisted” form of the Wronskian:
As before, we assume this oper is nondegenerate, i.e., \(q_-(z_m)\ne 0\) for all m; again, this implies that the zeros of \(q_-\) are simple.
To compute the Bethe ansatz equations, we observe that after multiplying (2.14) by \(-e^{-2a z}/(q_-(z))^2\), we obtain
Taking residues at each \(w_i\) now leads to the inhomogeneous Bethe equations
We thus obtain the following theorem:
Theorem 2.7
There is a one-to-one correspondence between the set of solutions of the inhomogeneous Bethe equations (2.16) and the set of nondegenerate \({{\,\mathrm{\mathrm {SL}}\,}}(2)\)-opers with regular singularities at the points \(z_1, \dots , z_L\) of weights \(k_1, \dots , k_L\) at the points \(z_1, \dots , z_L\) and with a double pole with 2-residue \(-\varvec{a}\).
Although we will not state it explicitly, there is a similar result for \({{\,\mathrm{\mathrm {SL}}\,}}(N)\). Applying the methods of Sect. 2.3, one shows that twisted Wronskians arise from nondegenerate \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-opers with regular singularities at fixed finite points and a double pole with regular semisimple 2-residue at infinity. One now obtains solutions of the Bethe equations from these twisted Wronskians.
We remark that for the opers considered in this section, there is no longer an entire flag variety of associated Miura opers. Indeed, the only line bundles \({\hat{\mathcal {L}}}\) preserved by \(d+\varvec{a}\, dz\) are those generated by \(e_1\) and \(e_2\). More generally, consider an \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-oper with underlying connection \(d+A\,dz\), where A is a diagonal matrix with distinct eigenvalues. The flags \({\hat{\mathcal {L}}}_\bullet \) preserved by this connection are precisely those generated by ordered bases obtained by permuting the standard basis. Hence, the associated Miura opers are parameterized by the Weyl group.
3 \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-Opers
3.1 Definitions
We now consider a q-deformation of the set-up in the previous section. It involves a difference equation version of connections and opers.
Fix \(q\in \mathbb {C}^*\). Given a vector bundle E over \(\mathbb {P}^1\), let \(E^q\) denote the pullback of E under the map \(z\mapsto qz\). We will always assume that E is trivializable. Consider a map of vector bundles \(A:E\longrightarrow E^q\). Upon picking a trivialization, the map A is determined by a matrix \(A(z)\in \mathfrak {gl}(N,\mathbb {C}(z))\) giving the linear map \(E_z\longrightarrow E_{qz}\) in the given bases. A change in trivialization by g(z) changes the matrix via
thus, q-gauge change is twisted conjugation. Let \(D_q:E\longrightarrow E^q\) be the operator that takes a section s(z) to s(qz). We associate the map A to the difference equation \(D_q(s)=As\).
Definition 3.1
A meromorphic \(({{\,\mathrm{\mathrm {GL}}\,}}(N),q)\)-connection over \(\mathbb {P}^1\) is a pair (E, A), where E is a (trivializable) vector bundle of rank N over \(\mathbb {P}^1\) and A is a meromorphic section of the sheaf \({{\,\mathrm{Hom}\,}}_{\mathcal {O}_{\mathbb {P}^1}}(E,E^q)\) for which A(z) is invertible, i.e. lies in \({{\,\mathrm{\mathrm {GL}}\,}}(N,\mathbb {C}(z))\). The pair (E, A) is called an \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-connection if there exists a trivialization for which A(z) has determinant 1.
For simplicity, we will usually omit the word ‘meromorphic’ when referring to q-connections.
Remark 3.2
More generally, if G is a complex reductive group, one can define a meromorphic (G, q)-connection over \(\mathbb {P}^1\) as a pair \((\mathcal {G},A)\) where \(\mathcal {G}\) is a principal G-bundle over \(\mathbb {P}^1\) and A is a meromorphic section of \({{\,\mathrm{Hom}\,}}_{\mathcal {O}_{\mathbb {P}^1}}(\mathcal {G},\mathcal {G}^q)\).
Next, we define a q-analogue of opers. In this section, we will restrict to type \(A_1\).
Definition 3.3
A \(({{\,\mathrm{\mathrm {GL}}\,}}(2),q)\)-oper on \(\mathbb {P}^1\) is a triple \((E,A,\mathcal {L})\), where (E, A) is a \(({{\,\mathrm{\mathrm {GL}}\,}}(2),q)\)-connection and \(\mathcal {L}\) is a line subbundle such that the induced map \({\bar{A}}:\mathcal {L}\longrightarrow (E/\mathcal {L})^q\) is an isomorphism. The triple is called an \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-oper if (E, A) is an \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-connection.
The condition that \({\bar{A}}\) is an isomorphism can be made explicit in terms of sections. Indeed, it is equivalent to
for s(z) any section generating \(\mathcal {L}\) over either of the standard affine coordinate charts.
From now on, we assume that q is not a root of unity. We want to define a q-analogue of the opers considered in Sect. 2.4. First, we introduce the notion of a q-oper with regular singularities. Let \(z_1,\dots ,z_L\ne 0,\infty \) be a collection of points such that \(q^\mathbb {Z}z_m\cap q^\mathbb {Z}z_n=\varnothing \) for all \(m\ne n\).
Definition 3.4
A \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-oper with regular singularities at the points \(z_1,\dots , z_L\ne 0, \infty \) with weights \(k_1, \dots k_L\) is a meromorphic \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-oper \((E,A,\mathcal {L})\) for which \({\bar{A}}\) is an isomorphism everywhere on \(\mathbb {P}^1{\setminus }\{0,\infty \}\) except at the points \(z_m\), \(q^{-1}z_m\), \(q^{-2}z_m\), ..., \(q^{-k_m+1}z_m\) for \(m\in \{1, \dots , L\}\), where it has simple zeros.
The second condition can be restated in terms of a section s(z) generating \(\mathcal {L}\) over \(\mathbb {P}^1{\setminus } \infty \): \(s(qz)\wedge A(z) s(z)\) has simple zeros at \(z_m\), \(q^{-1}z_m\), \(q^{-2}z_m\), ..., \(q^{-k_m+1}z_m\) for every \(m\in \{1, \dots , L\}\) and has no other finite zeros.
Next, we define twisted q-opers; these are q-analogues of the opers with a double pole singularity considered in Sect. 2.4. Let \(Z={{\,\mathrm{diag}\,}}(\zeta ,\zeta ^{-1})\) be a diagonal matrix with \(\zeta \ne \pm 1\).
Definition 3.5
A \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-oper \((E,A,\mathcal {L})\) with regular singularities is called a Z-twisted q-oper if A is gauge-equivalent to \(Z^{-1}\).
Finally, we will need the notion of a Miura q-oper. As in the classical case, this is a quadruple \((E,A,\mathcal {L},{\hat{\mathcal {L}}})\) where \((E,A,\mathcal {L})\) is a q-oper and \({\hat{\mathcal {L}}}\) is a line bundle preserved by A.
For the rest of Sect. 3, \((E, A,\mathcal {L})\) will be a Z-twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-connection with regular singularities at \(z_1,\dots , z_L\ne 0, \infty \) having (nonnegative) weights \(k_1, \dots k_L\).
3.2 The quantum Wronskian and the Bethe ansatz
Choose a trivialization for which the q-connection matrix is \(Z^{-1}\). Since \(\mathcal {L}\) is trivial on \(\mathbb {P}^1{\setminus }\infty \), it is generated by a section
where \(Q_+(z)\) and \(Q_-(z)\) are polynomials without common roots. The regular singularity condition on the q-oper becomes an explicit equation for the quantum Wronskian:
We can assume that \(\rho \) is monic, since we can multiply s by a nonzero constant. We are also free to perform a constant diagonal gauge transformation, since this leaves the q-connection matrix unchanged. Thus, we may assume that \(Q_-\) is monic, say \(Q_-(z)=\prod _{i=1}^{l-} (z-w_i)\).
We now restrict attention to nondegenerate q-opers. This means the \(q^\mathbb {Z}\)-lattices generated by the roots of \(\rho \) and \(Q_{-}\) do not overlap, i.e., \(q^\mathbb {Z}z_m\cap q^\mathbb {Z}w_i=\varnothing \) for all m and i. Note that this condition implies that \(w_j\ne qw_i\) for all i, j; if \(w_j=q w_i\), then (3.3) shows that \(w_i\) would be a common zero of \(\rho \) and \(Q_{-}\).
Evaluating (3.3) at \(q^{-1}z\) gives \(\rho (q^{-1}z)=\zeta ^{-1}Q_{+}(q^{-1}z)Q_{-}(z)-\zeta Q_{+}(z)Q_{-}(q^{-1}z)\). If we divide (3.3) by this equation and evaluate at the zeros of \(Q_-\), we obtain the following constraints:
or more explicitly, setting \(k=\sum {k_m}\),
Rewriting this equation, we obtain the \(\mathfrak {sl}_2\) XXZ Bethe equations (see e.g. [R1]):
We call a solution of the Bethe equations nondegenerate if the \(q^\mathbb {Z}\) lattices generated by the \(w_i\)’s and \(z_m\)’s are disjoint for all i and m. We have proven the following theorem:
Theorem 3.6
There is a one-to-one correspondence between the set of nondegenerate solutions of the \(\mathfrak {sl}_2\) XXZ Bethe equations (3.6) and the set of nondegenerate Z-twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-opers with regular singularities at the points \(z_1,\dots , z_L\ne 0, \infty \) with weights \(k_1, \dots k_L\).
3.3 The q-Miura transformation and the transfer matrix
We now consider the q-Miura transformation which puts the q-connection matrix into a form analogous to (2.10) in the classical setting. As we will see, the eigenvalue of the transfer matrix for the XXZ model will appear explicitly in the q-connection matrix.
First, we consider the gauge change by
which takes the section s(z) into \(g(z)s(z)=\left( {\begin{matrix}0\\ 1 \end{matrix}}\right) \). In this gauge, the q-connection matrix has the form
where \(\rho \) is the quantum Wronskian.
Before proceeding, we recall that every eigenvalue of the transfer matrix for the XXZ model has the form (see [R2], Sect. 4 and references therein)
For ease of notation, we set \(a(z)=\zeta ^{-1}Q_-(qz)Q^{-1}_-(z)\), so that \(A(z)=\left( {\begin{matrix}a(z)&{} \rho (z)\\ 0 &{} a^{-1}(z) \end{matrix}}\right) \) and \(T(z)=a(z)\rho (q^{-1}z)+a^{-1}(q^{-1}z)\rho (z)\). We now apply the gauge transformation by the matrix \(\left( {\begin{matrix} 1 &{} 0\\ a(z)/\rho (z) &{} 1 \end{matrix}}\right) \); this brings the q-connection into the form
If \(\left( {\begin{matrix} f_1\\ f_2 \end{matrix}}\right) \) is a solution of the corresponding difference equation, then we have \(D_q(f_1)=\rho (z)f_2\) and \(D_q(f_2)= -\rho ^{-1}(z)f_1+ T(qz)\rho ^{-1}(qz)f_2\). Simplifying, we see that \(f_1\) is a solution of the second-order scalar difference equation
Summing up, we have
Theorem 3.7
Nondegenerate Z-twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-opers with regular singularities at the points \(z_1,\dots , z_n\ne 0, \infty \) with weights \(k_1, \dots k_n\) may be represented by meromorphic q-connections of the form (3.10) or equivalently, by the second-order scalar difference operators (3.11).
3.4 Embedding of the tRS model into q-opers
We now explain a connection between nondegenerate twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-opers and the two particle trigonometric Ruijsenaars–Schneider model. More precisely, we show that the integrals of motion in the tRS model arise from nondegenerate twisted opers with two regular singularities of weight one and with \(Q_{-}\) linear.
Consider Z-twisted opers with two regular singularities \(z_\pm \), both of weight one, so \(\rho =(z-z_+)(z-z_-)\). For generic q, the degree of the quantum Wronskian equals \(\deg (Q_+)+\deg (Q_-)\). Here, we will only look at q-opers for which \(\deg (Q_\pm )=1\), say \(Q_{-}=z-p_{-}\) and \(Q_{+}=c(z-p_{+})\). Here, c is a nonzero constant for which the quantum Wronskian is monic; an easy calculation shows that \(c=q^{-1}(\zeta ^{-1}-\zeta )^{-1}\).
Setting the quantum Wronskian equal to \(\rho \) gives us the equation
Comparing powers of z on both sides, we obtain
Upon introducing coordinates \(\zeta _{+}, \zeta _-\) such that \(\zeta =\zeta _+/\zeta _-\) and viewing \(\zeta _{\pm }, p_{\pm }\) as the positions and momenta in the two particle tRS model, we see that (3.13) are just the trigonometric Ruijsenaars–Schneider equations [KPSZ]. In fact, the set of Z-twisted opers with weight one singularities at \(z_{\pm }\) is just the intersection of two Lagrangian subspaces of the two particle tRS phase space: the subspace determined by (3.13) and the subspace with the \(\zeta _{\pm }\) fixed constants satisfying \(\zeta =\zeta _+/\zeta _-\). As we will see in Sect. 7, this construction can be generalized to higher rank.
4 \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-Opers
4.1 Definitions
We now discuss the generalization of \(({{\,\mathrm{\mathrm {SL}}\,}}(2),q)\)-opers to \({{\,\mathrm{\mathrm {SL}}\,}}(N)\).
Definition 4.1
A \(({{\,\mathrm{\mathrm {GL}}\,}}(N),q)\)-oper on \(\mathbb {P}^1\) is a triple \((E,A,\mathcal {L}_\bullet )\), where (E, A) is a \(({{\,\mathrm{\mathrm {GL}}\,}}(N),q)\)-connection and \(\mathcal {L}_\bullet \) is a complete flag of subbundles such that A maps \(\mathcal {L}_i\) into \(\mathcal {L}_{i+1}^q\) and the induced maps \({\bar{A}}_i:\mathcal {L}_i/\mathcal {L}_{i-1}\longrightarrow \mathcal {L}^q_{i+1}/\mathcal {L}^q_i\) are isomorphisms for \(i=1,\dots ,N-1\). The triple is called an \({{\,\mathrm{\mathrm {SL}}\,}}(N)\)-oper if (E, A) is an \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-connection.
To make this definition more explicit, consider the determinants
for \(i=1,\dots , N\), where s is a local section of \(\mathcal {L}_1\). Then \((E,A,\mathcal {L}_\bullet )\) is a q-oper if and only if at every point, there exists local sections for which each \(\mathcal {W}_i(s)(z)\) is nonzero. It will be more convenient to consider determinants with the same zeros as those in (4.1), but with no q-shifts:
As in the classical setting, we need to relax these conditions to allow for regular singularities. Fix a collection of L points \(z_1,\dots ,z_L\ne 0,\infty \) such that the \(q^\mathbb {Z}\)-lattices they generate are pairwise disjoint. We associate a dominant integral weight \(\lambda _m=\sum l_m^i\omega _i\) to each \(z_m\). Set \(\ell _m^{i}=\sum _{j=1}^{i} l_m^{j}\).
Definition 4.2
An \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-oper with regular singularities at the points \(z_1, \dots , z_L\ne 0,\infty \) with weights \(\lambda _1, \dots \lambda _L\) is a meromorphic \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-oper such that each \({\bar{A}}_i\) is an isomorphism except at the points \(q^{-\ell ^{i-1}_m}z_m, q^{-\ell ^{i-1}_m+1}z_m, \dots , q^{-\ell ^{i}_m+1}z_m\) for each m, where it has simple zeros.
Weight of the singularity \(z_n\) as q-monodromy around the cylinder (\(\mathbb {P}^1\) with 0 and \(\infty \) removed)
In order to express the locations of the roots of the \(\mathcal {W}_i(s)\)’s, it is convenient to introduce the polynomials
with zeros precisely where \({\bar{A}}_i\) is not an isomorphism. We also set
We introduce the notation \(f^{(j)}(z)=D_q^j(f)(z)=f(q^j z)\). The zeros of \(\mathcal {W}_k(s)\) coincide with those of the polynomial
We now define twisted q-opers. Let \(Z={{\,\mathrm{diag}\,}}(\zeta _1,\dots ,\zeta _N)\in {{\,\mathrm{\mathrm {SL}}\,}}(N,\mathbb {C})\) be a diagonal matrix with distinct eigenvalues.
Definition 4.3
An \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-oper \((E,A,\mathcal {L}_\bullet )\) with regular singularities is called a Z-twisted q-oper if A is gauge-equivalent to \(Z^{-1}\).
As in the \({{\,\mathrm{\mathrm {SL}}\,}}(2)\) case, this is a deformed version of opers with irregular singularities that arise in the inhomogeneous version of the Gaudin model introduced in [FFTL, FFR2].
4.2 Miura q-opers and quantum Wronskians
Given a q-oper with regular singularities \((E,A,\mathcal {L}_\bullet )\), we can define the associated Miura q-opers as quadruples \((E,A,\mathcal {L}_\bullet ,{\hat{\mathcal {L}}}_\bullet )\) where \({\hat{\mathcal {L}}}_\bullet \) is a complete flag preserved by the q-connection, i.e., A maps \({\hat{\mathcal {L}}}_i\) into \({\hat{\mathcal {L}}}^q_i\) for all i. Again, we will primarily be interested in nondegenerate Miura q-opers. This means that the flags \((\mathcal {L}_\bullet (z),{\hat{\mathcal {L}}}_\bullet (z))\) are in general position at all but a finite number of points \(\{w_j\}\); moreover, at each \(w_j\), the relative position is \(w_0 \sigma _k\) for some simple reflection \(\sigma _k\). Finally, we assume that \(q^\mathbb {Z}w_i\cap q^\mathbb {Z}w_j=\varnothing \) if \(i\ne j\) and also that the \(q^\mathbb {Z}\) lattices generated by the \(z_m\)’s and \(w_j\)’s do not intersect. (We remark that these last conditions are stronger than necessary; for example, one may instead specify that \(w_j\ne q^i z_m\) for all j and m and for \(|i|\le n\), where n is a positive integer that may be computed explicitly from the weights.)
We now specialize to the case where \((E,A,\mathcal {L}_\bullet )\) is a Z-twisted q-oper. Here, there are only a finite number of possible associated Miura q-opers. Indeed, if we consider the gauge where the matrix of the q-connection is the regular semisimple diagonal matrix \(Z^{-1}\), we see that the only possibilities for \({\hat{\mathcal {L}}}_\bullet \) are the N! flags given by the permutations of the standard ordered basis \(e_1,\dots ,e_N\). (This is analogous to the classical situation. The Miura opers lying above a given oper with regular singularities and trivial monodromy are parametrized by the flag manifold. However, there are only N! Miura opers associated to an oper with regular singularities on \(\mathbb {P}^1{\setminus }\infty \) whose underlying connection if \(d+h\, dz\), where \(h\in \mathfrak {gl}(N,\mathbb {C})\) is regular semisimple.) It suffices to consider Miura q-opers for the standard flag; indeed, if not, we can gauge change to one where \({\hat{\mathcal {L}}}_\bullet \) is the standard flag, but where Z is replaced by a Weyl group conjugate.
Let \(s(z)=(s_1(z),\dots ,s_N(z))\) be a section generating \(\mathcal {L}_1\), where the \(s_a\)’s are polynomials. We now show that the nondegeneracy of the Miura q-oper may be expressed in terms of quantum Wronskians. Consider the zeros of the determinants
for \(k=1,\dots ,N\). The arguments of Sect. 2.3 show that for our q-oper to be nondegenerate, we need the zeros of \(\mathcal {D}_k(s)\) in \(\cup _m q^\mathbb {Z}z_m\) to coincide with those of \(\mathcal {W}_k(s)\). Moreover, we want the other roots of \(\mathcal {D}_k(s)\) to generate disjoint \(q^\mathbb {Z}\) lattices. To be more explicit, for \(k=1,\dots ,N\), we have nonzero constants \(\alpha _k\) and polynomials
for which
moreover \(q^\mathbb {Z}v_{k,a}\) is disjoint from every other \(q^\mathbb {Z}v_{i,b}\) and each \(q^\mathbb {Z}z_m\). Since \(\mathcal {D}_N(s)=\mathcal {W}_N(s)\), we have \(\mathcal {V}_N=1\). We also set \(\mathcal {V}_0=1\); this is consistent with the fact that (4.6) also makes sense for \(k=0\), giving \(\mathcal {D}_0=e_1\wedge \dots \wedge e_N\).
We can also rewrite (4.8) as
where \(i,j = 1,\dots ,k\).
We remark that the nonzero constants \(\alpha _1,\dots ,\alpha _N\) are normalization constants for the section s and may be chosen arbitrarily by first multiplying s by a nonzero constant and then applying constant gauge changes by diagonal matrices in \({{\,\mathrm{\mathrm {SL}}\,}}(N)\).
4.3 \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-Opers and the XXZ Bethe ansatz
We are now ready to state and prove our main theorem which relates twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-opers to solutions of the XXZ Bethe ansatz equations for \(\mathfrak {sl}_N\).
The Bethe equations for the general \(\mathfrak {sl}_N\) XXZ spin chain depend on an anisotropy parameter \(q\in \mathbb {C}^*\) and twist parameters \(\kappa _1,\dots ,\kappa _N\) satisfying \(\prod \kappa _i=1\). The equations can be written in the following form
for \(k=1,\dots N-1\), \(a=1,\dots ,r_k\). (See, for example, [R1].) The constants \(\ell ^i_m\) are determined by the dominant weights \(\lambda _1,\dots ,\lambda _{L}\) as in Sect. 4.1. We use the convention that \(r_0=r_N=0\), so one of the products in the first and last equations is empty.
We remark that there exist many different normalizations of the XXZ Bethe equations in the literature depending on the scaling of the twist parameters. The present normalization is designed to match the formulas obtained from q-opers.
We say that a solution of the Bethe equations is nondegenerate if \(z_s\notin q^{\frac{1-k}{2}}q^{\mathbb {Z}}u_{k,a}\) for all k and a and also that \(u_{k,a}\notin q^{\frac{k-k'}{2}}q^{\mathbb {Z}}u_{k',a'}\) unless \(k=k'\) and \(a=a'\).
Theorem 4.1
Suppose that \(\kappa _1,\dots ,\kappa _N\) generate disjoint \(q^\mathbb {Z}\)-lattices. Then, there is a one-to-one correspondence between nondegenerate solutions of the \(\mathfrak {sl}_N\) XXZ Bethe ansatz equations (4.10) with twist parameters \(\kappa _i\) and nondegenerate Z-twisted \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-opers with regular singularities at \(z_1,\dots ,z_L\) with dominant weight \(\lambda _1,\dots ,\lambda _L\) provided that
for \(k=1,\dots , N\). Moreover, the q-oper equations (4.8) become identical to the Bethe equations if one normalizes the section s via
For the computations to follow, it will be convenient to introduce the Baxter polynomialsFootnote 4
where the normalization constants \(p_k =q^{(\frac{k}{2}-1)\sum _{m=1}^L l_{k}}\) are chosen so that \(\Pi _k=\Lambda ^{\tiny {(\frac{k}{2}}-1)}_{k}\). The Bethe equations (4.10) can then be written as
where we recall that \(f^{(p)}(z)=f(q^p z)\).
We observe that the Baxter polynomials are remarkably similar to the polynomials \(\Lambda _k\) and \(\mathcal {V}_k\) (see (4.3) and (4.7)) which we used to describe the zeros of the quantum Wronskians arising from twisted q-opers. Our main theorem makes this connection precise.
In order to prove the theorem, we will need four lemmas.
Lemma 4.2
Suppose that \(\kappa _{k}\notin q^{\mathbb {N}_0}\kappa _{k+1}\) for all k. Then, the system of equations (4.14) is equivalent to the existence of auxiliary polynomials \({\widetilde{Q}}_k(z)\) satisfying the following system of equations
for \( k= 1,\dots N-1\). Moreover, these polynomials are unique.
Proof
Set \(g(z)=\widetilde{Q}_k(z)/Q_k(z)\) and \(f(z)=(\kappa _{k+1}-\kappa _k)Q_{k-1}^{(\frac{1}{2})}Q_{k+1}^{(\frac{1}{2})}\Pi _k^{(\frac{1}{2})}\), so that (4.15) may be rewritten as
We then have the partial fraction decompositions
where h(z) and \(\tilde{g}_k(z)\) are polynomials. In order for the residues at each \(u_{k,a}\) to match on the two sides of (4.16), one needs
The second equality is merely the Bethe equations (4.14) in the alternate form
or
Next, to solve for the polynomial \(\tilde{g}_k(z)\), set \(\tilde{g}_k(z)=\sum r_iz^i\) and \(h(z)=\sum s_i z^i\). We then obtain the equations \(r_i(\kappa _{k+1}q^i -\kappa _k )=s_i\). Our assumptions on the \(\kappa _j\)’s imply that these equations are always solvable. Thus, there exist polynomials \(\widetilde{Q}_k(z)\) satisfying (4.15) if and only if the Bethe equations hold. The uniqueness statement holds since the solutions for the residues \(d_a\) and the coefficients of the polynomial \(\tilde{g}_k(z)\) are unique. \(\quad \square \)
Lemma 4.3
The system of equations (4.15) is equivalent to the set of equations
for the polynomials
where \(F_k=W^{(\tiny {\frac{1-k}{2}})}_{k}\).
Proof
The \(F_k\)’s are solutions to the functional equation
Indeed, since \(W_k=P^{\tiny {(k-2)}}_{k-1}W_{k-1}\) and \(P_k=\Lambda _k P_{k-1}\), we have
The equivalence of (4.15) and (4.21) follows easily from this fact. \(\quad \square \)
Let \(V(\gamma _1,\dots ,\gamma _k)\) denote the \(k\times k\) Vandermonde matrix \((\gamma _i^j)\). We recall that this determinant is nonzero if and only if the \(\gamma _i\)’s are distinct.
Lemma 4.4
Suppose that \(\gamma _1,\dots ,\gamma _{k-1}\) are nonzero complex numbers such that \(\gamma _j\notin q^{\mathbb {N}_0}\gamma _{k}\) for \(j<k\). Let \(f_1,\dots ,f_{k-1}\) be polynomials that do not vanish at 0, and let g be an arbitrary polynomial. Then there exists a unique polynomial \(f_k\) satisfying
Moreover, if \(g(0)\ne 0\), then \(f_k(0)\ne 0\).
Proof
Set \(f_j(z)=\sum a_{ji} z^i\) and \(g(z)=\sum b_i z^i\), and let F denote the matrix in (4.25). We show that we can solve for the \(a_{ki}\)’s recursively. Expanding by minors along the bottom row, we get \(g=\sum _{j=1}^k (-1)^{k+j}\det F_{k,j}f_k^{(j-1)}\). First, we equate the constant terms. This gives
Since the \(\gamma _j\)’s are distinct, the Vandermonde determinant is nonzero. Moreover, \(a_{j0}\ne 0\) for \(j=1,\dots , k-1\). Thus, we can solve uniquely for \(a_{k0}\). In particular, if \(b_0=0\), then \(a_{k0}=0\).
For the inductive step, assume that we have found unique \(a_{kr}\) for \(r<s\) such that the polynomial equation (4.25) has equal coefficients up through degree \(s-1\). We now look at the coefficient of \(z^s\). The only way that \(a_{ks}\) appears in this coefficient is through the constant terms of the minors \(F_{k,j}\). To be more explicit, equating the coefficient of \(z^s\) in (4.25) expresses \(c a_{ks}\) as a polynomial in known quantities, where
Again, our condition on the \(\gamma _j\)’s implies that the Vandermonde determinant is nonzero, so there is a unique solution for \(a_{ks}\). \(\quad \square \)
In the following lemma, we consider matrices
where \(\mathfrak {q}_1,\dots ,\mathfrak {q}_N\) are polynomials. We also set \(V_{i_1,\dots ,i_j}=V(\kappa _{N+1-i_1},\dots ,\kappa _{N+1-i_j})\).
Lemma 4.5
Assume that the lattices \(q^\mathbb {Z}\kappa _k\) are disjoint for distinct k. Given polynomials \(\mathscr {D}_k, \widetilde{\mathscr {D}}_k\) for \(k=1,\dots ,N-1\) satisfying (4.21), there exist unique polynomials \(\mathfrak {q}_1,\dots ,\mathfrak {q}_N\) such that
For future reference, we note that the first relations from (4.27) can be rewritten as
Proof
We begin by observing that since \(W_k\) and \(Q_k\) do not vanish at 0, \(\mathscr {D}_k(0)\ne 0\) for all k. This implies that \({\widetilde{\mathscr {D}}}_k(0)\ne 0\) for all k as well; otherwise, by (4.21), either \(\mathscr {D}_{k-1}\) or \(\mathscr {D}_{k+1}\) would vanish at 0.
Now, set \(\mathfrak {q}_N=\mathscr {D}_1\) and \(\mathfrak {q}_{N-1}={\widetilde{\mathscr {D}}}_1\). It is obvious that these are the unique polynomials satisfying (4.27) for \(k=1\) and that \(\mathfrak {q}_N(0), \mathfrak {q}_{N-1}(0)\ne 0\). Also, (4.21) gives
so \(\mathscr {D}_2=M_{N-1,N}/V_{N-1,N}\).
Next, suppose that for \(2\le k\le N-1\), we have shown that there exist unique polynomials \(\mathfrak {q}_N,\dots ,\mathfrak {q}_{N-k+1}\) such the formulas for \(\mathscr {D}_j\) (resp. \({\widetilde{\mathscr {D}}}_j\)) in (4.27) hold for \(1\le j\le k\) (resp. \(1\le j\le k-1\)). Furthermore, assume that none of these polynomials vanish at 0. We will show that there exists a unique \(\mathfrak {q}_{N-k}\) such that the formulas for \(\mathscr {D}_{k+1}\) and \({\widetilde{\mathscr {D}}}_{k}\) hold and that \(\mathfrak {q}_{N-k}(0)\ne 0\). This will prove the lemma.
We use Lemma 4.4 to define \(\mathfrak {q}_{N-k}\). In the notation of that lemma, set \(f_j=\mathfrak {q}_{N+1-j}^{(\frac{1-k}{2})}\) and \(\gamma _j=\kappa _j\) for \(1\le j\le k-1\), and set \(g=(-1)^{\frac{k(k-1)}{2}}(\det V_{N-k,N-k+2,\dots ,N}){\widetilde{\mathscr {D}}}_{k}\). (The sign factor occurs because we have written the rows in reverse order to apply the lemma.) By hypothesis, \(f_j(0)\ne 0\) for \(1\le j\le k-1\), so there exists a unique \(f_{k}\) satisfying (4.25). Moreover, \(g(0)\ne 0\), so \(f_{k}\ne 0\). It is now clear that \(\mathfrak {q}_{N-k}=f_{k-1}^{(\frac{k-1}{2})}\) is the unique polynomial satisfying the formula in (4.27) for \({\widetilde{\mathscr {D}}}_{k}\). Of course, \(\mathfrak {q}_{N-k}\ne 0\).
To complete the inductive step, it remains to show that the formula for \(\mathscr {D}_{k+1}\) is satisfied. We make use of the Desnanot-Jacobi/Lewis Carroll identity for determinants. Given a square matrix M, let \(M^i_j\) denote the square submatrix with row i and column j removed; similarly, let \(M^{i,i'}_{j,j'}\) be the submatrix with rows i and \(i'\) and columns j and \(j'\) removed. We will apply this identity in the form
Set \(M=M_{N-k,\dots ,N}\). All of the matrices appearing in (4.30) are obtained from matrices of the form (4.26) via q-shifts, multiplication of each row by an appropriate \(\kappa _i\), or both. In particular, \(M^1_{k+1}=M_{N-k+1,\dots ,N}^{(-\frac{1}{2})}\) and \(M^{2}_{k+1}=M_{N-k,N-k+2,\dots ,N}^{(-\frac{1}{2})}\) while the determinants of the other three are given by
Upon substituting into (4.30) and dividing by \(\prod _{j=1}^{k-1}\kappa _j\), we obtain
Finally, dividing both sides by \(V_{N-k+1,\dots ,N}V_{N-k,N-k+2,\dots ,N}\) and applying the inductive hypothesis gives (4.22) multiplied by \(-1\). This is obvious for the left-hand sides. To see that the other sides match, one need only observe that
\(\square \)
We are finally ready to prove Theorem 4.1.
Proof of Theorem 4.1
We have shown that a solution to the Bethe equations is uniquely determined by polynomials \(\mathfrak {q}_k\) satisfying (4.28). We will show that after matching the parameters as in the statement and normalizing the section s(z) generating the q-oper, the components \(s_k\) also satisfy these equations, so \(s_k=\mathfrak {q}_k\) for all k. Since the twisted q-oper is uniquely determined by s, we obtain the desired correspondence.
After shifting (4.28) by \(\frac{k-1}{2}\) and using the definition of \(\mathscr {D}_k\) from (4.22), we obtain the equivalent form
On the other hand, rewriting the q-oper relations (4.9) for convenience, we have
If we set \(q^{\frac{1-k}{2}}u_{k,a} = v_{k,a}\), then the roots of \(\mathcal {V}\) and \(Q_k^{(\frac{k-1}{2})}\) coincide; moreover, the leading terms of the polynomials on the right are the same if one takes \(\alpha _k = q^{\frac{k-1}{2}r_k}\det V(\kappa _k,\dots ,\kappa _1)\). Thus, if one sets \(\zeta _{k} = \kappa _{N+1-k}\), the two equations are identical.
It only remains to observe that the notions of nondegeneracy are preserved by the transformation (4.11). \(\quad \square \)
5 Explicit Equations for \(({{\,\mathrm{\mathrm {SL}}\,}}(3),q)\)-Opers
5.1 A canonical form
In this section, we illustrate the general theory in the case of \({{\,\mathrm{\mathrm {SL}}\,}}(3)\). In particular, we show that the underlying q-connection can be expressed entirely in terms of the Baxter polynomials and the twist parameters.
We start in the gauge where the connection is given by the diagonal matrix \({{\,\mathrm{diag}\,}}(\zeta _1^{-1},\zeta _2^{-1},\zeta _3^{-1})\) and the section generating the line bundle \(\mathcal {L}_1\) is \(s=(s_1,s_2,s_3)\). We now apply a q-gauge change by a certain matrix g(z) mapping s to the standard basis vector \(e_3\):
where \(\alpha (z)=\zeta ^{-1}_1 s_1^{(-1)}s_2-\zeta ^{-1}_2 s_1 s_2^{(-1)}\) and \(\beta (z)=\frac{1}{s_2}(\zeta ^{-1}_2 s_2^{(-1)} s_3 -\zeta ^{-1}_3 s_2 s_3^{(-1)})\). Applying the q-change formula (3.1) leads to a matrix all of whose entries are expressible in terms of minors of the matrix
By (4.8), the relations between the Baxter polynomials and these determinants are given by
A further diagonal q-gauge change by \({{\,\mathrm{diag}\,}}(\alpha _3^{-2/3}(\Lambda ^{(-1)}_1)^{-1},\alpha _3^{1/3}\Lambda ^{(-1)}_1,\alpha _3^{1/3})\) brings us to our desired form:
where
Note that the singularities of the oper and the Bethe roots can be determined from the zeros of the superdiagonal and the diagonal respectively.
5.2 Scalar difference equations and eigenvalues of transfer matrices
The first-order system of difference equations \(f(qz)=A(z)f(z)\) determined by (5.4) can be expressed as a third-order scalar difference equation. This is accomplished by the q-Miura transformation: a q-gauge change by a lower triangular matrix which reduces A(z) to companion matrix form. (This procedure appears as part of the difference equation version of Drinfeld-Sokolov reduction introduced in [FRSTS, FR, SS].)
In the XXZ model, the eigenvalues of the \({{\,\mathrm{\mathrm {SL}}\,}}(3)\)-transfer matrices for the two fundamental weights are [BHK, FH1]
Just as for \({{\,\mathrm{\mathrm {SL}}\,}}(2)\), these eigenvalues appear in the coefficients of the scalar difference equation associated to our twisted q-oper. Indeed, a simple calculation shows that the system
is equivalent to
6 Scaling Limits: From q-Opers to Opers
In this section, we consider classical limits of our results. We will take the limit from q-opers to opers in two steps. The first will give rise to a correspondence between the spectra of a twisted version of the XXX spin chain and a twisted analogue of the discrete opers of [MV2]. By taking a further limit, we recover the relationship between opers with an irregular singularity and the inhomogeneous Gaudin model [FFTL, FFR2].
First, we introduce an exponential reparameterization of q, the singularities, and the Bethe roots: \(q=e^{R\upepsilon }\), \(z_s = e^{R \upsigma _s}\), and \(v_{k,a}= e^{R \upupsilon _{k,a}}\). We also set \({\tilde{\ell }}_s^k = \ell _s^k +\frac{k}{2}-\frac{3}{2}\). We now take the limit of the XXZ Bethe equations (4.10) as R goes to 0. This limit brings us to the XXX Bethe equations
Geometrically, we identify \(\mathbb {C}^*\) with an infinite cylinder of radius \(R^{-1}\) and view this cylinder as the base space of our twisted q-oper. We then send the radius to infinity, thereby arriving at a twisted version of the discrete opers of Mukhin and Varchenko [MV2].
The second limit takes us from the XXX spin chain to the Gaudin model. In order to do this, we set \(\kappa _i=e^{\upepsilon \upkappa _i}\), and let \(\upepsilon \) go to 0. As expected, we obtain the Bethe equations for the inhomogeneous Gaudin model, i.e., the higher rank analogues of (2.16):
Note that the difference of the twists \(\upkappa _i\) can be identified with the monodromy data of the connection A(z) at infinity.
We have thus established the following hierarchy between integrable spin chain models and oper structures.
We are currently working on extending this picture to include the XYZ spin chain in this hierarchy. We expect that the appropiate difference opers to consider are “twisted elliptic opers” on elliptic curves.
7 Quantum K-Theory of Nakajima Quiver Varieties and q-Opers
7.1 The quantum K-theory ring for partial flag varieties
As we discussed in the introduction, integrable models play an important role in enumerative geometry. For example, consider the XXZ spin chain for \(\mathfrak {sl}_N\) where the dominant weights at the marked points \(z_m\) all correspond to the defining representation, i.e., \(\lambda _m=(1,0,0,\dots , 0)\). Recall that cotangent bundles to partial flag varieties are particular case of quiver varieties of type A (see Fig. 2).
The cotangent bundle to the partial flag variety \(T^*\mathbb {F}l_{\varvec{\mu }}\)
It follows from work of Nakajima [N1] that the space of localized equivariant K-theory of such a cotangent bundle can be identified with an appropriate weight space in the corresponding XXZ model; moreover, the span of all such weight spaces for partial flag varieties of \({{\,\mathrm{\mathrm {SL}}\,}}(N)\) is endowed with a natural action of the quantum group \(U_{q}(\mathfrak {sl}_N)\).
In [KPSZ], it was established that the Bethe algebra for this XXZ model—the algebra generated by the Q-operators of the XXZ spin chain—can be entirely described in terms of enumerative geometry. The equivariant quantum K-theoryFootnote 5 of the cotangent bundle to a partial flag variety has generators which are quantum versions of tautological bundles. It is shown in [KPSZ] that the eigenvalues of these quantum tautological bundles are the symmetric functions in the Bethe roots, so that the quantum K-theory may be identified with the Bethe algebra. Moreover, the twist parameters \(\kappa _{i+1}/\kappa _{i}\) and the inhomogeneity (or evaluation) parameters \(z_m\) are identified with the Kähler parameters of the quantum deformation and the equivariant parameters respectively.
In the case of complete flag varieties, the authors of [KPSZ] found another set of generators which allows the identification of the quantum K-theory ring with the algebra of functions on a certain Lagrangian subvariety in the phase space for the trigonometric Ruijsenaars–Schneider model. The formulas used to establish this (see Proposition 4.4 of [KPSZ]) are strikingly similar to the equations (4.8) describing nondegenerate twisted q-opers. Let us normalize the section s(z) in the definition of a twisted q-oper so that all of its components are monic polynomials:
If we restrict to the space of q-opers for which all these polynomials have degree one, then their roots may be viewed as coordinates. These coordinates may be identified with the momenta of the dual tRS model whereas the coordinates of the tRS model correspond bijectively to the twist (Kähler) parameters \(\kappa _{i+1}/\kappa _{i}\) [KPSZ].
We remark that that the Bethe algebra of the XXX model has a similar enumerative description: it is the equivariant quantum cohomology ring of the cotangent bundle of a partial flag variety [GRTV]. This led Rimanyi, Tarasov and Varchenko to conjecture the analogous statement for the XXZ Bethe algebra [RTV, Conjecture 13.15]. Moreover, they found a particular set of generators for the Bethe algebra, involving determinantal formulas, which for complete flag varieties can be identified with the generators for the quantum K-theory ring discussed in the previous paragraph.
Their generators can be given a geometric interpretation as coordinates on an appropriate space of q-opers, and this leads to the following description of the quantum K-theory ring:
Theorem 7.1
Let X be the cotangent bundle of the \({{\,\mathrm{\mathrm {GL}}\,}}(L)\) partial flag variety \(T^*\mathbb {F}l_{\varvec{\mu }}\) labeled by the vector \(\varvec{\mu }=(r_{N-1}-r_{N-2},\dots ,r_1-r_2, L-r_{1})\) where \(r_1,\dots ,r_{N-1}\) and L are the dimensions of the vector spaces corresponding to the nodes of the \(A_{N-1}\) quiver and the framing on the first node in Fig. 2 respectively. Let T be a maximal torus in \({{\,\mathrm{\mathrm {GL}}\,}}(L)\).
Then the T-equivariant quantum K-theory of X is given by the algebra
where \(\varvec{\kappa } = (\kappa _1,\dots ,\kappa _N)\) are the quantum deformation parameters, \(\varvec{a}=(a_1,\dots ,a_L)\) are the equivariant parameters of the action of T on X,
are the coefficients of the polynomials
where \(\rho _k=r_k-r_{k-1}\), \(\Pi (z) = \prod \limits _{s=1}^L (z-a_s)\), and the matrix M is given by
The ideal in (7.2) depends on the auxiliary variable z, and both sides of the equation are polynomials of degree L in z. Thus, the quantum K-theory ring is determined by L relations.
The case where X is a complete flag variety, so that \(L = N\) and \(\rho _1=\dots =\rho _{N-1}=1\), was investigated in [KPSZ]. Here, the determinantal relation in (7.2) yields the equations of motion of the N-body trigonometric Ruijsenaars–Schneider model.
We would like to emphasize that the space of q-opers which is described by the system of equations (4.8) contains the K-theory of X (7.2) as a subspace. In particular, one identifies the singularities \(z_1,\dots ,z_L\) of the q-oper with the equivariant parameters \(a_1,\dots , a_L\) of the action of the maximal torus of \({{\,\mathrm{\mathrm {GL}}\,}}(L)\) on X, so that \(\Pi =W_{N}=\Lambda _1\). (For \(s>1\), \(l^k_s=0\), so \(\Lambda _s=1\).)
Proof
We prove this by combining two theorems. First, we will use Theorem 3.4 in [KPSZ], where the quantum K-theory of Nakajima quiver varieties was defined using quasimaps [CFKM, O] from the base curve of genus zero to the quiver variety. The second ingredient is Theorem 4.1 from this paper in the special case when the dominant weights at all oper singularities correspond to the defining representation, so that \(l_s^1=1\) for all s and the other \(l_s^k\) vanish. Here, the Bethe ansatz equations (4.10) are given by
where \(r_k=\rho _1+\dots +\rho _k\) and k runs from 2 to \(N-2\) in the middle equation.
The system (7.6) coincides with the Bethe equations from Theorem 3.4 of [KPSZ] up to the identification of Bethe roots and twists. This latter set of Bethe equations describes the relations in the quantum K-theory of X, where the Bethe roots \(v_{k,a}\) are the Chern roots of the k-th tautological bundle over X and the other variables are identified with the geometry of X as in the statement of the theorem.
We have proven in Theorem 4.1 that equations (7.6) can be written as (4.28). For \(k=N\) and for the dominant weights above, this gives
This statement completes the proof. \(\quad \square \)
7.2 The trigonometric RS model in the dual frame
The trigonometric Ruijsenaars–Schneider model enjoys bispectral duality. This may be described in geometric language as follows. For a given quiver variety X of type A, there are two dual realizations of the tRS model. The first was explained for \({{\,\mathrm{\mathrm {SL}}\,}}(2)\) in Sect. 3.4. Here, the twist variables \(\varvec{\kappa }\) play the role of particle positions; their conjugate momenta \(p_{\varvec{\kappa }}=(p_{\kappa _1},\dots ,p_{\kappa _N})\) are defined as
where \(\mathcal {Y}\) is the so-called Yang–Yang function which depends on the Bethe roots \(v_{k,a}\) as well as all other parameters. The Yang–Yang function serves as a potential for equations (7.6) [NS2, NS1], i.e., the k-th equation is given by
(See [GK, BKK, KPSZ] for more details.)
The other realization—the 3d-mirror or spectral/symplectic dual description—involves a mirror quiver variety \(X^\vee \) and the associated dual Yang–Yang function \(\mathcal {Y}^\vee \). (For a mathematical introduction, see [N2]. The construction of the mirror is discussed in [GK].) Under the mirror map, the Kähler parameters \(\varvec{\kappa }\) are interchanged with the equivariant parameters \(\varvec{a}\); the same holds for the conjugate momenta \(p_{\varvec{\kappa }}\) and \(p_{\varvec{a}}\). Therefore, the variables \(\varvec{a},p_{\varvec{a}}\) can be viewed as the canonical degrees of freedom in the dual tRS model; this has been studied in the context of enumerative geometry in [KZ, BLZZ]. In particular, such a duality was demonstrated between the XXZ spin chain whose Bethe equations describe the equivariant quantum K-theory of the quiver variety from Fig. 2 and the L-body tRS model whose coordinates are the equivariant parameters \((a_1,\dots ,a_L)\) of the maximal torus for \({{\,\mathrm{\mathrm {GL}}\,}}(L)\). This result allows us to construct a natural embedding of the intersection of two Lagrangian cycles inside the tRS phase space into the space of q-opers with the first fundamental weight at each regular singularity.
Notes
We remark that the Gaudin algebra is a subalgebra of the center of the universal enveloping algebra at the critical level, and this center was characterized in [FFR1]. Ding and Etingof have found generators of the center of the quantum affine algebra at the critical level, thereby proving the q-analogue of this statement [DE].
The authors and E. Frenkel have subsequently solved this problem in the general case using different techniques [FKSZ].
We refer the reader to Sect. 4 of [GK] for more information on quantum/classical duality and additional references.
The terminology comes from the analogy between these polynomials and the polynomials determining the eigenvalues of the Baxter operators arising from transfer matrices [R2].
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Acknowledgements
P.K. and A.M.Z. are grateful to the 2018 Simons Summer Workshop for providing a wonderful working atmosphere in the early stages of this Project. P.K. thanks the organizers of the program “Exactly Solvable Models of Quantum Field Theory and Statistical Mechanics” at the Simons Center for Geometry and Physics, where part of this work was done. We are also indebted to N. Nekrasov, A. Schwarz, and Y. Soibelman for stimulating discussions and suggestions. D.S.S. was partially supported by NSF Grant DMS-1503555 and Simons Collaboration Grant No. 637367, and A.M.Z. was partially supported by Simons Collaboration Grant No. 578501. P.K. is partly supported by an AMS-Simons Grant.
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Koroteev, P., Sage, D.S. & Zeitlin, A.M. \(({{\,\mathrm{\mathrm {SL}}\,}}(N),q)\)-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality. Commun. Math. Phys. 381, 641–672 (2021). https://doi.org/10.1007/s00220-020-03891-1
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DOI: https://doi.org/10.1007/s00220-020-03891-1