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REMARKS on the GAUDIN MODEL MODULO P ⋆Department Of REMARKS ON THE GAUDIN MODEL MODULO p ALEXANDER VARCHENKO ? ?Department of Mathematics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250, USA Abstract. We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional sl2-modules over the field Fp with p elements, where p is a prime number. 0 0 We define the Bethe ansatz equations and show that if (t1; : : : ; tk) is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin 0 0 Hamiltonians. We characterize solutions (t1; : : : ; tk) of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials Fp[x]. We consider the case when the number of parameters k equals 1. In that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation. If k = 1 and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map. In memory of Egbert Brieskorn (1936{2013) Contents 1. Introduction 2 2. sl2 Gaudin model 2 2.1. sl2 Gaudin model over C 2 2.2. Irreducible sl2-modules 3 2.3. Bethe ansatz on SingL⊗mjmj − 2k over C 4 2.4. Proof of Theorem 2.1 5 ⊗m 2.5. Bethe ansatz on SingL jmj − 2k over Fp 6 3. Two-dimensional spaces of polynomials 6 3.1. Two-dimensional spaces of polynomials over C 6 3.2. Two-dimensional spaces of polynomials over Fp 8 4. Example: the case k = 1 10 4.1. Gaudin model on SingL⊗m[jmj − 2] 10 4.2. Bethe ansatz equation and algebra A(z0; m) 11 4.3. Isomorphism of A(z0; m) and B(z0; m) 12 4.4. Eigenvectors of B(z0; m) and the polynomial P (t) 13 4.5. Algebra C(T ) 14 4.6. Wronski map 15 References 15 ?E-mail: [email protected], supported in part by NSF grant DMS-1665239 2 ALEXANDER VARCHENKO 1. Introduction The Gaudin model is a certain collection of commuting linear operators on the tensor n product V = ⊗i=1Vi of representations of a Lie algebra g. The operators are called the Gaudin Hamiltonians. The Bethe ansatz is a method used to construct common eigenvectors and eigenvalues of the Gaudin operators. One looks for an eigenvector in a certain form W (t), where W (t) is a V -valued function of some parameters t = (t1; : : : ; tk). One introduces a system of equations on the parameters, called the Bethe ansatz equations, and shows that if t0 is a solution of the system, then the vector W (t0) is an eigenvector of the Gaudin Hamiltonians, see for example [B, G, FFR, MV1, MV2, MTV1, MTV4, RV, SchV, SV1, V1, V2, V3]. The Gaudin model has strong relations with the Schubert calculus and real algebraic geometry, see for example [MTV2, MTV3, So]. All that is known in the case when the Lie algebra g is defined over the field C of complex numbers. In the paper we consider the case of the field Fp with p elements, where p is a prime number, cf. [SV3]. We carry out the first steps of the Bethe ansatz, the deeper parts of the Gaudin model over a finite field remain to be developed. We consider the case of the Lie algeba sl2, where the notations and constructions are shorter and simpler. It is known that over C, the Gaudin model is a semi-classical limit of the KZ differential equations of conformal field theory, and the construction of the multidimensional hyperge- ometric solutions of the KZ differential equations leads, in that limit, to the Bethe ansatz construction of eigenvectors of the Gaudin Hamiltonians, see [RV]. The Fp-analogs of the hypergeometric solutions of the KZ differential equations were constructed recently in [SV3], see also [V5]. Thus the constructions of this paper may be thought of as a semi-classical limit of the constructions in [SV3]. 0 0 In Section 2 we define the Bethe ansatz equations and show that if (t1; : : : ; tk) is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the 0 0 Gaudin Hamiltonians. In Section 3 we characterize solutions (t1; : : : ; tk) of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials Fp[x]. In Section 4 we consider the case in which the number k of the parameters equals 1. In that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation, see Theorem 4.2. If k = 1 and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map, see Corollary 4.9. The author thanks W. Zudilin for useful discussions and the Hausdorff Institute for Math- ematics in Bonn for hospitality in May-July of 2017. 2. sl2 Gaudin model 2.1. sl2 Gaudin model over C. Let e; f; h be the standard basis of the complex Lie algeba sl2 with [e; f] = h,[h; e] = 2e,[h; f] = −2f. The element 1 (2.1) Ω = e ⊗ f + f ⊗ e + h ⊗ h 2 sl ⊗ sl 2 2 2 (i;j) ⊗n is called the Casimir element. Given n, for 1 6 i < j 6 n let Ω 2 (U(sl2)) be the element equal to Ω in the i-th and j-th factors and to 1 in other factors. Let z0 = REMARKS ON THE GAUDIN MODEL MODULO p 3 0 0 Cn (z1; : : : ; zn) 2 have distinct coordinates. For s = 1; : : : ; n introduce X Ω(s;l) (2.2) H (z0) = 2 (U(sl ))⊗n; s z0 − z0 2 l6=s s l the Gaudin Hamiltonians, see [G]. For any s; l, we have 0 0 (2.3) Hs(z );Hl(z ) = 0; and for any x 2 sl2 and s we have 0 (2.4) [Hs(z ); x ⊗ 1 ⊗ · · · ⊗ 1 + ··· + 1 ⊗ · · · ⊗ 1 ⊗ x] = 0: n Let V = ⊗i=1Vi be a tensor product of sl2-modules. The commutative subalgebra of 0 End(V ) generated by the Gaudin Hamiltonians Hi(z ), i = 1; : : : ; n, and the identity opera- tor Id is called the Bethe algebra of V . If W ⊂ V is a subspace invariant with respect to the Bethe algebra, then the restriction of the Bethe algebra to W is called the Bethe algebra of W , denoted by B(W ). The general problem is to describe the Bethe algebra, its common eigenvectors and eigen- values. 2.2. Irreducible sl2-modules. For a nonnegative integer i denote by Li the irreducible i k k i + 1-dimensional module with basis vi; fvi; : : : ; f vi and action h:f vi = (i − 2k)f vi for k = k k+1 i k k−1 0; : : : ; i; f:f vi = f vi for k = 0; : : : ; i − 1, f:f vi = 0; e:vi = 0, e:f vi = k(i − k + 1)f vi for k = 1; : : : ; i. Zn ⊗m For m = (m1; : : : ; mn) 2 >0, denote jmj = m1 + ··· + mn and L = Lm1 ⊗ · · · ⊗ Lmn . Zn 6 For J = (j1; : : : ; jn) 2 >0, with js ms for s = 1; : : : ; n, the vectors j1 jn (2.5) fJ vm := f vm1 ⊗ · · · ⊗ f vmn form a basis of L⊗m. We have n X f:fJ vm = fJ+1s vm; h:fJ vm = (jmj − 2jJj)fJ vm; s=1 n X e:fJ vm = js(ms − js + 1)fJ−1s vm: s=1 For λ 2 Z, introduce the weight subspace L⊗m[λ] = f v 2 L⊗m j h:v = λvg and the singular weight subspace SingL⊗m[λ] = f v 2 L⊗m[λ] j h:v = λv; e:v = 0g. We have the weight ⊗m jmj ⊗m decomposition L = ⊕k=0L [jmj − 2k]. Denote Zn 6 Ik = fJ 2 >0 j jJj = k; js ms; s = 1; : : : ; ng: ⊗m The vectors (fJ v)J2Ik form a basis of L [jmj − 2k]. The Bethe algebra B(L⊗m) preserves each of the subspaces L⊗m[jm|−2k] and SingL⊗m[jm|− ⊗m 2k] by (2.4). If w 2 L is a common eigenvector of the Bethe algebra, then for any x 2 sl2 the vector x:w is also an eigenvector with the same eigenvalues. These observations show that in order to describe B(L⊗m), its eigenvectors and eigenvalues it is enough to describe for all k the algebra B(SingL⊗m[jmj − 2k]), its eigenvectors and eigenvalues. 4 ALEXANDER VARCHENKO ⊗m 2.3. Bethe ansatz on SingL jmj − 2k over C. Given k; n 2 Z>0, m = (m1; : : : ; mn) 2 Zn 0 0 0 Cn >0. Let z = (z1; : : : ; zn) 2 have distinct coordinates. The system of the Bethe ansatz equations is the system of equations n X 2 X ms (2.6) − 0 = 0; i = 1; : : : ; k; ti − tj ti − z j6=i s=1 s Ck 0 0 0 0 Ck+n on t = (t1; : : : ; tk) 2 . If (t1; : : : ; tk; z1; : : : ; zn) 2 p has distinct coordinates, denote k X msml=2 X ms (2.7) λ (t0; z0) = − ; s = 1; : : : ; n: s z0 − z0 z0 − t0 l6=s s l i=1 s i For any function or differential form F (t1; : : : ; tk), denote X X jσj Symt[F (t1; : : : ; tk)] = F (tσ1 ; : : : ; tσk ); Antt[F (t1; : : : ; tk)] = (−1) F (tσ1 ; : : : ; tσk ): σ2Sk σ2Sk For J = (j1; : : : ; jn) 2 Ik define the weight function " n js # 1 Y Y 1 (2.8) W (t; z) = Sym : J j ! : : : j ! t t − z 1 n s=1 i=1 j1+···+js−1+i s For example, 1 1 1 W(1;0;:::;0) = ;W(2;0;:::;0) = ; t1 − z1 t1 − z1 t2 − z1 1 1 1 1 W(1;1;0;:::;0) = + : t1 − z1 t2 − z2 t2 − z1 t1 − z2 The function X (2.9) Wk;n;m(t; z) = WJ (t; z)fJ vm J2Ik is the L⊗m[jmj − 2k]-valued vector weight function.
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