Remarks on the Gaudin Model Modulo $ P$

arXiv:1708.06264v4 [math.AG] 22 Feb 2018 ⋆ E-mail .Introduction 1. ..Wosimap References Wronski 4.6. Algebra of 4.5. Eigenvectors algebra of and 4.4. Isomorphism equation ansatz Bethe Sing 4.3. on model Gaudin 4.2. case the 4.1. over Example: polynomials of spaces 4. over Two-dimensional polynomials of spaces 3.2. Two-dimensional polynomials of spaces 3.1. Two-dimensional Sing on ansatz 3. Bethe Theorem of 2.5. Proof Sing on ansatz 2.4. Bethe 2.3. Irreducible 2.2. 2.1. 2. sl [email protected],spotdi atb S rn DMS-1665239 grant NSF by part in supported [email protected], : h adnHmloin,i smrhct h ler ffntoso t on functions of If algebra equation. the ansatz to Bethe isomorphic the by is Hamiltonians, Gaudin the ntefie fasial rnk map. Wronski suitable a isomorphic of also fiber is the algebra on Bethe the then representations, vector finite-dimensional Abstract. ubro parameters of number w-iesoa usae ftesaeo polynomials of space the of subspaces two-dimensional edfieteBteast qain n hwta f( if that show and equations ansatz Bethe the define We nazeutos hntecrepnigBtevco sa eige an is vector Bethe ( solutions corresponding characterize We the Hamiltonians. then equations, ansatz 2 sl ⋆ adnmodel Gaudin 2 eateto ahmtc,Uiest fNrhCrln at Carolina North of University Mathematics, of Department adnmdlover model Gaudin EAK NTEGUI OE MODULO MODEL GAUDIN THE ON REMARKS C ( T edsusteBteast nteGui oe ntetno prod tensor the on model Gaudin the in ansatz Bethe the discuss We sl ) 2 -modules sl B A k 2 ( ( mdlsoe h field the over -modules z 2.1 z 1 = 0 k 0 m , m , L L qas1 nta aew hwta h eh ler,generated algebra, Bethe the that show we case that In 1. equals hplHl,N 79-20 USA 27599-3250, NC Hill, Chapel ⊗ ⊗ C L n h polynomial the and ) and ) m m ⊗ LXNE VARCHENKO ALEXANDER m | | m m [ | k m − | − | B n nadto h esrpouti h rdc of product the is product tensor the addition in and 1 = − | ( z 2 2 0 k k m , 2] Contents A over over t 1 0 ) ( t , . . . , z F 0 p m , F C nmmr fEbr rekr (1936–2013) Brieskorn Egbert of memory In with p k 0 ) F C fteBteast qain scertain as equations ansatz Bethe the of ) P p p ( t lmns where elements, F ) t p 1 0 [ x t , . . . , .W osdrtecs hnthe when case the consider We ]. ⋆ k 0 saslto fteBethe the of solution a is ) oteagbao functions of algebra the to vco fteGaudin the of nvector p sapienumber. prime a is eshm defined scheme he hplHill Chapel p c of uct by 15 15 13 13 12 11 10 10 5 8 6 6 6 4 3 2 2 2 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 hypergeometric solutions of the KZ differential equations lead, 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 ∈ sl ⊗ sl 2 2 2 (i,j) ⊗n is called the Casimir element. Given n, for 1 6 i < j 6 n let Ω ∈ (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) ∈ have distinct coordinates. For s =1,...,n introduce Ω(s,l) (2.2) H (z0)= ∈ (U(sl ))⊗n, s z0 − z0 2 l s s l X6= the Gaudin Hamiltonians, see [G]. For any s,l, we have 0 0 (2.3) Hs(z ),Hl(z ) =0,

and for any x ∈ 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 operator 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 eigenvalues.

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) ∈ >0, denote |m| = m1 + ··· + mn and L = Lm1 ⊗···⊗ Lmn . Zn For J =(j1,...,jn) ∈ >0, with js 6 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

f.fJ vm = fJ+1s vm, h.fJ vm =(|m| − 2|J|)fJ vm, s=1 X n

e.fJ vm = js(ms − js + 1)fJ−1s vm. s=1 X For λ ∈ Z, introduce the weight subspace L⊗m[λ]= { v ∈ L⊗m | h.v = λv} and the singular weight subspace SingL⊗m[λ] = { v ∈ L⊗m[λ] | h.v = λv, e.v = 0}. We have the weight ⊗m |m| ⊗m decomposition L = ⊕k=0L [|m| − 2k]. Denote Zn Ik = {J ∈ >0 | |J| = k, js 6 ms, s =1,...,n}. ⊗m The vectors (fJ v)J∈Ik form a basis of L [|m| − 2k]. The Bethe algebra B(L⊗m) preserves each of the subspaces L⊗m[|m|−2k] and SingL⊗m[|m|− ⊗m 2k] by (2.4). If w ∈ L is a common eigenvector of the Bethe algebra, then for any x ∈ 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[|m| − 2k]), its eigenvectors and eigenvalues. 4 ALEXANDER VARCHENKO

⊗m 2.3. Bethe ansatz on SingL |m| − 2k over C. Given k, n ∈ Z>0, m =(m1,...,mn) ∈ Zn 0 0 0 Cn >0. Let z = (z1 ,...,zn) ∈ have distinct coordinates. The system of the Bethe ansatz equations is the system of equations n 2 ms (2.6) − 0 =0, i =1, . . . , k, ti − tj ti − z j i s=1 s X6= X Ck 0 0 0 0 Ck+n on t =(t1,...,tk) ∈ . If(t1,...,tk, z1 ,...,zn) ∈ p has distinct coordinates, denote k m m /2 m (2.7) λ (t0, z0)= s l − s , s =1, . . . , n. s z0 − z0 z0 − t0 l s s l i=1 s i X6= X For any function or diﬀerential form F (t1,...,tk), denote

|σ| Symt[F (t1,...,tk)] = F (tσ1 ,...,tσk ), Antt[F (t1,...,tk)] = (−1) F (tσ1 ,...,tσk ). σ S σ S X∈ k X∈ k For J =(j1,...,jn) ∈ Ik deﬁne the weight function n j 1 s 1 (2.8) WJ (t, z)= Symt . j ! ...jn! tj1 j −1 i − zs 1 "s=1 i=1 +···+ s + # Y Y 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

(2.9) Wk,n,m(t, z)= WJ (t, z)fJ vm J X∈Ik is the L⊗m[|m| − 2k]-valued vector weight function.

0 0 0 0 0 0 Theorem 2.1 ([RV, B], cf. [SV1]). If (t , z )=(t1,...,tk, z1 ,...,zn) is a solution of the 0 0 ⊗m Bethe ansatz equations (2.6), then the vector Wk,n,m(t , z ) lies in SingL [|m| − 2k] and is an eigenvector of the Gaudin Hamiltonians, moreover, 0 0 0 0 0 0 0 (2.10) Hi(z ).Wk,n,m(t , z )= λi(t , z )Wk,n,m(t , z ), i =1, . . . , n.

0 0 The eigenvector Wk,n,m(t , z ) is called the Bethe eigenvector. On the Bethe eigenvectors see, for example, [SchV, MV1, MV2, V1, V2, V3]. 0 0 ⊗m The fact that Wk,n,m(t , z ) in Theorem 2.1 lies in SingL [|m| − 2k] may be reformulated as follows. For any J ∈ Ik−1, we have n 0 0 (2.11) (js + 1)(ms − js)WJ+1s (t , z )=0, s=1 X 0 0 where we set WJ+1s (t , z ) = 0 if J + 1s ∈/ Ik. REMARKS ON THE GAUDIN MODEL MODULO p 5

2.4. Proof of Theorem 2.1. We sketch the proof following [SV1]. The intermediate statements in this proof will be used later when constructing eigenvectors of the Bethe algebra over Fp. The proof is based on the following cohomological relations. Given k, n ∈ Z>0 and a multi-index J =(j1,...,jn) with |J| 6 k, introduce a diﬀerential form

1 d(t1 − z1) d(tj1 − z1) d(tj1+1 − z2) ηJ = Antt ∧···∧ ∧ ∧ ... j1! ··· jn! t1 − z1 tj1 − z1 tj1+1 − z2 h d(t 1 −1 − z ) d(t − z ) ∧ j +···+jn +1 n ∧···∧ j1+···+jn n , tj1+···+jn−1+1 − zn tj1+···+jn − zn i which is a logarithmic diﬀerential form on Cn × Ck with coordinates z, t. If |J| = k, then for any z0 ∈ Cn we have on {z0} × Ck the identity 0 (2.12) ηJ {z0}×Ck = WJ (t, z )dt1 ∧···∧ dtk. Example 2.1. For k = n = 2 we have d(t1 − z1) d(t2 − z1) η(2,0) = ∧ , t1 − z1 t2 − z1 d(t1 − z1) d(t2 − z2) d(t2 − z1) d(t1 − z2) η(1,1) = ∧ − ∧ . t1 − z1 t2 − z2 t2 − z1 t1 − z2 Introduce the logarithmic diﬀerential 1-forms n k mimj d(zi − zj) d(ti − tj) d(ti − zs) α = + 2 − ms , 2 zi − zj ti − tj ti − zs 16i

(i,j) d(zi − zj) (2.14) α ∧ ηJ fJ vm = Ω ∧ ηJ fJ vm. zi − zj J i

⊗m 2.5. Bethe ansatz on SingL |m| − 2k over Fp. Given k, n ∈ Z>0, m =(m1,...,mn) ∈ Zn F >0, let p be a prime number. Consider the Lie algebra sl2 as an algebra over the field p F 0 0 0 Fn and the sl2-modules Lms , s = 1,...,n, over p. Let z = (z1 ,...,zn) ∈ p have distinct 0 coordinates. The Gaudin Hamiltonians Hs(z ) of formula (2.2) define commuting Fp-linear F ⊗m n operators on the p-vector space L = ⊗s=1Lms . By formula(2.4) the Gaudin Hamiltonians ⊗m preserve the Fp-subspaces SingL [|m| − 2k] and we may study eigenvectors of the Gaudin Hamiltonians on a subspace SingL⊗m[|m| − 2k]. Consider the system of Bethe ansatz equations n 2 ms (2.15) − 0 =0, i =1, . . . , k, ti − tj ti − z j i s=1 s X6= X Fk 0 0 0 0 Fk+n as a system of equations on t =(t1,...,tk) ∈ p. If(t1,...,tk, z1 ,...,zn) ∈ p has distinct coordinates, denote k m m /2 m (2.16) λ (t0, z0)= s l − s ∈ F , s =1, . . . , n. s z0 − z0 z0 − t0 p l s s l i=1 s i X6= X 0 Fk Theorem 2.3. Let p be a prime number and p > |m|. Let t ∈ p be a solution of the 0 0 Bethe ansatz equations (2.15). Then the vector Wk,n,m(t , z ) is well-defined and lies in the subspace SingL⊗m[|m| − 2k], that is, the equations n 0 0 (2.17) (js + 1)(ms − js)WJ+1s (t , z )=0, s=1 X 0 0 hold, also the vector Wk,n,m(t , z ) satisfies the equations 0 0 0 0 0 0 0 (2.18) Hs(z ).Wk,n,m(t , z )= λs(t , z )Wk,n,m(t , z ), s =1, . . . , n. Proof. The proof of Theorem 2.3 is the same as the proof of Theorem 2.1 since identities (2.13) and (2.14) hold over half integers and can be projected to Fp.

3. Two-dimensional spaces of polynomials 3.1. Two-dimensional spaces of polynomials over C. For a function g(x) denote g′ = dg dx . For functions g(x), h(x) deﬁne the Wronskian Wr(g(x), h(x)) = g′(x)h(x) − g(x)h′(x). Z Zn 0 0 Theorem 3.1 ([SchV], cf. [MV2]). Let k ∈ >0, m = (m1,...,mn) ∈ >0. Let (t , z ) ∈ Ck+n have distinct coordinates. Denote k n 0 0 ms (3.1) y(x)= (x − ti ), T (x)= (x − zs ) . i=1 s=1 Y Y We have the following two statements. (i) If (t0, z0) is a solution of the Bethe ansatz equations (2.6), then k 6 |m| +1, 2k =6 |m| +1, and there exists a polynomial y˜(x) ∈ C[x] of degree |m| +1 − k such that (3.2) Wr(˜y(x),y(x)) = T (x). REMARKS ON THE GAUDIN MODEL MODULO p 7

(ii) If there exists a polynomial y˜(x) satisfying equation (3.2), then k 6 |m| +1, 2k =6 |m| +1 and (t0, z0) is a solution of the Bethe ansatz equations (2.6). Proof. We will use the proof below in the proof of the p-version of Theorem 3.1. Equation (3.2) is a ﬁrst order diﬀerential equation with respect toy ˜(x). Then y˜(x) ′ T (x) (3.3) = y(x) y(x)2 and T (x) T (x) (3.4) y˜(x)= y(x) dx = y(x) dx. y(x)2 k (x − t0)2 Z Z i=1 i k 0 2 C We have the unique presentation T (x)= Q(x) i=1(Qx − ti ) + R(x) with P (x), Q(x) ∈ [x] |m|−2k such that Q(x) = 0 if 2k > |m| and Q(x) = a|m|−2kx + ··· + a0 is of degree |m| − 2k otherwise; deg R(x) < 2k. We have the uniqueQ presentation

k R(x) a a (3.5) = i,2 + i,1 , k 0 2 (x − t0)2 x − t0 i=1(x − ti ) i=1 i i X where Q T (x) d T (x) (3.6) ai,2 = , ai,1 = . k 0 2 k 0 2 (x − t ) x t0 dx (x − t ) x t0 j6=i j = i j6=i j ! = i

We have Q Q n d T (x) m 2 T (t0) = s − i . k 0 2 0 0 0 0 k 0 0 2 dx (x − t ) x t0 ti − zs ti − tj (t − t ) j6=i j ! = i s=1 j6=i ! j6=i i j X X 0 0 Since (t , z )Q has distinct coordinates we conclude that ai,1 = 0 forQ i =1,...,k, if and only if (t0, z0) is a solution of (2.6). Let (t0, z0) be a solution of (2.6). By formula (3.4) we have

k k a (3.7) y˜(x)= (x − t0) c − i,2 , if 2k> |m|, i x − t0 i=1 i=1 i ! Yk X k a m k a y˜(x)= (x − t0) | |−2 x|m|−2k+1 + ··· + a x + c − i,2 , i |m| − 2k +1 0 x − t0 i=1 i=1 i ! Y X if 2k 6 |m|, where c ∈ C is an arbitrary number. In each of the two cases we may choose c so that degy ˜(x) =6 deg y(x). Using the identity (3.8) Wr(xα, xβ)=(α − β)xλ+β−1 we obtain in this case that (3.9) degy ˜(x) + deg y(x)= |m| +1. Hence k 6 |m| + 1 and k =6 |m| +1 − k. The ﬁrst part of the theorem is proved. 8 ALEXANDER VARCHENKO

Let there exist a polynomialy ˜(x) satisfying equation (3.2). Adding toy ˜(x) the polynomial y(x) with a suitable coeﬃcient if necessary we may assume that degy ˜(x) =6 deg y(x). Then (3.9) implies k 6 |m| + 1 and k =6 |m| +1 − k. By formula (3.3) we have k y˜(x) ′ a a (3.10) = a x|m|−2k + ··· + a + i,2 + i,1 if 2k 6 |m| y(x) |m|−2k 0 (x − t0)2 x − t0 i=1 i i X and k y˜(x) ′ a a (3.11) = i,2 + i,1 if 2k > |m|. y(x) (x − t0)2 x − t0 i=1 i i X y˜(x) The function y(x) has a unique decomposition into the sum of a polynomial and simple fractions. The term by term derivative of that decomposition equals the right-hand side of (3.10) or (3.11). Hence all of coeﬃcients ai,1 must be zero. Hence the roots of y(x) satisfy the Bethe ansatz equations. Remark. This construction assigns to a solution (t0, z0) of the Bethe ansatz equations the two-dimensional subspace hy˜(x),y(x)i of the space of polynomials C[x] such that deg y(x)= k, degy ˜(x)= |m| − k +1, Wr(y(x), y˜(x)) = T (x). That subspace is a point of the Grassman- nian of two-dimensional subspaces of C[x].

3.2. Two-dimensional spaces of polynomials over Fp. Z Zn Theorem 3.2. Let k ∈ >0, m = (m1,...,mn) ∈ >0. Let p > |m| +1, p > n + k. Let 0 0 Fk+n (t , z ) ∈ p have distinct coordinates. Denote k n 0 0 ms F (3.12) y(x)= (x − ti ), T (x)= (x − zs ) ∈ p[x]. i=1 s=1 Y Y We have the following two statements. (i) If (t0, z0) is a solution of the Bethe ansatz equations (2.15), then k 6 |m| +1, 2k =6 |m| +1, and there exists a polynomial y˜(x) ∈ Fp[x] of degree |m| +1 − k such that (3.13) Wr(˜y(x),y(x)) = T (x).

(ii) If there exists a polynomial y˜(x) ∈ Fp[x] satisfying equation (3.13), then k 6 |m| +1, 2k =6 |m| +1 and (t0, z0) is a solution of the Bethe ansatz equations (2.15). Proof.

Lemma 3.3. Let p be a prime number. Let d1,...,dk ∈ Z>0 with di 6 2 for all i. Let 0 0 F F t1,...,tk ∈ p be distinct and T (x) ∈ p[x]. Then there exists a unique presentation

k di T (x) ai,j (3.14) k = Q(x)+ 0 j , 0 di (x − t ) i=1(x − ti ) i=1 j=1 i X X F where Q(x) ∈ p[x] andQ dj−1 T (x) (3.15) ai,j = . j−1 k 0 d dx (x − t ) l x t0 l6=i l ! = i

Q REMARKS ON THE GAUDIN MODEL MODULO p 9

0 0 1 1 Proof. The uniqueness is clear. Let us show the existence. Lift t1,...,tk, T (x) to t1,...,tk ∈ Z, T 1(x) ∈ Z[x]. We have

1 k di 1 T (x) 1 ai,j (3.16) k = Q (x)+ 1 j , 1 di (x − t ) i=1(x − ti ) i=1 j=1 i X X where Q1(x) ∈ Z[x] andQ

dj−1 T 1(x) (3.17) a1 = . i,j j−1 k 1 d dx (x − t ) j x t0 j6=i j ! = i

1 It is easy to see that for j =1, 2 and all Qi the coeﬃcient a i,j has a well-deﬁned projection to Fp. By projecting (3.16) to Fp we obtain a presentation of (3.14). The proof of Theorem 3.2 is based on Lemma 3.3 and is analogous to the proof of Theorem 3.1. If(t0, z0) is a solution of (2.15), then

k y˜(x) ′ T (x) a = = Q(x)+ i,2 , y(x) k 0 2 (x − t0)2 i=1(x − t ) i=1 i X |m|−2k where ai,2 are given by (3.15); QQ(x) ∈ Fp[x], Q(x) = 0 if 2k > |m| and Q(x)= a|m|−2kx + ··· + a0 is of degree |m| − 2k + 1 if 2k 6 |m|, see Section 3.1. If 2k 6 |m|, then

k k a m k a y˜(x)= (x − t0) | |−2 x|m|−2k+1 + ··· + a x − i,2 i |m| − 2k +1 0 x − t0 i=1 i=1 i ! Y X is a polynomial of degree |m| − k + 1 satisfying (3.2). Notice that the polynomial in the brackets is well-deﬁned since p> |m| +1. If 2k > |m|, then

k k a y˜(x)= − (x − t0) i,2 i x − t0 i=1 i=1 i ! Y X is a polynomial satisfying (3.13) of degree k +n imply (3.9). The ﬁrst part of Theorem 3.2 is proved. Let there exist a polynomialy ˜(x) satisfying equation (3.13). Adding toy ˜(x) a suitable p polynomial of the form c(x )y(x) for some c(x) ∈ Fp[x] if necessary, we may assume that degy ˜(x) − deg y(x) 6≡ 0 mod p. Then (3.9) holds, k 6 |m| + 1 and k =6 |m| +1 − k. By formula (3.3) and Lemma 3.3 we have

k y˜(x) ′ a a (3.18) = a x|m|−2k + ··· + a + i,2 + i,1 if 2k 6 |m| y(x) |m|−2k 0 (x − t0)2 x − t0 i=1 i i X and k y˜(x) ′ a a (3.19) = i,2 + i,1 if 2k > |m|. y(x) (x − t0)2 x − t0 i=1 i i X 10 ALEXANDER VARCHENKO

y˜(x) The function y(x) has a unique decomposition into the sum of a polynomial and simple fractions. The term by term derivative of that decomposition equals the right-hand side of (3.18) or (3.19). Hence all of coeﬃcients ai,1 must be zero. Hence the roots of y(x) satisfy the Bethe ansatz equations. Remark. This construction assigns to a solution (t0, z0) of the Bethe ansatz equations (2.15) the two-dimensional subspace hy˜(x),y(x)i of the space of polynomials Fp[x] such that deg y(x)= k, degy ˜(x)= |m| − k +1, Wr(y(x), y˜(x)) = T (x). That subspace is a point of the Grassmannian of two-dimensional subspaces in Fp[x]. 4. Example: the case k =1 ⊗m Zn 4.1. Gaudin model on SingL [|m| − 2]. Let m =(m1,...,mn) ∈ >0 and p> |m| + 1. ⊗m Consider the Gaudin model on SingL [|m| − 2] over Fp . That means that k = 1 in the notations of the previous sections. A basis of L⊗m[|m| − 2] is formed by the vectors (s) (4.1) f = vm1 ⊗···⊗ vs−1 ⊗ fvms ⊗ vs+1 ⊗···⊗ vmn , s =1, . . . , n. We have n n ⊗m (s) (4.2) SingL [|m| − 2] = csf | cs ∈ Fp and mscs =0 . n s=1 s=1 o X ⊗m X For s =1,...,n, deﬁne the vectors ws ∈ SingL [|m| − 2] by the formula n m (4.3) w = f (s) − s f (l). s |m| l X=1 We have

(4.4) w1 + ··· + wn =0. By [V4, Lemma 4.2], any n − 1 of these vectors form a basis of SingL⊗m[|m| − 2]. 0 0 0 Fn Let z = (z1 ,...,zn) ∈ p have distinct coordinates. For i = 1,...,n, the Gaudin 0 ⊗m Hamiltonian Hi(z ) acts on L [|m| − 2] by the formulas: m m /2 1 (4.5) f (s) 7→ i j f (s) + (m f (i) − m f (s)), s =6 i, z0 − z0 z0 − z0 s i j i i j i s X6= m m /2 1 f (i) 7→ i j f (i) + (m f (j) − m f (i)). z0 − z0 z0 − z0 i j j i i j j i i j X6= X6= Hence m m /2 1 (4.6) w 7→ i j w + (m w − m w ), s =6 i, s z0 − z0 s z0 − z0 s i i s j i i j i s X6= m m /2 1 w 7→ i j w + (m w − m w ). i z0 − z0 i z0 − z0 i j j i j i i j j i i j X6= X6= Recall that the Bethe algebra of SingL⊗m[|m|−2] is the subalgebra of End(SingL⊗m[|m|−2]) 0 generated by the Gaudin Hamiltonians Hi(z ), i = 1,...,n, and the identity operator. We denote it by B(z0, m). REMARKS ON THE GAUDIN MODEL MODULO p 11

0 Zn 4.2. Bethe ansatz equation and algebra A(z , m). Let m = (m1,...,mn) ∈ >0 and 0 0 0 p> |m| +1. Let z =(z1 ,...,zn) have distinct coordinates. The Bethe ansatz equations of SingL⊗m[|m| − 2] is the single equation m1 mn (4.7) 0 + ··· + 0 =0. t − z1 t − zn Write

m1 mn P (t) (4.8) 0 + ··· + 0 = n 0 , t − z1 t − zn s=1(t − zs ) where Q n 0 0 (4.9) P (t)= P (t, z , m)= ms (t − zl ). s=1 l s X Y6= A F A 0 0 Let Fp be the aﬃne line over p with coordinate t. Denote U = Fp −{z1 ,...,zn}. Let A O(U) be the ring of rational functions on the aﬃne line Fp regular on U. Introduce the algebra 0 0 (4.10) A(z , m)= O(U)/(P (t)), dimFp A(z , m)= n − 1. 0 Here (P (t)) is the ideal generated by P (t). Let us ∈A(z , m), s =1,...,n, be the image of ms 0 0 0 in A(z , m). The elements us span A(z , m) as a vector space and t−zs

(4.11) u1 + ··· + un =0. We have 1 (4.12) uius = 0 0 (msui − mius), s =6 i, zi − zs 1 u u = (m u − m u ). i i z0 − z0 i j j i j i i j X6= For a function g(t) ∈ O(U) denote [g(u)] its image in A(z0, m). The elements [1], [t],..., [tn−2] 0 0 form a basis of A(z , m) over Fp. The deﬁning relation in A(z , m) is P ([t]) = 0. The fol- i lowing formulas express the elements [t ] in terms of the elements us. Lemma 4.1. We have −1 (4.13) [1] = (z0u + ··· + z0u ), |m| 1 1 n n n n n 1 −1 [t] = z0m z0u + (z0)2u , |m|2 s s s s |m| s s s=1 ! s=1 ! s=1 ! i Xn X n X −1 −1 [ti] = (z0)jm [ti−j]+ (z0)i+1u , i > 0. |m| s s |m| s s j=1 s=1 s=1 X X X

These formulas are related to formulas for the sl2-action on tensor products of modules dual to Verma modules, see [SV2] and in particular to formula (11) in [SV2]. b 12 ALEXANDER VARCHENKO

4.3. Isomorphism of A(z0, m) and B(z0, m). Deﬁne the isomorphism of vectors spaces

0 ⊗m (4.14) α : A(z , m) → SingL [|m| − 2], us 7→ ws, s =1, . . . , n, in particular, we have −1 (4.15) h1i := α([1]) = (z0w + ··· + z0 w ). |m| 1 1 n n Theorem 4.2. The map m m /2 (4.16) [1] 7→ Id, u 7→ H (z0) − s j Id, s =1, . . . , n, s s z0 − z0 j s s j X6= extends to an algebra isomorphism (4.17) β : A(z0, m) → B(z0, m), such that α(gh)= β(g).α(h) for any g, h ∈A(z0, m). Proof. The proof follows from comparing (4.6) and (4.12). Remark. Theorem 4.2 says that the isomorphism α of vector spaces and the isomorphism β of algebras establish an isomorphism between the B(z0, m)-module SingL⊗m[|m| − 2] and the regular representation of the algebra A(z0, m). Example 4.1. Theorem 4.2 in particular says that if P (t) is irreducible then B(z0, m) =∼ n−1 Fpn−1 , where Fpn−1 is the ﬁeld with p elements. 0 2 0 0 0 0 0 0 0 0 0 For example, if n = 3, m = (1, 1, 1), then P (t, z )=3t −2(z1 +z2 +z3 )t+z1 z2 +z1 z3 +z2 z3 . 0 F 0 0 0 F 0 ∼ F If p = 5, then P (t, z ) is irreducible in 5[t] for all distinct z1 , z2 , z3 ∈ 5 and B(z , m) = 25. Corollary 4.3. We have

0 (4.18) dimFp B(z , m)= n − 1. Corollary 4.4. The operators β([1]) = Id, β([ti]), i =1,...,n−2, form a basis of the vector 0 space B(z , m) over Fp. The operator

n n 1 m m /2 (4.19) {t} := β([t]) = z0m z0 H (z0) − s j Id |m|2 s s s s z0 − z0 s=1 ! s=1 j s s j !! X X X6= n −1 m m /2 + (z0)2 H (z0) − s j Id |m| s s z0 − z0 s=1 j s s j !! X X6= generates B(z0, m) as an algebra with deﬁning relation P ({t})=0. Corollary 4.5. We have

m m /2 (4.20) H (z0) − s j Id .h1i = w , s =1, . . . , n. s z0 − z0 s j s s j ! X6= REMARKS ON THE GAUDIN MODEL MODULO p 13

4.4. Eigenvectors of B(z0, m) and the polynomial P (t). The elements of the algebra 0 A(z , m) have the form Q([t]), where Q(t) ∈ Fp[t], deg Q(t) < n − 1. An element Q([t]) is an eigenvector of all multiplication operators of A(z0, m) if and only if Q([t]) is an eigenvector 0 0 of the multiplication by [t]. If t ∈ Fp is the eigenvalue, then ([t] − t )Q([t]) = 0, that is, 0 (4.21) (t − t )Q(t) = const P (t), const ∈ Fp. Hence the set of eigenlines of all multiplication operators of A(z0, m) is in one-to-one correspondence with the set of distinct roots of the polynomial P (t), namely, a root t0 with decomposition (t − t0)Q(t)= P (t) corresponds to the line generated by the element Q([t]). Corollary 4.6. The set of eigenlines of B(z0, m) are in one-to-one correspondence with the set of distinct roots of the polynomial P (t), namely, a root t0 with decomposition (t−t0)Q(t)= P (t) for some Q(t) ∈ Fp[t] corresponds to the line generated by the vector (4.22) ω(t0, z0) := Q({t}).h1i ∈ SingL⊗m[|m| − 2]. Thus we have two ways to construct the eigenlines of B(z0, m) from roots t0 of the polynomial P (t). The ﬁrst is given by Theorem 2.1 and the eigenline is generated by the vector n n 0 0 1 (s) 1 (4.23) W1,n,m(t , z )= 0 f = 0 ws. t − zs t − zs s=1 s=1 X X The second is given by Corollary 4.6 and the eigenline is generated by the vector Q({t}).h1i. Theorem 4.7. The two eigenlines coincide, more precisely, we have 0 0 (4.24) W1,n,m(t , z ) = const Q({t}).h1i, const ∈ Fp. 0 −1 0 0 0 Proof. We need to show that ([t] − t )α (W1,n,m(t , z )) = 0 in A(z , m). Indeed n 0 n n 0 −1 0 0 ms t − t ms ms ([t] − t )α (W1,n,m(t , z )) = 0 = 0 [1] − =0 t − zs t − zs t − zs t − zs s=1 s=1 s=1 X h i X X h i due to the Bethe ansatz equation (4.7) and formula (4.11). 4.5. Algebra C(T ). In this section, p is a prime number, p > n+1. Fix a monic polynomial n n−1 n−2 (4.25) T (x)= x + σ1x + σ2x + ··· + σn ∈ Fp[x].

We consider the two-dimensional subspaces V ⊂ Fp[x] consisting of polynomials of degree n and 1 such that Wr(g1(x),g2(x)) = const T (x), where g1(x),g2(x) is any basis of V and const ∈ Fp. Such a subspace V has a unique basis of the form n n−1 2 (4.26) g1(x)= x + a1x + ··· + an−2x + a0, g2 = x − t with

(4.27) Wr(g1(x),g2(x))=(n − 1)T (x). Equation (4.27) is equivalent to the system of equations

(4.28) (n − r − 1)ar − (n − r + 1)ar−1t − (n − 1)σr =0, r =1,...,n − 1,

an − (n − 1)σn =0, 14 ALEXANDER VARCHENKO

where a0 = 1. Expressing a1 from the ﬁrst equation in terms of t, then expressing a2 from the ﬁrst and second equations in terms of t and so on, we can reformulate system (4.28) as the system of equations n − 1 (4.29) a − (ntr +(n − 1)σ tr−1 + ··· +(n − r)σ )=0, r =1,...,n − 2, r 2 1 r n−1 n−2 (4.30) nt +(n − 2)σ1t + ··· +2σ2t + σ1 =0,

(4.31) an + σn =0.

dT Notice that equation (4.30) is the equation dt (t) = 0, where T (x) is deﬁned in (4.25). Let I ⊂ Fp[t, a1,...,an−2, an] be the ideal generated by n polynomials staying in the left-hand sides of the equations of the system (4.28). Deﬁne the algebra

(4.32) C(T )= Fp[t, a1,...,an−2, an]/I. F dT Let J ⊂ p[t] be the ideal generated by dt (t). Deﬁne the algebra

(4.33) C˜(T )= Fp[t]/J. Lemma 4.8. We have an isomorphism of algebras

(4.34) C˜(T ) → C(T ), [t] 7→ [t].

0 Zn 0 0 0 Fn Let m = (1,..., 1) ∈ >0. Let z =(z1 ,...,zn) ∈ p be a point with distinct coordinates. 0 The Bethe ansatz for SingL⊗m [|m0| − 2] has the form

1 1 R(t) (4.35) 0 + ··· + 0 = =0, t − z1 t − zn T (t) where

n dT (4.36) T (t)= (t − z0), R(t)= (t). s dt s=1 Y Hence for this T (x) we have

(4.37) C˜(T )= A(z0, m0).

Corollary 4.9. For T (t) and R(t) as in (4.36) we have

(4.38) A(z0, m0) ∼= B(z0, m0) ∼= C(T )

0 and the B(z0, m)-module SingL⊗m [|m0| − 2] is isomorphic to the regular representation of the algebra C(T ). REMARKS ON THE GAUDIN MODEL MODULO p 15

4.6. Wronski map. Let Xn be the affine space of all two-dimensional subspaces V ⊂ Fp[x], each of which consists of polynomials of degree n and 1. The space Xn is identified with the space of pairs of polynomials given by formula (4.26). Let Fp[x]n ⊂ Fp[x] be the affine subspace of monic polynomials of degree n. Introduce the Wronski map 1 (4.39) W : X → F [x] , hg (x),g (x)i 7→ Wr(g (x),g (x)), n n p n 1 2 n − 1 1 2 cf. [MTV3]. The algebra C(T ) is the algebra of functions on the fiber W −1(T ) of the Wronski map. 3 2 −1 Example 4.2. Let n = 3 and T (x)= x + σ1x + σ2x + σ3. Then W3 (T ) consists of one 2 dT −1 point if the discriminant σ1 − 3σ2 of dx (x) equals zero; W3 (T ) consists of two points if the 2 discriminant is a nonzero square, and is empty otherwise. Thus, p points of X3 have one p−1 2 p−1 2 preimage, 2 p points have two preimages, and 2 p points have none. Cf. Example 4.1.

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