An overview for this section

6.1.1. In §5, we have proved that for each positive integer n, for every solution u of the mean field equation

(6.1.1) u+e^u = 8πn·δ0 on C/Λ

there exists a set a = {a1, . . . , an} of n complex numbers which satisfies

is a normalized type II developing map of u. Moreover every set a = {a1, . . . , an} of complex numbers satisfying conditions (5.6.1), (5.6.2) and (5.7.6) gives rise to a solution of the above mean field equation.

6.1.2. In this section we will leave the Green equation (5.6.1) alone and consider thosea={a₁, . . . , a_n} satisfy only the equations (5.6.2) under the constraint (5.7.6), that is, we consider ain the set Xn defined in (0.6.6) in the introduction. We would like to characterize a∈X_n in terms of certain Lam´e equations.

6.1.3. We will make use of the following addition formulas freely:

℘(z) in (C/Λ){[0]}, possibly with multiplicity. Define a meromorphic function f_[a](z) on Cby sym-metric product Symⁿ(C/Λ{[0]}) and not on the choice of representatives

ai ∈[ai]. Becauseζ(z; Λ) = _dz^d logσ(z; Λ), we get from (6.1.4) an equivalent definition

(6.1.8) f_[a](z) := (−1)ⁿ·e^2zni=1ζ(ai)· n i=1

σ(z−a_i) σ(z+a_i). Note also that f_[a](0) = 1 and f_[a](−z)·f_[a](z) = 1 for all z.

Definition 6.1.5. Leta={a₁, . . . , a_n}be an unordered list of elements of C Λ. The Hermite-Halphen ansatz functionw_a(z) attached to the list ais the meromorphic function on Cdefined by

(6.1.9) w_a(z) =w_a(z; Λ) :=e^zζ(ai) n i=1

σ(z−ai) σ(z) .

Remark. (a) In classical literature the functions w_a(z) arise as explicit solutions of the Lam´e equation

(6.1.10) w=

n(n+ 1)℘(z) +B w;

see [29, I–VII], [27, p. 495–497] and also [67,§23.7].

(b) Clearly we have

f_[a](z) = w_a(z) w_−a(z),

where −a is the list {−a₁, . . . ,−a_n} and [a] is the list{[a₁], . . . ,[a_n]}. (c) If b={b1, . . . , bn}is a list such that bi−ai ∈Λ for alli= 1, . . . , n, then

wb

wa ∈C^×, a non-zero constant.

Lemma 6.1.6. If a list [a] ={[a1], . . . ,[an]} of nelements of(C/Λ){[0]} satisfies (5.6.1),(5.6.2)and the non-degeneracy condition (5.7.6), then there exists a constantB =B_[a]such that the Schwarzian derivative off_[a]satisfies

S(f_[a]) =−2

n(n+ 1)℘(z; Λ) +B_[a]

.

Proof. By Theorem 5.6, f_[a] is a normalized developing map for the mean field equation (6.1.1), and the assertion follows from (3.1.4).

6.1.7. The constant B_[a] in Lemma6.1.6can be evaluated by a straight-forward computation; the answer is

(6.1.11) B_[a]= (2n−1) n i=1

℘(ai; Λ).

On the other hand there is a proof of the formula (6.1.11) for B_[a] without resorting to messy computations, via Lam´e’s differential equation (6.1.10) because f_[a] can be written as the ratio of two linearly independent solu-tions of (6.1.10). The idea is this: use the Hermite–Halphen ansatz funcsolu-tions wa(z) to find solutions to Lam´e equations, and the constant B can be com-puted from the ansatz solutions wa. Thenf_[a]a=wa/w₋a has the expected Schwarzian derivative by ODE theory.

We take this approach since it requires less computation to prove the formula (6.1.11) forB, and it leads to a characterization of the setY_ndefined in (0.5.2) as the set of all unordered lists a={[a₁], . . . ,[a_n]} of nelements in (C/Λ){[0]} such that w_a(z; Λ) satisfies a Lam´e equation (6.1.10) for someB ∈ C, see Theorem 6.2. We then move back to characterize the set Xndefined in (0.6.6) as the set of alla’s such that ordz=0f_[a] (z) = 2n, which is the highest possible value of ord_z=0f_[a](z); see Theorem6.5. This leads to the important consequence that for a∈ Yn,a ∈Xn if and only if a =−a, and a characterization ofXn via the Schwarzian derivative.

The following result is known in the literature, see e.g. [27]. We reproduce its proof here for the sake of completeness.

Theorem 6.2 (Characterization of Y_n). Let n ≥ 1 be a positive integer.

Let a = {a₁, . . . , a_n} be an unordered list of n elements in C Λ. Let w_a be defined as in (6.1.9). Let [a] be the unordered list {[a1], . . . ,[an]}, where [ai] :=aimod Λ∈C/Λ for each i.

(1) There exists a constantB∈Csuch that the meromorphic function w_a on C satisfies the Lam´e equation (6.1.10) if and only if the following conditions hold.

– [ai]= [aj]whenever i=j, and – the a_i’s satisfy

(6.2.1)

j=i

ζ(ai−aj; Λ)−ζ(ai; Λ) +ζ(aj; Λ)

= 0, i= 1, . . . , n.

In other words the necessary and sufficient condition is that a is a point of the varietyYn in the notation of (0.6.6).

(2) If the system of equations (6.2.1) holds for a, then wa satisfies the Lam´e equation (6.1.10) whose accessary parameter B of the equation is identi-cally zero, a contradiction. We have shown that the [a_i]’s must be mutually distinct if w_a(z) is a solution of (6.1.10).

The logarithmic derivative w_a(z; Λ)

w_a(z; Λ) =

i

ζ(a_i; Λ) +ζ(z−a_i; Λ)−ζ(zΛ)

of w_a is an elliptic function onC/Λ. Applying _dz^d again, we get w_a

of (6.1.5) and (6.1.6) to add up the first two sums after the second equality sign in (6.2.2) to get the last expression of ^w_w^a

a in (6.2.2).

The sum in the last line of (6.2.2) is an elliptic function on C/Λ with a double pole atz= 0 with Laurent expansion ⁿ_z²2+O(1); denote this function by F_a(z). Therefore w_a satisfies a Lam´e equation (6.1.10) for some B ∈ C if and only if Fa(z) has no pole outside of [0]∈C/Λ.

Suppose [ai0] appears in the list [a] ={[a1], . . . ,[an]} r times withr≥2 for somei0 ∈ {1, . . . , n}. Then Fa(z) has a double pole atz=ai0, where it has a Laurent expansion

F_a(z) =r(r−1)(z−a_i0)⁻²+O

(z−a_i0)⁻¹ .

We have shown that if Fa(z) is holomorphic outside Λ, then [ai] = [aj] whenever i=j.

Under the assumption that [a1], . . . ,[an] are mutually distinct, the func-tionFa(z) is holomorphic on (C/Λ){[z1], . . . ,[an]}and has at most simple poles at [z₁], . . . ,[z_n]. Therefore F_a(z) is holomorphic outside Λ if and only if its residue atz=a_i is zero for i= 1, . . . , n, which means that

j=i

ζ(a_j; Λ) +ζ(a_i−a_j; Λ)−ζ(a_i; Λ)

= 0, ∀1≤i≤n.

This proves the statement (1) of Theorem6.2.

We know that there a constantsB₁ ∈C such that (6.2.3) Fa(z) =n(n−1)℘(z; Λ) +B1,

because F_a(z) is holomorphic on C/Λ{[0]} and its Laurent expansion at z = 0 is n(n−1)·z⁻² +O(1). To determine B₁, we need to compute its Laurent expansion atz= 0 modulo O(z). From

ζ(z−ai; Λ) =−ζ(ai; Λ)−℘(ai; Λ)z+O(z²), we get

F_a(z) =

i=j

−1

z −℘(a_i; Λ)z+O(z²) −1

z −℘(a_j; Λ)z+O(z²)

=n(n−1)1

z² + 2(n−1)

i

℘(ai; Λ) +O(z).

In particularB₁ = 2(n−1). From (6.2.2) we get w_a

w_a =n(n+ 1)℘(z; Λ) + (2n−1) n

i=1

℘(ai; Λ).

We have proved the statement (2).

Remark 6.2.1. (a) Clearly that the necessary and sufficient condition in Theorem6.2(1), which defines the variety Yn, depends only on the list [a] = {[a1], . . . ,[an]} of elements in (C/Λ){[0]} determined bya.

(b) It is also clear that a lista={a1, . . . , an}satisfies the condition in6.2(1) if and only if the list −a={−a1, . . . ,−an}does.

Proposition 6.3. Let a={a1, . . . , an} be an unordered list of elements in C Λ, n≥1.

(1) The function w_a(z) is a common eigenvector for the translation action by elements of Λ:

wa(z+ω)

wa(z) =e^{ω·ni=1ζ(ai;Λ)−η(ω;Λ)·ni=1ai} ∀ω∈Λ.

This “eigenvalue package” attached to w_a is the homomorphism χa: Λ→C^×, ω→e^{ω·ni=1ζ(ai;Λ)−η(ω;Λ)·ni=1ai} ∀ω∈Λ, which depends only on the list [a] ={[a₁], . . . ,[a_n]}.

(2) If w_a satisfies a Lam´e equation (6.1.10), then so does w_−a.

(3) For any unordered list b = {b₁, . . . , b_n} of elements in C Λ, the functionsw_aandw_b are linearly dependent if and only if either[b] = [a]

or [b] = [−a], where [−a] is the unordered list {[−a₁], . . . ,[−a_n]} of elements in C/Λ.

(4) The homomorphisms χ_a and χ_−a are equal if and only if there exists an element ω ∈Λ such that

(6.3.1)

n i=1

ζ(a_i; Λ) = η(ω; Λ)

2 and

n i=1

a_i = ω 2, in which case Im(χ_a)⊆ {±1}.

(5) Suppose thatw_aandw_−aare two solutions of a Lam´e equation(6.1.10), and[a]= [−a]. Then χ_a=χ_−a. MoreoverC·w_aandC·w_−aare char-acterized by the monodromy representation of (6.1.10)as the two one-dimensional subspaces of solutions which are stable under the mon-odromy.

Proof. The statement (1) is immediate from the transformation formula for the Weierstrassσfunction. The statements (2) and (3) are obvious and easy respectively. The statement (4) is a consequence of the Legendre relation for the quasi-periods.

Suppose that [a]= [−a] andχa=χ₋a. By (3) the monodromy represen-tation of the Lam´e equation (6.1.10) is isomorphic to the direct sumχ_a⊕χ_b, and the characterχ_ahas order at most 2 by (4). Consider the algebraic form (6.3.2) p(x)d²y

dx² +1

2p(x)dy dx−

n(n+ 1)x+B y= 0

of the Lam´e equation (6.1.10). The monodromy groupM of (6.3.2) contains the monodromy group (6.1.10) as a normal subgroup of index at most 2, therefore M is a finite abelian group of order dividing 4. In particular the monodromy representation of the algebraic Lam´e equation (6.3.2) is com-pletely reducible. However one knows from [65, Thm. 4.4.1] or [4, Thm. 3.1]

that the monodromy representation of (6.3.2) is not completely reducible, a contradiction. We have proved the first part of (5). The second part of (5) follows from the first part of (5).

Proposition 6.4. Suppose that[a] ={[a1], . . . ,[an]}and[b] ={[b1], . . . ,[bn]} are two points of Yn, n ≥ 1. If _n

i=1℘(ai; Λ) = _n

i=1℘(bi; Λ), the either [a] = [b]or [a] = [−b].

Proof. Pick representatives ai ∈ [ai] and bi ∈ [bi] for each i = 1, . . . , n.

Suppose that [b] = [a] and [b] = [−a]. The functions wa(z) and wb(z) are linearly independent by Proposition6.3(3) because [b]= [a], and they satisfy the same Lam´e differential equation because B_[a] = B_[b]. By either [65, Thm. 4.4.1] or [4, Thm. 3.1], that image of the monodromy representation of the Lam´e equation ^d_d²2^wz −(n(n+ 1)℘(z; Λ) +B_[a])w= 0 is not contained in C^×I₂, for otherwise the monodromy group of the algebraic form of the above Lam´e equation onP¹(C) is contained in the product ofC^×I₂ with a subgroup of order two in GL₂(C). SoC·w_aand C·w_b are the two distinct common eigenspaces of the monodromy representation of the above Lam´e equation onC/Λ. If follows thatC·w₋a=C·wa andC·w_−b =C·w_−b, i.e.

[a] = [−a] and [b] = [−b]. Therefore the cardinality of the monodromy group of the above Lam´e equation divides 4, and the cardinality of the monodromy group of the algebraic form of the same Lam´e equation divides 8, which again contradicts [65, Thm. 4.4.1] and [4, Thm. 3.1].

Theorem 6.5 (Characterization of X_n by ord_z=0f_[a] (z)). Let n ≥ 1 be a positive integer. Let a = {[a1], . . . ,[an]} be an unordered list of n non-zero points on the elliptic curve C/Λ. Let a1, . . . , an be representatives of [a1], . . . ,[an]in C Λ.

(0) f_[a] is a constant if and only if[a] = [−a], where[−a]is the unordered list{[−a1], . . . ,[−an]}ofnnon-zero elements in the elliptic curveC/Λ.

(1) If [a]= [−a], then ordz=0f_[a] (z)≤2n.

(2) Assume that [a] = [−a]. Then ord_z=0f_[a] (z) = 2n if and only if the Weierstrass coordinates (℘(a_i; Λ), ℘(a_i; Λ)) of [a₁], . . . ,[a_n] in C/Λ satisfy the following system of polynomial equations.

(6.5.1)

n i=1

℘(ai; Λ)·℘^k(ai; Λ) = 0, for k= 0, . . . , n−2.

Moreover if the above equivalent conditions hold, then – ℘(a_i; Λ)= 0 for alli= 1, . . . , n, and

– ℘(ai; Λ)=℘(aj; Λ) whenever i=j.

In other words [a]is a point of the variety X_n defined in (0.6.6).

Proof. The divisor div(f_[a]) of the meromorphic functionf_[a] on Cis stable under translation by Λ, and div(f_[a]) mod Λ is the formal sum (or 0-cycle)

n

and f_[a] is equal to the constant function 1. We have proved statement (0).

Let (x_i, y_i) := (℘(a_i; Λ), ℘(a_i; Λ)). We have

• ordz=0f_[a](z) = 2n if and only if _n

i=1yix^k_i = 0 for 0 ≤ k ≤ n−2 whilen

i=1yixⁿ_i⁻¹= 0.

Suppose that ord_z=0f_[a] (z)>2n, i.e._n

i=1y_ix^k_i = 0 fork= 0, . . . , n−1.

Ifx₁, . . . , x_nare distinct, we get from the non-vanishing of the Vandermonde determinant that y₁ = · · · = y_n = 0, meaning that [a₁], . . . ,[a_n] are all 2-torsion points. That contradicts the assumption that [a]= [−a]. So we know thatx₁, . . . , x_nare not all distinct. Apply the argument in Remark5.8.6(b):

let{s₁, . . . , s_m}={x₁, . . . , x_n} as sets without multiplicity, let

z_j :=

all i s.t.si=tj

y_i for i= 1, . . . m,

and we havez₁ =· · ·=z_m = 0. Note that for eachj, they_i’s which appear in the sum definingz_j differ from each other at most by a sign±1, so that the sum z_j is either 0 or a non-zero multiple of a y_i. Note that we have cancelled a number of pairs ([a_i1],[−a_i2]) in forming the reduced system of equations

m j=1

z_js^k_j = 0 for k= 0, . . . , n−1.

That z₁ = · · · = z_m = 0 means that, after removing a number of pairs ([a_i1],[−a_i2]) from the unordered list [a], we are left with another unordered list [b] with [b] = [−b]. So again we have [a] = [−a], a contradiction with proves the statement (1). The first part of statement (2) follows.

It remains to prove the second part of (2). We are assuming that [a]= [−a] and _n

i=1y_ix^k_i = 0 for k = 0, . . . , n−2. If there exists i₁, i₂ between 1 andn such that x_i1 =x_i2, the same argument in the previous paragraph produces a contradiction that [a] = [−a]. Therefore x₁, . . . , x_n are mutually distinct. If there is an i0 such that yi0 = 0, then y1 = · · · = yn = 0 by Remark5.8.6(a), contradicting the assumption that [a]= [−a].

We have seen in Proposition 5.8.5 that Xn ⊂ Yn, where Xn is defined in (0.6.6) andYn is defined in (0.5.2). The following proposition, which is a consequence of Theorem6.5, describes the complement of Xn inYn. Proposition 6.6. Let [a] ={[a1], . . . ,[an]} be a point of Yn, i.e. [ai]= [0]

for each i, [ai]= [aj] whenever i= j and the equations (6.2.1) hold. Then [a]∈Xn if and only if [a]= [−a].

Proof. The “only if” part is part of the definition of Xn. Assume that [a]= [−a]. We mush show thata∈ Xn. We know from Theorem 6.2 and 6.5(0)

that wa and w₋a are linearly independent solutions of the Lam´e equation (6.1.10). If a∈ X_n, then the lists [a] and [−a] have common members. So either (A) there exist two indices i₁, i₂ such that i₁ =i₂ and [a_i1] = [−a_i2], or (B) there exists an index i₃ such that [a_i3] = [−a_i3]. We start with a non-canonical process to reduce the length of the list [a] while keeping the associated functions w_aandw_−aunchanged up to some non-zero constants:

First remove all a_i’s such that [a_i] = [−a_i] from the list a. In the resulting reduced list, remove bothai1 andai2from the list ifi1 =i2and [ai1] = [−ai2].

Keep doing so until we get a sublist b={b1, . . . , bm} ofa such that [b] and [−b] have no common members,m < n, and there exists a non-zero constant C ∈C^× such that

f_[b]= w_b w₋b

=C· w_a w₋a

=C·f_[a]. The Schwarzian derivative S(f_[a]) satisfies

(6.6.1) S(f_[a]) =−2(n(n+ 1)℘(z) +B_[a]).

Let

2η:= ordz=0f_[a] (z).

Theorem 6.5(1) tells us that

2η= ord_z=0f_[b] (z)≤2m <2n.

But then

S(f_[a]) = f_[a]

f_[a] −3 2

f_[a]

f_[a]

2

=−2η(η+ 1)1

z² +O(1), which contradicts (6.6.1). Therefore [a]∈X_n.

The characterization ofXnin terms of the Schwarzian derivative follows similarly:

Corollary 6.7 (Characterization of Xn by S(f)). Let n ≥1 be a positive integer. Let a1, . . . , an be complex numbers inC Λ, let [a]be the unordered list{[a1], . . . ,[an]}and let[−a]be the unordered list{[−a1], . . . ,[−an]}. Then [a]∈X_n if and only if [a]= [−a]and

(6.7.1) S(f_[a]) =−2(n(n+ 1)℘(z; Λ) + (2n−1) n

i=1

℘(a_i; Λ)).

Proof. If [a]∈Xn then [a]= [−a] by definition, and a∈Yn because Xn ⊂ Y_n. It follows thatf_[a]=w_a/w_−a is a quotient of two linearly independent solutions of a Lam´e equation (6.1.10) and the formula forS(f_a) follows from Theorem6.2and the standard ODE theory.

Conversely, if (6.7.1) holds then ord_z=0f_a(z) = 2n. Hence a ∈ X_n by Theorem6.5.

Remark 6.8. We would like to point out the striking similarity between the solution wa to the Lam´e equation and the defining power series of complex elliptic genera in the Weierstrass form studied in [66]. In a certain context of topological field theory, complex elliptic genera serves as the genus one partition function. In contrast to it, the mean field equation studies local yet very precise analytic behavior of a genus one curve. It would be very interesting to uncover a good reason that will account for the similarity between these two theories.

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