text: of spanning trees with speciﬁc boundary conditions. The diﬀerence is that we here consider + boundary conditions instead of free ones, and more importantly, we use the boundary trick of Chelkak and Smirnov [CS12] allowing us to remove all contributions from dual spanning trees, which we could not do in [dT16]. Z 6.2 Full Z-invariant case when the isoradial graph G is 2 The goal of th(cid:0)is (cid:1)example is to r(cid:0)ela(cid:1)te our results t(cid:0)o th(cid:1)e papers [Mes06, BDC12]. When the isoradial graph G = Z2 in its regular embedding, all rhombus half-angles θ¯ are equal to π. e 4 (cid:48) 1 (cid:48) Using that sn K2 = 1(cid:48) 1 , cn K2 = (k )(cid:48)2 1 , dn K2 = (k )12 [AS64, 16.5.2], we obtain the (1+k )2 (1+k )2 (cid:32) (cid:33) following parametrization of the Ising coupling constants: (cid:48) ∀ ∈ (cid:48) 1 1 + (1 + k )12 e E, J = J (k ) = ln , e e 2 (k(cid:48))12 C∞ ∞ ∞ which is a -bijection from (0, ) to ( , 0). ∀ ∈ (cid:48) 1 The masses of the Z-massive Laplacian ∆m are(cid:0)equ(cid:1)al to [BdTR17b], v V, mv = 21(+k k)(cid:48)2 , and the survival probabilities s of the random walk associated to the Z-invariant conductances ρ and masses m are (cid:0) (cid:1) (cid:48) ∀ v ∈ V, s = 4 sc K2 = 2(k )12(cid:48) . v 4 sc K + m 1 + k 2 v As in the papers [Mes06, BDC12] the survival probabilities satisfy ∀ ∈ 2 v V, s = − . v sinh(2J ) + sinh 1(2J ) e e − In the inﬁnite case, by Theorem 55 and Corollary 48, the coeﬃcient (KF) 1 of the inverse Kasteleyn operator (KF)−1 is(cid:16)equ(cid:0)al to, (cid:1)a(cid:2)s long as a and a do(cid:3)not b(cid:0)elong t(cid:1)o(cid:2) the¯as,aame decoratio(cid:3)(cid:17)n, ¯ | − | (cid:48) − (cid:48) − − (cid:48) − − (cid:48) ∗ − ∗ (KF)¯a,1a = (k ) 1(1 + (k ))12 cn βf 2 βi G¯mv,v2 (k )12 G¯mv,v1 sn βf 2 βi (k )12 G¯mf,f,2 G¯mf,f,1 . A similar expression holds in the ﬁnite case, see Corollary 50 and Example 30. According to [Dub11, NK85], this coeﬃcient is the fermionic spinor observable of [KC71] and, up to nor- malization, the FK-spin observable of [Smi10, CS12]. Now, the proof of [BDC12] for showing the occurrence of large deviation estimates of the massive random walk in the correlation length of the spin correlations [Mes06] consists in proving that, in the super-critical regime, spin correlations can be approximated by the FK-spin observable, and then using massive harmonicity of the latter to relate it to the massive random walk. Our explicit expression for | − | (KF) 1 in the ﬁnite case gives a direct explanation of the occurrence of these large deviation a,a ¯ 92 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: spanning

text: trees

text: with

text: speciﬁc

text: boundary

text: conditions.

text: The

text: diﬀerence

text: is

text: that

text: we

text: here

text: consider

text: boundary

text: conditions

text: instead

text: of

text: free

text: ones,

text: and

text: more

text: importantly,

text: we

text: use

text: the

text: boundary

text: trick

text: of

text: Chelkak

text: and

text: Smirnov

text: allowing

text: us

text: to

text: remove

text: all

text: contributions

text: from

text: dual

text: spanning

text: trees,

text: which

text: we

text: could

text: not

text: do

text: in

text: Z

label: section

text: Full

text: Z-invariant

text: case

text: when

text: the

text: isoradial

text: graph

text: G

text: is

text: The

text: goal

text: of

text: is

text: to

text: our

text: results

text: papers

text: When

text: the

text: isoradial

text: graph

text: G

text: in

text: its

text: regular

text: embedding,

text: all

text: rhombus

text: half-angles

text: θ¯

text: are

text: equal

text: to

text: π.

text: e

text: Using

text: that

text: sn

text: cn

text: (k

text: dn

text: (k

text: we

text: obtain

text: the

text: following

text: parametrization

text: of

text: the

text: Ising

text: coupling

text: constants:

text: k

text: e

text: E,

text: J

text: J

text: (k

text: ln

text: e

text: e

text: C∞

text: which

text: is

text: a

text: -bijection

text: from

text: to

text: The

text: masses

text: of

text: the

text: Z-massive

text: Laplacian

text: ∆m

text: to

text: v

text: V,

text: mv

text: and

text: the

text: survival

text: probabilities

text: s

text: of

text: the

text: random

text: walk

text: associated

text: to

text: the

text: Z-invariant

text: conductances

text: ρ

text: and

text: masses

text: m

text: are

text: v

text: V,

text: s

text: sc

text: v

text: sc

text: K

text: m

text: k

text: v

text: As

text: in

text: the

text: papers

text: the

text: survival

text: probabilities

text: satisfy

text: v

text: V,

text: s

text: v

text: sinh

text: e

text: e

text: In

text: the

text: inﬁnite

text: case,

text: by

text: Theorem

text: and

text: Corollary

text: the

text: coeﬃcient

text: (KF)

text: of

text: the

text: inverse

text: Kasteleyn

text: operator

text: to,

text: long

text: as

text: a

text: and

text: a

text: the¯as,aame

text: (k

text: (k

text: cn

text: βf

text: βi

text: (k

text: sn

text: βf

text: βi

text: (k

text: A

text: similar

text: expression

text: holds

text: in

text: the

text: ﬁnite

text: case,

text: see

text: Corollary

text: and

text: Example

text: According

text: to

text: this

text: coeﬃcient

text: is

text: the

text: fermionic

text: spinor

text: observable

text: of

text: and,

text: up

text: to

text: nor-

text: malization,

text: the

text: FK-spin

text: observable

text: of

text: Now,

text: the

text: proof

text: of

text: for

text: showing

text: the

text: occurrence

text: of

text: large

text: deviation

text: estimates

text: of

text: the

text: massive

text: random

text: walk

text: in

text: the

text: correlation

text: length

text: of

text: the

text: spin

text: correlations

text: consists

text: in

text: proving

text: that,

text: in

text: the

text: super-critical

text: regime,

text: spin

text: correlations

text: can

text: be

text: approximated

text: by

text: the

text: FK-spin

text: observable,

text: and

text: then

text: using

text: massive

text: harmonicity

text: of

text: the

text: latter

text: to

text: relate

text: it

text: to

text: the

text: massive

text: random

text: walk.

text: Our

text: explicit

text: expression

text: for

text: (KF)

text: in

text: the

text: ﬁnite

text: case

text: gives

text: a

text: direct

text: explanation

text: of

text: the

text: occurrence

text: of

text: these

text: large

text: deviation

text: a,a

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text: Let us suppose that M and N are gauge equivalent and that the associated digraph D is ∈ C × C strongly connected. Deﬁne the function q V V on pairs of vertices of V taking values in as follows. For vertices x, y such that (x, y) is an edge of D, set N x,y q = . x,y M x,y (cid:89) For vertices x, y of D, since the digraph is strongly connected, there exists a di-path γ from x to y; set q = q (cid:48) (cid:48). x,y x ,y (cid:48) (cid:48) ∈ e=(x ,y ) γ Note that if y = x, then γ is a di-cycle and we have q = 1. x,x Remark 63. The function q is well deﬁned, i.e., independent of the choice of path from x to (cid:54) y. If y = x, then q = 1 independently of the choice of di-cycle from x to x. If y = x, x,x consider two di-paths γ , γ from x to y. Since the digraph D is strongly connected, there 1 2 exists a(cid:16)simp(cid:89)le di-path γ˜(cid:17)(cid:16)from(cid:89)y to x. Th(cid:17)en, γ1 (r(cid:16)esp. (cid:89)γ2) followed(cid:17)(cid:16)by γ˜(cid:89)is a di-cycl(cid:17)e and by deﬁnition of gauge equivalence we have, q (cid:48) (cid:48) q (cid:48) (cid:48) = 1 = q (cid:48) (cid:48) q (cid:48) (cid:48) , x ,y x ,y x ,y x ,y (cid:48) (cid:48) ∈ (cid:48) (cid:48) ∈ (cid:48) (cid:48) ∈ (cid:48) (cid:48) ∈ e=(x ,y ) γ1 e=(x ,y (cid:89)) γ˜ e(cid:89)=(x ,y ) γ2 e=(x ,y ) γ˜ implying that q (cid:48) (cid:48) = q (cid:48) (cid:48), x ,y x ,y (cid:48) (cid:48) ∈ (cid:48) (cid:48) ∈ e=(x ,y ) γ1 e=(x ,y ) γ2 and the function q is thus well deﬁned. The following lemma proves that if the associated digraph is strongly connected, gauge equiv- alence amounts to having the two matrices related through a diagonal matrix. Lemma 64. Let M, N be two matrices having the same associated digraph D and suppose that D is strongly connected. Then M and N are gauge equivalent if and only if there exists an invertible diagonal matrix D such that, − M = DND 1. Proof. Suppose that M and N are gauge equivalent. Fix a vertex x0 of D, and deﬁne Dx0 to be the diagonal matrix whose diagonal coeﬃcient Dxx,0x corresponding to the vertex x is qx0,x. Since the digraph D is strongly connected, the matrix Dx0 is invertible. Let us prove that − M = Dx0N(Dx0) 1. Non zero coeﬃcients of M and N correspond to edges of D; let (x, y) (cid:48) ∪ { } be such an edge. Consider a di-path γ from x to x, then γ = γ (x, y) is a di-path from (cid:48) 0 x to y. Using γ to compute q and γ to compute q , we deduce that, 0 x0,x x0,y − q 1 M (Dx0N(Dx0) 1) = x0,x N = N = x,y N = M . x,y x,y x,y x,y x,y q q N x0,y x,y x,y 94 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

label: paragraph

text: us

text: suppose

text: that

text: M

text: and

text: N

text: are

text: gauge

text: equivalent

text: and

text: that

text: the

text: associated

text: digraph

text: D

text: is

text: C

text: C

text: strongly

text: connected.

text: Deﬁne

text: the

text: function

text: q

text: V

text: V

text: on

text: pairs

text: of

text: vertices

text: of

text: V

text: taking

text: values

text: in

text: as

text: follows.

text: For

text: vertices

text: x,

text: y

text: such

text: that

text: (x,

text: y)

text: is

text: an

text: edge

text: of

text: D,

text: set

text: N

text: x,y

text: q

text: x,y

text: M

text: x,y

text: For

text: vertices

text: x,

text: y

text: of

text: D,

text: since

text: the

text: digraph

text: is

text: strongly

text: connected,

text: there

text: exists

text: a

text: di-path

text: γ

text: from

text: x

text: to

text: y;

text: set

text: q

text: q

text: x,y

text: x

text: ,y

text: e=(x

text: ,y

text: γ

text: Note

text: that

text: if

text: y

text: x,

text: then

text: γ

text: is

text: a

text: di-cycle

text: and

text: we

text: have

text: q

text: x,x

text: Remark

text: The

text: function

text: q

text: is

text: well

text: deﬁned,

text: i.e.,

text: independent

text: of

text: the

text: choice

text: of

text: path

text: from

text: x

text: to

text: y.

text: If

text: y

text: x,

text: then

text: q

text: independently

text: of

text: the

text: choice

text: of

text: di-cycle

text: from

text: x

text: to

text: x.

text: If

text: y

text: x,

text: x,x

text: consider

text: two

text: di-paths

text: γ

text: γ

text: from

text: x

text: to

text: y.

text: Since

text: the

text: digraph

text: D

text: is

text: strongly

text: connected,

text: there

text: exists

text: di-path

text: to

text: x.

text: γ1

text: a

text: and

text: by

text: deﬁnition

text: of

text: gauge

text: equivalence

text: we

text: have,

text: q

text: q

text: q

text: q

text: x

text: ,y

text: x

text: ,y

text: x

text: ,y

text: x

text: ,y

text: e=(x

text: ,y

text: γ1

text: e=(x

text: ,y

text: γ˜

text: ,y

text: γ2

text: e=(x

text: ,y

text: γ˜

text: implying

text: that

text: q

text: q

text: x

text: ,y

text: x

text: ,y

text: e=(x

text: ,y

text: γ1

text: e=(x

text: ,y

text: γ2

text: and

text: the

text: function

text: q

text: is

text: thus

text: well

text: deﬁned.

text: The

text: following

text: lemma

text: proves

text: that

text: if

text: the

text: associated

text: digraph

text: is

text: strongly

text: connected,

text: gauge

text: equiv-

text: alence

text: amounts

text: to

text: having

text: the

text: two

text: matrices

text: related

text: through

text: a

text: diagonal

text: matrix.

text: Lemma

text: Let

text: M,

text: N

text: be

text: two

text: matrices

text: having

text: the

text: same

text: associated

text: digraph

text: D

text: and

text: suppose

text: that

text: D

text: is

text: strongly

text: connected.

text: Then

text: M

text: and

text: N

text: are

text: gauge

text: equivalent

text: if

text: and

text: only

text: if

text: there

text: exists

text: an

text: invertible

text: diagonal

text: matrix

text: D

text: such

text: that,

text: M

text: DND

text: Proof.

text: Suppose

text: that

text: M

text: and

text: N

text: are

text: gauge

text: equivalent.

text: Fix

text: a

text: vertex

text: of

text: D,

text: and

text: deﬁne

text: to

text: be

text: the

text: diagonal

text: matrix

text: whose

text: diagonal

text: coeﬃcient

text: corresponding

text: to

text: the

text: vertex

text: x

text: is

text: Since

text: the

text: digraph

text: D

text: is

text: strongly

text: connected,

text: the

text: matrix

text: is

text: invertible.

text: Let

text: us

text: prove

text: that

text: M

text: Non

text: zero

text: coeﬃcients

text: of

text: M

text: and

text: N

text: correspond

text: to

text: edges

text: of

text: D;

text: let

text: (x,

text: y)

text: be

text: such

text: an

text: edge.

text: Consider

text: a

text: di-path

text: γ

text: from

text: x

text: to

text: x,

text: then

text: γ

text: γ

text: (x,

text: y)

text: is

text: a

text: di-path

text: from

text: x

text: to

text: y.

text: Using

text: γ

text: to

text: compute

text: q

text: and

text: γ

text: to

text: compute

text: q

text: we

text: deduce

text: that,

text: q

text: M

text: N

text: N

text: x,y

text: N

text: M

text: x,y

text: x,y

text: x,y

text: x,y

text: x,y

text: q

text: q

text: N

text: x,y

text: x,y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: KP THEORY, PLANE-BIPARTITE NETWORKS IN THE DISK, AND RATIONAL DEGENERATIONS OF M–CURVES SIMONETTA ABENDA AND PETR G. GRINEVICH Abstract. We associate real and regular algebraic–geometric data to the class of multi–line soli- ton solutions of Kadomtsev-Petviashvili II (KP) equation whose asymptotic behavior in space– time has been recently classiﬁed using the combinatorial structure of the totally non–negative part of real Grassmannians GrTNN(k, n). In our construction we directly relate real and reg- ular KP divisors on reducible M–curves to the classiﬁcation of networks parametrizing a given positroid cell in GrTNN(k, n). 2010 MSC. 37K40; 37K20, 14H50, 14H70. Keywords. Total positivity, KP hierarchy, real solitons, M-curves, Le–diagrams, planar bipar- tite networks in the disk, Baker–Akhiezer function. Contents 1. Introduction 1 2. The totally nonnegative Grassmannian 8 3. KP–II multi-line solitons 19 4. Algebraic-geometric approach for KP soliton data in GrTNN(k, n) 27 5. A system of vectors on the Le–network 43 N 6. Proof of Theorem 4.1 on Γ( ) 52 T 7. Systems of vectors for general trivalent perfect bipartite oriented networks 68 8. The eﬀect of moves and reductions on the system of vectors 87 N 9. The construction of the KP divisor on Γ( ) 93 10. The combinatorial characterization of the reality and regularity conditions for both the vacuum and the KP divisors. 115 11. The eﬀect of Postnikov moves and reductions on the KP divisor 127 This research has been partially supported by GNFM-INDAM and RFO University of Bologna, by the Russian Foundation for Basic Research, grant 17-01-00366, by the program “Fundamental problems of nonlinear dynamics”, Presidium of RAS. Partially this research was fulﬁlled during the visit of the second author (P.G.) to IHES, Universit´e Paris-Saclay, France in November 2017. 1 ##LTLine##

text: KP

text: THEORY,

text: PLANE-BIPARTITE

text: NETWORKS

text: IN

text: THE

text: DISK,

text: AND

text: RATIONAL

text: DEGENERATIONS

text: OF

text: M–CURVES

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: Abstract.

text: We

label: abstract

text: associate

text: real

text: and

text: regular

text: algebraic–geometric

text: data

text: to

text: the

text: class

text: of

text: multi–line

text: soli-

text: ton

text: solutions

text: of

text: Kadomtsev-Petviashvili

text: II

text: (KP)

text: equation

text: whose

text: asymptotic

text: behavior

text: in

text: space–

text: time

text: has

text: been

text: recently

text: classiﬁed

text: using

text: the

text: combinatorial

text: structure

text: of

text: the

text: totally

text: non–negative

text: part

text: of

text: real

text: Grassmannians

text: GrTNN(k,

text: n).

text: In

text: our

text: construction

text: we

text: directly

text: relate

text: real

text: and

text: reg-

text: ular

text: KP

text: divisors

text: on

text: reducible

text: M–curves

text: to

text: the

text: classiﬁcation

text: of

text: networks

text: parametrizing

text: a

text: given

text: positroid

text: cell

text: in

text: GrTNN(k,

text: n).

text: MSC.

text: Keywords.

text: Total

text: positivity,

text: KP

text: hierarchy,

text: real

text: solitons,

text: M-curves,

text: Le–diagrams,

text: planar

text: bipar-

text: tite

text: networks

text: in

text: the

text: disk,

text: Baker–Akhiezer

text: function.

text: Contents

text: Introduction

text: The

text: totally

text: nonnegative

text: Grassmannian

text: KP–II

text: multi-line

text: solitons

text: Algebraic-geometric

text: approach

text: for

text: KP

text: soliton

text: data

text: in

text: GrTNN(k,

text: n)

text: A

text: system

text: of

text: vectors

text: on

text: the

text: Le–network

text: N

text: Proof

text: of

text: Theorem

text: on

text: Γ(

text: T

text: Systems

text: of

text: vectors

text: for

text: general

text: trivalent

text: perfect

text: bipartite

text: oriented

text: networks

text: The

text: eﬀect

text: of

text: moves

text: and

text: reductions

text: on

text: the

text: system

text: of

text: vectors

text: N

text: The

text: construction

text: of

text: the

text: KP

text: divisor

text: on

text: Γ(

text: The

text: combinatorial

text: characterization

text: of

text: the

text: reality

text: and

text: regularity

text: conditions

text: for

text: both

text: the

text: vacuum

text: and

text: the

text: KP

text: divisors.

text: The

text: eﬀect

text: of

text: Postnikov

text: moves

text: and

text: reductions

text: on

text: the

text: KP

text: divisor

text: This

text: research

text: has

text: been

text: partially

text: supported

text: by

text: GNFM-INDAM

text: and

text: RFO

text: University

text: of

text: Bologna,

text: by

text: the

text: Russian

text: Foundation

text: for

text: Basic

text: Research,

text: grant

text: by

text: the

text: program

text: “Fundamental

text: problems

text: of

text: nonlinear

text: dynamics”,

text: Presidium

text: of

text: RAS.

text: Partially

text: this

text: research

text: was

text: fulﬁlled

text: during

text: the

text: visit

text: of

text: the

text: second

text: author

text: (P.G.)

text: to

text: IHES,

text: Universit´e

text: Paris-Saclay,

text: France

text: in

text: November

text: ##LTLine##


text: 102 SIMONETTA ABENDA AND PETR G. GRINEVICH A change of gauge ray direction may eﬀect the signs of the vacuum and of the dressed vertex wavefunctions at the given vertex V (Proposition 9.1), but not the vacuum and the dressed network divisor numbers thanks to the linear relations at V (Corollary 9.1). N (cid:48) Every change of orientation in (either induced by a change of base in the matroid or due to a change of orientation in closed cycles) corresponds to a well deﬁned change of the local CP coordinate of each copy if 1, and the dressed network divisor numbers change according to such coordinate transformation (Corollary 9.2). Therefore, all the positions of the corresponding divisor points on Γ remain invariant (Theorem 4.2). N (cid:48) The dependence of the vacuum network divisor numbers on changes of orientation in is more complicated: any change of orientation just due to changes of orientation in closed cycles corresponds to a well deﬁned change of the of the vacuum network divisor number in agreement with the corresponding change of coordinate in the component of the curve (Lemma 7.3). On the contrary, a change of base in the matroid untrivially changes the vacuum network divisor numbers and the position of the corresponding vacuum divisor points in Γ. Proposition 9.1. The dependence of the vacuum v.w. and of the dressed v.w. on N (cid:48) the gauge ray directions and on the orientation of the network Let be a network ∈ O representing [A] GrTNN(k, n) and let , l , s = 1, 2, respectively be two perfect orientations s s N (cid:48) K and two gauge ray directions of . Let be a system of phases. Let f(r)((cid:126)t) be a system of heat K hierarchy solutions generating the Darboux transformation D(k) for the soliton data ( , [A]). ∈ N (cid:48) Then, for any ﬁxed (cid:126)t and e , (1) the vacuum v.w. Φ and the dressed v.w. Ψ may only change of sign if we change the gauge ray direction: ± ± (89) Φ O ((cid:126)t) = Φ O ((cid:126)t), Ψ O ((cid:126)t) = Ψ O ((cid:126)t); e, ,l2 e, ,l1 e, ,l2 e, ,l1 N (cid:48) O O ∈ N (cid:48) (2) if we change the perfect orientation in from to , then, for any edge e there 1 2 ∈ exist real constants α , cr, r [k], such that the vacuum v.w. Φ and the dressed v.w. Ψ e e satisfy

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: A

text: change

text: of

text: gauge

text: ray

text: direction

text: may

text: eﬀect

text: the

text: signs

text: of

text: the

text: vacuum

text: and

text: of

text: the

text: dressed

text: vertex

text: wavefunctions

text: at

text: the

text: given

text: vertex

text: V

text: (Proposition

text: but

text: not

text: the

text: vacuum

text: and

text: the

text: dressed

text: network

text: divisor

text: numbers

text: thanks

text: to

text: the

text: linear

text: relations

text: at

text: V

text: (Corollary

text: N

text: Every

text: change

text: of

text: orientation

text: in

text: (either

text: induced

text: by

text: a

text: change

text: of

text: base

text: in

text: the

text: matroid

text: or

text: due

text: to

text: a

text: change

text: of

text: orientation

text: in

text: closed

text: cycles)

text: corresponds

text: to

text: a

text: well

text: deﬁned

text: change

text: of

text: the

text: local

text: CP

text: coordinate

text: of

text: each

text: copy

text: if

text: and

text: the

text: dressed

text: network

text: divisor

text: numbers

text: change

text: according

text: to

text: such

text: coordinate

text: transformation

text: (Corollary

text: Therefore,

text: all

text: the

text: positions

text: of

text: the

text: corresponding

text: divisor

text: points

text: on

text: Γ

text: remain

text: invariant

text: (Theorem

text: N

text: The

text: dependence

text: of

text: the

text: vacuum

text: network

text: divisor

text: numbers

text: on

text: changes

text: of

text: orientation

text: in

text: is

text: more

text: complicated:

text: any

text: change

text: of

text: orientation

text: just

text: due

text: to

text: changes

text: of

text: orientation

text: in

text: closed

text: cycles

text: corresponds

text: to

text: a

text: well

text: deﬁned

text: change

text: of

text: the

text: of

text: the

text: vacuum

text: network

text: divisor

text: number

text: in

text: agreement

text: with

text: the

text: corresponding

text: change

text: of

text: coordinate

text: in

text: the

text: component

text: of

text: the

text: curve

text: (Lemma

text: On

text: the

text: contrary,

text: a

text: change

text: of

text: base

text: in

text: the

text: matroid

text: untrivially

text: changes

text: the

text: vacuum

text: network

text: divisor

text: numbers

text: and

text: the

text: position

text: of

text: the

text: corresponding

text: vacuum

text: divisor

text: points

text: in

text: Γ.

text: Proposition

text: The

text: dependence

text: of

text: the

text: vacuum

text: v.w.

text: and

text: of

text: the

text: dressed

text: v.w.

text: on

text: N

text: the

text: gauge

text: ray

text: directions

text: and

text: on

text: the

text: orientation

text: of

text: the

text: network

text: Let

text: be

text: a

text: network

text: O

text: representing

text: [A]

text: GrTNN(k,

text: n)

text: and

text: let

text: l

text: s

text: respectively

text: be

text: two

text: perfect

text: orientations

text: s

text: s

text: N

text: K

text: and

text: two

text: gauge

text: ray

text: directions

text: of

text: Let

text: be

text: a

text: system

text: of

text: phases.

text: Let

text: be

text: a

text: system

text: of

text: heat

text: K

text: hierarchy

text: solutions

text: generating

text: the

text: Darboux

text: transformation

text: D(k)

text: for

text: the

text: soliton

text: data

text: [A]).

text: N

text: Then,

text: for

text: any

text: ﬁxed

text: and

text: e

text: the

text: vacuum

text: v.w.

text: Φ

text: and

text: the

text: dressed

text: v.w.

text: Ψ

text: may

text: only

text: change

text: of

text: sign

text: if

text: we

text: change

text: the

text: gauge

text: ray

text: direction:

text: Φ

text: O

text: Φ

text: O

text: Ψ

text: O

text: Ψ

text: O

text: e,

text: e,

text: e,

text: e,

text: N

text: O

text: O

text: N

text: if

text: we

text: change

text: the

text: perfect

text: orientation

text: in

text: from

text: to

text: then,

text: for

text: any

text: edge

text: e

text: there

text: exist

text: real

text: constants

text: α

text: cr,

text: r

text: [k],

text: such

text: that

text: the

text: vacuum

text: v.w.

text: Φ

text: and

text: the

text: dressed

text: v.w.

text: Ψ

text: e

text: e

text: satisfy


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 105 so that − − γ O = 1 + (−1)wind(e(32),e(22))Ψe(22),O2,l  1 = 1 + (−1)wind(e(32),e(22))α1Ψe(11),O1,l  1 dr,V, 2 (−1)wind(e(32),e(12))Ψe(12),O2,l (−1)wind(e(32),e(12))α3Ψe(31),O1,l = (cid:32)1 + α1(−α13)ww3i(n−d1(e)(3w2)i,ne(2d2)()e+(32i)n,et(1O2)1)(e(31)) (−1)windO1(e(31),e(11))γdr,V,O1(cid:33)−1 − − (cid:3) = (1 γ O ) 1. dr,V, 1 Remark 9.6. In next Section we show that the transformation of the network divisor numbers CP of Corollary 9.2 is due to the change of coordinates on the copy of 1 associated with this white vertex. Corollary 9.3. The dependence of the vacuum network divisor numbers on the change O O O O N (cid:48) of orientation Let = (I) and = (I) be two perfect orientations for network 1 1 2 2 associated to the same base I. Then at any trivalent white vertex V (93)–(95) hold true with γ O , s = 1, 2 instead of γ O . vac,V, s dr,V, s The proof follows using Lemma 7.3 and (76). Remark 9.7. Vacuum network divisor numberss and changes of base If the change of M orientation is induced by a change in the base of , then the rule for the transformation of the vacuum network divisor numbers is more complicated and we do not write it explicitly since we shall not use it in the following. ∈ Remark 9.8. Relation with the construction in [2]. Let [A] GrTP(k, n). In [2] we N construct a vacuum and a dressed divisor in Γ(ξ) which may be obtained from Γ( ) with a T convenient desingularization at certain pair of double points (see Remark 4.2). It is possible to verify that the dressed divisor points in Γ(ξ) at leading order in ξ may be obtained from the dressed network divisor numbers of Deﬁnition 9.2 using the change of coordinates associated ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: so

text: that

text: γ

text: O

text: 1

text: 1

text: dr,V,

text: γ

text: O

text: dr,V,

text: Remark

text: In

text: next

text: Section

text: we

text: show

text: that

text: the

text: transformation

text: of

text: the

text: network

text: divisor

text: numbers

text: CP

text: of

text: Corollary

text: is

text: due

text: to

text: the

text: change

text: of

text: coordinates

text: on

text: the

text: copy

text: of

text: associated

text: with

text: this

text: white

text: vertex.

text: Corollary

text: The

text: dependence

text: of

text: the

text: vacuum

text: network

text: divisor

text: numbers

text: on

text: the

text: change

text: O

text: O

text: O

text: O

text: N

text: of

text: orientation

text: Let

text: (I)

text: and

text: (I)

text: be

text: two

text: perfect

text: orientations

text: for

text: network

text: associated

text: to

text: the

text: same

text: base

text: I.

text: Then

text: at

text: any

text: trivalent

text: white

text: vertex

text: V

text: hold

text: true

text: with

text: γ

text: O

text: s

text: instead

text: of

text: γ

text: O

text: vac,V,

text: s

text: dr,V,

text: s

text: The

text: proof

text: follows

text: using

text: Lemma

text: and

text: Remark

text: Vacuum

text: network

text: divisor

text: numberss

text: and

text: changes

text: of

text: base

text: If

text: the

text: change

text: of

text: M

text: orientation

text: is

text: induced

text: by

text: a

text: change

text: in

text: the

text: base

text: of

text: then

text: the

text: rule

text: for

text: the

text: transformation

text: of

text: the

text: vacuum

text: network

text: divisor

text: numbers

text: is

text: more

text: complicated

text: and

text: we

text: do

text: not

text: write

text: it

text: explicitly

text: since

text: we

text: shall

text: not

text: use

text: it

text: in

text: the

text: following.

text: Remark

text: Relation

text: with

text: the

text: construction

text: in

text: Let

text: [A]

text: GrTP(k,

text: n).

text: In

text: we

text: N

text: construct

text: a

text: vacuum

text: and

text: a

text: dressed

text: divisor

text: in

text: Γ(ξ)

text: which

text: may

text: be

text: obtained

text: from

text: Γ(

text: with

text: a

text: T

text: convenient

text: desingularization

text: at

text: certain

text: pair

text: of

text: double

text: points

text: (see

text: Remark

text: It

text: is

text: possible

text: to

text: verify

text: that

text: the

text: dressed

text: divisor

text: points

text: in

text: Γ(ξ)

text: at

text: leading

text: order

text: in

text: ξ

text: may

text: be

text: obtained

text: from

text: the

text: dressed

text: network

text: divisor

text: numbers

text: of

text: Deﬁnition

text: using

text: the

text: change

text: of

text: coordinates

text: associated

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 110 SIMONETTA ABENDA AND PETR G. GRINEVICH ∈ denote φˆ (P,(cid:126)t) the restriction of φˆ to Γ . Then, for any P Γ , φˆ (P,(cid:126)t) takes the value l l l l of the normalized vertex function Φˆ ((cid:126)t), where e is one of the edges at V : e,I l (cid:48) ∀ φˆ (ζ(P ),(cid:126)t) = Φˆ ((cid:126)t), (cid:126)t; l e,I (4) Let Γ be the component of Γ corresponding to a trivalent white vertex with normalized l (cid:54) ∈ vacuum v.w. taking distinct values on the edges at V for some (cid:126)t = (cid:126)t (i.e. γ l 0 vac,Vl D N(cid:48) is a non trivial vacuum network divisor numbers). Let us denote φˆ (P,(cid:126)t) the vac, ,I l ∈ restriction of φˆ to Γ . Then φˆ is meromorphic of degree one in P Γ and is uniquely l l l identiﬁed by the conditions ∈ ∀ φˆ (ζ(P (m)),(cid:126)t) = Φˆ ((cid:126)t), m [3], (cid:126)t, l l em,I ∈ where e , m [3] are the edges at V : m l − Φˆ ((cid:126)t)ζ γ Φˆ ((cid:126)t) (98) φˆ (ζ(P ),(cid:126)t) = e3,I − vac,Vl e1,I , l ζ γ vac,V l with γ as in (87). vac,V l D ≥ − D By construction, for any (cid:126)t, (φ (P,(cid:126)t))+ N 0 since the n k divisor points N(cid:48) at vac vac,Γ( ) vac, D the Darboux sink points are trivial. Moreover by construction N is contained in the union vac,Γ( ) of all ovals. In the following Theorem we summarize the properties of the vacuum wavefunction directly following from its deﬁnition and provide the ﬁrst step of the proof of Theorem 4.1 on N Γ( ). D Theorem 9.1. Characterization of the vacuum divisor of φˆ on Γ(N). Let φˆ, N(cid:48), vac, D N ∈ − N on Γ( ) be as in Deﬁnition 9.5. Then on each Γ , l [g + n k] corresponding to a vac,Γ( ) l trivalent white vertex the vacuum wavefunction satisﬁes − − − φˆl(ζ(P ),(cid:126)t) = ((−1()w1i)nwdi(ne3d,(ee13),Φe1e)1Φ((cid:126)te01)((cid:126)t+)(ζ(−1)1w) i+nd((e31,e)2w)Φined2((e(cid:126)t30,)e2))(Φζ e−2(γ(cid:126)t)vζac,Vl), ∀P ∈ Γl, ∀(cid:126)t. \{ } Moreover φˆ satisﬁes the following properties of Deﬁnition 4.4 on Γ P . 0 ∈ \{ } (1) At (cid:126)t = (cid:126)t φˆ(P,(cid:126)t ) = 1 at all points P Γ P ; 0 0 0 \{ } (2) The restriction of φˆ to Γ P coincides with the following normalization of Sato’s 0 0 − (cid:80) wavefunction, φˆ(ζ(P ),(cid:126)t) = eθ(ζ,(cid:126)t (cid:126)t0), where θ(ζ,(cid:126)t) = ≥ t ζl; l 1 l ##LTLine## ##LTLine##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

label: list

text: denote

text: φˆ

text: the

text: restriction

text: of

text: φˆ

text: to

text: Γ

text: Then,

text: for

text: any

text: P

text: Γ

text: φˆ

text: takes

text: the

text: value

text: l

text: l

text: l

text: l

text: of

text: the

text: normalized

text: vertex

text: function

text: Φˆ

text: where

text: e

text: is

text: one

text: of

text: the

text: edges

text: at

text: V

text: e,I

text: l

text: φˆ

text: (ζ(P

text: Φˆ

text: l

text: e,I

text: Let

text: Γ

text: be

text: the

text: component

text: of

text: Γ

text: corresponding

text: to

text: a

text: trivalent

text: white

text: vertex

text: with

text: normalized

text: l

font: default

text: vacuum

text: v.w.

text: taking

text: distinct

text: values

text: on

text: the

text: edges

text: at

text: V

text: for

text: some

text: (i.e.

text: γ

text: l

text: vac,Vl

text: D

text: is

text: a

text: non

text: trivial

text: vacuum

text: network

text: divisor

text: numbers).

text: Let

text: us

text: denote

text: φˆ

text: the

text: vac,

text: ,I

text: l

text: restriction

text: of

text: φˆ

text: to

text: Γ

text: Then

text: φˆ

text: is

text: meromorphic

text: of

text: degree

text: one

text: in

text: P

text: Γ

text: and

text: is

text: uniquely

text: l

text: l

text: l

text: identiﬁed

text: by

text: the

text: conditions

text: φˆ

text: (ζ(P

text: Φˆ

text: m

text: l

text: l

text: em,I

text: where

text: e

text: m

text: are

text: the

text: edges

text: at

text: V

text: m

text: l

text: Φˆ

text: γ

text: Φˆ

text: φˆ

text: (ζ(P

text: vac,Vl

text: l

text: ζ

text: γ

text: vac,V

text: l

text: with

text: γ

text: as

text: in

text: vac,V

text: l

text: D

text: D

text: By

text: construction,

text: for

text: any

text: (φ

text: N

text: since

text: the

text: n

text: k

text: divisor

text: points

text: at

text: vac

text: vac,Γ(

text: vac,

text: D

text: the

text: Darboux

text: sink

text: points

text: are

text: trivial.

text: Moreover

text: by

text: construction

text: N

text: is

text: contained

text: in

text: the

text: union

text: vac,Γ(

text: of

text: all

text: ovals.

text: In

text: the

text: following

text: Theorem

text: we

text: summarize

text: the

text: properties

text: of

text: the

text: vacuum

text: wavefunction

text: directly

text: following

text: from

text: its

text: deﬁnition

text: and

text: provide

text: the

text: ﬁrst

text: step

text: of

text: the

text: proof

text: of

text: Theorem

text: on

text: N

text: Γ(

text: D

text: Theorem

text: Characterization

text: of

text: the

text: vacuum

text: divisor

text: of

text: φˆ

text: on

text: Γ(N).

text: Let

text: φˆ,

text: vac,

text: D

text: N

text: N

text: on

text: Γ(

text: be

text: as

text: in

text: Deﬁnition

text: Then

text: on

text: each

text: Γ

text: l

text: [g

text: n

text: k]

text: corresponding

text: to

text: a

text: vac,Γ(

text: l

text: trivalent

text: white

text: vertex

text: the

text: vacuum

text: wavefunction

text: satisﬁes

text: φˆl(ζ(P

text: ∀P

text: Γl,

text: Moreover

text: φˆ

text: satisﬁes

text: the

text: following

text: properties

text: of

text: Deﬁnition

text: on

text: Γ

text: P

text: At

text: at

text: all

text: points

text: P

text: Γ

text: P

text: The

text: restriction

text: of

text: φˆ

text: to

text: Γ

text: P

text: coincides

text: with

text: the

text: following

text: normalization

text: of

text: Sato’s

text: wavefunction,

text: φˆ(ζ(P

text: where

text: t

text: ζl;

text: l

text: l

text: ##LTLine##

text: ##LTLine##


text: 122 SIMONETTA ABENDA AND PETR G. GRINEVICH ≤ sources or sinks and belonging to F , and let us denote ρ m the total number of changes of s s s signs of the wavefunction occurring at the edges of F ending at the boundary of the disk. Then s ∈ (1) For any s [g] (ﬁnite oval) (cid:88)ns (cid:88)ns µ + (cid:15) (f , g ) + (cid:15) (f , g ) = 1 (mod 2), s,si wind l l cd l l l=1 l=1 (104) (cid:80) Vl white µ + ns (cid:15) (f , g ) = 0 (mod 2), s,sou l=1 int l l (cid:80) ρ + ns (cid:15) (f , g ) = 0 (mod 2). s l=1 cs l l (2) If s = 0 (inﬁnite oval) (cid:88)n0 (cid:88)n0 µ + (cid:15) (f , g ) + (cid:15) (f , g ) = 0 ( mod 2), 0,si wind l l cd l l l=1 l=1 (105) V white (cid:80) l µ + k + n0 (cid:15) (f , g ) = 0 (mod 2), 0,sou l=1 int l l (cid:80) ρ + n0 (cid:15) (f , g ) = 0 (mod 2). 0 l=1 cs l l Proof. The only untrivial statement is the ﬁrst one in both (104) and (105). We prove the statement by induction in the number of trivalent white vertexes changing direction. N (cid:48) First of all let F be an internal face of the network ,i.e. a face whose boundary does not s contain portions of the boundary of the disk. If there are no internal sources or sinks in F and s no changes of directions at the vertexes bounding F then the statement holds true since the s total winding number of the face is an odd number (cid:88)ns (cid:15) (f , g ) = 1 ( mod 2). wind l l l=1 Now suppose to insert a change of direction in F along the path π = (e , e , . . . , e − , e ) s i i+1 j 1 j ∈ − changing all directions of the edges e , r [i + 1, j 1] (see Figure 52[left]). Then the winding r of F will change by one: s − (cid:88)j 1 (cid:0) − (cid:48) (cid:48) (cid:1) (cid:15) (e , e ) (cid:15) (e , e ) = 1( mod 2). wind r r+1 wind r r+1 r=i

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: sources

text: or

text: sinks

text: and

text: belonging

text: to

text: F

text: and

text: let

text: us

text: denote

text: ρ

text: m

text: the

text: total

text: number

text: of

text: changes

text: of

text: s

text: s

text: s

text: signs

text: of

text: the

text: wavefunction

text: occurring

text: at

text: the

text: edges

text: of

text: F

text: ending

text: at

text: the

text: boundary

text: of

text: the

text: disk.

text: Then

text: s

text: For

text: any

text: s

text: [g]

text: (ﬁnite

text: oval)

label: equation

text: µ

text: (f

text: g

text: (f

text: g

text: (mod

text: s,si

text: wind

text: l

text: l

text: cd

text: l

text: l

text: Vl

text: white

text: µ

text: ns

text: (f

text: g

text: (mod

text: s,sou

text: int

text: l

text: l

text: ρ

text: ns

text: (f

text: g

text: (mod

text: s

text: cs

text: l

text: l

text: If

text: s

text: (inﬁnite

text: oval)

text: µ

text: (f

text: g

text: (f

text: g

text: mod

text: wind

text: l

text: l

text: cd

text: l

text: l

text: V

text: white

text: l

text: µ

text: k

text: (f

text: g

text: (mod

text: int

text: l

text: l

text: ρ

text: (f

text: g

text: (mod

text: cs

text: l

text: l

text: Proof.

text: The

text: only

text: untrivial

text: statement

text: is

text: the

text: ﬁrst

text: one

text: in

text: both

text: and

text: We

text: prove

text: the

text: statement

text: by

text: induction

text: in

text: the

text: number

text: of

text: trivalent

text: white

text: vertexes

text: changing

text: direction.

text: N

text: First

text: of

text: all

text: let

text: F

text: be

text: an

text: internal

text: face

text: of

text: the

text: network

text: ,i.e.

text: a

text: face

text: whose

text: boundary

text: does

text: not

text: s

text: contain

text: portions

text: of

text: the

text: boundary

text: of

text: the

text: disk.

text: If

text: there

text: are

text: no

text: internal

text: sources

text: or

text: sinks

text: in

text: F

text: and

text: s

text: no

text: changes

text: of

text: directions

text: at

text: the

text: vertexes

text: bounding

text: F

text: then

text: the

text: statement

text: holds

text: true

text: since

text: the

text: s

text: total

text: winding

text: number

text: of

text: the

text: face

text: is

text: an

text: odd

text: number

text: (f

text: g

text: mod

text: wind

text: l

text: l

text: Now

text: suppose

text: to

text: insert

text: a

text: change

text: of

text: direction

text: in

text: F

text: along

text: the

text: path

text: π

text: (e

text: e

text: e

text: e

text: s

text: i

text: j

text: j

text: changing

text: all

text: directions

text: of

text: the

text: edges

text: e

text: r

text: [i

text: j

text: (see

text: Figure

text: Then

text: the

text: winding

text: r

text: of

text: F

text: will

text: change

text: by

text: one:

text: s

text: (e

text: e

text: (e

text: e

text: mod

text: wind

text: r

text: wind

text: r

text: r=i


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 123 If the oval F intersects the boundary of the disk, then each boundary sink edge in F increases s s by one the counter of the trivalent white vertexes changing of direction and keeps the winding of F invariant. If we move a boundary source edge from F to the other face, then the overall s s contribution from the edges at such vertex due to winding and changes of direction is the same at each face before and after such move (see Figure 52[middle]). Finally it is straightforward to check that internal black sources behave like boundary Darboux sources (see Figure 52[right]). The proof in the case s = 0 (inﬁnite oval) goes along the same lines with the only diﬀerence that in the simplest case where there are neither internal sources nor sinks in F nor changes of 0 directions at the internal vertexes bounding F then 0 (cid:88)n0 (cid:15) (f , g ) = 0 ( mod 2). wind l l l=1 (cid:3)

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: If

text: the

text: oval

text: F

text: intersects

text: the

text: boundary

text: of

text: the

text: disk,

text: then

text: each

text: boundary

text: sink

text: edge

text: in

text: F

text: increases

text: s

text: s

text: by

text: one

text: the

text: counter

text: of

text: the

text: trivalent

text: white

text: vertexes

text: changing

text: of

text: direction

text: and

text: keeps

text: the

text: winding

text: of

text: F

text: invariant.

text: If

text: we

text: move

text: a

text: boundary

text: source

text: edge

text: from

text: F

text: to

text: the

text: other

text: face,

text: then

text: the

text: overall

text: s

text: s

text: contribution

text: from

text: the

text: edges

text: at

text: such

text: vertex

text: due

text: to

text: winding

text: and

text: changes

text: of

text: direction

text: is

text: the

text: same

text: at

text: each

text: face

text: before

text: and

text: after

text: such

text: move

text: (see

text: Figure

text: Finally

text: it

text: is

text: straightforward

text: to

text: check

text: that

text: internal

text: black

text: sources

text: behave

text: like

text: boundary

text: Darboux

text: sources

text: (see

text: Figure

text: The

text: proof

text: in

text: the

text: case

text: s

text: (inﬁnite

text: oval)

text: goes

text: along

text: the

text: same

text: lines

text: with

text: the

text: only

text: diﬀerence

text: that

text: in

text: the

text: simplest

text: case

text: where

text: there

text: are

text: neither

text: internal

text: sources

text: nor

text: sinks

text: in

text: F

text: nor

text: changes

text: of

text: directions

text: at

text: the

text: internal

text: vertexes

text: bounding

text: F

text: then

text: (f

text: g

text: mod

text: wind

text: l

text: l


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 125 D ⊂ N ∈ For simplicity, we use the same symbol Γ( ) to denote the set of divisor points P Γ , l l ∈ − l [g+n k], such that ζ(P ) = γ , with ζ the coordinate on Γ associated to the given orientation l l l of the network. ∈ N D In every oval Ω , s [0, g], of Γ( ), ν denotes the number of divisor points of belonging s s to the oval: D ∩ ν = #( ∂Ω ), s s µ , µ and µ respectively denote the total number of Darboux univalent vertexes, of Darboux s s,sou s,si source univalent vertexes and of Darboux sink univalent vertexes belonging to F . s ∩ CP To count the number of divisor points on Γ Ω , where Γ is the 1 component corresponding 0 s 0 ∈ to the boundary of the disk, we use the index ρ , s [g]. ρ is simply the total number of s s changes of signs of the vertex wavefunction on the edges at the boundary vertexes belonging to the intersection of F with the boundary of the disk. We remark that, by construction, ρ = 0 for s s ∈ D all s [0, g], if is the vacuum network divisor. Theorem 10.2. The number of divisor points in each oval (Theorem 4.1, items (1b)- N (cid:48) D ∈ (1c). Let , Γ, , ν , µ , µ µ and ρ , s [g] be as above. Then s s s,sou s,si s  ∈  1 ( mod 2) if s [g], (106) ν + µ + ρ = s s s  k ( mod 2) if s = 0. ∈ Proof. The proof follows applying Theorem 10.1 and Lemmas 10.1–10.2. For any ﬁxed s [g] ∈ (s) let (V ; f , g ), r [n ], be the set of pair of edges f , g at internal vertexes V bounding F , r r r s r r r s where [n ] is the total number of pair of edges bounding F . We remark that a given vertex V s s may appear twice if two pairs of edges at V belong to the boundary of F (see Figure 49). s ∈ D ∩ By construction, a divisor point P belongs to Γ Ω if and only if Γ corresponds to a l l s l (s) (s) (s) white trivalent vertex V carrying an odd index at the edges (V , f , g ) bounding F (see l l l l s D ∩ Theorem 10.1). Therefore the number ν of divisor points in Ω satisﬁes s s

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: D

text: N

text: For

text: simplicity,

text: we

text: use

text: the

text: same

text: symbol

text: Γ(

text: to

text: denote

text: the

text: set

text: of

text: divisor

text: points

text: P

text: Γ

text: l

text: l

text: l

text: [g+n

text: k],

text: such

text: that

text: ζ(P

text: γ

text: with

text: ζ

text: the

text: coordinate

text: on

text: Γ

text: associated

text: to

text: the

text: given

text: orientation

text: l

text: l

text: l

text: of

text: the

text: network.

text: N

text: D

text: In

text: every

text: oval

text: Ω

text: s

text: g],

text: of

text: Γ(

text: ν

text: denotes

text: the

text: number

text: of

text: divisor

text: points

text: of

text: belonging

text: s

text: s

text: to

text: the

text: oval:

text: D

text: ν

text: ∂Ω

text: s

text: s

text: µ

text: µ

text: and

text: µ

text: respectively

text: denote

text: the

text: total

text: number

text: of

text: Darboux

text: univalent

text: vertexes,

text: of

text: Darboux

text: s

text: s,sou

text: s,si

text: source

text: univalent

text: vertexes

text: and

text: of

text: Darboux

text: sink

text: univalent

text: vertexes

text: belonging

text: to

text: F

text: s

text: CP

text: To

text: count

text: the

text: number

text: of

text: divisor

text: points

text: on

text: Γ

text: Ω

text: where

text: Γ

text: is

text: the

text: component

text: corresponding

text: s

text: to

text: the

text: boundary

text: of

text: the

text: disk,

text: we

text: use

text: the

text: index

text: ρ

text: s

text: [g].

text: ρ

text: is

text: simply

text: the

text: total

text: number

text: of

text: s

text: s

text: changes

text: of

text: signs

text: of

text: the

text: vertex

text: wavefunction

text: on

text: the

text: edges

text: at

text: the

text: boundary

text: vertexes

text: belonging

text: to

text: the

text: intersection

text: of

text: F

text: with

text: the

text: boundary

text: of

text: the

text: disk.

text: We

text: remark

text: that,

text: by

text: construction,

text: ρ

text: for

text: s

text: s

text: D

text: all

text: s

text: g],

text: if

text: is

text: the

text: vacuum

text: network

text: divisor.

text: Theorem

text: The

text: number

text: of

text: divisor

text: points

text: in

text: each

text: oval

text: (Theorem

text: items

text: N

text: D

text: Let

text: Γ,

text: ν

text: µ

text: µ

text: µ

text: and

text: ρ

text: s

text: [g]

text: be

text: as

text: above.

text: Then

text: s

text: s

text: s,sou

text: s,si

text: s

text: mod

text: if

text: s

text: [g],

text: ν

text: µ

text: ρ

text: s

text: s

text: s

text: k

text: mod

text: if

text: s

text: Proof.

text: The

text: proof

text: follows

text: applying

text: Theorem

text: and

text: Lemmas

text: For

text: any

text: ﬁxed

text: s

text: [g]

text: (s)

text: let

text: (V

text: f

text: g

text: r

text: [n

text: be

text: the

text: set

text: of

text: pair

text: of

text: edges

text: f

text: g

text: at

text: internal

text: vertexes

text: V

text: bounding

text: F

text: r

text: r

text: r

text: s

text: r

text: r

text: r

text: s

text: where

text: [n

text: is

text: the

text: total

text: number

text: of

text: pair

text: of

text: edges

text: bounding

text: F

text: We

text: remark

text: that

text: a

text: given

text: vertex

text: V

text: s

text: s

text: may

text: appear

text: twice

text: if

text: two

text: pairs

text: of

text: edges

text: at

text: V

text: belong

text: to

text: the

text: boundary

text: of

text: F

text: (see

text: Figure

text: s

text: D

text: By

text: construction,

text: a

text: divisor

text: point

text: P

text: belongs

text: to

text: Γ

text: Ω

text: if

text: and

text: only

text: if

text: Γ

text: corresponds

text: to

text: a

text: l

text: l

text: s

text: l

text: (s)

text: (s)

text: (s)

text: white

text: trivalent

text: vertex

text: V

text: carrying

text: an

text: odd

text: index

text: at

text: the

text: edges

text: (V

text: f

text: g

text: bounding

text: F

text: (see

text: l

text: l

text: l

text: l

text: s

text: D

text: Theorem

text: Therefore

text: the

text: number

text: ν

text: of

text: divisor

text: points

text: in

text: Ω

text: satisﬁes

text: s

text: s


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 129 The proof is omitted. In Figure 53 we show the position of the poles before and after the square move using the formulas of the example in Section . In such case α(1) α(1) α(1)(1 + α(2)c((cid:126)t )) 1 γ(1) = 2 , γ(1) = 1 , γ(2) = 4 4 0 , γ(2) = , 1 1 + α(2)c((cid:126)t ) 2 α(1) + c((cid:126)t ) 1 α(1) + c((cid:126)t ) 2 1 + α(2)c((cid:126)t ) 4 0 4 0 4 0 4 0 where c((cid:126)t ) = Φ ((cid:126)t )/Φ ((cid:126)t ) is the ratio of the dressed wavefunction at the edges corresponding 0 U4 0 U3 0 to vectors U and U . The colors of poles correspond to the following conditions on c((cid:126)t ): 3 4 0 Table 2. The square move and the eﬀective vacuum or dressed divisor Position of poles in Γ(1) Position of poles in Γ(2) Color used Range of parameter ∈ ∈ ∈ ∈ γ(1) Ω(1), γ(1) Ω(1) γ(2) Ω(2), γ(2) Ω(2) Green c((cid:126)t ) > 0 1 5 2 4 1 5 2 4 0 ∈ ∈ ∈ ∈ − γ(1) Ω(1), γ(1) Ω(1) γ(2) Ω(2), γ(2) Ω(2) Red α(1) < c((cid:126)t ) < 0 1 5 2 2 1 2 2 5 4 0 ∈ ∈ ∈ ∈ − − − γ(1) Ω(1), γ(1) Ω(1) γ(2) Ω(2), γ(2) Ω(2) Blue (α(2)) 1 < c((cid:126)t ) < α(1) 1 1 2 5 1 1 2 5 4 0 4 ∈ ∈ ∈ ∈ − − γ(1) Ω(1), γ(1) Ω(1) γ(2) Ω(2), γ(2) Ω(2) Pink c((cid:126)t ) < (α(2)) 1 1 3 2 5 1 5 2 3 0 4 The ﬂip move at black vertexes leaves the divisor invariant. The ﬂip move at white vertexes may transform trivial divisor points into untrivial divisor points. In the following lemma we label ∈ Ω , l [4], the ovals involved in the ﬂip move. l N (cid:48) Lemma 11.2. The eﬀect of the ﬂip move at white vertexes (M2) on the divisor If 2 N (cid:48) D N (cid:48) is obtained from via a ﬂip move move (M2) at a pair of trivalent white vertexes. Let ( ) 1 l either be the vacuum or dressed network divisor for l = 1, 2. Then (1) the number of ovals is invariant: g(2) = g(1); (2) the number of untrivial divisor points is invariant in every oval except possibly at the ∈ ovals involved in the move. In the ovals Ω , l [4] the parity of the number of divisor l − ∈ (2) (2) points before and after the move is invariant: ν ν = 0 mod(2), l [4]; (cid:16) (cid:17) l l D N (cid:48) D N (cid:48) \{ } ∪ { } (1) (1) (2) (2) (s) (s) (3) ( ) = ( ) γ , γ γ , γ , where γ , l = 1, 2, is the pole on Γ , the 2 1 1 2 1 2 l l CP N (cid:48) copy of 1 associated to each white vertex involved in the ﬂip move on , s = 1, 2; s The proof is omitted. In Figure 54 we show the position of the poles before and after the ﬂip move [top-left], in function of the relative signs of the value of the (vacuum or dressed) wave ≡ ∈ function Ψ(1) Ψ(1)((cid:126)t ), l [4], at the double points of Γ. Faces f correspond to ovals Ω , the l l 0 l l orientation of the edges in the graph at each vertex induces the local coordinates at each copy of ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: The

text: proof

text: is

text: omitted.

text: In

text: Figure

text: we

text: show

text: the

text: position

text: of

text: the

text: poles

text: before

text: and

text: after

text: the

text: square

text: move

text: using

text: the

text: formulas

text: of

text: the

text: example

text: in

text: Section

text: In

text: such

text: case

text: where

text: Φ

text: )/Φ

text: is

text: the

text: ratio

text: of

text: the

text: dressed

text: wavefunction

text: at

text: the

text: edges

text: corresponding

text: to

text: vectors

text: U

text: and

text: U

text: The

text: colors

text: of

text: poles

text: correspond

text: to

text: the

text: following

text: conditions

text: on

text: Table

text: The

text: square

text: move

text: and

text: the

text: eﬀective

text: vacuum

text: or

text: dressed

text: divisor

text: Position

text: of

text: poles

text: in

text: Position

text: of

text: poles

text: in

text: Color

text: used

text: Range

text: of

text: parameter

text: Green

text: Red

text: Blue

text: Pink

text: The

text: ﬂip

text: move

text: at

text: black

text: vertexes

text: leaves

text: the

text: divisor

text: invariant.

text: The

text: ﬂip

text: move

text: at

text: white

text: vertexes

text: may

text: transform

text: trivial

text: divisor

text: points

text: into

text: untrivial

text: divisor

text: points.

text: In

text: the

text: following

text: lemma

text: we

text: label

text: Ω

text: l

text: the

text: ovals

text: involved

text: in

text: the

text: ﬂip

text: move.

text: l

text: N

text: Lemma

text: The

text: eﬀect

text: of

text: the

text: ﬂip

text: move

text: at

text: white

text: vertexes

text: on

text: the

text: divisor

text: If

text: N

text: D

text: N

text: is

text: obtained

text: from

text: via

text: a

text: ﬂip

text: move

text: move

text: at

text: a

text: pair

text: of

text: trivalent

text: white

text: vertexes.

text: Let

text: l

text: either

text: be

text: the

text: vacuum

text: or

text: dressed

text: network

text: divisor

text: for

text: l

text: Then

text: the

text: number

text: of

text: ovals

text: is

text: invariant:

text: the

text: number

text: of

text: untrivial

text: divisor

text: points

text: is

text: invariant

text: in

text: every

text: oval

text: except

text: possibly

text: at

text: the

text: ovals

text: involved

text: in

text: the

text: move.

text: In

text: the

text: ovals

text: Ω

text: l

text: the

text: parity

text: of

text: the

text: number

text: of

text: divisor

text: l

text: points

text: before

text: and

text: after

text: the

text: move

text: is

text: invariant:

text: ν

text: ν

text: l

text: l

text: l

text: D

text: N

text: D

text: N

text: (s)

text: (s)

text: γ

text: γ

text: γ

text: γ

text: where

text: γ

text: l

text: is

text: the

text: pole

text: on

text: Γ

text: the

text: l

text: l

text: CP

text: N

text: copy

text: of

text: associated

text: to

text: each

text: white

text: vertex

text: involved

text: in

text: the

text: ﬂip

text: move

text: on

text: s

text: s

text: The

text: proof

text: is

text: omitted.

text: In

text: Figure

text: we

text: show

text: the

text: position

text: of

text: the

text: poles

text: before

text: and

text: after

text: the

text: ﬂip

text: move

text: [top-left],

text: in

text: function

text: of

text: the

text: relative

text: signs

text: of

text: the

text: value

text: of

text: the

text: (vacuum

text: or

text: dressed)

text: wave

text: function

text: l

text: at

text: the

text: double

text: points

text: of

text: Γ.

text: Faces

text: f

text: correspond

text: to

text: ovals

text: Ω

text: the

text: l

text: l

text: l

text: l

text: orientation

text: of

text: the

text: edges

text: in

text: the

text: graph

text: at

text: each

text: vertex

text: induces

text: the

text: local

text: coordinates

text: at

text: each

text: copy

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 132 SIMONETTA ABENDA AND PETR G. GRINEVICH ∈ ∩ (1) (1) (1) (3) The trivial eﬀective divisor point γ Γ Ω is removed; red red red (4) The positions of all other eﬀective divisor points are not eﬀected by the reduction. In Figure 55 we show the eﬀect of the parallel edge reduction. The relations between the edge vectors in the initial conﬁguration − − (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) E = w E , E = w E , E = w (E E ), 2 2 4 3 3 4 1 1 2 3 correspond to the following relations of the wavefunction in the initial conﬁguration − − ∀ Ψ(1)((cid:126)t) = w(1)Ψ(1)((cid:126)t), Ψ(1)((cid:126)t) = w(1)Ψ(1)((cid:126)t), Ψ(1)((cid:126)t) = w(1)(Ψ(1)((cid:126)t) Ψ(1)((cid:126)t)), (cid:126)t, 2 2 4 3 3 4 1 1 2 3 (1) so that the divisor point in Γ is trivial red − Ψ(1)((cid:126)t ) w(1) γ(1) = 3 − 0 = 3 . red Ψ(1)((cid:126)t ) Ψ(1)((cid:126)t ) w(1) + w(1) 2 0 3 0 2 3 (1) (1) On both Γ and Σ the normalized wavefunction is independent of the spectral parameter red red Ψ(1)((cid:126)t) ≡ Ψ(1)((cid:126)t) and takes the value 1 4 . After the reduction Ψ(11)((cid:126)t0) Ψ(41)((cid:126)t0) E(2) = E(1), Ψ(2)((cid:126)t) = Ψ(1)((cid:126)t), 1 1 1 1 so that, for any (cid:126)t, the normalized wavefunction keeps the same value after the reduction at the double point corresponding to such edge. The dipole reduction (R2) corresponds to the removal of an edge carrying a null vector and leaves invariant the number of ovals and the position of the divisor point. In the leaf reduction (R3) the only non-trivial case corresponds to the situation where the faces f , f are distinct in the initial conﬁguration. In this case we eliminate one oval and one 1 2 divisor point. N (cid:48) N (cid:48) Lemma 11.5. The eﬀect of (R3) on the divisor If is obtained from via reduction 2 1 D N (cid:48) (R3) and the faces f , f be distinct. Let ( ) either be the vacuum or dressed network divisor 1 2 l for l = 1, 2. Then − (1) the number of ovals diminishes by 1: g(2) = g(1) 1; (2) D(N (cid:48)) = D(N (cid:48))\{γ(1)}, where γ(1) is the pole on Γ(1) – the copy of CP1 associated to the 2 1 1 1 1 white vertex involved in the reduction. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: The

text: trivial

text: eﬀective

text: divisor

text: point

text: γ

text: Γ

text: Ω

text: is

text: removed;

text: red

text: red

text: red

text: The

text: positions

text: of

text: all

text: other

text: eﬀective

text: divisor

text: points

text: are

text: not

text: eﬀected

text: by

text: the

text: reduction.

text: In

text: Figure

text: we

text: show

text: the

text: eﬀect

text: of

text: the

text: parallel

text: edge

text: reduction.

text: The

text: relations

text: between

text: the

text: edge

text: vectors

text: in

text: the

text: initial

text: conﬁguration

text: E

text: w

text: E

text: E

text: w

text: E

text: E

text: w

text: (E

text: E

text: correspond

text: to

text: the

text: following

text: relations

text: of

text: the

text: wavefunction

text: in

text: the

text: initial

text: conﬁguration

text: so

text: that

text: the

text: divisor

text: point

text: in

text: Γ

text: is

text: trivial

text: red

text: red

text: On

text: both

text: Γ

text: and

text: Σ

text: the

text: normalized

text: wavefunction

text: is

text: independent

text: of

text: the

text: spectral

text: parameter

text: red

text: red

text: and

text: takes

text: the

text: value

text: After

text: the

text: reduction

text: so

text: that,

text: for

text: any

text: the

text: normalized

text: wavefunction

text: keeps

text: the

text: same

text: value

text: after

text: the

text: reduction

text: at

text: the

text: double

text: point

text: corresponding

text: to

text: such

text: edge.

text: The

text: dipole

text: reduction

text: corresponds

text: to

text: the

text: removal

text: of

text: an

text: edge

text: carrying

text: a

text: vector

text: and

text: leaves

text: invariant

text: the

text: number

text: of

text: ovals

text: and

text: the

text: position

text: of

text: the

text: divisor

text: point.

text: In

text: the

text: leaf

text: reduction

text: the

text: only

text: non-trivial

text: case

text: corresponds

text: to

text: the

text: situation

text: where

text: the

text: faces

text: f

text: f

text: are

text: distinct

text: in

text: the

text: initial

text: conﬁguration.

text: In

text: this

text: case

text: we

text: eliminate

text: one

text: oval

text: and

text: one

text: divisor

text: point.

text: N

text: N

text: Lemma

text: The

text: eﬀect

text: of

text: on

text: the

text: divisor

text: If

text: is

text: obtained

text: from

text: via

text: reduction

text: D

text: N

text: and

text: the

text: faces

text: f

text: f

text: be

text: distinct.

text: Let

text: either

text: be

text: the

text: vacuum

text: or

text: dressed

text: network

text: divisor

text: l

text: for

text: l

text: Then

text: the

text: number

text: of

text: ovals

text: diminishes

text: by

text: D(N

text: D(N

text: where

text: is

text: the

text: pole

text: on

text: the

text: copy

text: of

text: associated

text: to

text: the

text: white

text: vertex

text: involved

text: in

text: the

text: reduction.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 26 SIMONETTA ABENDA AND PETR G. GRINEVICH Ω1 Ω2 Ω1 Ω2 Ω Ω 0 0 Figure 8. Left: a regular M-curve, g = 2, 3 real ovals (painted green), involution σ is reﬂection, orthogonal to the blue plane. Right: degeneration of a genus 2 M-curve, 3 real ovals (painted green). Soliton KP solutions can be obtained from the ﬁnite-gap ones by proper degenerations of the spectral curve [40], [15]. The suﬃciency part of the Theorem from [16] remains valid if the spectral curve Γ degenerates in such a way that the divisor remains in the ﬁnite ovals at a ﬁnite distance from the essential singularity [16]. But this condition is not necessary for degenerate curves. Let us recall that in our construction the real part of the rational M-curve is represented by planar diagram. We assume that each real oval corresponds to a face of this diagram (see [2] for additional details). In this paper we construct the rational M-curves in Section from the graphs associated with the given point of the totally-nonnegative Grassmannian GrTNN(k, n). Each face of the graph corresponds to real oval. Having this in mind, we deﬁne the Baker-Akhiezer function associated with a given soliton data. Deﬁnition 3.3. Real regular algebraic-geometrical data associated with a given soli- ton solution. Assume that we have a reducible connected curve Γ with a marked point P , a 0 D \ local parameter 1/ζ near P , a non-special divisor on Γ P and a normalized Baker-Ahkiezer 0 0 K function ψˆ associated to such data. Assume also that we have ﬁxed soliton data ( , [A]), where K ∈ is a collection of real phases κ < κ < . . . < κ , [A] GrTNN(k, n). 1 2 n K ψˆ is the KP wave function on (Γ, P , ζ) associated to the soliton data ( , [A]) if the irreducible 0 component Γ of Γ containing P is CP1, and the restriction of ψˆ to Γ coincides with Sato 0 0 0 K normalized dressed wave function for the soliton data ( , [A]). ##LTFigure## ##LTFigure##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: Ω1

text: Ω2

text: Ω1

text: Ω2

text: Ω

text: Ω

text: Figure

text: Left:

label: caption

text: a

text: regular

text: M-curve,

text: g

text: real

text: ovals

text: (painted

text: green),

text: involution

text: σ

text: is

text: reﬂection,

text: orthogonal

text: to

text: the

text: blue

text: plane.

text: Right:

text: degeneration

text: of

text: a

text: genus

text: M-curve,

text: real

text: ovals

text: (painted

text: green).

text: Soliton

text: KP

text: solutions

text: can

text: be

text: obtained

text: from

text: the

text: ﬁnite-gap

text: ones

text: by

text: proper

text: degenerations

text: of

text: the

text: spectral

text: curve

text: The

text: suﬃciency

text: part

text: of

text: the

text: Theorem

text: from

text: remains

text: valid

text: if

text: the

text: spectral

text: curve

text: Γ

text: degenerates

text: in

text: such

text: a

text: way

text: that

text: the

text: divisor

text: remains

text: in

text: the

text: ﬁnite

text: ovals

text: at

text: a

text: ﬁnite

text: distance

text: from

text: the

text: essential

text: singularity

text: But

text: this

text: condition

text: is

text: not

text: necessary

text: for

text: degenerate

text: curves.

text: Let

text: us

text: recall

text: that

text: in

text: our

text: construction

text: the

text: real

text: part

text: of

text: the

text: rational

text: M-curve

text: is

text: represented

text: by

text: planar

text: diagram.

text: We

text: assume

text: that

text: each

text: real

text: oval

text: corresponds

text: to

text: a

text: face

text: of

text: this

text: diagram

text: (see

text: for

text: additional

text: details).

text: In

text: this

text: paper

text: we

text: construct

text: the

text: rational

text: M-curves

text: in

text: Section

text: from

text: the

text: graphs

text: associated

text: with

text: the

text: given

text: point

text: of

text: the

text: totally-nonnegative

text: Grassmannian

text: GrTNN(k,

text: n).

text: Each

text: face

text: of

text: the

text: graph

text: corresponds

text: to

text: real

text: oval.

text: Having

text: this

text: in

text: mind,

text: we

text: deﬁne

text: the

text: Baker-Akhiezer

text: function

text: associated

text: with

text: a

text: given

text: soliton

text: data.

text: Deﬁnition

text: Real

text: regular

text: algebraic-geometrical

text: data

text: associated

text: with

text: a

text: given

text: soli-

text: ton

text: solution.

text: Assume

text: that

text: we

text: have

text: a

text: reducible

text: connected

text: curve

text: Γ

text: with

text: a

text: marked

text: point

text: P

text: a

text: D

text: local

text: parameter

text: near

text: P

text: a

text: non-special

text: divisor

text: on

text: Γ

text: P

text: and

text: a

text: normalized

text: Baker-Ahkiezer

text: K

text: function

text: ψˆ

text: associated

text: to

text: such

text: data.

text: Assume

text: also

text: that

text: we

text: have

text: ﬁxed

text: soliton

text: data

text: [A]),

text: where

text: K

text: is

text: a

text: collection

text: of

text: real

text: phases

text: κ

text: κ

text: κ

text: [A]

text: GrTNN(k,

text: n).

text: n

text: K

text: ψˆ

text: is

text: the

text: KP

text: wave

text: function

text: on

text: (Γ,

text: P

text: ζ)

text: associated

text: to

text: the

text: soliton

text: data

text: [A])

text: if

text: the

text: irreducible

text: component

text: Γ

text: of

text: Γ

text: containing

text: P

text: is

text: and

text: the

text: restriction

text: of

text: ψˆ

text: to

text: Γ

text: coincides

text: with

text: Sato

text: K

text: normalized

text: dressed

text: wave

text: function

text: for

text: the

text: soliton

text: data

text: [A]).

text: ##LTFigure##

text: ##LTFigure##


text: 28 SIMONETTA ABENDA AND PETR G. GRINEVICH D The dressing operator, deﬁnes the real k point divisor (Sato divisor) (k)((cid:126)t). To ﬁx ideas, let S K ∈ us suppose that the soliton data ( , [A]) be regular and irreducible and that [A] GrTNN(k, n) | | belongs to a positroid cell of dimension D (see [52] and Section ). Irreducibility implies that | | ≥ { − } D max k, n k . As a consequence, in general, the Sato divisor does not parametrize the whole positroid cell to which [A] belongs to, so that, in general, we cannot expect to reconstruct the soliton data [A] from the Sato divisor. In the present paper, we do the following Main construction Assume we are given a real regular bounded multiline KP soliton solution generated by the following soliton data: K { · · · } (1) A set of n real ordered phases = κ < κ < < κ ; 1 2 n ∈ S ⊂ S | | (2) A point [A] MTNN GrTNN(k, n), where MTNN is a positroid stratum of dimension D . Our purpose is to associate an algebro-geometric data to this soliton data. N As the ﬁrst step we choose , a planar connected orientable bipartite trivalent perfect network in the disk in Postnikov equivalence class for [A]. Of course, this choice is not unique, and this is in perfect agreement with the fact that the same soliton solution can be obtained from diﬀerent degenerate algebro-geometrical data. In particular, following Postnikov [52], one can choose the N N maximally reduced network = , but any other network from the Postnikov equivalence T class for [A] can be used as well. At the second step we associate the following algebraic-geometrical objects to each triple K N ( , [A], ): N (1) A reducible curve Γ = Γ( ) which is the rational degeneration of a smooth M–curve of ≥ | | N genus g D , where g + 1 is the number of faces of . In our approach, the curve Γ is 0 one of the irreducible components of Γ. The marked point P belongs to the intersection 0 of Γ with an oval (inﬁnite oval); 0 D K (2) A unique real and regular degree g non–special KP divisor ( , [A]) such that any KP,Γ D K ∩ ﬁnite oval contains exactly one divisor point and ( , [A]) Γ coincides with Sato KP,Γ 0 divisor; (3) A unique KP wave–function ψˆ as in Deﬁnition 3.3 such that \{ } (a) Its restriction to Γ P coincides with the normalized Sato wave function (15); 0 0 ≤ D K (b) Its pole divisor has degree d g and is contained in ( , [A]). KP,Γ N N N In particular, if = is the trivalent bipartite Le–network, then Γ( ) is the rational T T | | | | | | degeneration of on M–curve of minimal genus D and has exactly D + 1 ovals and d = g = D .

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: D

text: The

text: dressing

text: operator,

text: deﬁnes

text: the

text: real

text: k

text: point

text: divisor

text: (Sato

text: divisor)

text: To

text: ﬁx

text: ideas,

text: let

text: S

text: K

text: us

text: suppose

text: that

text: the

text: soliton

text: data

text: [A])

text: be

text: regular

text: and

text: irreducible

text: and

text: that

text: [A]

text: GrTNN(k,

text: n)

text: belongs

text: to

text: a

text: positroid

text: cell

text: of

text: dimension

text: D

text: (see

text: and

text: Section

text: Irreducibility

text: implies

text: that

text: D

text: max

text: k,

text: n

text: k

text: As

text: a

text: consequence,

text: in

text: general,

text: the

text: Sato

text: divisor

text: does

text: not

text: parametrize

text: the

text: whole

text: positroid

text: cell

text: to

text: which

text: [A]

text: belongs

text: to,

text: so

text: that,

text: in

text: general,

text: we

text: cannot

text: expect

text: to

text: reconstruct

text: the

text: soliton

text: data

text: [A]

text: from

text: the

text: Sato

text: divisor.

text: In

text: the

text: present

text: paper,

text: we

text: do

text: the

text: following

text: Main

text: construction

text: Assume

text: we

text: are

text: given

text: a

text: real

text: regular

text: bounded

text: multiline

text: KP

text: soliton

text: solution

text: generated

text: by

text: the

text: following

text: soliton

text: data:

text: K

text: A

text: set

text: of

text: n

text: real

text: ordered

text: phases

text: κ

text: κ

text: κ

text: n

text: S

text: S

text: A

text: point

text: [A]

text: MTNN

text: GrTNN(k,

text: n),

text: where

text: MTNN

text: is

text: a

text: positroid

text: stratum

text: of

text: dimension

text: D

text: Our

text: purpose

text: is

text: to

text: associate

text: an

text: algebro-geometric

text: data

text: to

text: this

text: soliton

text: data.

text: N

text: As

text: the

text: ﬁrst

text: step

text: we

text: choose

text: a

text: planar

text: connected

text: orientable

text: bipartite

text: trivalent

text: perfect

text: network

text: in

text: the

text: disk

text: in

text: Postnikov

text: equivalence

text: class

text: for

text: [A].

text: Of

text: course,

text: this

text: choice

text: is

text: not

text: unique,

text: and

text: this

text: is

text: in

text: perfect

text: agreement

text: with

text: the

text: fact

text: that

text: the

text: same

text: soliton

text: solution

text: can

text: be

text: obtained

text: from

text: diﬀerent

text: degenerate

text: algebro-geometrical

text: data.

text: In

text: particular,

text: following

text: Postnikov

text: one

text: can

text: choose

text: the

text: N

text: N

text: maximally

text: reduced

text: network

text: but

text: any

text: other

text: network

text: from

text: the

text: Postnikov

text: equivalence

text: T

text: class

text: for

text: [A]

text: can

text: be

text: used

text: as

text: well.

text: At

text: the

text: second

text: step

text: we

text: associate

text: the

text: following

text: algebraic-geometrical

text: objects

text: to

text: each

text: triple

text: K

text: N

text: [A],

text: N

text: A

text: reducible

text: curve

text: Γ

text: Γ(

text: which

text: is

text: the

text: rational

text: degeneration

text: of

text: a

text: smooth

text: M–curve

text: of

text: N

text: genus

text: g

text: D

text: where

text: g

text: is

text: the

text: number

text: of

text: faces

text: of

text: In

text: our

text: approach,

text: the

text: curve

text: Γ

text: is

text: one

text: of

text: the

text: irreducible

text: components

text: of

text: Γ.

text: The

text: marked

text: point

text: P

text: belongs

text: to

text: the

text: intersection

text: of

text: Γ

text: with

text: an

text: oval

text: (inﬁnite

text: oval);

text: D

text: K

text: A

text: unique

text: real

text: and

text: regular

text: degree

text: g

text: non–special

text: KP

text: divisor

text: [A])

text: such

text: that

text: any

text: KP,Γ

text: D

text: K

text: ﬁnite

text: oval

text: contains

text: exactly

text: one

text: divisor

text: point

text: and

text: [A])

text: Γ

text: coincides

text: with

text: Sato

text: KP,Γ

text: divisor;

text: A

text: unique

text: KP

text: wave–function

text: ψˆ

text: as

text: in

text: Deﬁnition

text: such

text: that

text: (a)

text: Its

text: restriction

text: to

text: Γ

text: P

text: coincides

text: with

text: the

text: normalized

text: Sato

text: wave

text: function

text: D

text: K

text: (b)

text: Its

text: pole

text: divisor

text: has

text: degree

text: d

text: g

text: and

text: is

text: contained

text: in

text: [A]).

text: KP,Γ

text: N

text: N

text: N

text: In

text: particular,

text: if

text: is

text: the

text: trivalent

text: bipartite

text: Le–network,

text: then

text: Γ(

text: is

text: the

text: rational

text: T

text: T

text: degeneration

text: of

text: on

text: M–curve

text: of

text: minimal

text: genus

text: D

text: and

text: has

text: exactly

text: D

text: ovals

text: and

text: d

text: g

text: D


text: 32 SIMONETTA ABENDA AND PETR G. GRINEVICH Remark 4.2. Comparison with the construction in [2]. In [2], to any given soliton datum K ∈ CP ( , [A]), [A] GrTP(k, n), we associate a curve obtained gluing k + 1 copies of 1 at double points whose position is ruled by a parameter ξ >> 1. We then control the asymptotic leading behavior in ξ of the vacuum wavefunction via an algebraic construction using the positivity prop- CP erties of a speciﬁc representative matrix of [A]. In this approach the number of 1 components is much smaller, but we have to introduce extra parameters marking the positions of the glued points. In practice one can obtain such curve from the universal one by a proper desingularization of some double point. N K K { · · · Example 4.1. The degenerate rational curve Γ( ): Let ( , [A]), with = κ < < T 1 } ∈ S M κ and [A] MTNN a point in GrTNN(k, n) corresponding to the matroid . Moreover let n ≤ · · · ≤ M 1 i < < i n be the pivot indexes (i.e the lexicographically minimal base of ) and, 1 k ∈ ≤ · · · ≤ for any r [k], let 1 j < j < < j n be the non–pivot indexes of the boxes B with 1 2 Nr ir,js ∈ CP index χir = 1, s [N ], as in (4). Take 2n + k + 1 copies of 1 and denote them as Γ , Σ , js r 0 irjs ∈ ∈ CP Γ , Γ , for r [k], s [N ]. Orient clockwise the real part of each copy of 1 and ﬁx the irjs ir r following marked points: · · · (1) Γ has n ordered real marked points κ < < κ and a marked point P such that 0 1 n 0 − ζ 1(P ) = 0; 0 ∈ ∈ (2) For any r [k] and s [N ]: r ∈ (1) (2) (a) Σ has 2 real marked points Q , Q , either if r = 1 for all s [N ], or r > 1 irjs irjs irjs 1 and the j –th column of the Le–diagrams contains only zeroes above the box B ; s irjs (1) (2) (3) otherwise Σ has 3 real ordered marked points which we denote Q , Q , Q irjs irjs irjs irjs (see Figure 10[top, left]); (1) (2) (3) (b) each copy Γ has 3 real ordered marked points which we denote P , P , P irjs irjs irjs irjs (see Figure 10[bottom, left]) ; (1) (2) (3) (c) each copy Γ has 3 real ordered marked points which we denote P , P , P (see ir ir ir irj Figure 10[bottom, right]). CP Then Γ is the connected union of all these copies of 1, (cid:32) (cid:32) (cid:33)(cid:33) (cid:71)k (cid:116) (cid:71)Nr (cid:116) Γ = Γ Γ Γ Σ 0 ir irjs irjs r=1 s=1

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: Remark

text: Comparison

text: with

text: the

text: construction

text: in

text: In

text: to

text: any

text: given

text: soliton

text: datum

text: K

text: CP

text: [A]),

text: [A]

text: GrTP(k,

text: n),

text: we

text: associate

text: a

text: curve

text: obtained

text: gluing

text: k

text: copies

text: of

text: at

text: double

text: points

text: whose

text: position

text: is

text: ruled

text: by

text: a

text: parameter

text: ξ

text: We

text: then

text: control

text: the

text: asymptotic

text: leading

text: behavior

text: in

text: ξ

text: of

text: the

text: vacuum

text: wavefunction

text: via

text: an

text: algebraic

text: construction

text: using

text: the

text: positivity

text: prop-

text: CP

text: erties

text: of

text: a

text: speciﬁc

text: representative

text: matrix

text: of

text: [A].

text: In

text: this

text: approach

text: the

text: number

text: of

text: components

text: is

text: much

text: smaller,

text: but

text: we

text: have

text: to

text: introduce

text: extra

text: parameters

text: marking

text: the

text: positions

text: of

text: the

text: glued

text: points.

text: In

text: practice

text: one

text: can

text: obtain

text: such

text: curve

text: from

text: the

text: universal

text: one

text: by

text: a

text: proper

text: desingularization

text: of

text: some

text: double

text: point.

text: N

text: K

text: K

text: Example

text: The

text: degenerate

text: rational

text: curve

text: Γ(

text: Let

text: [A]),

text: with

text: κ

text: T

text: S

text: M

text: κ

text: and

text: [A]

text: MTNN

text: a

text: point

text: in

text: GrTNN(k,

text: n)

text: corresponding

text: to

text: the

text: matroid

text: Moreover

text: let

text: n

text: M

text: i

text: i

text: n

text: be

text: the

text: pivot

text: indexes

text: (i.e

text: the

text: lexicographically

text: minimal

text: base

text: of

text: and,

text: k

text: for

text: any

text: r

text: [k],

text: let

text: j

text: j

text: j

text: n

text: be

text: the

text: non–pivot

text: indexes

text: of

text: the

text: boxes

text: B

text: with

text: Nr

text: ir,js

text: CP

text: index

text: χir

text: s

text: [N

text: as

text: in

text: Take

text: k

text: copies

text: of

text: and

text: denote

text: them

text: as

text: Γ

text: Σ

text: js

text: r

text: irjs

text: CP

text: Γ

text: Γ

text: for

text: r

text: [k],

text: s

text: [N

text: Orient

text: clockwise

text: the

text: real

text: part

text: of

text: each

text: copy

text: of

text: and

text: ﬁx

text: the

text: irjs

text: ir

text: r

text: following

text: marked

text: points:

text: Γ

text: has

text: n

text: ordered

text: real

text: marked

text: points

text: κ

text: κ

text: and

text: a

text: marked

text: point

text: P

text: such

text: that

text: n

text: ζ

text: For

text: any

text: r

text: [k]

text: and

text: s

text: [N

text: r

text: (a)

text: Σ

text: has

text: real

text: marked

text: points

text: Q

text: Q

text: either

text: if

text: r

text: for

text: all

text: s

text: [N

text: or

text: r

text: irjs

text: irjs

text: irjs

text: and

text: the

text: j

text: –th

text: column

text: of

text: the

text: Le–diagrams

text: contains

text: only

text: zeroes

text: above

text: the

text: box

text: B

text: s

text: irjs

text: otherwise

text: Σ

text: has

text: real

text: ordered

text: marked

text: points

text: which

text: we

text: denote

text: Q

text: Q

text: Q

text: irjs

text: irjs

text: irjs

text: irjs

text: (see

text: Figure

text: left]);

text: (b)

text: each

text: copy

text: Γ

text: has

text: real

text: ordered

text: marked

text: points

text: which

text: we

text: denote

text: P

text: P

text: P

text: irjs

text: irjs

text: irjs

text: irjs

text: (see

text: Figure

text: left])

text: (c)

text: each

text: copy

text: Γ

text: has

text: real

text: ordered

text: marked

text: points

text: which

text: we

text: denote

text: P

text: P

text: P

text: (see

text: ir

text: ir

text: ir

text: irj

text: Figure

text: right]).

text: CP

text: Then

text: Γ

text: is

text: the

text: connected

text: union

text: of

text: all

text: these

text: copies

text: of

text: Γ

text: Γ

text: Γ

text: Γ

text: Σ

text: ir

text: irjs

text: irjs


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 37 e3 ζ( P (i 3j )) = ζ( Q ( i 2j )) =1 ζ( Q ( i 3j )) = a3 a2 e1 Vij e2 Γij Vi’j Σij a1 ζ( P (i 2j )) = 1 ζ( P (i 1j )) = 0 ζ( Q ( i 1j )) = 0 Figure 13. Local coordinates on the components Γ and Σ (reﬂection w.r.t. the ij ij ∈ vertical line): the marked point P Γ corresponds to the edge e at the white vertex s ij s ∈ (cid:48) V and the marked point Q Γ corresponds to the edge a at the black vertex V . ij s ij s ij N CP Deﬁnition 4.2. Local aﬃne coordinate on Γ( ) On each copy of 1 the local coordinate ζ is uniquely identiﬁed by the following properties: − · · · (1) On Γ , ζ 1(P ) = 0 and ζ(κ ) < < ζ(κ ). To abridge notations, we identify the 0 0 1 n ∈ ζ–coordinate with the marked points κ = ζ(κ ), j [n]; j j (2) On the component Γ corresponding to the internal white vertex V : ij ij ∞ (1) (2) (3) ζ(P ) = 0, ζ(P ) = 1, ζ(P ) = , ij ij ij (cid:48) while on the component Σ corresponding to the internal black vertex V : ij ij ∞ (1) (2) (3) ζ(Q ) = 0, ζ(Q ) = 1, ζ(Q ) = , ij ij ij see Figure 13. We start deﬁning the desired properties for both the vacuum divisor and the vacuum wave- N function on Γ( ) in the following Deﬁnitions. N Deﬁnition 4.3. Real and regular vacuuum divisor on Γ = Γ( ), compatible with a D \ base I. Let be a degree g divisor on Γ Γ . Let Ω be the inﬁnite oval containing the vac,Γ 0 0 ∈ { } marked point P Γ and let us call ﬁnite the remaining g ovals of Γ. Let I = i , . . . , i , 0 0 1 k ≤ ≤ ∈ 1 i < i < . . . < i n, and P (ir), r [k], be the corresponding Darboux source points in Γ. 1 2 k 3 D Then, we call a real and regular vacuum divisor compatible with I if: vac,Γ D (1) is contained in the union of all the ovals of Γ; vac,Γ (2) The total number of vacuum divisor poles plus the number of Darboux source points contained in any ﬁnite oval is 1 mod 2; ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: ζ(

text: P

text: (i

text: ζ(

text: Q

text: i

text: ζ(

text: Q

text: i

text: Vij

text: Γij

text: Vi’j

text: Σij

text: ζ(

text: P

text: (i

text: ζ(

text: P

text: (i

text: ζ(

text: Q

text: i

text: Figure

text: Local

text: coordinates

text: on

text: the

text: components

text: Γ

text: and

text: Σ

text: (reﬂection

text: w.r.t.

text: the

text: ij

text: ij

text: vertical

text: line):

text: the

text: marked

text: point

text: P

text: Γ

text: corresponds

text: to

text: the

text: edge

text: e

text: at

text: the

text: white

text: vertex

text: s

text: ij

text: s

text: V

text: and

text: the

text: marked

text: point

text: Q

text: Γ

text: corresponds

text: to

text: the

text: edge

text: a

text: at

text: the

text: black

text: vertex

text: V

text: ij

text: s

text: ij

text: s

text: ij

text: N

text: CP

text: Deﬁnition

text: Local

text: aﬃne

text: coordinate

text: on

text: Γ(

text: On

text: each

text: copy

text: of

text: the

text: local

text: coordinate

text: ζ

text: is

text: uniquely

text: identiﬁed

text: by

text: the

text: following

text: properties:

text: On

text: Γ

text: ζ

text: and

text: ζ(κ

text: ζ(κ

text: To

text: abridge

text: notations,

text: we

text: identify

text: the

text: n

text: ζ–coordinate

text: with

text: the

text: marked

text: points

text: κ

text: ζ(κ

text: j

text: [n];

text: j

text: j

text: On

text: the

text: component

text: Γ

text: corresponding

text: to

text: the

text: internal

text: white

text: vertex

text: V

text: ij

text: ij

text: ζ(P

text: ζ(P

text: ζ(P

text: ij

text: ij

text: ij

text: while

text: on

text: the

text: component

text: Σ

text: corresponding

text: to

text: the

text: internal

text: black

text: vertex

text: V

text: ij

text: ij

text: ζ(Q

text: ζ(Q

text: ζ(Q

text: ij

text: ij

text: ij

text: see

text: Figure

text: We

text: start

text: deﬁning

text: the

text: desired

text: properties

text: for

text: both

text: the

text: vacuum

text: divisor

text: and

text: the

text: vacuum

text: wave-

text: N

text: function

text: on

text: Γ(

text: in

text: the

text: following

text: Deﬁnitions.

text: N

text: Deﬁnition

text: Real

text: and

text: regular

text: vacuuum

text: divisor

text: on

text: Γ

text: Γ(

text: compatible

text: with

text: a

text: D

text: base

text: I.

text: Let

text: be

text: a

text: degree

text: g

text: divisor

text: on

text: Γ

text: Γ

text: Let

text: Ω

text: be

text: the

text: inﬁnite

text: oval

text: containing

text: the

text: vac,Γ

text: marked

text: point

text: P

text: Γ

text: and

text: let

text: us

text: call

text: ﬁnite

text: the

text: remaining

text: g

text: ovals

text: of

text: Γ.

text: Let

text: I

text: i

text: i

text: k

text: i

text: i

text: i

text: n,

text: and

text: P

text: (ir),

text: r

text: [k],

text: be

text: the

text: corresponding

text: Darboux

text: source

text: points

text: in

text: Γ.

text: k

text: D

text: Then,

text: we

text: call

text: a

text: real

text: and

text: regular

text: vacuum

text: divisor

text: compatible

text: with

text: I

text: if:

text: vac,Γ

text: D

text: is

text: contained

text: in

text: the

text: union

text: of

text: all

text: the

text: ovals

text: of

text: Γ;

text: vac,Γ

text: The

text: total

text: number

text: of

text: vacuum

text: divisor

text: poles

text: plus

text: the

text: number

text: of

text: Darboux

text: source

text: points

text: contained

text: in

text: any

text: ﬁnite

text: oval

text: is

text: mod

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 50 SIMONETTA ABENDA AND PETR G. GRINEVICH Otherwise, let s¯ = min S. In this case moving downwards from the white vertex Vir−1l the ﬁrst (cid:48) black vertex that we meet is V and such edge has unit weight. Then, using (40), we have is¯l − − − − − − Eˆ(r 1)[l] = ( 1)s¯ r+1Eˆ(s¯)[l] = ( 1)s¯ r+1E(s¯)[l] = E(r 1)[l]. Finally let l = i . In such case r − − − (cid:88)n − Eˆ(r 1)[i ] = C[r 1,r][i ]Eˆ(r) = cˆrEˆ(r)[j] = A[r]. r r j j=1 − − ∈ ∩ since, by deﬁnition C[r 1,r],ir = cˆr and cˆr is non zero if and only if j = i or j I¯ ]i , n] is j j j r r such that χir = 1. j (cid:3) Remark 5.2. Comparison with the algebraic construction in [2] Lemma 5.1 and Theorem 5.1 are the natural generalization to any point in GrTNN(k, n) of the algebraic construction in [2] for points in GrTP(k, n). The main diﬀerence is in the choice of the representative matrix: here A is the RREF matrix for the lexicographically minimal base while in [2] we use a matrix in banded form. In Remark 9.3 we explain the change of orientation in the network which produces the recursive construction presented in [2]. In [2] the system of vectors and coeﬃcients rule the leading order asymptotics in the parameter ξ of the vacuum wavefunction at the marked points on the reducible rational curve Γ(ξ), here the system of vectors and coeﬃcients rules the exact behavior of the vacuum wavfunction at the N CP marked points of Γ( ) since in the present paper we use an excessive number of 1 components to avoid the introduction of a parameter (see also Remark 4.2). Example 5.1. Let’s compute recursively the matrix A in RREF for Example 2.2 using Theorem 5.1 and its bipartite network in Figure 6. Eˆ(4) = Id × , the only non–zero coeﬃcients associated 9 9 ≡ to the forth line of A are cˆ4 = 1, cˆ4 = w and clearly A[4] = cˆ4Eˆ(4). Then Eˆ(3) C[3,4], the 7 8 78 transition matrix from the forth to the third line and   Id × 0 ×  6 6 6 3   − −   1 w 0  C[3,4] =  − 78  ,  03×6 0 1 0    − 0 0 1

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: Otherwise,

text: let

text: s¯

text: min

text: S.

text: In

text: this

text: case

text: moving

text: downwards

text: from

text: the

text: white

text: vertex

text: the

text: ﬁrst

text: black

text: vertex

text: that

text: we

text: meet

text: is

text: V

text: and

text: such

text: edge

text: has

text: unit

text: weight.

text: Then,

text: using

text: we

text: have

text: is¯l

text: Eˆ(r

text: E(r

text: Finally

text: let

text: l

text: i

text: In

text: such

text: case

text: r

text: Eˆ(r

text: C[r

text: ]Eˆ(r)

text: cˆrEˆ(r)[j]

text: A[r].

text: r

text: r

text: j

text: since,

text: by

text: deﬁnition

text: C[r

text: cˆr

text: and

text: cˆr

text: is

text: non

text: zero

text: if

text: and

text: only

text: if

text: j

text: i

text: or

text: j

text: I¯

text: ]i

text: n]

text: is

text: j

text: j

text: j

text: r

text: r

text: such

text: that

text: χir

text: j

text: Remark

text: Comparison

text: with

text: the

text: algebraic

text: construction

text: in

text: Lemma

text: and

text: Theorem

text: are

text: the

text: natural

text: generalization

text: to

text: any

text: point

text: in

text: GrTNN(k,

text: n)

text: of

text: the

text: algebraic

text: construction

text: in

text: for

text: points

text: in

text: GrTP(k,

text: n).

text: The

text: main

text: diﬀerence

text: is

text: in

text: the

text: choice

text: of

text: the

text: representative

text: matrix:

text: here

text: A

text: is

text: the

text: RREF

text: matrix

text: for

text: the

text: lexicographically

text: minimal

text: base

text: while

text: in

text: we

text: use

text: a

text: matrix

text: in

text: banded

text: form.

text: In

text: Remark

text: we

text: explain

text: the

text: change

text: of

text: orientation

text: in

text: the

text: network

text: which

text: produces

text: the

text: recursive

text: construction

text: presented

text: in

text: In

text: the

text: system

text: of

text: vectors

text: and

text: coeﬃcients

text: rule

text: the

text: leading

text: order

text: asymptotics

text: in

text: the

text: parameter

text: ξ

text: of

text: the

text: vacuum

text: wavefunction

text: at

text: the

text: marked

text: points

text: on

text: the

text: reducible

text: rational

text: curve

text: Γ(ξ),

text: here

text: the

text: system

text: of

text: vectors

text: and

text: coeﬃcients

text: rules

text: the

text: exact

text: behavior

text: of

text: the

text: vacuum

text: wavfunction

text: at

text: the

text: N

text: CP

text: marked

text: points

text: of

text: Γ(

text: since

text: in

text: the

text: present

text: paper

text: we

text: use

text: an

text: excessive

text: number

text: of

text: components

text: to

text: avoid

text: the

text: introduction

text: of

text: a

text: parameter

text: (see

text: also

text: Remark

text: Example

text: Let’s

text: compute

text: recursively

text: the

text: matrix

text: A

text: in

text: RREF

text: for

text: Example

text: using

text: Theorem

text: and

text: its

text: bipartite

text: network

text: in

text: Figure

text: Id

text: the

text: only

text: non–zero

text: coeﬃcients

text: associated

text: to

text: the

text: forth

text: line

text: of

text: A

text: are

text: w

text: and

text: clearly

text: Then

text: the

text: transition

text: matrix

text: from

text: the

text: forth

text: to

text: the

text: third

text: line

text: and

text: Id

text: w


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 51 × where 0 × is the (i j) null matrix. The third line coeﬃcient vector is i j cˆ3 = (0, 0, 0, 1, w , w w , 0, w w w , w w w w ) 45 45 46 45 46 48 45 46 48 49 and A[3] = cˆ3Eˆ(3). The transition matrix from the third to the second line, the new basis vectors and the coeﬃcient vector of the second line respectively are   Id × 0 ×  3 3 3 6   − − − − −   1 w w w 0 w w w w w w w   45 45 46 45 46 48 45 46 48 49   − − − −   0 1 w 0 w w w w w   46 46 48 46 48 49   − − −  C[2,3] =  0 0 1 0 w w w  ,  06×3 − 48 48 49   0 0 0 1 0 0     − −   0 0 0 0 1 w   49  − 0 0 0 0 0 1   Id × 0 ×  3 3 3 6   − − −   1 w w w 0 w w w w w w w   45 45 46 45 46 48 45 46 48 49   − −   0 1 w 0 w w w w w   46 46 48 46 48 49  Eˆ(2) = C[2,3]Eˆ(3) =  0 0 −1 0 w w w  ,  0 × 48 48 49   6 3   0 0 0 1 w 0   78     0 0 0 0 1 w   49  0 0 0 0 0 1 cˆ2 = (0, 1, w , 0, w w , 0, 0, 0, 0) 23 23 25 and A[2] = cˆ2Eˆ(2). Finally, the transition matrix from the second to the ﬁrst line, the new basis vectors and the coeﬃcient vector of the ﬁrst line, respectively, are   1 0 0 0 0    − − −   0 1 w 0 w w   23 23 25   − −  C[1,2] =  0 0 1 −0 w25 05×4  ,  0 0 0 1 0     −   0 0 0 0 1    − 0 × Id × 5 1 4 4

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