elliptic projection method and sos model with domain wall ......sos model with dwbc analytical...

Partition functionProjection method

Calculation of the projections

Elliptic projection method and SOS model withDomain Wall Boundary Conditions

Vladimir Rubtsov(joint work with S. Pakulyak (JINR) and

A. Silantyev (JINR-LAREMA))J.Physics A, 2008

Bonn, Hausdorff Centrum - MPIM, July 21 2008

Theory Division, ITEP, Moscow, Russia; LAREMA, Universite d’Angers, France

BonnVladimir Rubtsov Elliptic projections

The main scope of this work

6-vertex model

with DWBC

Vladimir Rubtsov Elliptic projections

6-vertex model Izergin’s

with DWBC determinant formula

6-vertex model Izergin’s Projection

with DWBC determinant formula method for Uq(sl2)

SOS modelwith DWBC

SOS model No determinantwith DWBC formula

SOS model Rosengren’s Projection methodwith DWBC formula for elliptic current algebra

1 Partition function for an elliptic modelSolid-On-Solid model with Domain Wall Boundary ConditionsAnalytical properties of the partition function

2 Projection method for current algebrasCurrents and current algebrasQuantization of current algebrasProjections and quantization

3 Calculation of the projections of the product of total currentsKhoroshkin-Pakuliak method generalized to the elliptic caseExtracting of the kernel form integral formula for theprojections

SOS model with DWBCAnalytical properties

Definition of the model

Consider the (n + 1)× (n + 1) open lattice.

n . . . 1 0

Consider the (n + 1)× (n + 1) open lattice. It has

n . . . 1 0

−1n . . . 1 0 −1

(n + 2)× (n + 2) faces.

n . . . 1 0

−1n . . . 1

0 −1

(n + 2)× (n + 2) faces. On each face we put a height d .

n . . . 1 0

−1n . . . 1

0 −1

(n + 2)× (n + 2) faces. On each face we put a height d .

Six states around a vertex

d − 1

d d − 1

d − 2

a(z − w)

d d + 1

a(z − w)

d d − 1

b(z − w ; d)

d − 1

d d + 1

b(z − w ; d)

d d + 1

c(z − w ; d)

d − 1

d d − 1

c(z − w ; d)

− −

d − 1

d d − 1

d − 2

a(z − w)

d d + 1

a(z − w)

d d − 1

b(z − w ; d)

− −

d − 1

d d + 1

b(z − w ; d)

d d + 1

c(z − w ; d)

d − 1

d d − 1

c(z − w ; d)

− −

d − 1

d d − 1

d − 2

a(zi − wj)

d d + 1

a(zi − wj)

d d − 1

b(zi − wj ; d)

− −

d − 1

d d + 1

b(zi − wj ; d)

d d + 1

c(zi − wj ; d)

d − 1

d d − 1

c(zi − wj ; d)

Inhomogeneous model

We consider a model, where the vertex weights depend on a site inthe lattice via the variables zi attached to the columns and wj

attached to the rows:

zn . . . z1 z0

Boltzmann weights of the Solid-On-Solid model

Boltzmann weights of a vertex (i , j) for the Solid-On-Solid modelare expressed via theta-functions of zi , wj , ~, λ = ~d = ~dij :

Wij(d + 1, d + 2, d + 1, d) = a(zi − wj) = θ(zi − wj + ~),

Wij(d − 1, d − 2, d − 1, d) = a(zi − wj) = θ(zi − wj + ~),

Wij(d − 1, d , d + 1, d) = b(zi − wj ;λ) =θ(zi − wj)θ(λ+ ~)

θ(λ),

Wij(d + 1, d , d − 1, d) = b(zi − wj ;λ) =θ(zi − wj)θ(λ− ~)

θ(λ),

Wij(d − 1, d , d − 1, d) = c(zi − wj ;λ) =θ(zi − wj + λ)θ(~)

θ(λ),

Wij(d + 1, d , d + 1, d) = c(zi − wj ;λ) =θ(zi − wj − λ)θ(~)

θ(−λ).

Felder R-matrix

The matrix of Boltzmann weights:

R(z ;λ) =

a(z) 0 0 0

0 b(z ;λ) c(z ;λ) 00 c(z ;λ) b(z ;λ) 00 0 0 a(z)

Partition function:

Z =∑ n∏

i ,j=0

Wij(di ,j−1, di−1,j−1, di−1,j , dij) =

=∑ n∏

i ,j=0

R(zi − wj ;λ = ~dij)αijβij

γijδij.

αij = di−1,j − dij , βij = di−1,j−1 − di−1,j ,

γij = di−1,j−1 − di,j−1, δij = di,j−1 − dij .

Felder R-matrix

The matrix of Boltzmann weights:

R(z ;λ) =

a(z) 0 0 0

0 b(z ;λ) c(z ;λ) 00 c(z ;λ) b(z ;λ) 00 0 0 a(z)

Partition function:

Z =∑ n∏

i ,j=0

Wij(di ,j−1, di−1,j−1, di−1,j , dij) =

=∑ n∏

i ,j=0

R(zi − wj ;λ = ~dij)αijβij

γijδij.

αij = di−1,j − dij , βij = di−1,j−1 − di−1,j ,

γij = di−1,j−1 − di,j−1, δij = di,j−1 − dij .

Domain Wall Boundary Conditions

The boundary conditions in terms of the height differences:

The boundary conditions in terms of the height differences and theheight dnn:

dnn+ −

The boundary conditions in terms of the height differences and theheight dnn:

dnn+ −

Z ({zi}ni=0; {wj}nj=0;λ = ~dnn).

Elliptic polynomials

Let n be a positive integer and let χ(1) and χ(τ) be somenon-vanishing complex numbers. A holomorphic function withthe translation properties

φ(u + 1) = χ(1)φ(u), φ(u + τ) = χ(τ)e−2πinuφ(u)

is called elliptic polynomial of degree n with character χ. Thespace Θn(χ) of these functions has a dimensiondim Θn(χ) = n.

If two elliptic polynomials of degree n with the same characterχ coincide in n points then they are identical.

An example of an elliptic polynomial of degree one is theusual odd theta-function:

θ(u + 1) = −θ(u), θ(u + τ) = −e−2πiu−πiτθ(u), θ′(0) = 1.

Proposition 1: Character

Proposition

The partition function is an elliptic polynomial of degree n + 1 inthe variable zn with the character

χ(1) = (−1)n+1,

χ(τ) = (−1)n+1 exp(− πi(n + 1)τ + 2πi(n + 1)(λ+

n∑j=0

To prove it one can find an explicit dependence on zn:

Z ({z}, {w};λ) =n∑

n∏j=k+1

a(zn − wj) c(zn − wk ;λ+ (n − k~))

k−1∏j=0

b(zn − wj ;λ+ (n − j~))gk(zn−1, . . . , z0, {w};λ),

Proposition 2: A recurrent relation

Proposition

The considered n-th partition function evaluated in the pointzn = wn − ~ is expressed via (n − 1)-th partition function:

Z (zn = wn − ~, {zi}n−1i=0 ; wn, {wj}n−1

j=0 ;λ) =θ(λ+ (n + 1)~)θ(~)

θ(λ+ n~)n−1∏m=0

θ(wn − wm − ~)θ(zm − wn)Z ({zi}n−1i=0 ; {wj}n−1

j=0 ;λ).

One needs to find an explicit dependence on zn and wn, but therestriction zn = wn − ~ keeps only one state of n-th column andn-th row. This state gives DWBC for n × n lattice and thefollowing weights:

c(−~)n−1∏j=0

b(wn − wj − ~)n−1∏i=0

b(zi − wn)

Proposition 3: Symmetry

Proposition

The partition function is symmetric in each sets of variables {zi}and {wj}:

Z ({z}, {w};λ) = Z ({zi ↔ zl}; {w}, λ) = Z ({z}, {wj ↔ wk};λ).

The proof is based on the Dynamical Yang-Baxter Equation for theFelder R-matrix:

R(12)(u1 − u2;λ)R(13)(u1 − u3;λ+ ~H(2))R(23)(u2 − u3;λ) =

= R(23)(u2 − u3;λ+ ~H(1))R(13)(u1 − u3;λ)R(12)(u1 − u2;λ+ ~H(3)).

where H =

(1 00 −1

If the functions Z (n)({zi}ni=0; {wj}nj=0;λ) satisfy the conditions ofthe Propositions 1, 2, 3 and the initial condition

Z (0)(z0; w0;λ) = c(z0 − w0) =θ(z0 − w0 − λ)θ(~)

θ(−λ)

then the function Z (n)({zi}ni=0; {wj}nj=0;λ) coincides with the n-thpartition function.

Proof.

If these functions coincide for n− 1, then they coincide for n in thepoint zn = wn − ~. Due to the symmetry w.r.t. {w} they coincidein n + 1 points zn = wj − ~, j = 0, . . . , n. These are the same ellipticpolynomial of degree n + 1 with character χ. Therefore theycoincide identically.

CurrentsQuantizationProjections

Definitions

Let sl2 be a complex Lie algebra with Chevalley’s generatorsh, e, f and standard commutation relations [h, e] = 2e,[h, f ] = −2f , [e, f ] = h.

Let Σ be a Riemann surface and p ∈ Σ be its point.

Kp is a field of the complex functions defined in a vicinity ofthe point p.

g = sl2 ⊗Kp is a complex Lie algebra of the sl2-valuedfunctions x [s](z) = x ⊗ s(z) defined in a vicinity of the pointp. It has a basis h[εn], e[εn], f [εn], where {εn}n∈Z is a basis inKp.

Let us suppose that the field Kp is equipped with anon-degenerated scalar product 〈·, ·〉 : Kp ×Kp → Kp and{εn}n∈Z is a dual basis: 〈εn, εm〉 = δnm.

Definitions

Total currents

Lie algebra g = sl2 ⊗Kp can be described by total currents

h(u) =∑n∈Z

εn(u)h[εn],

e(u) =∑n∈Z

εn(u)e[εn], f (u) =∑n∈Z

εn(u)f [εn].

The commutation relations between the total currents:

[h(u), e(v)] = 2e(u)δ(u, v),

[h(u), f (v)] = −2f (u)δ(u, v),

[e(u), f (v)] = h(u)δ(u, v).

where δ(u, v) =∑

n∈Z εn(u)εn(v) is delta-function

corresponding to 〈·, ·〉: 〈δ(u, v), s(u)〉u = s(v).

Total currents

Lie algebra g = sl2 ⊗Kp can be described by total currents

h(u) =∑n∈Z

εn(u)h[εn],

e(u) =∑n∈Z

εn(u)e[εn], f (u) =∑n∈Z

εn(u)f [εn].

The commutation relations between the total currents:

[h(u), e(v)] = 2e(u)δ(u, v),

[h(u), f (v)] = −2f (u)δ(u, v),

[e(u), f (v)] = h(u)δ(u, v).

where δ(u, v) =∑

n∈Z εn(u)εn(v) is delta-function

corresponding to 〈·, ·〉: 〈δ(u, v), s(u)〉u = s(v).

Half currents

Let {εn}n∈Z, {εn;(e)}n∈Z, {εn;(f )}n∈Z be three basesand {εn}n∈Z, {εn(e)}n∈Z, {εn(f )}n∈Z be their dual ones.

Then the total currents can be split as

h(u) = h+(u)− h−(u),

e(u) = e+(u)− e−(u), f (u) = f +(u)− f −(u),

where the currents

h+(u) =∑n≥0

εn(u)h[εn], h−(u) = −∑n<0

εn(u)h[εn],

e+(u) =∑n≥0

εn(e)(u)e[εn;(e)], e−(u) = −∑n<0

εn(e)(u)e[εn;(e)],

f +(u) =∑n≥0

εn(f )(u)f [εn;(f )], f −(u) = −∑n<0

εn(f )(u)f [εn;(f )],

are called half-currents.

Half currents

Let {εn}n∈Z, {εn;(e)}n∈Z, {εn;(f )}n∈Z be three basesand {εn}n∈Z, {εn(e)}n∈Z, {εn(f )}n∈Z be their dual ones.

Then the total currents can be split as

h(u) = h+(u)− h−(u),

e(u) = e+(u)− e−(u), f (u) = f +(u)− f −(u),

where the currents

h+(u) =∑n≥0

εn(u)h[εn], h−(u) = −∑n<0

εn(u)h[εn],

e+(u) =∑n≥0

εn(e)(u)e[εn;(e)], e−(u) = −∑n<0

εn(e)(u)e[εn;(e)],

f +(u) =∑n≥0

εn(f )(u)f [εn;(f )], f −(u) = −∑n<0

εn(f )(u)f [εn;(f )],

are called half-currents.

Green kernels

The half-currents can be expressed through the total ones bythe formulae

h+(u) = 〈G +(u, v)h(v)〉v , h−(u) = 〈G−(u, v)h(v)〉v ,e+(u) = 〈G +

(e)(u, v)e(v)〉v , e−(u) = 〈G−(e)(u, v)e(v)〉v ,

f +(u) = 〈G +(f )(u, v)f (v)〉v , f −(u) = 〈G−(f )(u, v)f (v)〉v ,

with Green kernels defined as

G +(u, v) =∑n≥0

εn(u)εn(v), G−(u, v) = −∑n<0

εn(u)εn(v),

G +(e)(u, v) =

∑n≥0

εn(e)(u)εn;(e)(v), G−(e)(u, v) = −∑n<0

εn(e)(u)εn;(e)(v),

G +(f )(u, v) =

∑n≥0

εn(f )(u)εn;(f )(v), G−(f )(u, v) = −∑n<0

εn(f )(u)εn;(f )(v),

Green kernels

The half currents can be defined directly by Green kernels ifthe last one satisfy

G +(u, v)− G +(u, v) = G +(e)(u, v)− G−(e)(u, v) =

= G +(f )(u, v)− G−(f )(u, v) = δ(u, v).

Two quantizations

A quantization of a (co)algebra A is:(i). A deformation of multiplication µ : A⊗A → A, µ(a⊗b) = ab;(ii). A deformation of comultiplication ∆: A → A⊗A;(iii). The condition ∆(ab) = ∆(a)∆(b).

the non-quantized comultiplication of currents:∆0(x(u)) = x(u)⊗ 1 + 1⊗ x(u).

Requirement: a quantized comultiplication ∆~ of some set ofcurrents {x1(u), . . . , xm(u)} is expressed via x1(u), . . . , xm(u).

Quantization — the choice of some number of current sets.

There are no quantization for the set {h(u), e(u), f (u)}.There are two important quantizations:[I]: For the sets {h+(u), f (u)} and {h−(u), e(u)}. [Enriquez-R.][II]: For the sets {h+(u), e+(u), f +(u)} and {h−(u), e−(u), f −(u)}.Projection method : [I]7→[II]

Two quantizations

Enriquez-R. quantization

The quantization [I] of the current algebras U(g) is describedin terms of the currents h+(u), f (u) and h−(u), e(u) asfollows.

The comultiplication after the quantization takes the form

∆h+(u) = h+(u)⊗ 1 + 1⊗ h+(u),

∆h−(u) = h−(u)⊗ 1 + 1⊗ h−(u),

∆e(u) = e(u)⊗ 1 + K−(u)⊗ e(u),

∆f (u) = f (u)⊗ K +(u) + 1⊗ f (u),

K +(u) = e~Tuh+(u), K−(u) = e~h−(u).

Enriquez-R. quantization

The quantization [I] of the current algebras U(g) is describedin terms of the currents h+(u), f (u) and h−(u), e(u) asfollows.

The comultiplication after the quantization takes the form

∆h+(u) = h+(u)⊗ 1 + 1⊗ h+(u),

∆h−(u) = h−(u)⊗ 1 + 1⊗ h−(u),

∆e(u) = e(u)⊗ 1 + K−(u)⊗ e(u),

∆f (u) = f (u)⊗ K +(u) + 1⊗ f (u),

K +(u) = e~Tuh+(u), K−(u) = e~h−(u).

Commutation relations

The quantized multiplication is defined by the commutationrelations:

[K±(u),K±(v)] = [K +(u),K−(v)] = 0,

K±(u)e(v)K±(u)−1 = q(u, v)e(v),

K±(u)f (v)K±(u)−1 = q(v , u)f (v),

e(u)e(v) = q(u, v)e(v)e(u),

f (u)f (v) = q(v , u)f (v)f (u),

[e(u), f (v)] =1

~δ(u, v)

(K +(u)− K−(v)

where q(u, v) is a function depending on a choice of thecartan half-currents h+(u) and h−(u).

Example 1: Rational case

Rational case. Consider a ”rational curve” Σ = CP1, the localfield in the origin K0 with the scalar product

〈s(u), t(u)〉u =

2πis(u)t(u), (∗)

and the basis εn(z) = zn. In this case we have

q(u, v) =u − v + ~u − v − ~

The corresponding algebra we obtain is called Yangian Double.[Khoroshkin]

Example 2: Elliptic case

Elliptic case. Consider an elliptic curve (a complex torus)Σ = C/Γ, where Γ = Z + τZ, the local field in the origin K0 withthe same scalar product (∗) and the basis

εn(z) = zn, n ≥ 0;

εn(z) =d−n−1

dz−n−1

θ′(z)

θ(z), n < 0.

In this case

q(u, v) =θ(u − v + ~)

θ(u − v − ~).

This is Enriquez-Felder-R. elliptic algebra.

Example 3: Trigonometric case

Trigonometric case. Consider a ”trigonometric curve” (thecomplex cylinder) Σ = C/Z, the local field in the origin K0 withthe same scalar product (∗) and the basis

εn(z) = zn, n ≥ 0;

εn(z) = πd−n−1

dz−n−1ctg πz , n < 0.

In this case

q(u, v) =sinπ(u − v + ~)

sinπ(u − v − ~).

But this is not Uq(sl2).

Example 4: Trigonometric Uq(sl2) case

Uq(sl2) case. Consider again the rational curve Σ = CP1 and thelocal field in the origin K0 but with another scalar product

〈s(u), t(u)〉u =1

us(u)t(u).

The Cartan half-currents are defined in this case:

h+(u) = 〈G +(u/v)h(v)〉v , h−(u) = 〈G−(u/v)h(v)〉v ,

where the Green kernels is defined by the pairings:

〈G +(u/v), s(u)〉 =1

∮|u|>|v |

u − vs(u),

〈G−(u/v), s(u)〉 =1

∮|u|<|v |

u − vs(u).

Example 4: Trigonometric Uq(sl2) case

Then the commutation relations are defined by the function

q(u, v) =e~u − v

u − e~v=

qu − q−1v

q−1u − qv,

where q = e~/2 is a multiplicative quantization parameter usuallyused in the theory of the quantum affine algebras.

Elliptic half-currents

Restrict our attention to the elliptic case. Elliptic half-currents:

h+(u) = 〈G (u, v)h(v)〉v , h−(u) = −〈G (v , u)h(v)〉v ,e+λ (u) = 〈G +

λ (u, v)e(v)〉v , e−λ (u) = 〈G−λ (u, v)e(v)〉v ,f +λ (u) = 〈G +

−λ(u, v)f (v)〉v , f −λ (u) = 〈G−−λ(u, v)f (v)〉v .Parings of elliptic Green kernels:

〈G (u, v), s(u)〉u =

∮|u|>|v |

θ′(u − v)

θ(u − v)s(u),

〈G +λ (u, v), s(u)〉u =

∮|u|>|v |

θ(u − v + λ)

θ(u − v)θ(λ)s(u),

〈G−λ (u, v), s(u)〉u =

∮|u|<|v |

θ(u − v + λ)

θ(u − v)θ(λ)s(u).

Subalgebras of currents

The cartan half-current h+(u) and the total current f (u)generate the subalgebra AF ⊂ A. The projections are definedas the linear maps acting only on this subalgebra:

P+ : AF → AF , P− : AF → AF .

The map P+ is called positive projection and the map P− iscalled negative projection.

On the subalgebra AE ⊂ A generated by the Cartanhalf-current h−(u) and the total current e(u) another (dual)projections act. We shall not consider them.

P+ : AF → AF , P− : AF → AF .

Definition of the projections

The projections on the half-currents can be defined recursively:

(Xh+(u)

λ (X )h+(u), P−λ(Xh+(u)

(Xf +λ (u)

λ+2~(X)f +λ (u),

(f −λ+2n~(un)f εn−1

λ+2(n−1)~(un−1) · · · f ε0λ(u0)

P−λ(f −λ (u)X

)= f −λ (u)P−λ−2~

P−λ(f ε0

λ(u0) · · · f εn−1λ−2(n−1)~(un−1)f +

λ−2n~(un))

where X ∈ AF and εn−1, . . . , ε0 = ±.

Second quantization

The quantization [II] of the current algebras U(g) is describedin terms of the half-currents h+(u), f +

λ (u), e+λ (u) and h−(u),

f −λ (u), e−λ (u). In this case the algebra has the same

multiplication µ, but another comultiplication ∆. Thiscomultiplication was obtained using the projections byEnriquez and Felder and the result can be presented as follows.

The comultiplication can be written in terms of L-operators:

∆L+(u;λ) = L+(u;λ+ ~h(2))⊗ L+(u;λ),

∆L−(u;λ) = L−(u;λ+ ~h(2))⊗ L−(u;λ).

where L-operator L+λ (u) is consists of the positive

half-currents and L−λ (u) is consists of the negativehalf-currents. Here h(2) = h ⊗ 1, h = 〈h+(u), 1〉u.

Second quantization

The quantization [II] of the current algebras U(g) is describedin terms of the half-currents h+(u), f +

λ (u), e+λ (u) and h−(u),

f −λ (u), e−λ (u). In this case the algebra has the same

multiplication µ, but another comultiplication ∆. Thiscomultiplication was obtained using the projections byEnriquez and Felder and the result can be presented as follows.

The comultiplication can be written in terms of L-operators:

∆L+(u;λ) = L+(u;λ+ ~h(2))⊗ L+(u;λ),

∆L−(u;λ) = L−(u;λ+ ~h(2))⊗ L−(u;λ).

where L-operator L+λ (u) is consists of the positive

half-currents and L−λ (u) is consists of the negativehalf-currents. Here h(2) = h ⊗ 1, h = 〈h+(u), 1〉u.

Multiplication in terms of L-operators

The multiplication in this case does not change and thecommutation relations can be presented in terms ofL-operators:

R(12)(u − v ;λ)L+,(1)(u;λ+ ~H(2))L+,(2)(v ;λ) =

= L+,(2)(v ;λ+ ~H(1))L−,(1)(u;λ)R(12)(u − v ;λ+ ~h),

R(12)(u − v ;λ)L−,(1)(u;λ+ ~H(2))L−,(2)(v ;λ) =

= L−,(2)(v ;λ+ ~H(1))L+,(1)(u;λ)R(12)(u − v ;λ+ ~h),

R(12)(u − v ;λ)L+,(1)(u;λ+ ~H(2))L−,(2)(v ;λ) =

= L−,(2)(v ;λ+ ~H(1))L+,(1)(u;λ)R(12)(u − v ;λ+ ~h).

These relations are called Dynamical RLL-relations.

Multiplication in terms of L-operators

The multiplication in this case does not change and thecommutation relations can be presented in terms ofL-operators:

R(12)(u − v ;λ)L+,(1)(u;λ+ ~H(2))L+,(2)(v ;λ) =

= L+,(2)(v ;λ+ ~H(1))L−,(1)(u;λ)R(12)(u − v ;λ+ ~h),

R(12)(u − v ;λ)L−,(1)(u;λ+ ~H(2))L−,(2)(v ;λ) =

= L−,(2)(v ;λ+ ~H(1))L+,(1)(u;λ)R(12)(u − v ;λ+ ~h),

R(12)(u − v ;λ)L+,(1)(u;λ+ ~H(2))L−,(2)(v ;λ) =

= L−,(2)(v ;λ+ ~H(1))L+,(1)(u;λ)R(12)(u − v ;λ+ ~h).

These relations are called Dynamical RLL-relations.

Khoroshkin-Pakuliak methodExtracting of the kernel

Problem

Problem: to calculate the following expression in terms of thehalf-currents:

P+λ−n~

(f (zn)f (zn−1) · · · f (z1)f (z0)

The case n = 0 (one total current):

(f (z0)

)= f +

λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =

∮|z0|>|w0|

θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ)f (w0)

The kernel gives the initial condition:

Z (0)(z0; w0;λ) = θ(~)θ(z0 − w0)θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ).

Problem

P+λ−n~

(f (zn)f (zn−1) · · · f (z1)f (z0)

(f (z0)

)= f +

λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =

∮|z0|>|w0|

θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ)f (w0)

Z (0)(z0; w0;λ) = θ(~)θ(z0 − w0)θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ).

Problem

P+λ−n~

(f (zn)f (zn−1) · · · f (z1)f (z0)

(f (z0)

)= f +

λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =

∮|z0|>|w0|

θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ)f (w0)

Z (0)(z0; w0;λ) = θ(~)θ(z0 − w0)θ(z0 − w0−λ)

θ(z0 − w0)θ(−λ).

Splitting of the right total current

Splitting of the right total current:

P+λ−n~

(f (zn) · · · f (z1)f (z0)

= P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−n~(z0)− P+

λ−n~(f (zn) · · · f (z1)f −λ−n~(z0)

The second term:

P+λ−n~

(f (zn) · · · f (z1)f −λ−n~(z0)

n∑j=1

Qj(z0)Xj ,

Xj = P+λ−n~

(f (zn) · · · f (zj+1)Fλ−(n−2j+2)~(zj)f (zj−1) · · · f (z1)

Fλ(zj) =θ(~)

θ(λ+ ~)

(f +λ+2~(zj)f +

λ (zj)− f −λ+2~(zj)f −λ (zj)),

Qj(z0) =θ(zj − z0 + λ− (n − 2j + 1)~)

θ(zj − z0 + ~)

j−1∏k=1

θ(zk − z0 − ~)

θ(zk − z0 + ~).

Splitting of the right total current

Splitting of the right total current:

P+λ−n~

(f (zn) · · · f (z1)f (z0)

= P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−n~(z0)− P+

λ−n~(f (zn) · · · f (z1)f −λ−n~(z0)

The second term:

P+λ−n~

(f (zn) · · · f (z1)f −λ−n~(z0)

n∑j=1

Qj(z0)Xj ,

Xj = P+λ−n~

(f (zn) · · · f (zj+1)Fλ−(n−2j+2)~(zj)f (zj−1) · · · f (z1)

Fλ(zj) =θ(~)

θ(λ+ ~)

(f +λ+2~(zj)f +

λ (zj)− f −λ+2~(zj)f −λ (zj)),

Qj(z0) =θ(zj − z0 + λ− (n − 2j + 1)~)

θ(zj − z0 + ~)

j−1∏k=1

θ(zk − z0 − ~)

θ(zk − z0 + ~).

System of equations

Putting z0 = zi i = 1, . . . , n and using the relationf (u)f (u) = 0 we obtain the system of equations

P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−n~(zi ) =

n∑j=1

Qj(zi )Xj .

Resolving it we derive the following expression

P+λ−n~

(f (zn) · · · f (z1)f (z0)

= P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−(n−2m)~(z0; zn, . . . , z1),

f +λ−(n−2m)~(z0; zn, . . . , z1) =

n∏k=1

θ(zk − z0)

θ(zk − z0 + ~)×

n∑i=0

θ(zi − z0 + λ)

θ(λ)

n∏k=1

θ(zk − zi + ~)

n∏k=0,k 6=i

θ(zk − zi )f +λ−n~(zi ).

System of equations

Putting z0 = zi i = 1, . . . , n and using the relationf (u)f (u) = 0 we obtain the system of equations

P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−n~(zi ) =

n∑j=1

Qj(zi )Xj .

Resolving it we derive the following expression

P+λ−n~

(f (zn) · · · f (z1)f (z0)

= P+λ−(n−2)~

(f (zn) · · · f (z1)

)f +λ−(n−2m)~(z0; zn, . . . , z1),

f +λ−(n−2m)~(z0; zn, . . . , z1) =

n∏k=1

θ(zk − z0)

θ(zk − z0 + ~)×

n∑i=0

θ(zi − z0 + λ)

θ(λ)

n∏k=1

θ(zk − zi + ~)

n∏k=0,k 6=i

θ(zk − zi )f +λ−n~(zi ).

Final formula

Continuing this calculation by induction we obtain the final formula

P+λ−n~

(f (zn) · · · f (z1)f (z0)

←−∏n≥m≥0

f +λ−(n−2m)~(zm; zn, . . . , zm+1),

f +λ−(n−2m)~(zm; zn, . . . , zm+1) =

n∏k=m+1

θ(zk − zm)

θ(zk − zm + ~)×

n∑i=m

θ(zi − zm + λ+ m~)

θ(λ+ m~)

n∏k=m+1

θ(zk − zi + ~)

n∏k=m,k 6=i

θ(zk − zi )

f +λ−(n−2m)~(zi ).

Integral formula

The projection can be written in integral form:

P+λ−n~

(f (zn) · · · f (z1)f (z0)

∮|zi |>|wj |

K (n)({z}, {w};λ)f (wn) · · · f (w0)dwn

2πi· · · dw0

with the kernel

K (n)({z}, {w};λ) =

k,m=0k>m

θ(zk − zm)θ(zk − wm + ~)

θ(zk − zm + ~)θ(zk − wm)

n∏m=0

θ(zm − wm−λ−m~)

θ(zm − wm)θ(−λ−m~).

q-symmetric functions

The q-symmetric functions:

Y (u, v) = q(v , u)Y (v , u).

The examples of q-symmetric functions:

Y (u, v) = f (u)f (v), Y (u, v) = P+λ−~(f (u)f (v)

Y (u, v) =θ(u − v − ~)

θ(u − v), Y (u, v) =

θ(u − v)

θ(u − v + ~)

The ratio of q-symmetric functions is symmetric.

The examples of functions q-symmetric with respect to theneighbour variables zi and zi−1:

P+λ−n~

(f (zn) · · · f (z1)f (z0)

), K (n)({z}, {w};λ).

Y (u, v) = q(v , u)Y (v , u).

Y (u, v) =θ(u − v − ~)

θ(u − v), Y (u, v) =

θ(u − v)

θ(u − v + ~)

P+λ−n~

(f (zn) · · · f (z1)f (z0)

), K (n)({z}, {w};λ).

Y (u, v) = q(v , u)Y (v , u).

Y (u, v) =θ(u − v − ~)

θ(u − v), Y (u, v) =

θ(u − v)

θ(u − v + ~)

P+λ−n~

(f (zn) · · · f (z1)f (z0)

), K (n)({z}, {w};λ).

Y (u, v) = q(v , u)Y (v , u).

Y (u, v) =θ(u − v − ~)

θ(u − v), Y (u, v) =

θ(u − v)

θ(u − v + ~)

P+λ−n~

(f (zn) · · · f (z1)f (z0)

), K (n)({z}, {w};λ).

Inversely q-symmetric functions

The inversely q-symmetric functions:

Y (u, v) = q(v , u)−1Y (v , u).

The product of a q-symmetric function and an inverselyq-symmetric function is symmetric.

Arbitrary function Y (wn, . . . ,wn) can be inverselyq-symmetrized:

iq-Sym({w})Y (wn, . . . ,wn) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

θ(wl − wl ′ − ~)Y (wσ(n), . . . ,wσ(n))

Y (u, v) = q(v , u)−1Y (v , u).

iq-Sym({w})Y (wn, . . . ,wn) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

θ(wl − wl ′ − ~)Y (wσ(n), . . . ,wσ(n))

Y (u, v) = q(v , u)−1Y (v , u).

iq-Sym({w})Y (wn, . . . ,wn) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

θ(wl − wl ′ − ~)Y (wσ(n), . . . ,wσ(n))

Inversely q-symmetric kernel

We can extract only symmetrized integrand.The product of total currents is q-symmetric.The kernel K (n)({z}, {w};λ) should be inverselyq-symmetrized:

iq-Sym({w})K (n)({z}, {w};λ) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

θ(wl − wl ′ − ~)K (n)({z},wσ(n), . . . ,wσ(n);λ).

This expression is q-symmetric w.r.t. {z} and inverselyq-symmetric w.r.t. {w}.Formally this kernel extracting can be done using the Hopfpairing with the product of dual total currents:⟨·, e(wn) · · · e(w0)

⟩Hopf

iq-Sym({w})K (n)({z}, {w};λ) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

⟩Hopf

iq-Sym({w})K (n)({z}, {w};λ) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

⟩Hopf

iq-Sym({w})K (n)({z}, {w};λ) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

⟩Hopf

iq-Sym({w})K (n)({z}, {w};λ) =

=∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

⟩Hopf

The main result

Theorem

The function

Z (n)({z}; {w};λ) = (θ(~))n+1n∏

i ,j=0

θ(zi − wj)×

∏n≥k>m≥0

θ(zk − zm + ~)θ(wk − wm − ~)

θ(zk − zm)θ(wk − wm)×

iq-Sym({w})K (n)({z}, {w};λ)

satisfies to the conditions of the Propositions 1, 2, 3 and initialcondition and, therefore, this is a partition function for theSOS model with DWBC.

The explicit expression

The explicit expression for the partition function:

Z (n)({z}; {w};λ) =∏

n≥k>m≥0

θ(wk − wm − ~)

θ(wk − wm)×

∑σ∈Sn,0

∏l>l ′

σ(l)<σ(l ′)

θ(wl − wl ′ + ~)

θ−(wl − wl ′ − ~)

∏n≥k>m≥0

θ(zk − wσ(m) + ~)×

∏0≤k<m≤n

θ(zk − wσ(m))n∏

θ(zm − wσ(m)−λ−m~)θ(~)

θ(−λ−m~).

elliptic projection method and sos model with domain wall ......sos model with dwbc analytical...

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