Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical...
Transcript of Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical...
![Page 1: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/1.jpg)
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
![Page 2: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/2.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model
with DWBC
Vladimir Rubtsov Elliptic projections
![Page 3: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/3.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model Izergin’s
with DWBC determinant formula
Vladimir Rubtsov Elliptic projections
![Page 4: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/4.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model Izergin’s Projection
with DWBC determinant formula method for Uq(sl2)
Vladimir Rubtsov Elliptic projections
![Page 5: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/5.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model Izergin’s Projection
with DWBC determinant formula method for Uq(sl2)
SOS modelwith DWBC
Vladimir Rubtsov Elliptic projections
![Page 6: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/6.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model Izergin’s Projection
with DWBC determinant formula method for Uq(sl2)
SOS model No determinantwith DWBC formula
Vladimir Rubtsov Elliptic projections
![Page 7: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/7.jpg)
Partition functionProjection method
Calculation of the projections
The main scope of this work
6-vertex model Izergin’s Projection
with DWBC determinant formula method for Uq(sl2)
SOS model Rosengren’s Projection methodwith DWBC formula for elliptic current algebra
Vladimir Rubtsov Elliptic projections
![Page 8: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/8.jpg)
Partition functionProjection method
Calculation of the projections
Plan
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
Vladimir Rubtsov Elliptic projections
![Page 9: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/9.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Definition of the model
Consider the (n + 1)× (n + 1) open lattice.
n
. . .
1
0
n . . . 1 0
Vladimir Rubtsov Elliptic projections
![Page 10: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/10.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Definition of the model
Consider the (n + 1)× (n + 1) open lattice. It has
n
. . .
1
0
n . . . 1 0
n
. . .
1
0
−1n . . . 1 0 −1
(n + 2)× (n + 2) faces.
Vladimir Rubtsov Elliptic projections
![Page 11: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/11.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Definition of the model
Consider the (n + 1)× (n + 1) open lattice. It has
n
. . .
1
0
n . . . 1 0
n
. . .
1
0
−1n . . . 1
d
0 −1
(n + 2)× (n + 2) faces. On each face we put a height d .
Vladimir Rubtsov Elliptic projections
![Page 12: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/12.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Definition of the model
Consider the (n + 1)× (n + 1) open lattice. It has
n
. . .
1
0
n . . . 1 0
n
. . .
1
0
−1n . . . 1
d
d±1
0 −1
(n + 2)× (n + 2) faces. On each face we put a height d .
Vladimir Rubtsov Elliptic projections
![Page 13: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/13.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Six states around a vertex
d − 1
d d − 1
d − 2
a(z − w)
d + 1
d d + 1
d + 2
a(z − w)
d + 1
d d − 1
d
b(z − w ; d)
d − 1
d d + 1
d
b(z − w ; d)
d + 1
d d + 1
d
c(z − w ; d)
d − 1
d d − 1
d
c(z − w ; d)
Vladimir Rubtsov Elliptic projections
![Page 14: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/14.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Six states around a vertex
− −
−
−
d − 1
d d − 1
d − 2
a(z − w)
+ +
+
+
d + 1
d d + 1
d + 2
a(z − w)
+ +
−
−
d + 1
d d − 1
d
b(z − w ; d)
− −
+
+
d − 1
d d + 1
d
b(z − w ; d)
+ −
−
+
d + 1
d d + 1
d
c(z − w ; d)
− +
+
−
d − 1
d d − 1
d
c(z − w ; d)
Vladimir Rubtsov Elliptic projections
![Page 15: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/15.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Six states around a vertex
− −
−
−
d − 1
d d − 1
d − 2
a(zi − wj)
+ +
+
+
d + 1
d d + 1
d + 2
a(zi − wj)
+ +
−
−
d + 1
d d − 1
d
b(zi − wj ; d)
− −
+
+
d − 1
d d + 1
d
b(zi − wj ; d)
+ −
−
+
d + 1
d d + 1
d
c(zi − wj ; d)
− +
+
−
d − 1
d d − 1
d
c(zi − wj ; d)
Vladimir Rubtsov Elliptic projections
![Page 16: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/16.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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:
wn
. . .
w1
w0
zn . . . z1 z0
Vladimir Rubtsov Elliptic projections
![Page 17: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/17.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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).
Vladimir Rubtsov Elliptic projections
![Page 18: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/18.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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 .
Vladimir Rubtsov Elliptic projections
![Page 19: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/19.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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 .
Vladimir Rubtsov Elliptic projections
![Page 20: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/20.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Domain Wall Boundary Conditions
The boundary conditions in terms of the height differences:
+ −
+ −
+ −
+ −
−
+
−
+
−
+
−
+
Vladimir Rubtsov Elliptic projections
![Page 21: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/21.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Domain Wall Boundary Conditions
The boundary conditions in terms of the height differences and theheight dnn:
dnn+ −
+ −
+ −
+ −
−
+
−
+
−
+
−
+
Vladimir Rubtsov Elliptic projections
![Page 22: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/22.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Domain Wall Boundary Conditions
The boundary conditions in terms of the height differences and theheight dnn:
dnn+ −
+ −
+ −
+ −
−
+
−
+
−
+
−
+
Z ({zi}ni=0; {wj}nj=0;λ = ~dnn).
Vladimir Rubtsov Elliptic projections
![Page 23: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/23.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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.
Vladimir Rubtsov Elliptic projections
![Page 24: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/24.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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.
Vladimir Rubtsov Elliptic projections
![Page 25: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/25.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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.
Vladimir Rubtsov Elliptic projections
![Page 26: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/26.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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
wj)).
Vladimir Rubtsov Elliptic projections
![Page 27: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/27.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
To prove it one can find an explicit dependence on zn:
Z ({z}, {w};λ) =n∑
k=0
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),
Vladimir Rubtsov Elliptic projections
![Page 28: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/28.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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).
Vladimir Rubtsov Elliptic projections
![Page 29: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/29.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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)
Vladimir Rubtsov Elliptic projections
![Page 30: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/30.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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).
Vladimir Rubtsov Elliptic projections
![Page 31: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/31.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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
).
Vladimir Rubtsov Elliptic projections
![Page 32: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/32.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
Lemma
Lemma
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.
Vladimir Rubtsov Elliptic projections
![Page 33: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/33.jpg)
Partition functionProjection method
Calculation of the projections
SOS model with DWBCAnalytical properties
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.
Vladimir Rubtsov Elliptic projections
![Page 34: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/34.jpg)
Partition functionProjection method
Calculation of the projections
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.
Vladimir Rubtsov Elliptic projections
![Page 35: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/35.jpg)
Partition functionProjection method
Calculation of the projections
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.
Vladimir Rubtsov Elliptic projections
![Page 36: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/36.jpg)
Partition functionProjection method
Calculation of the projections
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.
Vladimir Rubtsov Elliptic projections
![Page 37: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/37.jpg)
Partition functionProjection method
Calculation of the projections
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.
Vladimir Rubtsov Elliptic projections
![Page 38: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/38.jpg)
Partition functionProjection method
Calculation of the projections
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.
Vladimir Rubtsov Elliptic projections
![Page 39: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/39.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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).
Vladimir Rubtsov Elliptic projections
![Page 40: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/40.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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).
Vladimir Rubtsov Elliptic projections
![Page 41: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/41.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 42: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/42.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 43: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/43.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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),
Vladimir Rubtsov Elliptic projections
![Page 44: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/44.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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).
Vladimir Rubtsov Elliptic projections
![Page 45: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/45.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 46: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/46.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 47: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/47.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 48: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/48.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 49: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/49.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 50: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/50.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 51: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/51.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 52: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/52.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 53: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/53.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 54: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/54.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 55: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/55.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 56: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/56.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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]
Vladimir Rubtsov Elliptic projections
![Page 57: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/57.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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),
where
K +(u) = e~Tuh+(u), K−(u) = e~h−(u).
Vladimir Rubtsov Elliptic projections
![Page 58: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/58.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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),
where
K +(u) = e~Tuh+(u), K−(u) = e~h−(u).
Vladimir Rubtsov Elliptic projections
![Page 59: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/59.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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).
Vladimir Rubtsov Elliptic projections
![Page 60: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/60.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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 =
∮C0
du
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]
Vladimir Rubtsov Elliptic projections
![Page 61: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/61.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 62: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/62.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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).
Vladimir Rubtsov Elliptic projections
![Page 63: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/63.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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
2πi
∮C0
du
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
2πi
∮|u|>|v |
du
u − vs(u),
〈G−(u/v), s(u)〉 =1
2πi
∮|u|<|v |
du
u − vs(u).
Vladimir Rubtsov Elliptic projections
![Page 64: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/64.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 65: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/65.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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 |
du
2πi
θ′(u − v)
θ(u − v)s(u),
〈G +λ (u, v), s(u)〉u =
∮|u|>|v |
du
2πi
θ(u − v + λ)
θ(u − v)θ(λ)s(u),
〈G−λ (u, v), s(u)〉u =
∮|u|<|v |
du
2πi
θ(u − v + λ)
θ(u − v)θ(λ)s(u).
Vladimir Rubtsov Elliptic projections
![Page 66: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/66.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 67: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/67.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 68: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/68.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 69: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/69.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
Definition of the projections
The projections on the half-currents can be defined recursively:
P+λ
(Xh+(u)
)= P+
λ (X )h+(u), P−λ(Xh+(u)
)= 0,
P+λ
(Xf +λ (u)
)= P+
λ+2~(X)f +λ (u),
P+λ
(f −λ+2n~(un)f εn−1
λ+2(n−1)~(un−1) · · · f ε0λ(u0)
)= 0
P−λ(f −λ (u)X
)= f −λ (u)P−λ−2~
(X),
P−λ(f ε0
λ(u0) · · · f εn−1λ−2(n−1)~(un−1)f +
λ−2n~(un))
= 0.
where X ∈ AF and εn−1, . . . , ε0 = ±.
Vladimir Rubtsov Elliptic projections
![Page 70: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/70.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 71: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/71.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 72: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/72.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 73: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/73.jpg)
Partition functionProjection method
Calculation of the projections
CurrentsQuantizationProjections
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.
Vladimir Rubtsov Elliptic projections
![Page 74: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/74.jpg)
Partition functionProjection method
Calculation of the projections
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):
P+λ
(f (z0)
)= f +
λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =
=
∮|z0|>|w0|
dw0
2πi
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).
Vladimir Rubtsov Elliptic projections
![Page 75: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/75.jpg)
Partition functionProjection method
Calculation of the projections
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):
P+λ
(f (z0)
)= f +
λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =
=
∮|z0|>|w0|
dw0
2πi
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).
Vladimir Rubtsov Elliptic projections
![Page 76: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/76.jpg)
Partition functionProjection method
Calculation of the projections
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):
P+λ
(f (z0)
)= f +
λ (z0) = 〈G +−λ(z0 − w0)f (w0)〉w0 =
=
∮|z0|>|w0|
dw0
2πi
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).
Vladimir Rubtsov Elliptic projections
![Page 77: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/77.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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 ,
where
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 + ~).
Vladimir Rubtsov Elliptic projections
![Page 78: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/78.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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 ,
where
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 + ~).
Vladimir Rubtsov Elliptic projections
![Page 79: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/79.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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 ).
Vladimir Rubtsov Elliptic projections
![Page 80: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/80.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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 ).
Vladimir Rubtsov Elliptic projections
![Page 81: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/81.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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),
where
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 ).
Vladimir Rubtsov Elliptic projections
![Page 82: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/82.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
2πi
with the kernel
K (n)({z}, {w};λ) =
=n∏
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~).
Vladimir Rubtsov Elliptic projections
![Page 83: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/83.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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).
Vladimir Rubtsov Elliptic projections
![Page 84: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/84.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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).
Vladimir Rubtsov Elliptic projections
![Page 85: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/85.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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).
Vladimir Rubtsov Elliptic projections
![Page 86: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/86.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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).
Vladimir Rubtsov Elliptic projections
![Page 87: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/87.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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))
Vladimir Rubtsov Elliptic projections
![Page 88: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/88.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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))
Vladimir Rubtsov Elliptic projections
![Page 89: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/89.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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))
Vladimir Rubtsov Elliptic projections
![Page 90: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/90.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
.
Vladimir Rubtsov Elliptic projections
![Page 91: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/91.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
.
Vladimir Rubtsov Elliptic projections
![Page 92: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/92.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
.
Vladimir Rubtsov Elliptic projections
![Page 93: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/93.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
.
Vladimir Rubtsov Elliptic projections
![Page 94: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/94.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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
.
Vladimir Rubtsov Elliptic projections
![Page 95: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/95.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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.
Vladimir Rubtsov Elliptic projections
![Page 96: Elliptic projection method and SOS model with Domain Wall ......SOS model with DWBC Analytical properties Inhomogeneous model We consider a model, where the vertex weights depend on](https://reader036.fdocuments.net/reader036/viewer/2022071112/5fe85ae56ef2845bb2721728/html5/thumbnails/96.jpg)
Partition functionProjection method
Calculation of the projections
Khoroshkin-Pakuliak methodExtracting of the kernel
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∏
m=0
θ(zm − wσ(m)−λ−m~)θ(~)
θ(−λ−m~).
Vladimir Rubtsov Elliptic projections