2 fermionic basis for the xxz model - research.kek.jp...
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Fermionic Basis for the XXZ model
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Fermionic Basis for the XXZ model
T. Miwa
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Fermionic Basis for the XXZ model
T. Miwa
joint work with
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo, F. Smirnov
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo, F. Smirnov, Y. Takeyama
1. Quantum XXZ Hamiltonian2. Quantum symmetry and integral formula3. Algebraic formula4. Quasi-local operators5. Annihilation operators6. Particle structure
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1. Quantum XXZ Hamiltonian
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1. Quantum XXZ Hamiltonian• quantum spin chain
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
12
1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ
17
1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functions
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functionsfor local operator O
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functionsfor local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous model
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous modelspectral parameters ξ1, . . . , ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous modelspectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
• inhomogeneous modelspectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉ξ1,...,ξn =ξ1,...,ξn
〈vac|O|vac〉ξ1,...,ξnξ1,...,ξn
〈vac|vac〉ξ1,...,ξn• inhomogeneous model
spectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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2. Quantum symmetry and inte-gral formula
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2e1, t1e0t
−11 = q−2e0
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2e1, t1e0t
−11 = q−2e0
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
),
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
t1 = t−10 =
(q
q−1
)
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
t1 = t−10 =
(q
q−1
)
• level 1 HWRs and intertwiners
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
• integral formula is obtained by bosoniza-tion
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2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
• integral formula is obtained by bosoniza-tion• leads to the qKZ equation ζi → q2ζi
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3. Algebraic formula
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
ω(ζ) =
∫ i∞−0
−i∞−0ζu sin
π(1−ν)u2
sin πu2 cos πνu
2du
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3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
ω(ζ) =
∫ i∞−0
−i∞−0ζu sin
π(1−ν)u2
sin πu2 cos πνu
2du
+ rational in q, ζ
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4. Quasi-local operators
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4. Quasi-local operators• disorder parameter α
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
⊗ (∗ ∗ ∗) ⊗
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
⊗ (∗ ∗ ∗) ⊗(
11
)⊗ · · ·
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operators
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
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4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
=
67
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
68
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
69
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X) =
70
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X)=
∏
k
1
qα/2 + q−α/2trVk
q−ασ3k/2
(X)
71
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X)=
∏
k
1
qα/2 + q−α/2trVk
q−ασ3k/2
(X)
Ω : nilpotent
72
5. Annihilation operators
73
5. Annihilation operators• decomposition of Ω
74
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξj
75
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
76
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
77
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
78
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
79
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
80
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
81
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1]) =
82
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
83
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1
84
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
85
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support property
86
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
87
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
88
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
89
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
90
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
91
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
92
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
93
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j ,
94
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
95
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
96
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
97
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR?
98
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO
99
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
100
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
101
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
102
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional
103
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
104
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
c±n,[1,n]
(Wα)[1,n] ⊂ (Wα∓1)[1,n−1]
105
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
c±n,[1,n]
(Wα)[1,n] ⊂ (Wα∓1)[1,n−1]
IMPLIES →
106
6. Particle structure
107
6. Particle structure• n = 1 case
108
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
109
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
110
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)
111
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= 1
112
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0)
113
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
114
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= qασ3
1
115
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
116
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state
117
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= σ±1
118
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
119
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
120
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
121
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
122
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1
123
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case
124
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case : (Wα)[1,2] is 42 dimensional
125
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
126
6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
127
6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
• other states
128
6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
129
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
130
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
131
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
132
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
133
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
134
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
135
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
∼ means ‘up to sign’
136
6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
∼ means ‘up to sign’
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
137
6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
138
6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basis
139
6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
140
6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
pj ∈ ±, 0, 0, pj =
0 if j << 0
0 if j >> 0
141
6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
pj ∈ ±, 0, 0, pj =
0 if j << 0
0 if j >> 0
exists, though not unique
142
1. Universal R matrix
main ingredient L operators= images of universal R matrix
L1,2 = (π1⊗π2)(R) where R ∈ Uq(b+)⊗Uq(b
−)
πi : Uq(b+) → ai Uq(b
+) = 〈e0, e1, t0, t1〉a1 : auxiliary space
a2 : quantum space = End(V1 ⊗ · · · ⊗ Vn)
143
2. Transfer matrix inEnd(V1 ⊗ · · · ⊗ Vn)
R matrix auxiliary space = Va ' C2
R(ζ) = (qζ − q−1ζ−1)
1β(ζ) γ(ζ)γ(ζ) β(ζ)
1
β(ζ) =(1 − ζ2)q
1 − q2ζ2, γ(ζ) =
(1 − q2)ζ
1 − q2ζ2.
t(N)(ζ) = tra (Ran(ζ) · · ·Ra1(ζ))
144
3. Commuting family
Yang-Baxter equation
R1,2R1,3R2,3 = R2,3R1,3R1,2
implies [t(N)(ζ1), t(N)(ζ2)] = 0
quantum Hamiltonian
t(N)(1)−1t(N)(ζ) = 1 + const(ζ − 1)H(N)
XXZ + · · ·inhomogeneous model
t(N)(ζ) = tra (Ran(ζ/ξn) · · ·Ra1(ζ/ξ1))
145
4. Adjoint action
transfer matrix acting on
X ∈ End(V1 ⊗ · · · ⊗ Vn)
Ta(ζ) = Ra,n(ζ/ξn) · · ·Ra,1(ζ/ξ1)
t(ζ, α)(X) = tra(q−ασ3aTa(ζ)−1XTa(ζ))
tautological reduction to the right
t[1,n](ζ, α)(X[1,n−1]) = t[1,n−1](ζ, α)(X[1,n−1])
analytic structure
tra(q−ασ3a∗) =
∑±q±α(rational in q, ζ, ξi)
simple poles at (ζ/ξi)2 = q±2
146
5. Baxter’s TQ relation
(Bazhanov-Lukyanov-Zamolodchikov)second order difference equation
t(ζ, α)Q±(ζ, α) = Q±(q−1ζ, α) + Q±(qζ, α)
n = 0 case
(qα + q−α)ζ±α = (q−1ζ)±α + (qζ)±α
147
6. q-Oscillator algebra
algebra Osc with generators and relations
qDaq−D = q−1a, qDa∗q−D = qa∗
aa∗ = 1 − q2D+2, a∗a = 1 − q2D
two morphisms
o±ζ : Uq(b+) → Osc
o±ζ (e0) =ζ
q − q−1
a∗
ao±ζ (e1) =
ζ
q − q−1
a
a∗
o±(t0) = q±2D, o±(t1) = q∓2D
two representations of Osc
W+ = ⊕k≥0C|k〉, W− = ⊕k≤−1C|k〉qD|k〉 = qk|k〉, a|k〉 = (1 − q2k)|k − 1〉,a∗|k〉 = (1 − δk,−1)|k + 1〉.
148
7. Q operators
L operators
L±(ζ) = iζ−1/2q−1/4
×(1 − ζa∗σ± − ζaσ∓ − ζ2q2D+2τ∓
)q±σ3D
σ+=
(1
0
), σ−=
(0
1
), τ+=
(1
0
), τ−=
(0
1
)
T±A (ζ) = L±
A,n(ζ/ξn) · · ·L±A,1(ζ/ξ1)
Q operators
Q±(ζ, α)(X) = ±(1 − q±2(α−S))ζ±(α−S)
×Tr±A
(q±2αDAT±
A(ζ)−1(X))
where T±A(ζ)−1(X) = T±
A (ζ)−1(X)T±A (ζ)
and S(X) = [S[∞], X ]
analytic structure
Tr±A(q±2αDAqmDA) = ±(1 − q±2α+m)−1
simple poles at (ζ/ξj)2 = 1
149
8. Triangularity
TQ relation follows from triangular de-composition
L+A,a,j(ζ) = (G+
A,a)−1L+A,j(ζ)Ra,j(ζ)G+
A,a
= (qζ − q−1ζ−1) ×L+
A,j(q−1ζ)q
−σ3j/2
0
∗ β(ζ)L+A,j(qζ)q
σ3j/2
a
G+A,a = q−σ3
aDA(1 + a∗Aσ+a )
150
9. Reduction to the left
q difference operator
∆q(F (ζ)) = F (qζ) − F (q−1ζ)
off diagonal element
±(1 − q±2(α−S))−1k±(ζ, α)(X)def= ζ±(α−S)
×Tr±Atra
(q±2(α∓1)DAσ±a Ta(ζ)−1T±
A(ζ)−1(X))
= ζ±(α−S)Tr±Atra
(q±2αDAσ±a T±
A,a(ζ)−1(X))
σ±a picks up the off-diagonal elements
151
10. Annihilation operators
Reduction modulo q exact form
k±[1,n](ζ,α)(qασ31X[2,n])=q(α∓1)σ3
1k±[2,n](ζ,α)(X[2,n])
+σ±1 ∆q
(q−q−1
ζ/ξ1−(ζ/ξ1)−1Q±[2,n](ζ, α ∓ 1)(X[2,n])
)
annihilation operators
c±(ζ, α)def= (normalization)×
n∑
j=1
singζ=ξj k±(ζ, α)
satisfies the reduction to the left
c±[1,n](ζ, α)(qασ31X[2,n])=q(α∓1)σ3
1c±[2,n](ζ, α)(X[2,n])
152
11. Creation operators
conjugate transfer matrix
t∗[1,n](ζ, α)(X) = tra(Ta(ζ)qασ3aXTa(ζ)−1)
filling Dirac sea
t∗[1,n](ξl, α)(X[1,j])=2sl−1 · · · s1(qασ3
1 · τ (X[1,j]))
where τ is shift operator
another TQ relations
Q∗±[1,n]
(ζ, α)(X) = ±(1 − q±2(α−S))ζ±(α−S)
×Tr±A
(T∓
A,[1,n](ζ)q±2αDA(X)
)
t∗[1,n](ζ, α)Q∗±[1,n]
(ζ, α)=Q∗±[1,n](q
−1ζ, α)+Q∗±[1,n](qζ, α)
153
off diagonal operators
k∗±(ζ, α)(X) = ±(1 − q±2(α±1−S))ζ±(α±1−S)
×Tr±Atra
(σ±a Ta(ζ)T∓
A(ζ)(qα(±2DA+σ3
a)∓2SX))
reduction to the right modulo q exact form
k∗±[1,n](ζ, α)(X[1,n−1]) = k∗±[1,n−1](ζ, α)(X[1,n−1])
+σ±n∆q
(q−q−1
ζ/ξ1−(ζ/ξ1)−1Q∗±[1,n−1](ζ, α)(q
∓2S[1,n−1]X[1,n−1]))
removing ∆q
f∗±[1,n](ζ, α)(X[1,n−1]) = f∗±[1,n−1](ζ, α)(X[1,n−1])
+σ±n κ(ζ/ξn)Q∗±[1,n−1](ζ, α)(q
∓2S[1,n−1]X[1,n−1])
154
creation operators
c∗±[1,n](ζ, α)def= f∗±[1,n](qζ, α) + f∗±[1,n](q
−1ζ, α)
−t∗[1,n](ζ, α)f∗±[1,n](ζ, α)
satisfies reduction property in a restricted sense
c∗±[1,n]
(ζ, α)(X[1,j]) (1 ≤ j < l ≤ n)
is regular at ζ = ξl and satisfies
c∗±[1,n]
(ξl, α)(X[1,j]) = c∗±[1,n−1]
(ξl, α)(X[1,j])
(1 ≤ j < l ≤ n − 1)
ConjectureA fermionic basis is created by the creation
operators c∗±[1,n]
(ξl, α)