Old roots in new equations Christian Marboe Based on [1608.06504] - - PowerPoint PPT Presentation
Old roots in new equations Christian Marboe Based on [1608.06504] + [1701.03704] + [17.] with Dmytro Volin [1706.02320] with Rouven Frassek, David Meidinger IGST 2017 1 / 32 The perturbative spectral problem in planar N = 4 SYM D O =
Christian Marboe
Based on [1608.06504] + [1701.03704] + [17–.—] with Dmytro Volin [1706.02320] with Rouven Frassek, David Meidinger
IGST 2017
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IGST05 Asymptotic Bethe Ansatz [Beisert, Staudacher]
j
x−
j
L = x−
j
− x+
k
x+
j − x− k
1 −
g2 x+
j x− k
1 −
g2 x−
j x+ k
e2 i θ(uj ,uk ) 1 x + x = u g
3 / 32
IGST05 Asymptotic Bethe Ansatz [Beisert, Staudacher]
j
x−
j
L = x−
j
− x+
k
x+
j − x− k
1 −
g2 x+
j x− k
1 −
g2 x−
j x+ k
e2 i θ(uj ,uk ) 1 x + x = u g
IGST09 Thermodynamic Bethe Ansatz
Ambjørn, Janik, Kristjansen Arutyunov, Frolov Gromov, Kazakov, Vieira, Kozak Bombardelli, Fioravanti, Tateo
log (Y a,s(u)) =
a,s
(u, v) log(1 + Y a′,s′(v))
3 / 32
IGST05 Asymptotic Bethe Ansatz [Beisert, Staudacher]
j
x−
j
L = x−
j
− x+
k
x+
j − x− k
1 −
g2 x+
j x− k
1 −
g2 x−
j x+ k
e2 i θ(uj ,uk ) 1 x + x = u g
IGST09 Thermodynamic Bethe Ansatz
Ambjørn, Janik, Kristjansen Arutyunov, Frolov Gromov, Kazakov, Vieira, Kozak Bombardelli, Fioravanti, Tateo
log (Y a,s(u)) =
a,s
(u, v) log(1 + Y a′,s′(v))
IGST13 Quantum Spectral Curve [Gromov, Kazakov, Leurent, Volin]
QQ = Q−Q+ − Q+Q−, ˜ Q(u) = Q(u⋆)
3 / 32
Tr[D4Z2XΨ]
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
Tr[D4Z2XΨ]
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
Tr[D4Z2XΨ]
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
Tr[D4Z2XΨ]
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
Tr[D4Z2XΨ]
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
4 / 32
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
4 / 32
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
4 / 32
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
✫✪ ✬✩
4 / 32
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
4 / 32
12g2 − 48g4 + 336g6 +g8 − 2496 + 576 ζ3 −1440 ζ5
15168 +6912 ζ3 − 5184 ζ2
3
−8640 ζ5 + 30240 ζ7
❅
4 / 32
What to put into the machine
5 / 32
b1, ..., bN a1, ..., aM f1, ..., fK [ai, a†
j ] = [bi, b† j ] = {fi, f† j } = δij
Emn = −b ˙
αb† ˙ β
−b ˙
αaβ
−b ˙
αfj
a†
αb† ˙ β
a†
αaβ
a†
αfj
f†
i b† ˙ β
f†
i aβ
f†
i fj
C =
Enn = −N − nb + na + nf
6 / 32
a1 f1 f2 Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C =
Enn = na1 + nf1 + nf2
7 / 32
Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C = na1 + nf1 + nf2 = 1
7 / 32
Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0
7 / 32
Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0 States: f†
1|0 ≡ Z
f†
2|0 ≡ X
a†
1|0 ≡ Ψ
7 / 32
Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0 States: f†
1|0 ≡ Z
f†
2|0 ≡ X
a†
1|0 ≡ Ψ
HWS: Emn|0 = 0, m < n
7 / 32
Emn = a†
1a1
a†
1f1
a†
1f2
f†
1a1
f†
1f1
f†
1f2
f†
2a1
f†
2f1
f†
2f2
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0 States: f†
1|0 ≡ Z
f†
2|0 ≡ X
a†
1|0 ≡ Ψ
HWS: Emn|0 = 0, m < n
✈ ✈ ❢ ❢
×
7 / 32
Emn = f†
1f1
f†
1a1
f†
1f2
a†
1f1
a†
1a1
a†
1f2
f†
2f1
f†
2a1
f†
2f2
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0 States: f†
1|0 ≡ Z
f†
2|0 ≡ X
a†
1|0 ≡ Ψ
HWS: Emn|0 = 0, m < n
✈ ✈ ❢ ❢
× ×
7 / 32
Emn = f†
1f1
f†
1f2
f†
1a1
f†
2f1
f†
2f2
f†
2a1
a†
1f1
a†
1f2
a†
1a1
C = na1 + nf1 + nf2 = 1 Fock vacuum: a|0 = f|0 = 0 States: f†
1|0 ≡ Z
f†
2|0 ≡ X
a†
1|0 ≡ Ψ
HWS: Emn|0 = 0, m < n
✈ ✈ ❢ ❢
×
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ, ZΨ, ΨZ, ΨΨ, ZX, XZ, XΨ, ΨX, XX
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
✈ ✈ ❢ ❢
×
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
✈ ✈ ❢ ❢
× ×
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
✈ ✈ ❢ ❢
×
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
✲ ✛ ✲ ✛ ✻ ❄ ⑦ ⑦ ⑦
na1 nf1 nf2
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
7 / 32
Tensor product spaces e.g. a†
1|0 ⊗ f1|0 ≡ ΨZ
ZZ ZΨ+ΨZ ZX +XZ XΨ+ΨX XX ZΨ−ΨZ ΨΨ ZX −XZ XΨ−ΨX
7 / 32
u(1|2)
8 / 32
u(3|3)
8 / 32
u(4|1)
8 / 32
u(0|3)
8 / 32
u(3) C = 1
← at each site
L = 8
✇
f†
1|0 ⊗ f† 1|0 ⊗ f† 1|0 ⊗ f† 1|0 ⊗ f† 2|0 ⊗ f† 2|0 ⊗ f† 2|0 ⊗ f† 3|0
8 / 32
⊗8 u(3) C = 1
← at each site
L = 8
✇
f†
1|0 ⊗ f† 1|0 ⊗ f† 1|0 ⊗ f† 1|0 ⊗ f† 2|0 ⊗ f† 2|0 ⊗ f† 2|0 ⊗ f† 3|0
8 / 32
u(4) C = 2 L = 8
✇
|0 → f†
0|˜
C → C + 1
f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
3f† 0|˜
8 / 32
⊗8 u(4) C = 2 L = 8
✇
|0 → f†
0|˜
C → C + 1
f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
1f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
2f† 0|˜
0 ⊗ f†
3f† 0|˜
8 / 32
u(5) C = 3 L = 8
✇
|0 → f†
0f†
˜ C → C + 2
8 / 32
b1, b2 a1, a2 f1, f2, f3, f4
9 / 32
b1, b2 a1, a2 f1, f2, f3, f4 nf + na − nb = 2
9 / 32
b1, b2 a1, a2 f1, f2, f3, f4 nf + na − nb = 2 Fαβ ≡ a†
αa† β|0
Ψαi ≡ a†
αf† i |0
Φij ≡ f†
i f† j |0
Dα ˙
α ≡ a† αb† ˙ α
¯ Ψ ˙
αi
≡ ǫijklb†
˙ αf† j f† kf† l |0
¯ F ˙
α ˙ β
≡ b†
˙ αb† ˙ βf† 1f† 2f† 3f† 4|0
9 / 32
✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✻ ❄ ✻ ❄ ✻ ❄ ✻ ❄ ⑥ ⑥ ⑥ ⑥ ⑥ ② ⑥ ⑥
nf4 nf3 nf2 nf1 nb1 nb2 na1 na2 [G¨ unaydin, Volin] pu(2, 2|4)
10 / 32
s
11 / 32
s
11 / 32
s
11 / 32
s ✉ ✉
u(2, 3|2) C = −1
11 / 32
s ✉ ✉
u(1, 4|12) C = 3
11 / 32
s ✉ ✉
u(9) C = 5
11 / 32
s ✉ ✉
u(2, 2|4) C = 0
Multiplet counting with so(N) characters (susy, non-compact)
Morales, Samtleben ’04
s ✉ ✉
u(9) C = 5
Multiplet counting with so(N) characters (susy, non-compact)
Morales, Samtleben ’04
u(N) characters
[CM, Volin ’17] 11 / 32
∆0 2 3 4
2× 2× 2×
5
4×
27 multiplets
12 / 32
∆0 2 3 4
2× 2× 2×
6
16×
144 multiplets
12 / 32
∆0 2 3 4
2× 2× 2×
7
74×
918 multiplets
12 / 32
∆0 2 3 4
2× 2× 2×
8
376×
6918 multiplets
12 / 32
Getting the machine started
13 / 32
256 Q’s
Q = Q(u)
multiplet
Q∅|∅ Q∅|i Q∅|ij Q∅|ijk Q∅|1234 Qa|∅ Qa|i Qa|ij Qa|ijk Qa|1234 Qab|∅ Qab|i Qab|ij Qab|ijk Qab|1234 Qabc|∅ Qabc|i Qabc|ij Qabc|ijk Qabc|1234 Q1234|∅ Q1234|i Q1234|ij Q1234|ijk Q1234|1234
14 / 32
Qa|i Qab|i Qac|i Qabc|i QQ = Q−Q+ − Q+Q−
Q± = Q(u ± i
2)
14 / 32
Qa|i Qa|ij Qa|ik Qa|ijk QQ = Q−Q+ − Q+Q−
Q± = Q(u ± i
2)
14 / 32
Qa|i Qa|ij Qab|i Qab|ij QQ = Q−Q+ − Q+Q−
Q± = Q(u ± i
2)
14 / 32
Boundary conditions g → 0 Q =
weights
up to factors of u±L
14 / 32
Q0,0 Q0,1 Q0,2 Q0,3 Q0,4 Q1,0 Q1,1 Q1,2 Q1,3 Q1,4 Q2,0 Q2,1 Q2,2 Q2,3 Q2,4 Q3,0 Q3,1 Q3,2 Q3,3 Q3,4 Q4,0 Q4,1 Q4,2 Q4,3 Q4,4 Distinguished Q-functions
≡
polynomials of lowest degree
14 / 32
15 / 32
Q∅|1 Q1|1 Q12|1 Q12|12 Q12|123 Q123|123 Q1234|123 Bethe/Baxter equations
15 / 32
Q∅|1 Q1|1 Q12|1 Q12|12 Q12|123 Q123|123 Q1234|123 Bethe/Baxter equations
Q1|1(u1,k + i
2 ) = Q1|1(u1,k − i 2 )
Q∅|1(u2,j+ i
2 )Q12(u2,j+ i 2 )
Q∅(u2,j− i
2 )Q12(u2,j− i 2 ) = − Q1(u2,j−i)
Q1(u2,j+i)
15 / 32
Q∅|1 Q1|1 Q12|1 Q12|12 Q12|123 Q123|123 Q1234|123 Bethe/Baxter equations
Q1|1(u1,k + i
2 ) = Q1|1(u1,k − i 2 )
Q∅|1(u2,j+ i
2 )Q12(u2,j+ i 2 )
Q∅(u2,j− i
2 )Q12(u2,j− i 2 ) = − Q1(u2,j−i)
Q1(u2,j+i)
15 / 32
Q∅|1 Q1|1 Q12|1 Q12|12 Q12|123 Q123|123 Q1234|123 Bethe/Baxter equations
Q1|1(u1,k + i
2 ) = Q1|1(u1,k − i 2 )
Q∅|1(u2,j+ i
2 )Q12(u2,j+ i 2 )
Q∅(u2,j− i
2 )Q12(u2,j− i 2 ) = − Q1(u2,j−i)
Q1(u2,j+i)
15 / 32
s s s s s s
Q∅|∅ Q∅|1 Q2|∅ Q1|∅ Q12|∅ Q2|1 Q1|1 Q12|1 u(1|2) again
16 / 32
s s s s s s
Q∅|∅ Q∅|1 Q2|∅ Q1|∅ Q12|∅ Q2|1 Q1|1 Q12|1 Bethe/Baxter equations
16 / 32
s s s s s s
Q∅|∅ Q∅|1 Q2|∅ Q1|∅ Q12|∅ Q2|1 Q1|1 Q12|1 Bethe/Baxter equations
How to impose polynomiality
16 / 32
s s s s s s
Q∅|∅ Q∅|1 Q2|∅ Q1|∅ Q12|∅ Q2|1 Q1|1 Q12|1 u(1|2)
17 / 32
s s s s s s
Q∅|∅ Q∅|1 Q2|∅ Q1|∅ Q12|∅ Q2|1 Q1|1 Q12|1
17 / 32
s s s s s s s s s s s s s s s
Q∅|∅ Q∅|i Q∅|ij Q∅|ijk Q∅|1234 Qa|∅ Qa|i Qa|ij Q1|ijk Q1|1234 Qab|∅ Qab|i Q12|12 Q123|∅ Q123|1
17 / 32
s s s s s s s s s s s s s s s
Q0,0 Q0,1 Q0,2 Q0,3 Q0,4 Q1,0 Q1,1 Q1,2 Q1,3 Q1,4 Q2,0 Q2,1 Q2,2 Q3,0 Q3,1 All distinguished Q’s polynomial ⇓ All Q’s polynomial
[CM, Volin ’16] 17 / 32
② ② ②
1
② ② ②
3
②
1
② ② ② ②
7
②
4
②
2
②
1
②
#roots = #boxes right/above
17 / 32
② ② ②
1
② ② ②
3
②
1
② ② ② ②
7
②
4
②
2
②
1
②
RECIPE (1) Make poly. ansatz on path u + c u3 + d2u2 + d1u + d0 u7
✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❅ ■ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮
17 / 32
❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ■ ❅ ❅ ■ ② ② ②
1
② ② ②
3
②
1
② ② ② ②
7
②
4
②
2
②
1
②
RECIPE (1) Make poly. ansatz on path u + c u3 + d2u2 + d1u + d0 (2) Generate rest by polynomial division Q ∝ Quotient
Q
❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ■ ❅ ❅ ■ ② ② ②
1
② ② ②
3
②
1
② ② ② ②
7
②
4
②
2
②
1
②
RECIPE (1) Make poly. ansatz on path u + c u3 + d2u2 + d1u + d0 (2) Generate rest by polynomial division Q ∝ Quotient
Q
Remainder
Q
17 / 32
✇ ✇
2
✇
4
②
11
✇ ✇ ✇
1
✇
2
✇
8
✇ ✇ ✇ ✇ ✇
5
✇ ✇ ✇ ✇ ✇
1
✇
2
✇
4
✇ ✇ ✇
2
✇
4
✇
3
✇ ✇ ✇
3
✇
6
✇
2
✇ ✇ ✇
4
✇
4
✇
1
✇ ✇ ✇ ✇
5
✇
2
✇ ✇ ✇ ✇
1
✇
7
✇
1
✇ ✇ ✇ ✇
2
✇
9
✇ ✇ ✇ ✇
1
✇
4
②
12
✇ ✇ ✇
2
✇
6
②
15
✇
same idea ∼ factors of u±L
18 / 32
✇ ✇
2
✇
4
②
11
✇ ✇ ✇
1
✇
2
✇
8
✇ ✇ ✇ ✇ ✇
5
✇ ✇ ✇ ✇ ✇
1
✇
2
✇
4
✇ ✇ ✇
2
✇
4
✇
3
✇ ✇ ✇
3
✇
6
✇
2
✇ ✇ ✇
4
✇
4
✇
1
✇ ✇ ✇ ✇
5
✇
2
✇ ✇ ✇ ✇
1
✇
7
✇
1
✇ ✇ ✇ ✇
2
✇
9
✇ ✇ ✇ ✇
1
✇
4
②
12
✇ ✇ ✇
2
✇
6
②
15
✇
same idea ∼ factors of u±L
18 / 32
Give generic code 15 minutes to solve per diagram
(on a worn-out MacBook Air)
∆0 # diagrams solved total # solutions found 2 1 / 1 1 / 1 3 1 / 1 1 / 1 4 7 / 7 10 / 10 5 13 / 13 27 / 27 5.5 12 / 12 36 / 36 6 39 / 39 144 / 144 6.5 36 / 36 276 / 276 7 68 / 77 600 / 918 7.5 54 / 84 694 / 2204 8 107 / 180 1395 / 6918
19 / 32
Engine architecture
20 / 32
256 Q’s Q∅|∅ Q∅|i Q∅|ij Q∅|ijk Q∅|1234 Qa|∅ Qa|i Qa|ij Qa|ijk Qa|1234 Qab|∅ Qab|i Qab|ij Qab|ijk Qab|1234 Qabc|∅ Qabc|i Qabc|ij Qabc|ijk Qabc|1234 Q1234|∅ Q1234|i Q1234|ij Q1234|ijk Q1234|1234
[Gromov, Kazakov, Leurent, Volin ’13,’14]
21 / 32
Cut structure
Qa|∅ Qabc|1234
u u −2g 2g
r r ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ r r r r r r r r r r
21 / 32
Cut structure
Q∅|i Q∅|ij Q∅|ijk Qa|i Qa|ij Qa|ijk Qab|i Qab|ij Qab|ijk Qabc|i Qabc|ij Qabc|ijk Q1234|i Q1234|ij Q1234|ijk
u u
r r r r r r ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ r r r r r r
21 / 32
Analytic continuation
Qa|∅ Qabc|1234
s s
u
21 / 32
Analytic continuation
Pa Pa
s s
u
21 / 32
Analytic continuation
Pa Pa
s s s s s s s s s s
˜ Pa = µabPb u
21 / 32
Analytic continuation
Pa Pa
s s s s s s s s s s
˜ Pa = µabPb µab = ωijQ−
ab|ij
u
21 / 32
22 / 32
Pµ, sl(2)
[CM,Volin ’14] 22 / 32
Pµ, sl(2)
[CM,Volin ’14]
Q-system, general
Gromov, Sizov Levkovich-Maslyuk’15
22 / 32
Pµ, sl(2)
[CM,Volin ’14]
Q-system, general
Gromov, Sizov Levkovich-Maslyuk’15
Pµ, general
[CM,Volin ’17] 22 / 32
Pa ˜ Pa
r r r r r r r r r r r r
u
23 / 32
Pa ˜ Pa
r r r r r r r r r r r r
u
r r ✫✪ ✬✩
x Pa ˜ Pa
r r rr r r r r r r r r r r r r r
x + 1
x = u g
P(x) =
∞
ck xk
23 / 32
Pa ˜ Pa
r r r r r r r r r r r r
u
r r ✫✪ ✬✩
x Pa ˜ Pa
r r rr r r r r r r r r r r r r r
x + 1
x = u g
P(x) =
∞
ck xk
P(u) = − 1
u + g2
− 1
u3 − c u2 − γ1 2u
finite # of constants at each loop
23 / 32
1 Pa Pa 1
24 / 32
solve Q−
a|i − Q+ a|i = PaPaQ+ b|i
Q-system based [Gromov, Levkovich-Maslyuk, Sizov ’15] 1 Pa Qa|i Pa 1
24 / 32
solve Q−
a|i − Q+ a|i = PaPaQ+ b|i
Q-system based [Gromov, Levkovich-Maslyuk, Sizov ’15] 1 Qi Pa Qa|i Pa 1
24 / 32
1 Qi Pa Qa|i Qa|i Pa Qi 1 Q-system based [Gromov, Levkovich-Maslyuk, Sizov ’15] Gluing ˜ Q• = ¯ Q•
r r
r r
24 / 32
1 Qi Pa Qa|i Qa|i Pa Qi 1 Gluing ˜ Q• = ¯ Q• Q-system based [Gromov, Levkovich-Maslyuk, Sizov ’15]
r r
r r ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅
24 / 32
1 Pa Pa 1 Pµ-system based [CM, Volin ’17]
24 / 32
1 Pa Qab|ij Pa 1 solve
Q−
ab|ij − Q+ ab|ij = P[aQ− b]c|ijPc
Pµ-system based [CM, Volin ’17]
24 / 32
1 Pa
µab
Pa 1 solve
µab − µ++
ab = P[aµb]cPc
Pµ-system based [CM, Volin ’17]
24 / 32
1 Pa
µab
Pa 1 Gluing ˜ P• = µ•P•
r r r r
24 / 32
25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 + g10 28188 − 288ζ3
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 + g10 28188 − 288ζ3
− 339012 + 7776ζ3 + 12096ζ5 − 18144ζ9
25 / 32
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 + g10 28188 − 288ζ3
− 339012 + 7776ζ3 + 12096ζ5 − 18144ζ9
4214268 − 39744ζ3 − 181440ζ5 + 57024ζ2
3 − 260064ζ7 − 34560ζ3ζ5 − 60480ζ9 − 8640ζ2 5 − 96768ζ3ζ7
+665280ζ11
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 + g10 28188 − 288ζ3
− 339012 + 7776ζ3 + 12096ζ5 − 18144ζ9
4214268 − 39744ζ3 − 181440ζ5 + 57024ζ2
3 − 260064ζ7 − 34560ζ3ζ5 − 60480ζ9 − 8640ζ2 5 − 96768ζ3ζ7
+665280ζ11
3 − 82944 ζ3 3 + 1664064 ζ5 − 1510272 ζ3 ζ5
−290304 ζ2
3 ζ5 + 250560 ζ2 5 + 4257792 ζ7 + 628992 ζ3 ζ7 + 1451520 ζ5 ζ7 + 4711968 ζ9 + 2903040 ζ3 ζ9
+ 11144736 5 ζ11 − 16061760 ζ13 − 124416 5 Z(2)
11
Exceptional solution
Q = u(u − i
2)(u + i 2)
su(2) sector Z3X 3
r
γ = 12g2 − 36g4 + 252g6 − 2484g8 + g10 28188 − 288ζ3
− 339012 + 7776ζ3 + 12096ζ5 − 18144ζ9
4214268 − 39744ζ3 − 181440ζ5 + 57024ζ2
3 − 260064ζ7 − 34560ζ3ζ5 − 60480ζ9 − 8640ζ2 5 − 96768ζ3ζ7
+665280ζ11
3 − 82944 ζ3 3 + 1664064 ζ5 − 1510272 ζ3 ζ5
−290304 ζ2
3 ζ5 + 250560 ζ2 5 + 4257792 ζ7 + 628992 ζ3 ζ7 + 1451520 ζ5 ζ7 + 4711968 ζ9 + 2903040 ζ3 ζ9
+ 11144736 5 ζ11 − 16061760 ζ13 − 124416 5 Z(2)
11
3 − 1119744 ζ3 3 − 248832 ζ4 3 − 502848 ζ5 + 25653888 ζ3 ζ5
+3836160 ζ2
3 ζ5 + 5987520 ζ2 5 + 6635520 ζ3 ζ2 5 − 45170784 ζ7 + 22037184 ζ3 ζ7 + 6676992 ζ2 3 ζ7
−5766336 ζ5 ζ7 − 16027200 ζ2
7 − 75035808 ζ9 + 10018944 ζ3 ζ9 − 38361600 ζ5 ζ9 − 79511328 ζ11
−58848768 ζ3 ζ11 − 273255552 5 ζ13 + 324324000 ζ15 + 311040 Z(2)
11 −
601344 5 Z(2)
13 + 145152 Z(3) 13
e.g. Z (2)
13
= −ζ5,3,5 + 11 ζ5 ζ3,5 + 5 ζ5 ζ8
appearance of Z conjectured in [Broadhurst, Kreimer ’95]
25 / 32
✇ ✇
1
✇
3
✇
5
✇ ✇ ✇
2
✇
5
✇
4
✇ ✇ ✇
3
✇
7
✇
3
✇ ✇ ✇
4
✇
5
✇
2
✇ ✇ ✇
5
✇
3
✇
1
✇ ✇ ✇
1
✇
3
✇
8
✇
4
✇ ✇
2
✇
5
✇
6
✇
3
✇ ✇
3
✇
7
✇
4
✇
2
✇ ✇
4
✇
4
✇
2
✇
1
✇ ✇
5
✇
1
✇ ✇
26 / 32
✇ ✇
1
✇
3
✇
5
✇ ✇ ✇
2
✇
5
✇
4
✇ ✇ ✇
3
✇
7
✇
3
✇ ✇ ✇
4
✇
5
✇
2
✇ ✇ ✇
5
✇
3
✇
1
✇ ✇ ✇
1
✇
3
✇
8
✇
4
✇ ✇
2
✇
5
✇
6
✇
3
✇ ✇
3
✇
7
✇
4
✇
2
✇ ✇
4
✇
4
✇
2
✇
1
✇ ✇
5
✇
1
✇ ✇
Q = (u + i
2)L(u − i 2)L 3 j=1(u − uj)
Q = (u + i
2)L(u − i 2)L 1 j=1(u − uj)
❥ ✐ ❤ ❥ ✐ ❤
26 / 32
✇ ✇
1
✇
3
✇
5
✇ ✇ ✇
2
✇
5
✇
4
✇ ✇ ✇
3
✇
7
✇
3
✇ ✇ ✇
4
✇
5
✇
2
✇ ✇ ✇
5
✇
3
✇
1
✇ ✇ ✇
1
✇
3
✇
8
✇
4
✇ ✇
2
✇
5
✇
6
✇
3
✇ ✇
3
✇
7
✇
4
✇
2
✇ ✇
4
✇
4
✇
2
✇
1
✇ ✇
5
✇
1
✇ ✇
Q = (u + i
2)4(u − i 2)4(u + i 2)(u − i 2)(u − u•)
Q = (u + i
2)5(u − i 2)5(u − u•)
❥ ✐ ❤ ❥ ✐ ❤
26 / 32
✇ ✇
1
✇
3
✇
5
✇ ✇ ✇
2
✇
5
✇
4
✇ ✇ ✇
3
✇
7
✇
3
✇ ✇ ✇
4
✇
5
✇
2
✇ ✇ ✇
5
✇
3
✇
1
✇ ✇ ✇
1
✇
3
✇
8
✇
4
✇ ✇
2
✇
5
✇
6
✇
3
✇ ✇
3
✇
7
✇
4
✇
2
✇ ✇
4
✇
4
✇
2
✇
1
✇ ✇
5
✇
1
✇ ✇
Q = (u + i
2)4(u − i 2)4(u + i 2)(u − i 2)(u − u•)
Q = (u + i
2)5(u − i 2)5(u − u•)
❥ ✐ ❤ ❥ ✐ ❤
26 / 32
27 / 32
27 / 32
28 / 32
For fixed quantum numbers Q = {Q(1), Q(2), Q(3)} solution ⇔ primary operator (HWS)
29 / 32
Q = Q(1) Q(2) Q(3)
29 / 32
Q = Q(1) Q(2) Q(3) ∞ ∞ ∞ ∞ ∞
29 / 32
Q = Q(1) Q(2) Q(3) Q(4) Q(5) Q(6) Q(7) Q(8)
29 / 32
Q = Q(1,1) Q(1,2) Q(1,3) Q(1,4) Q(1,5) Q(1,6) Q(1,7) Q(1,8) Q(2,1) Q(2,2) Q(2,3) Q(2,4) Q(2,5) Q(2,6) Q(2,7) Q(2,8) Q(3,1) Q(3,2) Q(3,3) Q(3,4) Q(3,5) Q(3,6) Q(3,7) Q(3,8) Q(4,1) Q(4,2) Q(4,3) Q(4,4) Q(4,5) Q(4,6) Q(4,7) Q(4,8) Q(5,1) Q(5,2) Q(5,3) Q(5,4) Q(5,5) Q(5,6) Q(5,7) Q(5,8) Q(6,1) Q(6,2) Q(6,3) Q(6,4) Q(6,5) Q(6,6) Q(6,7) Q(6,8) Q(7,1) Q(7,2) Q(7,3) Q(7,4) Q(7,5) Q(7,6) Q(7,7) Q(7,8) Q(8,1) Q(8,2) Q(8,3) Q(8,4) Q(8,5) Q(8,6) Q(8,7) Q(8,8)
29 / 32
Q(m,n)
ab...|ij... = ZX...X|Qab...|ij...|XΨ...Z
30 / 32
Q(m,n)
A|J
= ZX...X|QA|J|XΨ...Z
30 / 32
Q(m,n)
A|J
= ZX...X|TrA
ξξ R{1} A|J R{2} A|J · · · R{L} A|J
Bazhanov, Frassek, Lukowski, Meneghelli, Staudacher ’09-’15
twist phase
✻
R-matrix
acts on one physical site + auxiliary space
❄
auxiliary space oscillators
30 / 32
Q(m,n)
A|J
= TrA
ξξ Z|R{1} A|J |X X|R{2} A|J |Ψ · · · X|R{L} A|J |Z
Z|RA|J|X
30 / 32
Z|RA|J|X RA|J(u) ∼ eξE n¯
n Γ(u+iE ¯ n¯ n)
Γ(u)
e¯
ξE ¯
nn
n ∈ A|J E mn =
1
2
1
2
f†
1b† 1
f†
1b† 2
f†
1f1
f†
1f2
f†
1f3
f†
1f4
f†
1a1
f†
1a2
f†
2b† 1
f†
2b† 2
f†
2f1
f†
2f2
f†
2f3
f†
2f4
f†
2a1
f†
2a2
f†
3b† 1
f†
3b† 2
f†
3f1
f†
3f2
f†
3f3
f†
3f4
f†
3a1
f†
3a2
f†
4b† 1
f†
4b† 2
f†
4f1
f†
4f2
f†
4f3
f†
4f4
f†
4a1
f†
4a2
a†
1b† 1
a†
1b† 2
a†
1f1
a†
1f2
a†
1f3
a†
1f4
a†
1a1
a†
1a2
a†
2b† 1
a†
2b† 2
a†
2f1
a†
2f2
a†
2f3
a†
2f4
a†
2a1
a†
2a2
30 / 32
Z|RA|J|X RA|J(u) ∼ eξE n¯
n Γ(u+iE ¯ n¯ n)
Γ(u)
e
¯ ξE ¯
nn
n ∈ A|J E mn =
1
2
1
2
f†
1b† 1
f†
1b† 2
f†
1f1
f†
1f2
f†
1f3
f†
1f4
f†
1a1
f†
1a2
f†
2b† 1
f†
2b† 2
f†
2f1
f†
2f2
f†
2f3
f†
2f4
f†
2a1
f†
2a2
f†
3b† 1
f†
3b† 2
f†
3f1
f†
3f2
f†
3f3
f†
3f4
f†
3a1
f†
3a2
f†
4b† 1
f†
4b† 2
f†
4f1
f†
4f2
f†
4f3
f†
4f4
f†
4a1
f†
4a2
a†
1b† 1
a†
1b† 2
a†
1f1
a†
1f2
a†
1f3
a†
1f4
a†
1a1
a†
1a2
a†
2b† 1
a†
2b† 2
a†
2f1
a†
2f2
a†
2f3
a†
2f4
a†
2a1
a†
2a2
30 / 32
n Γ(u+iE ¯ n¯ n)
Γ(u)
e
¯ ξE ¯
nn
n ∈ A|J E mn =
1
2
1
2
f†
1b† 1
f†
1b† 2
f†
1f1
f†
1f2
f†
1f3
f†
1f4
f†
1a1
f†
1a2
f†
2b† 1
f†
2b† 2
f†
2f1
f†
2f2
f†
2f3
f†
2f4
f†
2a1
f†
2a2
f†
3b† 1
f†
3b† 2
f†
3f1
f†
3f2
f†
3f3
f†
3f4
f†
3a1
f†
3a2
f†
4b† 1
f†
4b† 2
f†
4f1
f†
4f2
f†
4f3
f†
4f4
f†
4a1
f†
4a2
a†
1b† 1
a†
1b† 2
a†
1f1
a†
1f2
a†
1f3
a†
1f4
a†
1a1
a†
1a2
a†
2b† 1
a†
2b† 2
a†
2f1
a†
2f2
a†
2f3
a†
2f4
a†
2a1
a†
2a2
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j(u) ∼ eξE n¯
n Γ(u+iE ¯ n¯ n)
Γ(u)
e¯
ξE ¯
nn 30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...)
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = Γ(...) Γ(...)Γ(...)
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = explicitly truncating sums
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = explicitly truncating sums
Z|R∅|j|X = u + N + N2 + ... u + N + ...
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = explicitly truncating sums
Z|R∅|j|X = u + N + N2 + ... u + N + ...
trace over auxiliary space Qa|∅, Q∅|j
s s s
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = explicitly truncating sums
Z|R∅|j|X = u + N + N2 + ... u + N + ...
trace over auxiliary space Qa|∅, Q∅|j
from QQ-relations Qabc...|ijk...
s s s s s s s s s s s s s s s s s s s s s s s s s
ր ր ր ր ր ր ր ր ր ր ր ր ր ր ր ր
30 / 32
[Frassek, Meidinger, CM ’17]
R’s with one index: R∅|j ∼
∞
(ξE n¯
n)θ(k)k · · · · · · · · · (¯
ξE ¯
nn)θ(−k)k
↑
3F2(N, N, k, u, ...) = explicitly truncating sums
Z|R∅|j|X = u + N + N2 + ... u + N + ...
trace over auxiliary space Qa|∅, Q∅|j
from QQ-relations Qabc...|ijk...
twisted QSC δQabc...|ijk...
[Volin, Kazakov, Leurent ’15] 30 / 32
◮ Automatic perturbative solution of the spectral problem
Analytic 8-10 loop results for the first thousands of states
31 / 32
◮ Automatic perturbative solution of the spectral problem
Analytic 8-10 loop results for the first thousands of states
◮ Patterns in the spectrum?
31 / 32
◮ Automatic perturbative solution of the spectral problem
Analytic 8-10 loop results for the first thousands of states
◮ Patterns in the spectrum? ◮ Perturbative Q-operators might tell us more about nature of
AdS/CFT integrability
31 / 32
32 / 32