Gluon scattering amplitudesand two-dimensional integrable systemsJHEP 1202(2012)003; arXiv:1211.6225, to appear in JHEP Y. Hatsuda (DESY), K. Ito (TIT) and Y.S.

Yuji Satoh

(University of Tsukuba)

Based on

Y. Hatsuda (DESY), K. Ito (TIT), K. Sakai (YITP) and Y.S.

JHEP 1004(2010)108; 1009(2010)064; 1104(2011)100 ;

Gluon scattering amplitudes

and two-dimensional integrable systems

JHEP 1202(2012)003; arXiv:1211.6225, to appear in JHEP Y. Hatsuda (DESY), K. Ito (TIT) and Y.S.

string theory in

dual⇥

1 (λ： coupling)

AdS space

4D maximal 1

1. Introduction

✩ gauge/string duality

_≈

AdS/CFT correspondence

[CFT: conformal field theory]

[AdS: anti-de Sitter]

super Yang-Mills

strong/weak

✩ discovery of integrability

⇒  

  opened up new dimensions

● gauge/string duality beyond susy sectors

● quantitative analysis of

gauge theory dynamics at strong coupling

● gauge/string duality beyond susy sectors

● quantitative analysis of

gauge theory dynamics at strong coupling

.. .

● insights into applications

● stimulating study of SYM

In fact,

● spectrum of planar AdS/CFT for arbitrary coupling

・ given by thermodynamic Bethe ansatz (TBA) eqs.

・ checked up to 5/6-loops of SYM

[Gromov-Kazakov-Vieira ’09, Bombardelli-Fioravanti-Tateo ’09,

[Bajnok-Hegedus-Janik-Lukowski ’09, ... ] Arutyunov-Frolov ’09, ...]

( finite size effects of 2D integrable models )

( 4 loops ≈ 100000 Feynman diagrams )

In fact,

● spectrum of planar AdS/CFT for arbitrary coupling

・ given by thermodynamic Bethe ansatz (TBA) eqs.

・ checked up to 5/6-loops of SYM

[Gromov-Kazakov-Vieira ’09, Bombardelli-Fioravanti-Tateo ’09,

[Bajnok-Hegedus-Janik-Lukowski ’09, ... ] Arutyunov-Frolov ’09, ...]

( finite size effects of 2D integrable models )

( 4 loops ≈ 100000 Feynman diagrams )

● other aspects ?

Also,

.. .

・

gluon scattering amplitude/Wilson loops

・

correlation fn.

・

quark - anti

^-

quark potential/cusp anomalous dim.

[this talk]

gluon scatt. amplitudes at strong coupling

minimal surfaces in AdS5 x S5

thermodynamic Bethe ansatz (TBA) equations

[ AdS/CFT ]

[ integrability ]

⇐

⇐

In this talk,

● discuss maximally helicity violating (MHV) amplitude/

underlying 2D integrable models and CFTs

● derive analytic expansions

around certain kinematic points

N = 4

Wilson loops of 4D maximal ( ) SYM at strong coupling

⇑

[ regular polygonal Wilson loops ]

Plan of talk

1. Introduction

2. Gluon scattering amplitudes at strong coupling 3. Minimal surfaces in AdS and integrability

4. Analytic expansion of amplitudes at strong coupling 5. Summary

[ amplitude → min. surface ] [ min. surface → Hitchin system, TBA ] [ AdS3 case, AdS4 case ]

2. Gluon scattering amplitudes at strong coupling

dual to

[Alday-Maldacena ’07]

amplitudes of SYM

at strong coupling minimal surfaces in AdS

=

AdS/CFT

・ null boundary at AdS boundary x^µ_i+1 x^µ_i = 2 k_i^µ

・ _M : scalar part of MHV amplitudes

n-pt. amplitude n-cusp min. surface

(momentum of particle)

・ λ : ’t Hooft coupling

M e

p

2⇡

(Area)

N = 4

x4

x3 x2

x1 k4

k2

4-cusp solution (4-pt. amplitude)

● AdS5 is parametrized in asR^2,4

Y · Y := Y ²₁ Y₀² + Y₁² + Y₂² + Y₃² + Y₄² = 1

● string e.o.m. ≈ eq. of min. surface

¯Y⇥ ( Y⇥ · ¯Y⇥ )Y⇥ = 0 ( Y⇥ )² = ( ¯Y⇥ )² = 0

● simple solution ⊂  AdS3  

⇣ Y ¹ + Y ⁴ Y ¹ + Y ⁰ Y ¹ Y ⁰ Y ¹ Y ⁴

⌘ = 1 p2

⇣ e^⌧+ e^⌧ e ^⌧+ e ^⌧

⌘ Y ² = Y ³ = 0, z = ⇥ + i

( = _z , ¯ = _z¯)

● Poincare coordinates

Y ^µ = x^µ

r ^{(µ = 0, 1, 2, 3)} Y ¹ + Y ⁴ = 1

r , Y ¹ Y ⁴ = r² + x^µx_µ r

) ds² = 1

r² dr² + dx^µdx_{µ [ r →} 0 : AdS boundary]

● boundary coord. of AdS3 : _x^± ₌ ^{Y 0 ± Y 1}

Y ¹ + Y ⁴

∞ Z ∞

1 2

3 4

● regularizing area, e.g., by _{4 ! 4 2✏}

M⁴ ⇥ M^div4 exp

 1

8 f( ) ln s t

2 + const.

f( ) = p

/⇥

・ f(λ) : exactly same as in spectral problem

・ precisely matches BDS conjecture [cusp anom. dim]

[all orders in perturbation]

[Bern-Dixon-Smirnov ’05]

・ s, t : Mandelstam variables

Insights into SYM

amplitudes ≈ min. surface ≈ Wilson loop

Remainder fn. : deviation from BDS formula

dual conformal symm.

⇒ remainder fn. = fn. of cross-ratios of cusp. coord. x^µ_a

Insights into SYM

amplitudes ≈ min. surface ≈ Wilson loop

Remainder fn. : deviation from BDS formula

dual conformal symm.

⇒ remainder fn. = fn. of cross-ratios of cusp. coord. x^µ_a

amplitude at

strong coupling ≈ geometrical object

cf. 2-loop 6-pt. remainder fn.

[Del Duca-Duhr-Smirnov ’09]

R⁽²⁾_{6,W L}(u₁, u₂, u₃) = (H.1)

1 24π²G

! 1

1−u1

, u2 −1 u1+u2−1; 1

"

+ 1 24π²G

! 1

u1

, 1 u1+u2

; 1

"

+ 1 24π²G

! 1

u1

, 1 u1 +u3

; 1

"

+ 1

24π²G

! 1

1−u2

, u3 −1 u2+u3−1; 1

"

+ 1 24π²G

! 1

u2

, 1 u1+u2

; 1

"

+ 1 24π²G

! 1

u2

, 1 u2 +u3

; 1

"

+ 1

24π²G

! 1

1−u3

, u1 −1 u1+u3−1; 1

"

+ 1 24π²G

! 1

u3

, 1 u1+u3

; 1

"

+ 1 24π²G

! 1

u3

, 1 u2 +u3

; 1

"

+ 3

2G

! 0,0, 1

u1

, 1 u1+u2

; 1

"

+ 3 2G

! 0,0, 1

u1

, 1 u1+u3

; 1

"

+ 3 2G

!

0,0, 1 u2

, 1 u1+u2

; 1

"

+ 3

2G

! 0,0, 1

u2

, 1 u2+u3

; 1

"

+ 3 2G

! 0,0, 1

u3

, 1 u1+u3

; 1

"

+ 3 2G

!

0,0, 1 u3

, 1 u2+u3

; 1

"

− 1

2G

! 0, 1

u1

,0, 1 u2

; 1

"

+G

! 0, 1

u1

,0, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u1

,0, 1 u3

; 1

"

+

G

! 0, 1

u1

,0, 1 u1+u3

; 1

"

− 1 2G

! 0, 1

u1

, 1 u1

, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u1

, 1 u1

, 1 u1+u3

; 1

"

− 1

2G

! 0, 1

u1

, 1 u2

, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u1

, 1 u3

, 1 u1+u3

; 1

"

− 1 2G

! 0, 1

u2

,0, 1 u1

; 1

"

+

G

! 0, 1

u2

,0, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u2

,0, 1 u3

; 1

"

+G

! 0, 1

u2

,0, 1 u2+u3

; 1

"

− 1

2G

! 0, 1

u2

, 1 u1

, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u2

, 1 u2

, 1 u1+u2

; 1

"

− 1 2G

! 0, 1

u2

, 1 u2

, 1 u2 +u3

; 1

"

− 1

2G

! 0, 1

u2

, 1 u3

, 1 u2+u3

; 1

"

+ 1 4G

!

0, u₂−1

u1+u2−1,0, 1 1−u1

; 1

"

+ 1

4G

!

0, u₂−1

u1+u2−1, 1 1−u1

,0; 1

"

− 1 4G

!

0, u₂−1

u1+u2 −1, 1 1−u1

,1; 1

"

+ 1

4G

!

0, u₂−1

u₁+u₂−1, 1

1−u₁, 1 1−u₁; 1

"

− 1 4G

!

0, u₂−1

u₁ +u₂−1, u₂−1

u₁+u₂−1, 1 1−u₁; 1

"

− 1

2G

! 0, 1

u3

,0, 1 u1

; 1

"

− 1 2G

! 0, 1

u3

,0, 1 u2

; 1

"

+G

! 0, 1

u3

,0, 1 u1+u3

; 1

"

+

G

! 0, 1

u₃,0, 1 u₂+u₃; 1

"

− 1 2G

! 0, 1

u₃, 1

u₁, 1 u₁+u₃; 1

"

− 1 2G

! 0, 1

u₃, 1

u₂, 1 u₂+u₃; 1

"

− 1

2G

! 0, 1

u₃, 1

u₃, 1 u₁+u₃; 1

"

− 1 2G

! 0, 1

u₃, 1

u₃, 1 u₂+u₃; 1

"

+ 1

4G

!

0, u₁−1

u₁+u₃−1,0, 1 1−u₃; 1

"

+ 1 4G

!

0, u₁−1

u₁+u₃ −1, 1

1−u₃,0; 1

"

− 1

4G

!

0, u₁−1

u₁+u₃−1, 1

1−u₃,1; 1

"

+ 1 4G

!

0, u₁−1

u₁+u₃ −1, 1

1−u₃, 1 1−u₃; 1

"

− 1

4G

!

0, u₁−1

u₁+u₃−1, u₁−1

u₁+u₃−1, 1 1−u₃; 1

"

+ 1 4G

!

0, u₃−1

u₂+u₃−1,0, 1 1−u₂; 1

"

+ 1

4G

!

0, u₃−1

u₂+u₃−1, 1

1−u₂,0; 1

"

− 1 4G

!

0, u₃−1

u₂+u₃ −1, 1

1−u₂,1; 1

"

+ 1

4G

!

0, u₃−1

u₂+u₃−1, 1

1−u₂, 1 1−u₂; 1

"

− 1 4G

!

0, u₃−1

u₂ +u₃−1, u₃−1

u₂+u₃−1, 1 1−u₂; 1

"

− 1

4G

! 1

1−u₁,1, 1 u₃,0; 1

"

+ 1 2G

! 1

1−u₁, 1

1−u₁,1, 1 1−u₁; 1

"

+

– 99 –

[ “heroic effort” ]

continues 17 pages ...

[Goncharov-Spradlin-Vergu-Voloich ’10]

w/ a little more heroic effort ...

R²-^loop

6 =

3

i=1

⇥L₄(x⁺_i , x_i ) 1

2Li₄(1 1/u_i)⇤ 1

8

⇥Li₂(1 1/u_i)⇤²

+ 1

24J⁴ + ²

12 J² + ⁴ 72

amplitude at

strong coupling ≈ sum up all-orders

3. Minimal surfaces and integrability

● difficult to construct min. surfaces w/ null bound. for n ≥ 5

● but possible to obtain A(area) w/o explicit solutions

[Alday-Maldacena ’09]

・ string e.o.m. ⇒  Hitchin system

[cf. special 6-cusp sol., Sakai-Satoh ’09]

・ “patching” 4-cusp sol.

● amplitudes ⇐   TBA eq.

[Alday-Gaiotto-Maldacena ’09, Hatsuda-Ito-Sakai-YS ’10 Alday-Maldacena-Sever-Vieira ’10]

Let us see this

● 4D external momenta in

● # of cusps : even in AdS3

● 2 light-cone coord. at ∂(AdS)

( X  ,  X   ) +1 1- ( X  ,  X   ) +2 2-

( X  ,  X   ) +2 1-

[mom. conservation]

for 2n-cusp min. surface in AdS

3

R^1,1

x

^±

Pohlmeyer reduction

● basis in

[evolution of moving frame]

q = (Y ,⇥ Y ,⇥ ¯⇥Y , N⇥ )^T

● string e.o.m.

⇔

evolution eq. of

q :

_{(d + U)q = 0}

●

introducing spectral parameter ζ R^2,2 AdS₃ :

⇒  linearized eq.

B_z = ⇣ ¹

2⇥ ¹_⇥ e

1

⇥ e p ¹₂⇥

⌘

B_z¯ = ⇣ ¹

2⇤¯ ⇥e p¯

⇥e ¹₂⇤¯

0 = ⇥ ⌘

d + B( ) ⇤

⇥ , so(4) ⇠ su(2) su(2)

N_a = ¹₂ e _abcdY ^b Y ^c ¯Y ^d, e² = ¹₂ Y · ¯Y

p = p(z) := 2 ²Y · N

● compatibility/flatness cond.

) D_{z¯ z} = D_{z z¯} = 0, F_zz¯ + [ _z, _z¯] = 0 B_z = A_z + ¹ _z , B_z¯ = A_z¯ + _z¯

[ su(2) Hitchin system ]

● original solution

Y_aa˙ = ⇣ Y ¹ + Y ² Y ¹ + Y ⁰ Y ¹ Y ⁰ Y ¹ Y ²

⌘ = ( = 1)M ( = i)

= ( ₁, ₂) ; M : certain matrix

⇔

_⇥⇥^¯ _e^2↵ _{+ |p(z)|}²_e ^2↵ _{= 0}

[ generalized sinh-Gordon ]

set

2n-cusp solution

● change of variables

dw = ppdz, ˆ = ¹₄ ln pp¯

) @@¯↵ˆ 2 sinh ˆ↵ = 0 [ sinh-Gordon ] ˆ

↵ = 0

● ⇒  4-cusp sol. in w-plane

● take _{p(z) = z}^{n 2} _{+ · · ·} [polynomial]

⇒

solution w/ _{↵ˆ ! 0} (|w| 1) : 2n-cusp sol. in z-plane ) w ⇠ z ^{n2 (}_|^z_| ¹⁾

Z W

n/2 times

1 2

3 4 [no explicit form]

Cross-ratios

[ how to obtain physical info. w/o explicit sol. ]

● in each (i-th) quadrant in w-plane

⇥( ; w) = b_i( ; w) + s_i( ; w)

b_i / s_i : big/small sol. ⇠ ^ew/₀^{+ ¯w} , _e (w/⁰ + ¯w )

● can show cross-ratios are given by Stokes data

x^±_ijx^±_kl

x^±_ikx^±_jl = (s_i ^ s_j)(s_k ^ s_l)

(s_i ^ s_k)(s_j ^ s_l)( ) [ = 1, ±i for +, ]

s_i ^ s_j := det (s_i, s_j) ; x^µ_i,i+1 := x^µ_i+1 x^µ_i = 2 k_i^µ

⇒

shape of min. surface ≈ cross-ratios/momenta

Finding cross-ratios

● define T- and Y- fn. by

T_2k+1 = (s _{k 1} ^ s_k+1), T_2k = (s _{k 1} ^ s_k)^[+]

Y_s = T_{s 1}T_s+1

● identities among det

(s_i ^ s_j)(s_k ^ s_l) = (s_i ^ s_k)(s_j ^ s_l) + (s_i ^ s_l)(s_k ^ s_j)

⇡ T_s^[+]T_s^{[ ]} = T_s+1T_{s 1} + 1

⇡ Y_s^[+]Y_s^{[ ]} = (1 + Y_{s 1})(1 + Y_s+1)

[T-system]

[Y-system]

(s = 1, ..., n-3 )

where f^[a]( ) := f(e^{a i/2} )

1 2 3 n-3

[≈ cross-ratios]

● asymptotics for ζ → 0,

∞

by WKB for Hitchin system

(⇣ ! 0) etc.

Z_s = H

s

pp dz

● (asymptotics) + (analyticity)

log Y_s = m_s cosh + X

r

K_sr ⇤ log(1 + Y_r)

s :

[ cycle on alg. curve]

[for real Zs]

K_sr = I_sr/ cosh ⇥ , I_sr = _s,r+1 + _{s,r 1} Y₀ = Y_{n 2} = 0

log Y_s Z_s

= e , m_s = 2Z_s

Computing area

_≈

amplitudes

● regularization of area

(Area) = 4 Z

d²z e² ! 4 Z

d²z (e² p

pp) + 4¯ Z

r ⇥

d²z p

pp¯

=: A_free + A_reg

A_free = X

s

Z d

2⇥ m_s cosh · log(1 + Y_s)

● remainder fn.

R (remainder fn.) = A A_BDS = A_free + · · ·

.. .

Summarizing so far,

・ ^Ys : (extended) cross-ratios of

● to compute AdS3 amplitudes (2n-pt.), need to solve log Y_s( ) = m_s cosh +

r

K_sr ⇥ log(1 + Y_r)

・ m_s : complex param.

≈ shape of surface

⇔

momenta

x^±_a

e.g. )

Y₁ i 2

⇥ = x⁺₁₅x⁺₆₇

x⁺₅₆x⁺₁₇ , Y₁ 0) = x₁₅x₆₇ x₅₆x₁₇ Y

Y

Y

1 Y2

3 4

1 1

2 2

3 3

4 4

5 5

6 6

7 7

n = 7

TBA eqs.

1 2 3 n-3 ・ ^{Ksr ∝ Isr} (incidence mat. of An-3)

Summarizing so far,

・ ^Ys : (extended) cross-ratios of

● to compute AdS3 amplitudes (2n-pt.), need to solve log Y_s( ) = m_s cosh +

r

K_sr ⇥ log(1 + Y_r)

・ m_s : complex (mass) param.

≈ shape of surface

⇔

momenta

x^±_a

e.g. )

Y₁ i 2

⇥ = x⁺₁₅x⁺₆₇

x⁺₅₆x⁺₁₇ , Y₁ 0) = x₁₅x₆₇ x₅₆x₁₇ Y

Y

Y

1 Y2

3 4

1 1

2 2

3 3

4 4

5 5

6 6

7 7

n = 7

⇒ TBA eq. of hom. sine-Gordon model

TBA eqs.

[Hatsuda-Ito-Sakai-YS ’10]

1 2 3 n-3 ・ ^{Ksr ∝ Isr} (incidence mat. of An-3)

Remainder function

A_periods = 1

4 m_r I_rs¹ m_s

c^±_i,j = x^±_i+2,i+1x^±_i+4,i+3 · · · x^±_j,i x^±_i+1,ix^±_i+3,i+2 · · · x^±_{j,j 1}

A_free =

n 3

s=1

⇥ _⇥

⇥

d

2⇥ m_s cosh log(1 + Y_s( ))

A_BDS = 1 4

n

i,j=1

log c⁺_i,j

c⁺_i,j+1 log c_{i 1,j} c_i,j ,

[overall coupling dependence : omitted]

● Once Y-fn. are obtained

1 2

3

4

5

6 7

C¹⁶

c₁₆ = x₂₃ x₄₅ x₁₆ x₁₂ x₃₄ x₅₆

⇒

non-trivial part : _{Afree , ABDS}

A ^ABDS

= 7

12 (n 2) + A_periods + A_BDS + A_free

[amp.]　- [BDS formula]

R_2n :=

free energy of TBA system

for given ms [ n odd ]

4D amplitude

at strong coupling

2D integrable systems

10D superstring

➚ ➚

⇒

● TBA system : solved numerically

m_s 0 : ^{CFT limit}

{

● exact solutions regular polygon

m_s ⇥

Solving TBA eqs.

● TBA system : solved numerically

m_s 0 : ^{CFT limit}

{

● exact solutions regular polygon

m_s ⇥

Solving TBA eqs.

momentum dependence : ms ↔ momentum

solutions to TBA system : not fully investigated

[but sometimes simple iteration fails]

● TBA system : solved numerically

m_s 0 : ^{CFT limit}

{

● exact solutions regular polygon

m_s ⇥

Solving TBA eqs.

momentum dependence : ms ↔ momentum

solutions to TBA system : not fully investigated

[but sometimes simple iteration fails]

Analytic results at strong coupling

⇒ we discuss expansions around CFT lim.

??

m_s ! 0 , 1

l = M L 0 ^{( m}s = M_sL = ˜M_sM L )

[ M: mass scale, L: length scale ]

except

4. Analytic expansion at strong coupling

[Hatsuda-Ito-Sakai-YS ’11, Hatsuda-Ito-YS ’11 ]

TBA for AdS3

2n-pt. amplitude

⇐

HSG from _[^{su(nb 2)2}

bu(1)]^{n 3}

[w/ imaginary resonance param.]

●

● relation between T-fn. and g-function (boundary entropy)

[Bazhanov-Lukyanov-Zamolodchikov ’94;

Dorey-Runkel-Tateo-Watts ’99; Dorey-Lishman-Rim-Tateo ’05]

[Hatsuda-Ito-Sakai-YS ’10]

Basis of expansion

Homogenous sine-Gordon (HSG) model

[Fernandez Pousa-Gallas-Hollowood-Miramontes ’96]

● start w/ coset CFT

[ for 2n-cusp AdS3 min. surface N = n-2, k=2 ]

● HSG model : integrable deformation of coset CFT

Φ : comb. of weight 0 adjoint op.

S_HSG = S_gWZNW + d²x b

su(N )_k/[bu(1)]^{N 1}

[multi-param. deformation]

● simplest case from _{su(2)b k/bu(1)}

⇒  complex sine-Gordon/minimal affine Toda_{Ak 1}

= ⇥ M ^{2(1 )}

= ¯ = N

N + k ,

● spectrum [copies of min. affine Toda]

M_s^a = sin _k^a

sin _k M_s , M_s = MM˜_s

1 2

1 2 k-1

N-1 3

s a

M_{N 1} M₂

M₁

min. surface ← 2D integrable model ← CFT

Expansion of A

^free

:

[ R_2n ⇠ A_free + A_BDS ]

● free energy around CFT limit

⇐

CFT perturbation

● for the case of 2n-cusp min. surface in AdS3

A_free =

6 c_n + f_n^bulk +

k=2

f_n^(k) l ^4kn

c_n = (n 2)(n 5)

n [central charge]

f_n^bulk = 1

4 m_r I_rs¹ m_s f_n^(k) ⇥ k-pt. fn.of

A

_BDS

:

^{( c±}_ij ⁾

● can show cross-ratios c^±_ij : nothing but T-fn.

where

c⁺_ij = T_|^[i+j_{i j|}^] ₁(0), c_ij = T_|^[i+j_{i j|}^+1]₁ (0)

)

T_s^[k]( ) := T_s + ⇥i 2 k

everything fits into language of 2D integrable model A_BDS =

[≈ trans. matrix]

A_BDS(T_s)

T-function ⇐ g-function

● integral eq. for g-fn. is known :

[Dorey-Lishman-Rim-Tateo ’05; Poszgay ’10; Woynarovich ’10]

● comparing this w/ TBA eq. following

⇒

similar to TBA eq. , including boundary contributions

●

g-fn.

_≈

Z_{⇤ | ⌅} = |e ^RH | ⇥ = ⇤^⇥

k=0

G_|^(k)_⌅(l)⇥2

e ^REk

[Dorey-Runkel-Tateo-Watts ’99; Dorey-Lishman-Rim-Tateo ’05]

log G_|⁽⁰⁾_{↵ R}

L

G_|⁽⁰⁾_s,C .

G_|⁽⁰⁾₁ = T_s i

2 C

[counts ground state degeneracy]

(boundary entropy)

|1 : trivial boundary

● boundaries _≈ reflection factors

[deforming factor]

● need to find

satisfying boundary bootstrap, unitarity, crossing symm.

・

corresponding precisely to

・

T

_s

(x) := sinh ¹₂ ( + i^⇥₂ x) sinh ¹₂ ( i^⇥₂ x)

[Sasaki ’93]

)

|s, C : ( R^|_r^s,C ( ) = R^|_r¹ ( ) Z_r^|s,C ( )

Z_r^|s,C

Z_r^|s,C ( ) := ⇣

(1 + C)_⇥(1 C)_⇥⌘ _sr

R ^|s,_r ^{C >}

Expansion of T-function

● periodicity

T_s( ) = ⇤^⇥

p,q=0

t^(p,q_s ⁾l^{(1 )(p+q)} cosh 2p n

⇥

● boundary CFT perturbation of g-fn.

t^(2,0)_s

t^(0,0)_s = ⁿG( ˜M_j)B(1 2 , ) 2(2⇥)^{1 2}

sin(^3(s+1)_n ) sin(^(s+1)_n )

s sin(_n ) sin(³_n )

ssin(³_n ) sin(_n)

!

[ given by modular S-matrix ]

t^(0,0)_s = sin ^(s+1)_n ⇥

sin(_n ) (z) (0)⇥

= G²( ˜M_s)

|z|⁴

,

t^(2,0)_s , t^(0,4)_s

T-system

nG( ˜M_s)

∝

● t^(2,0)₁

[Dorey-Runkel-Tateo-Watts ’99;

Dorey-Lishman-Rim-Tateo ’05]

[

⇐

T-system ]

R_2n = R_2n⁽⁰⁾ + l ⁿ⁸ R⁽⁴⁾_2n + O(l ¹²ⁿ )

:=

⇤ t^(2,0)_{s 1} t^(0,0)_{s 1}

⇥2

+ t^(2,0)_s t^(0,0)_s

⇥2⌅

cos 2 n

⇥ 2t^(2,0)_s