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Benjamin-Ono integrable systems in 2D conformal field theory

Alexey Litvinov L. D. Landau Institute for Theoretical Physics


Plan of talk:
1. Integrals of Motion in 2D CFT

2. Classical BO equation: review

3. Quantization of BO system: relation to CS mo del

4. Applications: conformal blo cks and AGT formula

5. Concluding remarks

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Integrals of Motion in 2D CFT:
· Lo cal Integrals of Motion were intro duced by Zamolo dchikov in 1987 · Consider CFT with the symmetry algebra A and define

Ik =
such that 1. [Ik , Il ] = 0 2. Ik has spin k

1 2

Gk

+1dx

3. The simultaneous sp ectrum of Ik is non-degenerate 4. Some other conditions: like appropriate semiclassical limit

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· The most known example is the system originally studied by Zamolo dchikov: A = Vir, T (z ) with the central charge c

G2 = T (z ),

G4 = (T (z ))2 ,

G6 = (T (z ))3 +

c+2 (T (z ))2, . . . 12

· In semiclassical limit c it reduces to KdV system · One of the advantages of IM's is that they may survive under the integrable p erturbation (1,3 p erturbation in this particular case) · Bazhanov, Lukyanov and Zamolo dchikov studied this system in great details. In particular they derived T and Q functions ...

· One of the impressive results is the so called IM/ODE corresp ondence (Dorey, Tateo, BLZ)

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Benjamin-Ono equation:
· App ears in hydro dynamics of stratified fluid (Benjamin­1967, Ono­ 1975) vt + 2vvx + Hvxx = 0,
2

x [0, 2 ]

where H is the op erator of Hilb ert transform defined by

HF (x) =

def 1

2 0

-

1 F (y ) cot (y - x) dy . 2

· Following Bo ck-Kruskal and Nakamura (1979) we define 1 (1 i H) 2 Substituting () into BO equation one arrives at v = (ew - 1) + i P+ wx, e + i P+ x
w

P± =

()

2 wt + 2vwx - iwx + Hwxx = 0

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· We can assume that the expression in angular brackets vanishes
2 wt + 2vwx - iwx + Hwxx = 0

· Substituting w = w+ + w- , where w± = P±w one finds i+ + ( + v)+ = -, x where t - ixx - 2x 2 P± vx = 0
+ + = e-w , - - = ew . ± ± ± ±

(*)

We note that this system is a Lax pair for BO equation.

· One can easily check that w is a conserved density

2 w(x, t) dx = 0, t 0 and hence its expansion in sp ectral parameter gives infinitely conserved densities for BO equation.

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d · Expanding at (here and b elow D = H dx ) we get

v3 1 v4 3 12 v2 , G3 = + v D v , G4 = + v 2 D v + vx , . . . G1 = v , G2 = 2 3 2 4 4 2 such that the quantities (classical Integrals of Motion) Ik = are conserved in time.
def 1 2

2 0

Gk+1 dx,

· We note that BO equation can b e written in a Hamiltonian form vt = {I2, v}, {v(x), v(y )} = (x - y ), {Ik , Il } = 0. and all the quantities Ik form a commutative Poisson algebra

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· All this can b e generalized for the n-matrix version of BO equation. For example for n = 2 we have Lax pair ( - iv + )2 + u + = 2-, i ± - i± - ±x 2± P±vx = 0, x x t 2 · The compatibility condition is equivalent to BO2 equation
u + v u + 2uv + x x t vt + ux + Hvxx + v 2
1v 2 xxx = 0, vx = 0,

· This equation is also Hamiltonian with H = 1/2 (uv +v Dv +1/3 v 3)dx 1 {u(x), u(y )} = (u(x) + u(y )) (x - y ) + (x - y ), 2 1 {v (x), v (y )} = (x - y ), {u(x), v (y )} = 0. 2

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Quantization of BO system: relation to CS mo del
· the field v(x) is the semiclassical limit of the U (1) current field v (z ) = P +
k =0

ak e-ikz ,

defined on a half-cylinder.

· The Fourier comp onents an satisfy [am, an ] = m m,-n. · The quantum counter part of the integral I2 has to b e chosen 1 2 : I2 = 2 0 iQ v3 + v Dv 3 2 : dx,

where Q = b + 1/b and the semiclassical limit b 0 is realized as v -ib-1v.
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· In comp onents we have

I2 = i Q
k >0

ka-k ak +

1 aiaj ak 3 i +j +k =0

· One can represent

i , k b pk where pk are p ower-sum symmetric p olynomials a-k = -ib pk , ak =
N

pk = pk (x) =
j =1

xk . j

· When acting on a space of symmetric functions the op erator I2 can b e rewritten as (Awata et al 1995) xi I2 xi i =1
N 2

xi + xj +g xi - xj xi - xj xi xj i
= HCS, 2

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· It is natural to assume that higher IM's Ik can b e expressed in terms of higher Calogero-Sutherland Hamiltonians:

Ik HC k

S

· This statement can b e checked by explicit computation on lower levels, but the general pro of as well as the explicit form of the relation b etween two integrable systems is still lacking

· In fact higher qBO systems are also related to CS mo del

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Applications: conformal blo cks and AGT formula:
· We consider CFT with symmetry algebra b eing A = Wn H v (z ), W (2)(z ) = T (z ), W (3)(z ), . . . W (n)(z )

· In universal enveloping of A we can construct system of IM's Ik :

Ik =

1 2

Gk

+1 v , W

(k )

, D dx,

[Ik , Il ] = 0

such that in semiclassical limit b , v b-1v, W (k) b-k uk
B Ik Ik On

· For example for n = 2 we have A = Vir H with c = 1 + 6Q2 1 I3 = 2 13 T v - iQ v Dv + v dx 3

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· We can find simultaneous eigenfunctions

Ik | P

= h (P )|P ,

(k )

where P = (P1, P2, . . . , Pn-1) and = (1, . . . , n ) - n partitions
(n ) |P = (-1)(n-1)||(W-1 )|| + . . . |P

· Then we define primary op erator where ± =

(nQ-a)/ n- ea/ n+ V W , Va = e a 1 -ikz and V W is the primary field in W theory n k± iak /ke a 1

· and matrix elements F (a, P, ; P , ) = P |Va|P

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· One can prove (Alba et al 2010, Fateev and A.L. 2011) that F (a, P, ; P , ) =
n

=
i,j =1 si

(Q - Ei,j (xj - x |s) - a/n) i

t j

(Ej ,i (x - xj |t) - a/n), i

where xj = (P, hj ) with hj = 1 - e1 - · · · - ej -1 and E,µ P s = P - b lµ(s) + b-1(a(s) + 1). where a(s) and lµ (s) are the arm length of the square s in the partition and the leg length of the square s in the partition µ

s

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Consider Wn CFT with c = (n - 1)(1 + n(n + 1)Q2). The primary fields V are parameterize by vector parameters = (1, . . . , n-1). Then we consider the conformal blo ck of sp ecial form:

a21

a31

ak-21

ak-11

1

P1

P2

Pk-4

Pk-3

k

It is also convenient to cho ose z1 = 0, zk-1 = 1, zk = and zi+1 = qiqi+1 . . . qk-3
def k -3 k -3 j =1 m=j

for

1 i k - 3.
j j j

The AGT relation claims that the function Z(q ) = (1 - qj . . . qm)a
j +1

(Q-am+2/n) F(q ) = 1 + j

3 q11 q22 . . . qkk-3 Zj , -

coincides the instanton part of the Nekrasov partition function.
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Concluding remarks:
· Multiplying Wn theory by additional free b oson we found an integrable system with nice prop erties. In particular, it helps with computation of the conformal blo cks.

· Can we find similar phenomena for different CFT's? predicted by (Shiraishi, Feigin et al 2011)

Yes, as was

· We find the generalization for N S algebra. Again BO system. (Belavin, Beshtein, Tarnop olsky, A.L. to app ear)

· It seems that we can do the same for parafermionic "NS" algebra... · The physical=conceptual understanding is far from b eing complete
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