Fermionic structure of form factors. Fedor Smirnov.. p.1/21

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1 Fermionic structure of form factors Fedor Smirnov p1/21

2 Preliminaries We consider sg model A sg = {[ 1 16π ( µϕ(x)) 2 + µ2 sinπβ 2e iβϕ(x)] + µ2 sinπβ 2eiβϕ(x)} d 2 x We shall use the parameter = 1 β 2, We are interested in local fields which do not change the topological charge, namely in the primary fields Φ α (z, z) = e iα 2 1 ϕ(z, z), and their descendants For irrational α, β the correspondence with CFT fields is fixed by dimensional considerations We shall consider only the fundamental particles: solitons and antisolitons p2/21

3 S-matrix Let β j be rapidities We shall use The two soliton S-matrix is given by and B j = e β j, b j = e 2 1 β j, q = e πi 1 1 S i,j (β i β j ) = S 0 (β i β j ) S i,j (b i /b j ), sin(2kβ) sinh((2 1)πk) S 0 (β) = exp i kcosh(πk)sinh(π(1 )k) dk, 0 S i,j (b i /b j ) = 1 2 (I i I j +σ 3 i σ 3 j)+ b i b j b i q 1 b j q 1 2 (I i I j σ 3 i σ 3 j) + b i b j q 1 q b i q 1 b j q (σ+ i σ j +σ i σ + j ) p3/21

4 Form factor axioms Symmetry axiom S j,j+1 (β j β j+1 )f Oα (β 1,,β j,β j+1,,β 2n ) = f Oα (β 1,,β j+1,β j,,β 2n ) Riemann-Hilbert problem axiom f Oα (β 1,,β 2n 1,β 2n +2πi) = e πi 1 ασ3 2n foα (β 2n,β 1,,,β 2n 1 ) Residue axiom 2πi res β2n =β 2n 1 +πif Oα (β 1,,β 2n 2,β 2n 1,β 2n ) = ( ) 1 e πi 1 ασ3 2n S2n 1,1 (β 2n 1 β 1 ) S 2n 1,2n 2 (β 2n 1 β 2n 2 ) f Oα (β 1,,β 2n 2 ) (e + 2n 1 e 2n +e 2n 1 e+ 2n ) p4/21

5 Important observation: The form factor equations for all the fields with α+2m 1 coincide The structure of the form factor: f Oα (β 1,,β 2n ) = w ǫ 1,,ǫ 2n (β 1,,β 2n ) F Oα,n(β I β I+ ), ǫ 1,,ǫ 2n =± where I ± = {j 1 j 2n, ǫ j = ±}, and the sum over ǫ j s is such that (I + ) = (I ), β I = {β i1,,β in } The basis satisfies S i,i+1 (β i β i+1 )w ǫ 1,,ǫ i,ǫ i+1,,ǫ 2n (β 1,,β i,β i+1,,β 2n ) =w ǫ 1,,ǫ i+1,ǫ i,,ǫ 2n (β 1,,β i+1,β i,,β 2n ) p5/21

6 Integrals We introduce integration variable σ We use the notations S = e σ, B j = e β j, Q = e πi1, A = e πiα s = e 2 1 σ, b j = e 2 1 β j, q = e πi 1 1, a = e πi 1 α Introduce the function χ(σ β 1,,β 2n ) which satisfies where χ(σ +2πi)p(sq 4 ) = χ(σ)p(sq 2 ) χ(σ + 1 πi)p(sq) = χ(σ)p( S), P(S) = 2n (S B j ), p(s) = 2n (s b j ) For β j R no singularities in the strip 0 > Im(σ) > π p6/21

7 Asymptotic behaviour for σ ± : χ(σ β 1,,β 2n ) σ e 2n 1 1 σ, χ(σ β 1,,β 2n ) σ = 1 Our main tool is the integral I α (β 1,,β 2n ) = R i0 χ(σ β 1,,β 2n )e α 1 σ dσ This integral converges for 0 < Re(α) < 2n, and can be continued analytically with respect to α The result is a meromorphic function in C with simple poles at α = 2(n+m)+(2n+l) 1, l,m 0; α = 2m 1 l, l,m 0 p7/21

8 Definition Consider two Laurent polynomials l(s) and L(S) We define their pairing(l,l) α by the following two requirements: 1 The pairing is bilinear 2 If l(s) = s m,l(s) = S l then q-exact forms and Q-exact forms (l,l) α = I α+2m+ 1 l For D a [z](s) = a 2 p(s)z(s) p(sq 2 )z(sq 4 ), (D a [z],l) α = 0, L For D A [Z](S) = Z(S)P(S) AZ(SQ)P( S), (l,d A [Z]) α = 0, l p8/21

9 For antisymmetric Laurent polynomials of k variables: l (k) (s 1, s k ) = ( l 1 l k ) (s1, s k ), define L (k) (S 1,,S k ) = ( L 1 L k )(S 1,,S k ) (l (k),l (k) ) α = det((l i,l j ) α ) i,,,k Form factors and BBS fermions We need F(β I,β I +) Introduce c I I +(t,s) = p I +(q 2 t)p I (s) ( a 1 q 2 ) t q 2 t s a 1 t t s +p I (t)p I +(q 2 s) + at t q 2 s p(q2 s) a 1 q 2 n 1 t q 2 t s p(s) := (q 2 t) n i l I I +,i(s), i=0 and ( a 1 t t s at ) t q 2 s l (n) I I + (s 1,,s n ) = ( l I I +,0 l I I +,n 1) (s1,,s n ) p9/21

10 Then the Riemann-Hilbert axiom is satisfied if F Oα,n(β I β I +) = (l (n) I I +, L (n) ) α, where L (n) = L (n) (S 1,,S n B 1,,B 2n ) is an arbitrary Laurent polynomial which is anti-symmetric in S i s and symmetric in B j s Definition of tower Polynomials constitute a tower if L ( ) = {L (n) (S 1,,S n B 1,,B 2n )} L (n) (S 1,,S n 1,B B 1,,B 2n 2,B, B) = ( 1) c B (B 2 Sp) L 2 (n 1) (S 1,,S n 1 B 1,,B 2n 2 ) n 1 p=1 p10/21

11 Every tower of charge 0 defines a local operator Descendants of the local integrals are obtained multiplying by i 2k 1 = C k M 2k 1 B 2k 1 j, ī 2k 1 = C k M 2k 1 B (2k 1) j The primary field Φ α corresponds to the tower M (n) 0 (S 1,,S n ) = Φ α S S 3 S 2n 1 It is possible to pass to odd degrees only introducing the polynomials C ±,n (S 1,S 2 ) = 1 2 ǫ 1,ǫ 2 =± ǫ 1 ǫ 2 P n (ǫ 1 S 1 )P n (ǫ 2 S 2 )τ ± (ǫ 2 S 2 /ǫ 1 S 1 ), where τ + (x) = l=0 ( x) l i 2 cot π 2 (α+ l ), τ (x) = 1 l= ( x) l i 2 cot π 2 (α+ l ) p11/21

12 Then the towers are created by action of four fermions ψ (Z) = ψ (Z) = Z 2j+1 ψ2j 1, χ (X) = X 2j+1 χ 2j 1, Z 2j 1 ψ 2j 1, χ (X) = X 2j 1 χ 2j 1 ( ψ (Z 1 ) ψ (Z p ) ψ (Z p+1 ) ψ (Z k ) χ (X k ) χ (X q+1 )χ (X q ) χ (X 1 )M (n) ) 0 (S1,,S n ) = Φ α ( ) 1 kk k k A B C D Pn (Z j )P n ( Z j ) Pn (X j )P n ( X j ), where A, B, C, D are respectively k k, k n, n k, n n matrices p12/21

13 C + (Z 1,S 1 ) C + (Z 1,S n ) B = C + (Z p,s 1 ) C + (Z p,s n ) C (Z p+1,s 1 ) C (Z p+1,s n ) C (Z k,s 1 ) C (Z k,s n ), C = X 1 X k X1 2n 1 X 2n 1 k, D = S 1 S n S1 2n 1 Sn 2n 1 χ (X k ), χ (X k ) create holes, ψ (Z k ), ψ (Z k ) crate particles p13/21

14 The matrix A is needed in order to avoid interference between two chiralities A = 0 0 C + (Z 1,X q+1 ) C + (Z 1,X k ) 0 0 C + (Z p,x q+1 ) C + (Z p,x k ) C (Z p+1,x 1 ) C (Z p+1,x q ) 0 0 C (Z k,x 1 ) C (Z k,x q ) 0 0 Take large n The fermions χ and ψ are doing something at the right end of Fermi zone, while χ and ψ are doing something at the left end There is an important integer (weight) m = 1 2 (#(ψ ) #(χ )+#( χ ) #( ψ )) For given weight m we describe descendants of Φ α+2m 1 p14/21

15 In particular, ψ1 ψ2m 1 χ 2m 1 χ 1 M ( ) 0 Φ α m = Φ α+2m 1 ( i )m cot π 2 ψ 1 ψ 2m 1χ 2m 1 χ 1 M ( ) 0 Φ α m = Later we shall use Φ α 2m 1 ( i )m cot π 2 (α +(2j 1)) M( ) m, (α (2j 1)) M( ) m ψ(z) = χ(z) = Z 2j 1 ψ 2j 1, ψ(z) = Z 2j+1 ψ2j 1, Z 2j 1 χ 2j 1, χ(z) = Z 2j+1 χ 2j 1, p15/21

16 Null-vectors Something special happens when α = m 1 which means e α 1 = S m Consider the antisymmetric polynomial c(s 1,s 2 ) = p(s 1 ) Suppose s 2 q 2 s 1 q 2 s 2 p(q 2 s 1 ) s 2 s 1 q 2 p(s s 1 q 2 2 ) +p(q 2 s s 2 s 2 q 2 2 ) s 1 s 1 s 2 q 2 s 1 c(s 1,s 2 ) = n 1 (r i (s 2 )s i (s 1 ) r i (s 1 )s i (s 2 )) Pairing: r i r j = s i s j = 0, r i s j = δ i,j p16/21

17 Quantum Riemann bilinear identity It can be checked that (m 1 m 2,C ± ) 0 = 2πi(m 1 m 2 ) l I I +,i l I I +,j = 0, which implies vanishing of certain local operators In particular, if we define C even = ψ dd (D)χ(D) 2πiD 3, then for α = 1 C even ψ I + χ I M ( ) 0 0, #(I + ) = #(I ) 2 Simplest case: ψ 1χ 2m 1Φ (2m 1) 1 = 0 p17/21

18 Integrable structure of CFT We claim that the space m= V α+2m 1 V α+2m 1, is created by action on Φ α (0) of i 2j 1, ī2j 1 and fermionic creation operators: β 2j 1, γ 2j 1, β 2j 1, γ 2j 1 What do we know about descendants of these fermions to Virasoro descendants? For technical reasons exact formulae are known only on the quotient space by i 2j 1, ī2j 1 which can be realized as created by l 2k p18/21

19 Introduce the notations I + = {2i + 1 1,,2i+ p 1}, β I + = β 2i β 2i + p 1, etc Then if #(I + ) = #(I ), β I +γ I Φ α (0) = 2j 1 I + D 2j 1 (α) 2j 1 I D 2j 1 (2 α) [ P even I +,I ({l 2k } α,c)+d α P odd I +,I ({l 2k } α,c) ] Φ α (0), where i 2j 1 D 2j 1 (α) = Γ() (1 ) 2j 1 2 d α = 1 6 (25 c)(24 α +1 c) Γ ( α (2j 1)) (j 1)!Γ ( α (2j 1)), Similarly for other chirality changing α 2 α p19/21

20 Our main theorem Consider β (ζ) = β 2j 1(ζ µ) 2j 1 + β 2j 1(ζ/µ) 2j 1 etc Then β (ζ 1 ) β (ζ m )γ (ξ m ) γ (ξ 1 )Φ α (0) sg R Φ α (0) sg R = detω sg R (ζ i,ξ j α), and the function ωr sg (ζ,ξ α) can be described through the TBA data This allows to compare with previously introduced fermions and to conclude that β 2k 1 = ψ 2k 1,γ 2k 1 = χ 2k 1, β 2k 1 = ψ 2k 1, γ 2k 1 = χ 2k 1 p20/21

21 Back to null vectors In the simplest case we have the singular vectors: β 1γ 2m 1Φ 1,2m = 0 What can be checked with data at hand? For φ 1,2 not a lot β 1γ 1Φ 1,2 l 2 Φ 1,2 (mod left action of i 2j 1,ī2j 1) Actual null-vector is contains l 2 1, but l 1 = i 1 For m = 2 we have β 1γ 3Φ 1,4 = 0 On the other hand we have ( 2c 32 β 1γ 3Φ α l ) +d α l 4 (mod left action of i 2j 1,ī2j 1) 3 For α = 3 1 this coincides with the CFT null-vector factorised by the left action of th integrals of motion! p21/21

Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra

Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra kalle.kytola@aalto.fi Department of Mathematics and Systems Analysis, Aalto University joint work with March 5,