Space of Local Fields in Integrable Field Theory

and Deformed Abelian Differentials

Feodor A. Smirnov^{1}

Abstract. In this talk I consider the space of local operators in integrable field theory. This space allows two different descriptions. The first of them is due to conformal field theory which provides a universal picture of local properties in quantum field theory. The second arises from counting solutions to form factors equations. Considering the example of the restricted Sine-Gordon model I show that these two very different descriptions give the same result. I explain that the formulae for the form factors are given in terms of deformed hyper-elliptic integrals. The properties of these integrals, in particular the deformed Riemann bilinear relation, are important for describing the space of local operators.

1 Quantum field theory in two dimensions.

Consider a massive relativistic quantum field theory (QFT) in two dimensional Minkowski space M^{2}. For x = (x_{0}, x_{1}) ∈ M^{2} we put x^{2} = x_{0}^{2} − x_{1}^{2}. Let us take for simplicity the case when there is only one stable particle of mass m in the spectrum. To this particle we associate the creation-annihilation operators a^{∗}(β), a(β) where the rapidity β parameterizes the energy-momentum of particle: p_{0}(β) = m cosh β, p_{1}(β) = m sinh β. The only non-vanishing commutator is

[a(β_{1}), a^{∗}(β_{2})] = δ(β_{1} − β_{2})

The space of states of the theory is the Fock space created by the action of an arbitrary finite number of operators a^{∗}(β) on the vacuum |0⟩ which is annihilated by a(β). We denote this space by H_{p}. The action of the operators of energy and momentum P_{μ} in H_{p} is defined by P_{μ}|0⟩ = 0, [P_{μ}, a^{∗}(β)] = p_{μ}(β)a^{∗}(β).

In local QFT there exist local operators O_{i}(x) = e^{iP_{μ}x_{μ}}O_{i}(0)e^{−iP_{μ}x_{μ}} acting in the space H_{p} and satisfying

[O_{i}(x), O_{j}(x′)] = 0  for  (x − x′)^{2} < 0

Obviously, these local operators create a linear space which will be denoted by H_{o}. The Lehmann-Symanzik-Zimmerman axiomatic requires the existence of two special local operators. One of them is the symmetric energy-momentum tensor T_{μν} such that ∂_{μ}T_{μν} = 0 and P_{μ} = ∫ T_{μ0}(x)dx_{1}. The other one is the interpolating field ϕ(x) weakly approaching, when x_{0} → ±∞, the free ”out” and ”in” fields constructed via the creationannihilation operators a_{out}^{∗}(β), a_{out}(β) , a_{in}^{∗}(β), a_{in}(β). It is required that they are unitary equivalent: a_{out}(β) = Sa_{in}(β)S^{−1} with the unitary operator S (S-matrix) which leaves invariant the vacuum and one particle state, and commutes with P_{μ}. We identify the original operator a(β) with a_{in}(β).

1Membre du CNRS

Documenta Mathematica · Extra Volume ICM 1998 · III · 183–192

This axiomatic has an obvious generalization to the case when the particle has internal (isotopic) degrees of freedom, and to the case of fermionic statistic or even generalized statistic, the latter case is also possible in two dimensions.

Let us consider in some details the space of local operators H_{o}. The general philosophy teaches us that in order to understand the structure of this space one has to consider the ultra-violet (short-distance) limit of the original QFT. At least intuitively this idea is quite natural. The ultra-violet limit of massive QFT is described by a certain conformal field theory (CFT). The spaces of local operators of two theories must coincide, and, since the CFT in two dimensions allows in many cases a complete solution [1], we get a description of ”universality classes” of two-dimensional QFT.

In the conformal case the theory essentially splits into two chiral sectors, which means that any operator O(x) can be rewritten as O^{−}(x_{−})O^{+}(x_{+}) where x_{±} = x_{0} ± x_{1} are light-cone coordinates. The space of local operators in CFT is described in terms of two Virasoro algebras with generators L_{n}^{±} satisfying

[L_{m}^{±}, L_{n}^{±}] = (m − n)L_{m+n}^{±} + c δ_{m,−n} \frac{n^{3}−n}{12}

where the central charge c is an important characteristic of the theory. These Virasoro algebras act on the space H_{o} which happens to be organized as follows. There are primary fields ϕ_{m} satisfying

L_{n}^{±}ϕ_{m} = 0, n < 0, L_{0}^{±}ϕ_{m} = Δ_{m}ϕ_{m}

where Δ_{m} is the scaling dimension of primary field. Different local operators are obtained by acting with L_{−n}^{±} on the primary fields. So, the one has

H_{o} = ⨁_{m} W_{m}^{−} ⊗ W_{m}^{+}

where W_{m} is a Verma module of the Virasoro algebra.

In this talk I shall consider a particular example of CFT with c < 1. The coupling constant ξ which we use is related to c as follows

c = 1 − \frac{6}{ξ(ξ+1)}

Considering the coupling constant in generic position we have infinitely many primary fields ϕ_{m} , m ≥ 0 with scaling dimensions Δ_{m} = −\frac{m}{2} + \frac{m}{2}(\frac{m}{2} + 1)ξ. We shall concentrate on one chirality considering only one Virasoro algebra with generators L_{n} ≡ L_{n}^{−}. The Verma module W_{m} has a singular vector on level m + 1. The irreducible representation of the Virasoro algebra is obtained by factorizing over the Verma submodule created over this singular vector. The vectors from this submodule are called ”null-vectors”. It must be emphasized that the process of factorizing over the null-vectors has the dynamical meaning of imposing the equations of motion. The latter statement can be clearly understood in the classical limit ξ → 0 when the chiral CFT gives the classical Kortewegde-Vris (KdV) hierarchy with second Poisson structure. The space of local operators turns into the space of functions on the phase space of KdV. It is shown in [2] that the null-vectors in that case provide all the equations of motion of the KdV hierarchy.

Let us return to massive QFT. Consider a local operator O(x) and define

f_{O}(β_{1}, ⋯, β_{n}) = ⟨0| O(0) a^{∗}(β_{1}) ⋯ a^{∗}(β_{n})|0⟩

where β_{1} < ⋯ < β_{n}. To other ranges of β’s the function f_{O} is continued analytically. The function f_{O} is called form factor. The matrix elements of a local operator between two arbitrary states of H_{p} can be obtained by certain analytical continuation of the form factors due to crossing symmetry. The dependence on x can be taken into account trivially because the matrix element is taken between the eigen-states of the energy-momentum. Thus the form factors define the local operator completely. On the other hand the set of form factors define a pairing between the spaces H_{o} and H_{p}.

2 Integrable field theory.

The problem of finding the form factors of local operators for any massive QFT looks rather hopeless. However, in the special case of integrable field theory (IFT) this problem can be solved. In IFT the scattering is factorizable which means that every scattering process is reduced to two-particle scattering [3]. The two-particle S-matrix S(β_{1} − β_{2}) depends analytically on the difference of rapidities. As it has been already said the particle can carry internal degrees of freedom lying in finite-dimensional isotopic space. In that case S(β_{1} − β_{2}) is an operator acting in the tensor product of the isotopic spaces attached to the particles scattered. The S-matrix must satisfy certain requirements, the most important of which being the Yang-Baxter equation [4].

Consider now the form factors. The first examples of exact form factors in IFT are given in [5]. I gave a complete solution of the problem in a series of papers (partly in collaboration with A.N. Kirillov) summarized in the monograph [6]. If the particles have internal degrees of freedom the form factor takes values in the tensor product of isotopic spaces. It is convenient to consider the form factors as row-vectors. Then we act from the right by the operators like S(β_{i}−β_{j}) (which act non-trivially only in the tensor product of i-th and j-th spaces). It has been shown that for the operator O to be local it is necessary and sufficient that the following requirements are satisfied [6]:

1. Analyticity. The form factor f_{O}(β_{1}, ⋯, β_{n}) is a meromorphic function of all its arguments in the finite part of the complex plane.

2. Symmetry.

f_{O}(β_{1}, ⋯ β_{i}, β_{i+1} ⋯, β_{n})S(β_{i} − β_{i+1}) = f_{O}(β_{1}, ⋯ β_{i+1}, β_{i} ⋯, β_{n}) (1)

3. Total Euclidean rotation.

f_{O}(β_{1}, ⋯, β_{n−1}, β_{n} + 2πi) = f_{O}(β_{n}, β_{1}, ⋯, β_{n−1})  (2)

3. Annihilation pole. In the absence of bound states there are no other singularities in variable β_{n} in the strip 0 < Imβ_{n} < 2π but simple poles at the points β_{n} = β_{j} + πi.

The residue at the one of them ( β_{n} = β_{n−1} + πi) is given below, other residues can be obtained from the symmetry property.

2πi resf_{O}(β_{1}, ⋯ β_{n−1}, β_{n}) =
=f_{O}(β_{1}, ⋯ β_{n−2}) ⊗ c_{n−1,n} (I − S(β_{n−1} − β_{1}) ⋯ S(β_{n−1} − β_{n−2}))  (3)

where c_{n−1,n} is a certain vector from the tensor product of n-th and (n − 1)-th isotopic spaces which is canonically related to the S-matrix.

Two comments are in order here. First, clearly the IFT is completely defined by the S-matrix in agreement with the general idea of Heisenberg. Second, the space H_{o} is in one-to-one correspondence with the space of solutions of the system of linear equations (1, 2, 3). So, we have to establish the relation of this description to the one given by CFT.

Let me consider my favorite example of IFT which is the restricted Sine-Gordon model (RSG)[7]. I will not give the traditional Lagrangian definition of the model, instead I shall present the S-matrix which, as it has been said, defines the IFT completely. The particles in RSG are two-component (soliton-antisoliton), so, the S-matrix is an operator acting in ℂ^{2} ⊗ ℂ^{2} as follows

S(β) = S_{0}(β) (e^{− \frac{β}{ξ}} R(q) − e^{\frac{β}{ξ}} R̂(q)^{−1})  (4)

where β = β_{1} − β_{2}, ξ is a coupling constant, S_{0}(β) is certain c-number multiplier which is not very relevant for our goals. The matrix R(q) is the R-matrix of the quantum group U_{q}(sl_{2}) [8] with q = exp(\frac{2πi}{ξ}) acting in the tensor product of two-dimensional representations:

R(q) = q^{\frac{1}{4} (σ^{3}⊗σ^{3}+1)} (I + (q^{\frac{1}{2}} − q^{− \frac{1}{2}} )σ^{+} ⊗ σ^{−}) ,

where σ^{3}, σ^{±} are Pauli matrices. Finally, R̂(q) = P R(q), where P is the operator of permutation. The S-matrix (4) gives the famous Sine-Gordon (SG) S-matrix found by Zamolodchikov [9].

The RSG model is a sector of SG model. Let us consider the isotopic spaces as spaces of two-dimensional representations of the quantum group. The S-matrix (4) is written in a manifestly U_{q}(sl_{2})-invariant form. If one introduces the action of U_{q}(sl_{2}) in the space of particles of the SG model, restriction to RSG corresponds to considering U_{q}(sl_{2})- invariant subspace. This restriction looks at the first glance as a kinematical one, but it has important dynamical consequences. The space H_{o} of RSG corresponds to U_{q}(sl_{2})- invariant solutions of the equations (1, 2, 3). From certain physical consideration we know that this space must coincide with the space of operators of CFT with c = 1 − \frac{6}{ξ(ξ+1)} defined above.

One remark should be made. I have said that particles in two dimensional QFT can have generalized statistics which means that their interpolating fields are quasi-local (some phases appear in the commutation relations on the space-like interval). In that case the equations (1, 2, 3) are satisfied for the operators which are not only local, but also mutually local with the interpolating fields, otherwise some minor modification is needed. This is the situation which takes place in RSG model: solitons are particles with generalized statistics. Only the primary fields ϕ_{2m} and their Virasoro descendents are mutually local with the interpolating fields of solitons. For simplicity we shall take for H_{o} of RSG the space span by these “truely local” operators.

3 Deformed hyper-elliptic differentials.

The formulae for the form factors of the RSG model are given in terms of deformed hyper-elliptic integrals. Let me explain what these integrals are. Consider a hyper-elliptic Riemann surface of genus n−1 defined by the equation c^{2} = p(a) with p(a) = ∏_{j=1}^{2n}(a− b_{j}). Take the abelian differentials regular everywhere except at the two points lying over the point a = ∞, and having no simple poles. Up to exact forms there are 2n − 2 such differentials, everyone of them is written in the form

ω = \frac{l(a)}{\sqrt{p(a)}}da  (5)

with some polynomial l(a). Introduce the intersection form

ω_{1} ∘ ω_{2} = ∑_{a=∞} res (ω_{1}Ω_{2})

where dΩ = ω. The basis of dual differentials can be constructed as follows. Consider the anti-symmetric polynomial of two variables:

c(a_{1}, a_{2}) = \sqrt{p(a_{1})}\frac{∂}{∂a_{1}} (\frac{\sqrt{p(a_{1})}}{a_{1} − a_{2}}) − \sqrt{p(a_{2})}\frac{∂}{∂a_{2}} (\frac{\sqrt{p(a_{2})}}{a_{2} − a_{1}})

For any decomposition of this polynomial of the form

c(a_{1}, a_{2}) = ∑_{i=1}^{n−1}(r_{i}(a_{1})s_{i}(a_{2}) − r_{i}(a_{2})s_{i}(a_{1}))

the differentials η_{i} and ζ_{i} defined by using r_{i} and s_{i} respectively in equation (5) are dual:

η_{i} ∘ η_{j} = 0, ζ_{i} ∘ ζ_{j} = 0, η_{i} ∘ ζ_{j} = δ_{ij}

Consider the canonical homology basis with a-cycles α_{i} and b-cycles β_{i}. The Riemann bilinear relation (as A. Nakayashiki pointed out to me, the hyper-elliptic case was found by Weierstrass) says that the matrix of periods

P = (\begin{matrix}∫_{α} η , & ∫_{β} η \\ ∫_{α} ζ , & ∫_{β} ζ \end{matrix})

belongs to the symplectic group Sp(2n − 2).

Now I am going to describe a deformation of these abelian differentials which is needed for the description of RSG form factors. Obviously only tensor product of even number of two-dimensional representations can have a U_{q}(sl_{2})-invariant subspace. So, we have only form factors with even number of particles with rapidities β_{1}, ⋯, β_{2n}. Let us introduce the notations b_{j} = exp(\frac{2β_{j}}{ξ}), B_{j} = exp(β_{j}). Consider two polynomials l(a) and L(A) which can depend respectively on b_{j} and B_{j} as parameters. We define the following pairing for these polynomials [10]:

⟨l, L⟩ = ∫_{−∞}^{∞} Φ(α)l(a)L(A)dα  (6)

where a = exp(\frac{2α}{ξ}), A = exp(α). The function Φ(α) satisfies the equations

p(aq)Φ(α + 2πi) = q^{2}p(a)Φ(α),  p(a) = ∏(a − b_{j})
P (−AQ)Φ(α + iξ) = QP (A)Φ(α),  P (A) = ∏(A + B_{j})  (7)

where Q = e^{iξ}. We require that Φ(α) is regular for 0 < Imα < π, that it behaves as aA when α → −∞, and that it has the following asymptotics when α → +∞:

Φ(α) ∼ f(a)F (A),

f(a) = a^{−(n−1)}(1 + ∑_{k>0} c_{k}a^{−k}), F (A) = A^{−(2n−1)}(1 + ∑_{k>0} C_{k}A^{−k})

where the coefficients c_{k}, C_{k} can be found using (7). These requirements fix the function Φ(α) completely, the explicit formula is available, but we shall not need it. The following functionals are defined for arbitrary polynomials l(a) and L(A):

rl ≡ res_{a=∞}(a^{−1}l(a)f(a)),  RL ≡ res_{A=∞}(A^{−1}L(A)F (A))  (8)

What is the relation between the pairing ⟨l, L⟩ and the hyper-elliptic integrals? Take the limit when ξ → ∞ keeping b_{j} finite. In RSG model this is the strong coupling limit. In this limit the integral (6) goes asymptotically to the period of the differential defined by l(a) (5) over a cycle which is fixed by the polynomial L(A). Thus we have a deformation of hyper-elliptic integrals in which the differentials and the cycles enter in much more symmetric way than they do classically. We shall call l and L respectively q-form and q-cycle. The striking feature of this deformation is that it preserves all the important properties of classical hyper-elliptic integrals. Let me explain this point.

After appropriate regularization [6], the pairing ⟨l, L⟩ can be defined for every pair of polynomials l and L satisfying rl = 0, RL = 0. However, only a finite number of them give really different results because it can be shown that the value of the integral does not change if we add to l or L polynomials of the form

d[h](a) ≡ a^{−1} (p(a)h(a) − p(aq^{−1})h(aq^{−2}))
D[H](A) ≡ A^{−1} (P (A)H(A) − P (AQ)H(−A))  (9)

where the polynomials h and H are arbitrary. The first polynomial from (9) can be considered as an exact q-form and the second one as a q-boundary. It is easy to see that modulo (9) we have 2n − 2 q-forms and 2n − 2 q-cycles, so, the dimensions of cohomologies and homologies do not change after the deformation.

Consider now two anti-symmetric polynomials:

c(a_{1}, a_{2}) =\frac{p(a_{1})}{a_{1}(a_{1}q − a_{2})} − \frac{p(a_{1}q^{−1})}{a_{1}(a_{1}q^{−1} − a_{2})} −
−\frac{p(a_{2})}{a_{2}(a_{2}q − a_{1})} + \frac{p(a_{2}q^{−1})}{a_{2}(a_{2}q^{−1} − a_{1})}
C(A_{1}, A_{2}) = \frac{1}{A_{1}A_{2}} ( \frac{A_{1} − A_{2}}{A_{1} + A_{2}} (P (A_{1})P (A_{2}) − P (−A_{1})P (−A_{2})) +
+ (P (−A_{1})P (A_{2}) − P (A_{1})P (−A_{2})))  (10)

Documenta Mathematica · Extra Volume ICM 1998 · III · 183–192

Suppose that modulo exact q-forms and q-boundaries we have the decompositions:

c(a_{1}, a_{2}) = ∑_{i=1}^{n−1} (r_{i}(a_{1})s_{i}(a_{2}) − r_{i}(a_{2})s_{i}(a_{1}))

C(A_{1}, A_{2}) = ∑_{i=1}^{n−1} (R_{i}(A_{1})S_{i}(A_{2}) − R_{i}(A_{2})S_{i}(A_{1}))

then the following deformed Riemann bilinear relation holds [10]. Proposition. The matrix

P = (\begin{matrix}⟨r, R⟩, & ⟨r, S⟩ \\ ⟨s, R⟩, & ⟨s, S⟩ \end{matrix})

belongs to the symplectic group Sp(2n − 2).

4 Exact form factors and space of operators.

The quantum group invariance means that the 2n-particle form factors in the RSG model belong to U_{q}(sl_{2})-invariant subspace of the tensor product (ℂ^{2})^{⊗2n}. The dimension of this invariant subspace equals \binom{2n}{n} − \binom{2n}{n−1}. There is a nice coincidence of dimensions

\binom{2n}{n} − \binom{2n}{n−1} = \binom{2n−2}{n−1} − \binom{2n−2}{n−3}

where the RHS gives the dimension of (n − 1)-th fundamental irreducible representation of Sp(2n − 2) (explicitly described later). This representation is naturally related to the construction of the previous section.

Consider the space 𝔥_{k} of anti-symmetric polynomials of k variables a_{1}, ⋯, a_{k}. We can define the following operators acting between different 𝔥_{k}:

1. The operator r acting from 𝔥_{k} to 𝔥_{k−1} by applying the “residue” (8) to one argument, obviously r^{2} = 0.

2. For every h ∈ 𝔥_{1} (a polynomial of one variable) define the operator d[h] acting from 𝔥_{k−1} to 𝔥_{k} by

(d[h]l_{k−1})(a_{1}, ⋯, a_{k}) = ∑_{i=1}^{k}(−1)^{i}d[h](a_{i}) l_{k−1}(a_{1}, ⋯, â_{i}, ⋯, a_{k})

3. The operator c acts from 𝔥_{k−2} to 𝔥_{k} by

(cl_{k−2})(a_{1}, ⋯, a_{k}) = ∑_{i<j}^{k}(−1)^{i+j}c(a_{i}, a_{j}) l_{k−2}(a_{1}, ⋯, â_{i}, ⋯, â_{j} ⋯, a_{k})

Denote by 𝔥̂_{k} the following subspace of 𝔥_{k}:

𝔥̂_{k} = Ker(r |_{𝔥_{k}→𝔥_{k+1}})/ ⨁_{h∈𝔥_{1}} Im(d[h] |_{𝔥_{k−1}→𝔥_{k}})

The space 𝔥̂_{k} is finite-dimensional of dimension \binom{2n−2}{k}. The action of the operator c can be restricted to the spaces 𝔥_{k}. We denote by 𝔥_{k}^{0} the subspace:

𝔥_{k}^{0} = 𝔥̂_{k}/Im(c |_{𝔥̂_{k−2}→𝔥̂_{k}})

which is isomorphic to Sp(2n − 2)-irreducible submodule of maximal dimension in the space of anti-symmetric tensors of rank k. We are interested in the biggest possible: 𝔥_{n−1}^{0}.

The construction of form factors starts by describing a certain linear isomorphism:

(ℂ^{2})_{inv}^{⊗2n} ≃ 𝔥_{n−1}^{0}  (11)

of which we shall not need the explicit form. Using this isomorphism we identify every e ∈ (ℂ^{2})_{inv}^{⊗2n} with a polynomial l[e]_{n−1} ∈ 𝔥_{n−1}^{0}.

Consider now the spaces ℌ_{k} of anti-symmetric polynomials L_{k}(A_{1}, ⋯, A_{k}). The action of operators R, D[H], C is defined in exactly the same way as the action of r, d[h], c using the formulae (8) and (9). For L_{n} ∈ ℌ_{n} and for l_{n−1} ∈ 𝔥_{n−1}^{0} define the pairing:

⟨l_{n−1}, L_{n}⟩ = ∫_{−∞}^{∞} dα_{1} ⋯ ∫_{−∞}^{∞} dα_{n−1} ∏_{i=1}^{n−1} Φ(α_{i})
× l_{n−1}(a_{1}, ⋯, a_{n−1})(RL_{n})(A_{1}, ⋯, A_{n})  (12)

The requirement l_{n−1} ∈ 𝔥_{n−1}^{0} together with the existence of q-boundaries and of deformed Riemann bilinear relation leads to the following remarkable consequence. For arbitrary H ∈ ℌ_{1}, L_{n−1} ∈ ℌ_{n−1} and L_{n−2} ∈ ℌ_{n−2}

D[H]L_{n−1} ≃ 0,  CL_{n−2} ≃ 0  (13)

where L_{n} ≃ 0 means that for such L_{n} the integrals (12) vanish for any l_{n−1} ∈ 𝔥_{n−1}^{0}.

The form factors must satisfy three equations (1), (2), (3). Consider first the equations (1) and (2) only. Obviously, one can multiply any solution of these two equations by a quasi-constant, i.e. 2πi-periodic symmetric function of β_{j} which is the same as symmetric Laurent polynomial of B_{j}. We have Proposition. To every L_{n} ∈ ℌ_{n} corresponds a solution to (1), (2) belonging to (ℂ^{2})_{inv}^{⊗2n}:

f^{L_{n}}(β_{1}, ⋯, β_{2n}) = ∑_{e}⟨l_{n−1}[e], L_{n}⟩e

where the sum is taken over a basis of (ℂ^{2})_{inv}^{⊗2n}. These solutions span a vector space over the ring of quasiconstants, and the only possible linear dependence of solutions arises from relations (13).

Let me appeal to the strong coupling limit for explaining the meaning of this construction. In this limit the equations (1), (2) turn into certain linear differential equations. These linear differential equations are solved in terms of hyper-elliptic integrals (b_{j} are the branch points) and, naturally, different solutions are counted by different cycles. So, it is not a wonder that after the deformation the solutions are counted by L_{n} which have the meaning of deformed cycles as explained above.

With every local operator O we identify an infinite tower of polynomials L[O]_{n} such that

f_{O}(β_{1}, ⋯, β_{n}) = f^{L_{n}[O]}(β_{1}, ⋯, β_{2n})

The polynomials L_{n}[O] must be related for different n in order to satisfy the remaining equation (3). We have Proposition. The form factors f_{O}(β_{1}, ⋯, β_{n}) satisfy (3) if and only if the antisymmetric polynomials L_{n}[O](A_{1}, ⋯, A_{n}), which are at the same time symmetric Laurent polynomials of the parameters B_{1}, ⋯, B_{2n}, satisfy the recurrence relations:

L_{n}[O](A_{1}, ⋯, A_{n}|B_{1}, ⋯, B_{2n})|_{B_{2n}=−B_{2n−1}} ≡ ∑_{i=1}^{n−1}(−1)^{i} ∏_{j≠i}(A_{j}^{2} − B_{2n}^{2})

× L_{n−1}[O](A_{1}, ⋯, Â_{i}, ⋯, A_{n}|B_{1}, ⋯, B_{2n−2})  (mod ∏_{j=1}^{n}(A_{j}^{2} − B_{2n}^{2})) (14)

i.e. the difference between LHS and RHS is divisible by ∏(A_{j}^{2} − B_{2n}^{2}) as polynomial of A_{j}.

Recall that the equation (3) concerns the residue at the pole β_{2n} = β_{2n−1} + πi which corresponds to B_{2n} = −B_{2n−1} and b_{2n} = qb_{2n−1}. In the strong coupling limit the branch points b_{2n} and b_{2n−1} approach each other, so, we arrive at a singularity of the moduli space. Thus the geometrical analogy of our construction is as follows. With every 2n-particle space we associate the moduli space of hyper-elliptic curves, the lower moduli space is embedded into the upper one as its singularity. Equation (14) gives a rule for embedding of deformed homologies.

The solution to the relation (14) which describes the primary field ϕ_{2m} is

L_{n}[ϕ_{2m}](A_{1}, ⋯, A_{n}|B_{1}, ⋯, B_{2n}) = ∏_{i<j}(A_{i}^{2} − A_{j}^{2}) ∏ A_{i}^{2m} ∏ B_{j}^{−m}

One can multiply L_{n}(ϕ_{2m}) by the polynomials

I_{2k−1}(B) = ∑ B_{j}^{2k−1}, J_{2k}(A|B) = ∑ A_{i}^{2k} − \frac{1}{2} ∑ B_{j}^{2k}

which does not spoil the relation (14). It corresponds to the action of operators I_{2k−1} and J_{2k} in the space H_{0}, for example, L_{n}[J_{2k}O](A|B) = J_{2k}(A|B)L_{n}[O](A|B). I put forward the following Conjecture. The space of operators span by I_{2k_{1}−1} ⋯ I_{2k_{p}−1}J_{2k_{1}} ⋯ J_{2k_{q}}ϕ_{2m} coincides with the Verma module of Virasoro algebra generated over the primary field.

Let me say a few words about the meaning of this construction. In RSG model there is an infinite number of local integrals of motion I_{2k−1} which can be written in the form I_{2k−1} = ∫ h_{2k}(x)dx_{1} with some local densities h_{2k}(x). For any local operator O we define an operator I_{2k_{1}−1}O = [I_{2k−1}, O] which is also local. The operator I_{2k−1} acting on H_{p} is the same as before because the eigen-value of I_{2k−1} on 2n-particle state equals I_{2k−1}(B). The operators J_{2k} describe certain transverse to the integrals of motion coordinates.

The Verma module of Virasoro algebra is span by the vectors L_{k_{1}} ⋯ L_{k_{l}}ϕ_{2m}. The operator L_{0} defines the grading in this space such that the degree of L_{k} equals k. It can be shown that the degrees of I_{2k−1} and J_{2k} with respect to the same grading equal respectively 2k − 1 and 2k. So, the characters of the two graded spaces coincide which makes the above conjecture very plausible. There are other arguments in favour of this conjecture which I cannot explain here.

There is a crucial check for the above conjecture. It has been said that the Verma module of the Virasoro algebra is reducible: there is a submodule of null-vectors which corresponds to the equations of motion of the model. The question is whether it is possible to find these null-vectors describing the space H_{o} in terms of I_{2k−1} and J_{2k}? This can be done because certain local operators vanishes due to the relations (13), moreover, the number of these operators is exactly the same as the number of null-vectors in the Verma module [2]. I think that this statement which links together two very different descriptions of the space H_{o} is a good point to finish this talk. Acknowledgement. I would like to thank O. Babelon and D. Bernard for fruitful collaboration and for help in preparing this talk.

References

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[2] O. Babelon, D. Bernard, F.A. Smirnov, Comm. Math. Phys. 186(1997) 601

[3] L.D.Faddeev, V.E.Korepin, Physics Reports 420 (1978) pp.1-78

[4] A.B.Zamolodchikov, Al.B.Zamolodchikov, Annals. Phys. 120 (1979) p.253; B. Berg, M. Karowski, P. Weisz, V. Kurak, Nucl. Phys. 134B (1979) 125

[5] M. Karowski, P. Weisz, Nucl. Phys. 139B (1978) 455

[6] F.A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory. Adv. Series in Math. Phys. 14, World Scientific, Singapore (1992)

[7] F.A. Smirnov, Int. J. Mod. Phys. A4 (1989) 4213, and Nucl. Phys. 337B (1990) 156; N.Yu Reshetikhin and F. Smirnov, Comm. Math. Phys. 131 (1990) 157 A. LeClair, Phys. Lett. 230B (1989) 103

[8] V.G. Drinfeld, Quantum Groups, Proc. of ICM, Berkeley, CA, New York: Academic Press (1986) 798

[9] A.B.Zamolodchikov, Pisma ZhETF 25 (1977) 499

[10] F.A. Smirnov, Lett.Math.Phys. 36 (1996) 267

Feodor A. Smirnov

Laboratoire de Physique Théorique et Hautes Energies,

Université Pierre et Marie Curie, Tour 16, 1-er étage, 4 place Jussieu, 75252 Paris, France, smirnov@lpthe.jussieu.fr On leave from Steklov Mathematical Institute, St. Petersburg, Russia