Higher Equations of Motion in N=2 Superconformal Liouville Field Theory

We present an infinite set of higher equations of motion in N=2 supersymmetric Liouville field theory. They are in one to one correspondence with the degenerate representations and are enumerated in addition to the U(1) charge \omega by the positive integers m or (m,n) respectively. We check that in the classical limit these equations hold as relations among the classical fields.

In ref. [1] it has been shown that in the Liouville field theory (LFT) an infinite set of relations holds for quantum operators. These equations relate different basic Liouville primary fields V α (z) (V α can be thought of as normal ordered exponential field exp(αφ) of the basic Liouville field φ). They are parameterized by a pair of positive integers (m, n) and are called conventionally "higher equations of motion" (HEM), because the first one (1,1) coincides with the usual Liouville equation of motion. The equations are derived on the basis of a conjecture of the vanishing of all singular vectors, imposed by the requirement of irreducibility of the corresponding representation. They are easily verified in the classical LFT. Higher equations turn out to be useful in practical calculations. In particular, in [2][3][4][5], they were used to derive general four-point correlation function in the minimal Liouville gravity.
Similar operator valued relations have been found also for N = 1 supersymmetric Liouville field theory (SLFT) [6] and for SL(2, R) Wess-Zumino-Novikov-Witten model [7,8] . Recently it was shown in [9] that such relations hold for the boundary operators in the LFT with conformal boundary.
It is the purpose of this note to reveal a similar set of higher equations of motion in N = 2 SLFT. The N = 2 SLFT has a wide variety of applications in string theory [10][11][12] . This theory is quite interesting because of the fact that it has actually few properties in common with the N = 0, 1 SLFTs. For example, unlike the Liouville theories with less supersymmetry, the N = 2 SLFT does not have a simple strong-weak coupling duality. In fact, under the change b → 1/b of the coupling constant, the N = 2 SLFT flows to another N = 2 supersymmetric theory as proposed in [13,14]. Another important difference between the N = 2 SLFT and the N = 0, 1 SLFTs is the spectrum of the degenerate representations [15][16][17] (see also [18,19] ). We will show below that the N = 2 SLFT still possesses higher equations of motion despite these differences.

N = 2 SLFT
The N = 2 SLFT is based on the Lagrangian: where (φ ± , ψ ∓ ) are the components of a chiral N = 2 supermultiplet, b is the coupling constant and µ is the cosmological constant. It is invariant under the N = 2 superconformal algebra: where L m , G ± r and J n are the modes of the corresponding conserved currents, the stressenergy tensor T (z), the super-current G(z) and the U (1) current J(z), and the central charge is: These are the left handed generators, there are in addition the right handed onesL n ,J n , G ± r closing the same algebra. The basic objects are the primary fields (normal ordered exponents): N α,ᾱ = e αφ + +ᾱφ − , the corresponding states being annihilated by the positive modes. These are the primary fields in the Neveu-Schwartz (NS) sector with r, s in (2) half-integer. There are in addition also Ramond (r, s -integer) primary fields R α,ᾱ but we will not be concerned with them in this paper. The conformal dimension and the U (1) charge of the primary fields are: Among the primary fields there is a series of degenerate fields of the N = 2 SLFT.
They are characterized by the fact that at certain level of the corresponding conformal family a new primary field (i.e. annihilated by all positive modes) appears. Such fields can be divided in three classes (see e.g. [18] ).
Class I degenerate fields are given by m, n are positive integers. N ω m,n is degenerate at level mn and relative U (1) charge zero. The irreducibility of the corresponding representations is assured by imposing the nullvector condition D ω m,n N ω m,n = 0,D ω m,n N ω m,n = 0, where D ω m,n is a polynomial of the generators in (2) of degree mn and has U (1) charge zero. It is normalized by choosing the coefficient in front of (L −1 ) mn to be 1. Let us give some examples of the corresponding null-operators: The second class of degenerate fields is denoted by N ω m and comes in two subclasses IIA and IIB: where Here m is an odd positive integer number and the level of degeneracy of N ω m is m 2 , relative charge ±1. In this case the operator D ω m is a polynomial of "degree" m/2, the coefficient is chosen to be 1. Analogously to the class I we have to impose Here are the first examples for class IIA fields: The null-operators for class IIB fields are obtained from (8) by changing G ± → G ∓ and ω → −ω.
A special case of Class IIA (B) fields are the chiral (antichiral) fields with m = 1.
The Class II fields having U (1) charge zero are classified in a separate Class III fields. The simplest m = 1 field here represents the identity operator.

Norms of the null-states
Let us now consider, for a further use, the norms of the states created by applying the null-operators on primary states |α . As explained above, such sates should vanish at α = α ω M . Taking the first terms in the corresponding Taylor expansion, we define: for both classes of representations, M = m or (m, n), where D ω M is the corresponding null-operator and D ω † M is defined as usual through L † n = L −n , J † n = J −n , (G ± r ) † = G ∓ −r . One can compute "by hand" the first few r's. With the use of the explicit form of the null-operators (5) we find for the class I fields: andr ω m,n = r ω m,n for all the examples above. Based on these expression we propose for the general form of r ω m,n : Similarly, from (8) we have for the class IIA: These expressions can be fitted in a general form of r ω m andr ω m : For the class IIB fields one obtainsr ω m = 0 and r ω m is asr ω m in (11) with the change ω → −ω.

Logarithmic fields and HEM
Let us now introduce the so called logarithmic fields. They are defined as: One can introduce also the logarithmic primary fields corresponding to degenerate fields by: with D ω M ,D ω M as in (5), (8) are again primary. The proof of this statement goes along the same lines as for N = 0, 1 SLFT ( [1], [6]) so we will not repeat it here.
Comparing the dimension and U (1) charge for class I fields:∆ m,n = ∆ m,n +mn,ω = ω we conclude that the fields (13) are proportional to N ω m,−n . Thus, we arrive at the higher equations of motion (HEM) for the class I fields: For class IIA (B) the dimension of the resulting primaries in (13) is∆ ω m = ∆ ω m + m 2 , the U (1) charges areω = ω + 1 (ω = ω − 1) respectively, and the HEMs in this case are: Computation of B ω m,n (B ω m,n ) and B ω m (B ω m ) is the final goal of this note. HEMs (14) and (15) are to be understood in an operator sense, i.e. they should hold for any correlation function. Here we will insert them into the simplest one-point function on the so called Poincaré disk [20]. In this case we have: The boundary state B 1 | corresponds to the identity boundary conditions on the Poincaré disc. It enjoys N = 2 superconformal invariance: (so called A-type boundary conditions, see e.g. [21]).
With the definition of r's in (9) the HEMs (14) and (15) take the form: r ω m,n U 1 (m, n; ω) = B ω m,n U 1 (m, −n; ω), r ω m,n U 1 (m, n; ω) =B ω m,n U 1 (m, −n; ω) for class I, and for class II. Here U 1 is the one-point function for "identity boundary conditions" of the corresponding field. In (17) the factor i's appear because the class II null-operators are fermionic, and + (−) refers to class IIA (IIB).
The one-point function on the Poincaré disk for identity boundary conditions in N = 2 SLFT was obtained in [18] and has a general form: .
With the specific values (4) the ratio of one-point functions of class I fields then is: and for the HEM coefficient we obtain: B ω m,n =B ω m,n = r ω m,n U 1 (m, n; ω) U 1 (m, −n; ω) where we impose that (k, l) = (0, 0) is excluded in the product.
Analogously for class IIA fields: and B ω m = 0, For class IIB B andB are exchanged and ω is replaced by −ω. Equalities (18) and (19) are the main results of this paper.
The holomophic currents are conserved by∂T =∂S ± =∂J = 0 on the equations of motion and similarly for the antiholomorphic ones. One has to introduce also the generators of N = 2 supersymmetry G ± andḠ ± : obeying the algebra: For the class IIA fields only the chiral fields, N ω 1 = e ωbφ + , has a classical limit. Their HEMs take the form: where B ω 1 = −2πiµb can be read from (19). In the classical limit along with the analogous HEMs for class IIB anti-chiral fields with ω = 0, these become: Together with (22) and the algebra (23) these relations encode the equations of motion (20).
From the class I fields only the series N ω 1,n has a classical limit, the simplest "classical null-operators" being: It is easy to check, using the algebra (23) and the explicit form of the currents (21), that the classical expressions of the corresponding null-vector conditions is: This is in a perfect agreement with (14) if we take into account that the classical limit, b → 0, of B ω 1,n =B ω 1,n from (18) is: B ω 1,n → (−1) n+1 ω + n ω − n n!(n − 1)! b −1 (πµb 2 ) 2n .
To conclude, we presented relations among primary fields, the higher equations of motion, in N = 2 supersymmetric Liouville field theory. We stress that, since in general the null-vectors of this theory are unknown, our results (18) and (19) should be understood as a proposal. Also, we were concerned in this note with primary fields from the NS sector only. Since the Ramond sector in N = 2 SLFT is not very different, in particular the degenerate fields fall into the same classes, we expect that very similar HEMs hold for them too.