Minimal string theory is logarithmic

Abstract

We study the simplest examples of minimal string theory whose worldsheet description is the unitary (p, q) minimal model coupled to twodimensional gravity (Liouville field theory). In the Liouville sector, we show that four-point correlation functions of 'tachyons' exhibit logarithmic singularities, and that the theory turns out to be logarithmic. The relation with Zamolodchikov's logarithmic degenerate fields is also discussed. Our result holds for generic values of (p, q).

Moduli integrals in liouville gravity

“…The (p, q) minimal models coupled to the quantized metric of 2D surface have drawn much attention in the literature (see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). Partially, it is because the models, or 2D minimal gravity, are nothing but noncritical string theories and serve as string laboratories, whose target space is two-dimensional in a naive sense.…”

“…One is the discrete approach, by which the conformal matter on random lattices is described by the dynamics of random matrices: matrix models (many references including [3,4]). The other is the continuum approach, where the theory is formulated in the field theory language (for example, see [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). The former can handle the free energy directly, summing over all topologies of the two-dimensional surface in question.…”

“…In this paper, we therefore focus on the continuum approach in the conformal gauge, that is, the Liouville field theory coupled to the (p, q) minimal models. In this approach, we consider the four-point correlation functions of 'tachyons' on a sphere that exhibit the logarithmic divergence [1,7], and its moduli integrals [2,6,8]. In particular, we explicitly show a full correlation function (ceee) in the Majorana fermion coupled to 2D gravity.…”

“…2 Tachyons and the Liouville correlation functions As given in [10], the action of the Liouville field theory on a sphere is i /d2zv/-~(~a~Oar162162247162 (1) where/~ is the two-dimensional scalar curvature with the metric ga~ and the renormalised cosmological constant #. The background charge is given by Q = -c~ -2/c~ with c~ = -x/~/P, and the central charge of the Liouville sector is CL = 1 + 3Q 2 [11].…”

“…Note that the divergence of the prefactor should be appropriately regularized, for example, by (p/27r) s F(-s) --~ (#/27r) ~ [(-1)s+l/F(s + 1)] lnp [12]. For the general expression of G(L ~) for s C Z_>0, see [1,7,24]. The general condition on fli for s E Z_>o is given by …”