Rational CFTs on Riemann surfaces

Abstract

The partition function of rational conformal field theories (CFTs) on Riemann surfaces is expected to satisfy ODEs of Gauss-Manin type. We investigate the case of hyperelliptic surfaces and derive the ODE system for the (2, 5) minimal model.Abusing notations, we shall simply write ϕ(z) where we actually mean ϕ z (p). (This will entail notations like φ(ẑ) instead of ϕ ẑ( p) etc.)We shall only consider bundles that lie in |Vec(F)|. Primary fieldsLet O C be the sheaf of germs U, f which are represented by pairs (U, f ) for some open set U ⊆ C and some conformal map f : U → C. Let O C,0 be the fiber of germs in O C which are defined at the origin in C, and letIt is easy to see that G is a group under pointwise composition, with identity element C, id . G is actually a Lie group [15, p. 267]. G is a real manifold that admits no complexification.The Lie algebra g of G can be identified with the Lie algebra of germs of holomorphic vector fields on C which vanish at the origin [3],

An algebraic approach to minimal models in CFTs

“…This section largely uses results obtained for arbitrary genus in [10,11] though Theorem 4 is proved independently using the methods introduced in Subsection 3.3. It is shown (Proposition 8) that the two formulations are equivalent for g = 1.…”

“…In [11], we defined a singular metric on Σ which is obtained by lifting a polyhedral metric on P 1 C , whose curvature is concentrated on the set of ramification points and equally distributed over this set. Let 1 be the 0-point function corresponding to the singular metric.…”

“…The leading coefficient can be read off from the equation for the singular vector (Lemma 4.3 in [17]) and only the specific value of the remaining coefficients in eq. (11) seem to be new. Rather than setting up a closed formula for α m , we shall outline the algorithm to determine these numbers, and leave the actual computation as an easy numerical exercise.…”

“…For a discussion of the relation with the Virasoro field T (z) addressed below, cf. [11]. 3 The change to complex coordinates is more intricate: We have dx 0 ∧ dx 1 = iG zz dz ∧ dz with G zz = 1 2 .…”

“…metric drops out. Yet we suggest to use a singular metric that is adapted to the specific problem [11]. On Σ, this metric is the lift of the flat metric…”