Logarithmic Heat Projective Operators

Let C → S be a proper flat family of stable curves, smooth over S0 = S r ∆. One can associate a flat family f : SUC(r, d) → S0 of moduli spaces SUCs(r, d) of semistable vector bundles of rank r and degree d with fixed determinant over the curves Cs, and also a line bundle Θ on SUC(r.d) such that its restriction to each fibre is the line bundle on SUCs(r, d) defined by the theta divisors. It is well known that for any positive integer k the direct image E0 := f∗Θ k is a vector bundle on S0 and have a flat projective (Hitchin) connection. A natural question suggested by conformal fields theory is that: Is the connection regular ? By my knowledge, this problem seems open, which is part of the motivation of this paper. On the other hand, let S be a smooth curve, ∆ = {s0} a closed point and Cs0 be irreducible, smooth but one node p0, we can associate a flat family f : UC(r, d) → S of moduli spaces UCs(r, d) of semistable torsion free sheaves of rank r and degree d over the curves Cs, and also a line bundle Θ on UC(r.d) such that its restriction to each fibre is the theta line bundle on UCs(r, d). It was proved in [NR] and [Su] a factorization of the space H(UCs(r, d),Θ k s0 ), which is unfortunately not canonical. One may ask for a canonical factorization, or ask further the questions: if there exists a logarithmic projective connection on E = f∗Θ k such that its restriction to (E0, S0) is the flat projective connection satisfying Hitchin’s requirements ? if the answer is yes, what is the relationship between the residue of the connection and the factorization one proved in [NR] and [Su] ? In other words, even if we knew that Hitchin’s connection is regular, it is still an interesting problem to know for what extension of E0 such that the connection extends. Thus we formulate the question as following