Worldsheet fermion correlators, modular tensors and higher genus integration kernels

The cyclic product of an arbitrary number of Szegö kernels for even spin structure \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> on a compact higher-genus Riemann surface \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Σ</mml:mi> </mml:math> may be decomposed via a descent procedure which systematically separates the dependence on the points z_i ∈ \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>Σ</mml:mi> </mml:mrow> </mml:math> from the dependence on the spin structure \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> . In this paper, we prove two different, but complementary, descent procedures to achieve this decomposition. In the first procedure, the dependence on the points z_i ∈ \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>Σ</mml:mi> </mml:mrow> </mml:math> is expressed via the meromorphic multiple-valued Enriquez kernels of e-print 1112.0864 while the dependence on \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> resides in multiplets of functions that are independent of z_i <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , locally holomorphic in the moduli of \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Σ</mml:mi> </mml:math> and generally do not have simple modular transformation properties. The \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> -dependent constants are expressed as multiple convolution integrals over homology cycles of \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Σ</mml:mi> </mml:math> , thereby generalizing a similar representation of the individual Enriquez kernels. In the second procedure, which was proposed without proof in e-print 2308.05044, the dependence on z_i <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is expressed in terms the single-valued, modular invariant, but non-meromorphic DHS kernels introduced in e-print 2306.08644 while the dependence on \delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> resides in modular tensors that are independent of z_i <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> and are generally non-holomorphic in the moduli of \Sigma <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Σ</mml:mi> </mml:math> . Although the individual building blocks of these decompositions have markedly different properties, we show that the combinatorial structure of the two decompositions is virtually identical, thereby extending the striking correspondence observed earlier between the roles played by Enriquez and DHS kernels. Both decompositions are further generalized to the case of linear chain products of Szegö kernels.