Worldsheet fermion correlators, modular tensors and higher genus integration kernels

Abstract

The cyclic product of an arbitrary number of Szegö kernels for even spin structure $δ$ on a compact higher-genus Riemann surface $Σ$ may be decomposed via a descent procedure which systematically separates the dependence on the points $z_i \in Σ$ from the dependence on the spin structure $δ$. In this paper, we prove two different, but complementary, descent procedures to achieve this decomposition. In the first procedure, the dependence on the points $z_i \in Σ$ is expressed via the meromorphic multiple-valued Enriquez kernels of e-print 1112.0864 while the dependence on $δ$ resides in multiplets of functions that are independent of $z_i$, locally holomorphic in the moduli of $Σ$ and generally do not have simple modular transformation properties. The $δ$-dependent constants are expressed as multiple convolution integrals over homology cycles of $Σ$, 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 $z_i$ is expressed in terms the single-valued, modular invariant, but non-meromorphic DHS kernels introduced in e-print 2306.08644 while the dependence on $δ$ resides in modular tensors that are independent of $z_i$ and are generally non-holomorphic in the moduli of $Σ$. 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.

Figures (3)

• Figure 1: The left panel shows a compact genus two Riemann surface $\Sigma$ and a choice of canonical homology cycles $\mathfrak{A}^1, \mathfrak{A}^2, \mathfrak{B}_1, \mathfrak{B}_2$ with a common base point $q$. A fundamental domain $D$, contained in the universal cover $\tilde{\Sigma}$ of $\Sigma$, for the action of $\pi_1(\Sigma, q)$ on $\Sigma$ is obtained in the right panel by cutting $\Sigma$ along the cycles in the left panel. The surface $\Sigma$ may be recovered from $D$ by pairwise identifying inverse boundary components with one another under the dashed arrows. The vertices $q_i \in \tilde{\Sigma}$ project to $q = \pi(q_i) \in \Sigma$ for $i=1,\cdots, 8$ under the canonical projection $\pi: \tilde{\Sigma} \to \Sigma$.
• Figure 2: The left panel depicts the cycles $\mathfrak{A}^L_\varepsilon$ and $\mathfrak{A}^L_{2 \varepsilon}$ for $\varepsilon >0$ as a small homotopic deformation of $\mathfrak{A}^L$ contained in the interior $D^o$ of $D$ (barring the end points). The right panel depicts the integration contours $\mathfrak{A}^L_{k \varepsilon}$ for $k=1, \cdots , p$ for the multiple integrals in (\ref{['4.DA.7']}) and coincident indices $I_{i_1} = I_{i_2} =\cdots =I_{i_p} = L$, with the associated integration variables $t_{i_1} , \cdots , t_{i_p}$. In both cases, any other arguments of the integrand, including $z_2 , \cdots ,z_n$, are assumed to be deeper inside$D^o$ than any of the curves $\mathfrak{A}^L_{k \varepsilon}$.
• Figure 3: Contour deformation that brings the cyclically permuted ordering of contour displacements with $t_1$ in the innermost position back to the original order of (\ref{['4.DA.7']}) with $t_1$ in the outermost position. The crossing of the $t_1$ contour with those of $t_2,\cdots,t_r$ through the contour deformation is homotopic to infinitesimal circles around $t_j$ drawn in red which only arise if $I_1 = I_j$. Moreover, the residue structure (\ref{['rescdelta']}) of the integrand implies that the circles around $t_3,\cdots,t_{r-1}$ in the second line of the figure do not contribute to (\ref{['4.lem.10']}).