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MiS Preprint

Compactness of $A_r$-spin equations

Huijun Fan, Tyler Jarvis and Yongbin Ruan


We intruduce the $W$-spin equations on a Riemann surface $\Sigma$ and give a precise defintion to the corresponding $W$-spin equations for $W$ being a quasi-homogeneous polynomial. When $W$ is the $A_r$-potential, then they correspond to the $r$-spin strucutres and the $r$-spin equations considered by E. Witten [W2]. If the number of the Ramond marked points on $\Sigma$ is at least 1, then Witten's lemma does not hold and the $W$-spin equations may have nontrivial solutions. An nontrivial solution of $r$-spin equation is given in this case. We demonstrate the "inner compactness" of the $W$-spin equations when $W$ is one of the superpotentials" $A_r, D_r, E_r$ or pure neveu-schwarz. Especially, if $W$ is$A_r$-potential, then the solution space of the $r$-spin equation is compact in sutable topology.

MSC Codes:
53D45, 53C27, 58J05
moduli space, generalized spin structure, $w$-spin equations, compactness

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2004 Repository Open Access
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Compactness of A[gamma]-spin equations