Compactness of Ar-spin equations
Huijun Fan, Tyler Jarvis,and Yongbin Ruan
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Submission date: 23. Jun. 2004
MSC-Numbers: 53D45, 53C27, 58J05
Keywords and phrases: moduli space, generalized spin structure, $w$-spin equations, compactness
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We intruduce the W-spin equations on a Riemann surface and give a precise defintion to the corresponding W-spin equations for W being a quasi-homogeneous polynomial. When W is the -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 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" or pure neveu-schwarz. Especially, if W is-potential, then the solution space of the r-spin equation is compact in sutable topology.