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arXiv 1111.2389hep-th
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弦场论中的缠绕数

Winding Number in String Field Theory

Hiroyuki Hata, Toshiko Kojita

备注：
20 pages, no figures; v2: references added, minor changes
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摘要：

基于三维空间中立方弦场论（CSFT）与Chern-Simons场论的相似性，我们研究了在取量子化值的CSFT中将N=（π^2/3）\int（U Q_B U^{-1}）^3解释为一种缠绕数的可能性.特别是，我们专注于表达的N作为一个BRST-精确量，N= \int Q_B A，它消失在天真的治疗相同的积分。为了实现非平凡N，我们需要对KBc代数中算子K的零本征值的偏差进行正则化。这种正则化必须同时违反N的被积函数的BRST-精确性。通过将K移动一个正无穷小的正则化，得到了对应于快子真空的U_tv的期望值N[（U_tv）^{ \pm 1}]= \mp 1.然而，我们发现N[（U_tv）^{ \pm 2}]不同于\mp 2，即根据N的加法定律所期望的值。这一结果可以从以下事实来理解：在强意义上，U=（U_tv）^{ \pm 2}的情况下，U =U Q_B U^{-1}不满足CSFT EOM，因此在我们的正则化中不是真正的纯规范。

英文摘要：

Motivated by the similarity between cubic string field theory (CSFT) and the Chern-Simons theory in three dimensions, we study the possibility of interpreting N=(π^2/3)\int(U Q_B U^{-1})^3 as a kind of winding number in CSFT taking quantized values. In particular, we focus on the expression of N as the integration of a BRST-exact quantity, N=\int Q_B A, which vanishes identically in naive treatments. For realizing non-trivial N, we need a regularization for divergences from the zero eigenvalue of the operator K in the KBc algebra. This regularization must at same time violate the BRST-exactness of the integrand of N. By adopting the regularization of shifting K by a positive infinitesimal, we obtain the desired value N[(U_tv)^{\pm 1}]=\mp 1 for U_tv corresponding to the tachyon vacuum. However, we find that N[(U_tv)^{\pm 2}] differs from \mp 2, the value expected from the additive law of N. This result may be understood from the fact that Ψ=U Q_B U^{-1} with U=(U_tv)^{\pm 2} does not satisfy the CSFT EOM in the strong sense and hence is not truly a pure-gauge in our regularization.

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