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arXiv 0812.2085hep-th
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R^3 * S^1上拓扑激发的指数定理和Chern-Simons理论

Index theorem for topological excitations on R^3 * S^1 and Chern-Simons theory

Erich Poppitz, Mithat Unsal

备注：
29 pages, refs added, published version
发表信息：
JHEP 0903:027,2009
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摘要：

我们在R^3 \times S^1几何的各种拓扑激发背景下导出了狄拉克算子的一个指数定理。该指数定理为R^4上的瞬子提供了比APS指数更精确的数据，并在去聚化极限下再现了它。在R^3极限下，它简化为卡利亚斯指标定理。该指数用拓扑荷和与边界狄拉克算子相关的η-不变量表示。拓扑荷和η-不变量都不是典型的整数，然而，非整数部分抵消以给出整数值指数。我们的推导是基于轴向电流非保守-一个确切的运营商身份有效的任何四个流形-和存在的中心对称，或近似中心对称，边界holonomy（威尔逊线）。我们希望指数定理能有效地应用于许多感兴趣的物理系统，例如规范理论的低温（大S^1，受限）相，具有矢量或手征物质的中心稳定杨-米尔斯理论（在任何大小的S^1处），以及具有超对称性保持边界条件的超对称规范理论（也在任何S^1处）。在类QCD和手征规范理论中，指数定理应该揭示导致手征对称性破缺和规范扇区中质量隙产生的拓扑激发的本质。我们还证明了在含有费米子的规范理论中施加手征扭曲边界条件会在红外区引入一个Chern-Simons项。这表明，某些类QCD规范理论在小S^1区应该具有拓扑陈-西蒙斯相的分量。

英文摘要：

We derive an index theorem for the Dirac operator in the background of various topological excitations on an R^3 \times S^1 geometry. The index theorem provides more refined data than the APS index for an instanton on R^4 and reproduces it in decompactification limit. In the R^3 limit, it reduces to the Callias index theorem. The index is expressed in terms of topological charge and the eta-invariant associated with the boundary Dirac operator. Neither topological charge nor eta-invariant is typically an integer, however, the non-integer parts cancel to give an integer-valued index. Our derivation is based on axial current non-conservation--an exact operator identity valid on any four-manifold--and on the existence of a center symmetric, or approximately center symmetric, boundary holonomy (Wilson line). We expect the index theorem to usefully apply to many physical systems of interest, such as low temperature (large S^1, confined) phases of gauge theories, center stabilized Yang-Mills theories with vector-like or chiral matter (at S^1 of any size), and supersymmetric gauge theories with supersymmetry-preserving boundary conditions (also at any S^1). In QCD-like and chiral gauge theories, the index theorem should shed light into the nature of topological excitations responsible for chiral symmetry breaking and the generation of mass gap in the gauge sector. We also show that imposing chirally-twisted boundary condition in gauge theories with fermions induces a Chern-Simons term in the infrared. This suggests that some QCD-like gauge theories should possess components with a topological Chern-Simons phase in the small S^1 regime.

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