The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is ext...The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [α^E], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α^L/A] induces a graded Lie algebra structure on HCE·(A,L/A[-2]), and the Atiyah class [α^E] of any A-module E induces a Lie algebra module structure on HCE(A,E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.展开更多
For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and ...For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e.,the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F■T_M on the other hand.展开更多
It is shown that a novel anomaly associated with transverse Waxd-Takahashi identity exists for a pseudo- tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiy...It is shown that a novel anomaly associated with transverse Waxd-Takahashi identity exists for a pseudo- tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiyah- Singer index theorem.展开更多
A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By using the properties of the Atiyah-Singer index theorem, the nomtrivial Jacobian factor of the i...A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By using the properties of the Atiyah-Singer index theorem, the nomtrivial Jacobian factor of the integration measure in the path-integral formulation of the theory is connected with the topological properties of the gauge field. The singularity of the fermion current related to the topological character can be correctly examined in a gauge background.展开更多
In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a c...In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.展开更多
文摘The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [α^E], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α^L/A] induces a graded Lie algebra structure on HCE·(A,L/A[-2]), and the Atiyah class [α^E] of any A-module E induces a Lie algebra module structure on HCE(A,E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.
基金supported by National Natural Science Foundation of China (Grant No. 11901221)。
文摘For any regular Lie algebroid A, the kernel K and the image F of its anchor map ρA, together with A itself fit into a short exact sequence, called the Atiyah sequence, of Lie algebroids. We prove that the Atiyah and Todd classes of dg manifolds arising from a regular Lie algebroid respect the Atiyah sequence, i.e.,the Atiyah and Todd classes of A restrict to the Atiyah and Todd classes of the bundle K of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution F■T_M on the other hand.
基金Supported by National Natural Science Foundation of China(10775059)
文摘It is shown that a novel anomaly associated with transverse Waxd-Takahashi identity exists for a pseudo- tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiyah- Singer index theorem.
文摘A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By using the properties of the Atiyah-Singer index theorem, the nomtrivial Jacobian factor of the integration measure in the path-integral formulation of the theory is connected with the topological properties of the gauge field. The singularity of the fermion current related to the topological character can be correctly examined in a gauge background.
基金Supported by National Natural Science Foundation of China(Grant No.11271062)Program for New Century Excellent Talents in University(Grant No.13-0721)
文摘In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed＇s odd dimensional index formula. Then we extend this theorem to the noncommuta- tive geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.
