In this paper, in Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In Section 2, we have described the possible mathematical applications, of Lebesgue...In this paper, in Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In Section 2, we have described the possible mathematical applications, of Lebesgue integration, in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory, precisely the two dimensional quantum Yang-Mills theory. In conclusion, in Section 3, we have described also the possible mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings.展开更多
The Chern-Simons theory in two-space one-time dimensions is quantized on the light-front under appropriate gauge-fixing conditions using the Hamiltonian, path integral and BRST formulations.
By generalizing the topological current of Abelian Chern Simons （CS） vortices, we present a topological tensor current of CS p-branes based on the φ-mapping topological current theory. It is revealed that CS p-bran...By generalizing the topological current of Abelian Chern Simons （CS） vortices, we present a topological tensor current of CS p-branes based on the φ-mapping topological current theory. It is revealed that CS p-branes are located at the isolated zeros of the vector field φ（x）, and the topological structure of CS p-branes is characterized by the winding number of the φ-mappings. Furthermore, the Nambu-Goto action and the equation of motion for multi CS p-branes are obtained.展开更多
In this paper, spinor and vector decompositions of SU（2） gauge potential are presented and their equivalence is constructed using a simply proposal. We also obtain the action of Faddeev nonlinear 0（3） sigma model ...In this paper, spinor and vector decompositions of SU（2） gauge potential are presented and their equivalence is constructed using a simply proposal. We also obtain the action of Faddeev nonlinear 0（3） sigma model from the SU（2） mass/ve gauge field theory, which is proposed according to the gauge invariant principle. At last, the knot structure in SU（2） Chern-Simons filed theory is discussed in terms of the Φ-mapping topological current theory, The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of Φ-mapping.展开更多
The Weyl double copy builds the relation between gauge theory and gravity theory, in particular the correspondence between gauge solutions and gravity solutions. In this paper, we obtain the slowly rotating charge sol...The Weyl double copy builds the relation between gauge theory and gravity theory, in particular the correspondence between gauge solutions and gravity solutions. In this paper, we obtain the slowly rotating charge solutions from the Weyl double copy for the Kerr black hole with small Chern-Simons correction. Based on the Weyl double copy relation, for the Petrov type D solution in Chern-Simons modified gravity, we find the additional correction to the electromagnetic field strength tensor of the rotating charge. For the Petrov type I solution, we find that the additional electromagnetic field strength tensors have external sources, while the total sources vanish at the leading order.展开更多
The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localizati...The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.展开更多
Based on the global canonical symmetry of the path integral in phase space, the quantal conserved angular momentum and BRS charge for non-Abelian Chern-Simons (CS) theories have been derived. The possibility of the oc...Based on the global canonical symmetry of the path integral in phase space, the quantal conserved angular momentum and BRS charge for non-Abelian Chern-Simons (CS) theories have been derived. The possibility of the occurrence of fractional spin properties for non-Abelian CS theories cannot be always preserved at the quantum level.展开更多
The (2+1)-dimensional Maxwell-Chern-Simons gravity with phantom dilaton field coupling is studied in this paper.It is shown that black hole solution to exist when electromagnetic coupled to dilaton field in the non-tr...The (2+1)-dimensional Maxwell-Chern-Simons gravity with phantom dilaton field coupling is studied in this paper.It is shown that black hole solution to exist when electromagnetic coupled to dilaton field in the non-trivial way.Moreover,asymptotic index and distribution parameter of current density are calculated by using black hole solution,some new features of this solution are briefly discussed.展开更多
This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and...This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.展开更多
With the Coulomb gauge, the Chern-Simons-Georgi-Glashow （CSGG） model is quantized in the Dirac formalism for the constrained system. Combining the Gauss law and Coulomb gauge consistency condition, the difference be...With the Coulomb gauge, the Chern-Simons-Georgi-Glashow （CSGG） model is quantized in the Dirac formalism for the constrained system. Combining the Gauss law and Coulomb gauge consistency condition, the difference between the Schwinger angular momentum and canonical angular momentum of the system is found to be an anomalous spin. The reason for this result lies in the fact that the Schwinger energy momentum tensor and the canonical one have different symmetry properties in the presence of the Chern-Simons term.展开更多
The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic...The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic （FS） path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.展开更多
We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter A 〉 0 has at lea...We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter A 〉 0 has at least two solutions （uλ^-, uλ^2） for A sufficiently large, which satisfy that uλ^1 - -u0 almost everywhere as λ →∞, and that uλ^2 →-∞ almost everywhere as λ→∞, where u0 is a （negative） Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ →∞, and prove that uλ^2 - uλ^2- converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary OM is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.展开更多
We review a 3d quantum gravity model, which incorporates massive spinning fields into the Euclidean path integral in a Chern-Simons formulation. Fundamental matter as defined in our previous preon model is recapped. B...We review a 3d quantum gravity model, which incorporates massive spinning fields into the Euclidean path integral in a Chern-Simons formulation. Fundamental matter as defined in our previous preon model is recapped. Both quantum gravity and the particle model are shown to be derivable from the supersymmetric 3d Chern-Simons action. Forces-Matter unification is achieved.展开更多
Chern-Simon theory and the holographic principle as well as scale relativity are used to find out the exact value of cosmic ordinary and dark energy density. The result agrees completely with previously obtained ones ...Chern-Simon theory and the holographic principle as well as scale relativity are used to find out the exact value of cosmic ordinary and dark energy density. The result agrees completely with previously obtained ones as well as with accurate cosmic measurements.展开更多
文摘In this paper, in Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In Section 2, we have described the possible mathematical applications, of Lebesgue integration, in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory, precisely the two dimensional quantum Yang-Mills theory. In conclusion, in Section 3, we have described also the possible mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings.
文摘The Chern-Simons theory in two-space one-time dimensions is quantized on the light-front under appropriate gauge-fixing conditions using the Hamiltonian, path integral and BRST formulations.
基金Project supported by the National Natural Science Foundation of China （Grant No 10475034）.
文摘By generalizing the topological current of Abelian Chern Simons （CS） vortices, we present a topological tensor current of CS p-branes based on the φ-mapping topological current theory. It is revealed that CS p-branes are located at the isolated zeros of the vector field φ（x）, and the topological structure of CS p-branes is characterized by the winding number of the φ-mappings. Furthermore, the Nambu-Goto action and the equation of motion for multi CS p-branes are obtained.
基金＊The project supported by National Natural Science Foundation of China and the Doctoral Foundation of the Ministry of Education of China
文摘In this paper, spinor and vector decompositions of SU（2） gauge potential are presented and their equivalence is constructed using a simply proposal. We also obtain the action of Faddeev nonlinear 0（3） sigma model from the SU（2） mass/ve gauge field theory, which is proposed according to the gauge invariant principle. At last, the knot structure in SU（2） Chern-Simons filed theory is discussed in terms of the Φ-mapping topological current theory, The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of Φ-mapping.
基金supported by the National Key Research and Development Program of China (No. 2023YFC2206200)the National Natural Science Foundation of China (No.12375059)the Fundamental Research Funds for the Central Universities。
文摘The Weyl double copy builds the relation between gauge theory and gravity theory, in particular the correspondence between gauge solutions and gravity solutions. In this paper, we obtain the slowly rotating charge solutions from the Weyl double copy for the Kerr black hole with small Chern-Simons correction. Based on the Weyl double copy relation, for the Petrov type D solution in Chern-Simons modified gravity, we find the additional correction to the electromagnetic field strength tensor of the rotating charge. For the Petrov type I solution, we find that the additional electromagnetic field strength tensors have external sources, while the total sources vanish at the leading order.
基金Project supported by the Cheung-Kong Scholarshipthe Key Laboratory of Pure MathematicsCombinatorics of the Ministry of Education of Chinathe 973 Project of the Ministry of Science and Technology of China.
文摘The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.
文摘Based on the global canonical symmetry of the path integral in phase space, the quantal conserved angular momentum and BRS charge for non-Abelian Chern-Simons (CS) theories have been derived. The possibility of the occurrence of fractional spin properties for non-Abelian CS theories cannot be always preserved at the quantum level.
基金Supported by Natural Science Foundation of Sichuan Education Committee under Grant No. 11ZA100Scientific Research Foundation for Graduate Student of Sichuan Normal University under Grant No. 20113
文摘The (2+1)-dimensional Maxwell-Chern-Simons gravity with phantom dilaton field coupling is studied in this paper.It is shown that black hole solution to exist when electromagnetic coupled to dilaton field in the non-trivial way.Moreover,asymptotic index and distribution parameter of current density are calculated by using black hole solution,some new features of this solution are briefly discussed.
基金Supported by NSFC (Grant Nos. 10531090 and 10229101)Chang Jiang Scholars Program
文摘This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.
基金Supported by National Natural Science Foundation of China (NSFC) (51032002, 11074017, 11174021)National Natural Science Fund of China (10875129, 11075166)+4 种基金IHLB (PHR 201007101)Beijing Nova Program (2008B10)Beijing Natural Science Foundation (1102006)Basic Research Foundation of Beijing University of TechnologyChina Postdoctoral Science Foundation Funded Project(2012M510548)
文摘With the Coulomb gauge, the Chern-Simons-Georgi-Glashow （CSGG） model is quantized in the Dirac formalism for the constrained system. Combining the Gauss law and Coulomb gauge consistency condition, the difference between the Schwinger angular momentum and canonical angular momentum of the system is found to be an anomalous spin. The reason for this result lies in the fact that the Schwinger energy momentum tensor and the canonical one have different symmetry properties in the presence of the Chern-Simons term.
文摘The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic （FS） path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.
基金Supported by National Natural Science Foundation of China (Grant Nos.10701064,10931001)XINXING Project of Zhejiang University
文摘We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter A 〉 0 has at least two solutions （uλ^-, uλ^2） for A sufficiently large, which satisfy that uλ^1 - -u0 almost everywhere as λ →∞, and that uλ^2 →-∞ almost everywhere as λ→∞, where u0 is a （negative） Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ →∞, and prove that uλ^2 - uλ^2- converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary OM is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.
文摘We review a 3d quantum gravity model, which incorporates massive spinning fields into the Euclidean path integral in a Chern-Simons formulation. Fundamental matter as defined in our previous preon model is recapped. Both quantum gravity and the particle model are shown to be derivable from the supersymmetric 3d Chern-Simons action. Forces-Matter unification is achieved.
文摘Chern-Simon theory and the holographic principle as well as scale relativity are used to find out the exact value of cosmic ordinary and dark energy density. The result agrees completely with previously obtained ones as well as with accurate cosmic measurements.
