Calculation of the pontrjagin class for \(U(1)\) instantons on non-commutative \(\mathcal R^{4}\). (English) Zbl 1226.81096
Summary: In non-commutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define “instanton number” by the size of the matrices \(B_{\alpha }\) in the ADHM construction. We show the analytical derivation of the non-commuatative \(U(1)\) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge non-commutative \(U(1)\) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes’ like theorem for the number operator representation. The theorem shows that instanton charge is given by some boundary sum. Using the ASD conditions, we conclude that the instanton charge is equivalent to the instanton number.
MSC:
81R60 | Noncommutative geometry in quantum theory |
58B34 | Noncommutative geometry (à la Connes) |
81T75 | Noncommutative geometry methods in quantum field theory |
14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |
References:
[4] | doi:10.1103/RevModPhys.73.977 · Zbl 1205.81126 · doi:10.1103/RevModPhys.73.977 |
[5] | doi:10.1103/PhysRevD.64.046009 · doi:10.1103/PhysRevD.64.046009 |
[6] | doi:10.1016/0375-9601(78)90141-X · Zbl 0424.14004 · doi:10.1016/0375-9601(78)90141-X |
[7] | doi:10.1016/0550-3213(78)90311-5 · doi:10.1016/0550-3213(78)90311-5 |
[8] | doi:10.1016/0550-3213(78)90312-7 · doi:10.1016/0550-3213(78)90312-7 |
[9] | doi:10.1007/s002200050490 · Zbl 0923.58062 · doi:10.1007/s002200050490 |
[12] | doi:10.1103/PhysRevE.63.0650XX · doi:10.1103/PhysRevE.63.0650XX |
[13] | doi:10.1016/S0550-3213(01)00576-4 · Zbl 0988.81123 · doi:10.1016/S0550-3213(01)00576-4 |
[15] | doi:10.1016/S0370-2693(01)01354-5 · Zbl 0982.81053 · doi:10.1016/S0370-2693(01)01354-5 |
[16] | doi:10.1016/S0370-2693(01)00846-2 · Zbl 0971.81079 · doi:10.1016/S0370-2693(01)00846-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.