Semiclassical Yang-Mills theory. I: Instantons. (English) Zbl 0792.53024
Summary: The partition functions of quantum Yang-Mills theory have an expansion in powers of the coupling constant; the leading order term in this expansion is called the semiclassical approximation. We study the semiclassical approximation for Yang-Mills theory on a compact Riemannian 4-manifold using geometric techniques, and do explicit calculations for the case when the manifold is the 4-sphere. This involves calculating the Riemannian measure and certain functional determinants on the moduli space of self-dual connections. The main result is that the contribution to the semiclassical partition functions coming from the \(k = 1\) connections on the 4-sphere is finite and calculable. We also discuss a renormalization procedure in which the radius of the 4-sphere is allowed to tend to infinity.
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
References:
| [1] | [AHDM] Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G., Manin, Yu.I.: Construction of instantons. Phys. Lett.65A, 185–187 (1978) · Zbl 0424.14004 |
| [2] | [AHS] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A362, 425–461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143 |
| [3] | [BP] Belavin, A.A., Polyakov, A.M.: Quantum fluctuations of pseudoparticles. Nucl. Phys. B123, 429–444 (1977) · Zbl 1015.81541 · doi:10.1016/0550-3213(77)90175-4 |
| [4] | [BV] Babelon, O., Viallet, C.M.: The geometrical interpretation of the Faddeev-Popov determinant. Phys. Lett.85B, 246–248 (1979) |
| [5] | [C] Coleman, S.: Aspects of Symmetry, Chap. 5: Secret Symmetry: An introduction to spontaneous symmetry breakdown and gauge fields, and Chap. 7: The Uses of Instantons. Cambridge: Cambridge University 1985 |
| [6] | [CVDN] Chadha, S., di Vecchia, P., D’Adda, A., Nicodemi, F.: {\(\zeta\)}-function regularization of the quantum fluctuations around the Yang-Mills pseudoparticle. Phys. Lett.72B, 103–108 (1977) |
| [7] | [DMM] Doi, H., Matsumoto, Y., Matumoto, T.: An explicit formula of the metric on the moduli space of BPST-instantons overS 4. A Fete of Topology, pp. 543–556. New York: Academic Press 1988 |
| [8] | [FU] Freed, D.S., Uhlenbeck, K.K.: Instantons and Four-Manifolds. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0559.57001 |
| [9] | [F] Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall 1971 |
| [10] | [G] Gilkey, P.: The spectral geometry of a Riemannian manifold. J. Differ. Geom.10, 601–618 (1973) · Zbl 0316.53035 |
| [11] | [GP1] Groisser, D., Parker, T.H.: The Riemannian geometry of the Yang-Mills moduli space. Commun. Math. Phys.112, 663–689 (1987) · Zbl 0637.53037 · doi:10.1007/BF01225380 |
| [12] | [GP2] Groisser, D., Parker, T.H.: The geometry of the Yang-Mills moduli space for definite manifolds. J. Differ. Geom.29, 499–544 (1989) · Zbl 0679.53024 |
| [13] | [Gr] Groisser, D.: The geometry of the moduli space ofCP 2 instantons. Invent. Math.99, 393–409 (1990) · doi:10.1007/BF01234425 |
| [14] | [H] ’tHooft, G.: Computation of the quantum efects due to a four-dimensional pseudoparticle. Phys. Rev. D14, 3432–3450 (1976) · doi:10.1103/PhysRevD.14.3432 |
| [15] | [L] Lawson, H.B.: The theory of gauge fields in four dimensions. Providence, RI: Am. Math. Soc. 1986 |
| [16] | [MSZ] Montgomery, D., Samelson, H., Zippin, L.: Exceptional orbits of highest dimension. Ann. Math.64, 131–141 (1956) · Zbl 0074.26001 · doi:10.2307/1969951 |
| [17] | [O] Osborn, H.: Semiclassical functional integrals for self-dual gauge fields. Ann. Phys.135, 373–415 (1981) · Zbl 0479.58022 · doi:10.1016/0003-4916(81)90159-7 |
| [18] | [P] Patodi, V.K.: Curvature and the eigenvalues of the Laplace operator. J. Differ. Geom.5, 233–249 (1971) · Zbl 0211.53901 |
| [19] | [PR] Parker, T.H., Rosenberg, S.: Invariants of conformal Laplacians. J. Differ. Geom.25, 199–222 (1987) · Zbl 0644.53038 |
| [20] | [RaSi] Ray, D.B., Singer, I.M.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145–210 (1971) · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4 |
| [21] | [RoSc] Romanov, V.N., Schwarz, A.S.: Anomalies and elliptic operators. Theor. Math. Phys. (English Translation)41, 967–977 (1979) · doi:10.1007/BF01028502 |
| [22] | [S] Seeley, R.: Complex powers of an elliptic operator. Proc. Symp. Pure Math.10, 288–307 (1973) |
| [23] | [Sc] Schwarz, A.S.: Instantons and fermions in the field of instanton. Commun. Math. Phys.64, 233–268 (1979) · Zbl 0406.58032 · doi:10.1007/BF01221733 |
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.
