Modular forms and generalized anomaly cancellation formulas
نویسندگان
چکیده
0. Introduction In [1], gravitational anomaly cancellation formulas are derived from direct computations. In particular, in dimension 12, the Alvarez-Gaumé and Witten ‘‘miraculous cancellation’’ formula can be written as {L(TX, ∇TX )}(12) = {8A(TX, ∇TX )ch(TCX, ∇TCX )}(12) − 32{A(TX, ∇TX )}(12), (0.1) where X is a twelve-dimensional Riemannian manifold, ∇TX is the associated Levi-Civita connection, TCX is the complexification of TX (with the induced Hermitian connection ∇TCX ) andL(TX, ∇TX ),A(TX, ∇TX ) are the Hirzebruch characteristic forms (see (1.1)). In [2], Liu generalizes (0.1) to general 8m + 4 dimensions by developing modular invariance properties of characteristic forms. Actually, in [2], Liu obtains a more general cancellation formula by including an auxiliary bundleW . More precisely, assume X to be 8m + 4 dimensional and W be a rank 2l Euclidean vector bundle over X with a Euclidean connection ∇W and curvature RW = ∇W ,2; if p1(TX, ∇TX ) = p1(W , ∇W ), then the following identity holds: A(TX, ∇TX ) det1/2 2 cosh√−1 4π RW (8m+4) = m r=0 2l+2m+1−6r A(TX, ∇TX )ch(br(TCX,WC, C2))(8m+4) , (0.2) where the br(TCX,WC, C2) are virtual complex vector bundleswith connections over X canonically determined by (TX, ∇TX ) and (W , ∇W ). In dimension 12, by direct computation, (0.2) becomes ∗ Corresponding author. E-mail addresses:[email protected] (F. Han), [email protected] (K. Liu), [email protected] (W. Zhang). 0393-0440/$ – see front matter© 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2012.01.016 F. Han et al. / Journal of Geometry and Physics 62 (2012) 1038–1053 1039 A(TX, ∇TX ) det1/2 2 cosh√−1 4π RW (12) = 2l−3 A(TX, ∇TX )ch(WC, ∇WC)(12) − 2l−2(l − 4) A(TX, ∇TX )(12) . (0.3) When (TX, ∇TX ) = (W , ∇W ), (0.2) gives 1 8 L(TX, ∇TX )(8m+4) = m r=0 26m−6r A(TX, ∇TX )ch(br(TCX, TCX, C2))(8m+4) . (0.4) We obtain, as an application [3], by the Atiyah–Hirzebruch divisibility [4], that (0.4) implies the Ochanine divisibility [5], which asserts that the signature of an 8k + 4-dimensional smooth closed spin manifold is divisible by 16. To study higher dimensional Rokhlin congruence, Han and Zhang [6,7] extend the ‘‘miraculous cancellation’’ formulas of Alvarez-Gaumé, Witten and Liu to a twisted version where an extra complex line bundle (or equivalently a rank 2 real oriented vector bundle) is involved. More precisely, if ξ is a rank 2 real oriented Euclidean vector bundle equipped with a Euclidean connection ∇ and c = e(ξ , ∇ ) is the associated Euler form, when p1(TX, ∇TX ) = p1(W , ∇W ), the following identity holds: A(TX, ∇TX ) det1/2 2 cosh √−1 4π RW cosh2 c 2 (8m+4)
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