G LOBAL GRAVITATIONAL ANOMALY CANCELLATION FOR FIVE - BRANES Samuel Monnier Universität Zürich, Institut für Mathematik String-Math 2013, SCGP, Stony Brook, June 18th 2013 I NTRODUCTION Anomalies occur when a global symmetry of a classical field theory is not realized in the corresponding quantum field theory. If the global symmetry is supposed to be gauged, an anomaly signals an inconsistency of the quantum theory. ⇒ Very useful constraints on quantum field theories. Gravitational anomalies: The global symmetry is the isometry group of spacetime. The associated gauge symmetry is the group of diffeomorphisms. I NTRODUCTION The low energy limit of a consistent theory of quantum gravity must have vanishing gravitational anomalies. The miraculous cancellation of local gravitational anomalies in string theory provided the first serious hint that string theory may be a fully consistent theory of quantum gravity. Global gravitational anomalies have been comparatively little studied. Recent results about the global gravitational anomaly of the self-dual field enable us to check the cancellation of global anomalies systematically. Today, we will check that the world-volume theory of the M5-brane is free of global gravitational anomalies. This is a new non-trivial consistency test of M-theory. P LAN OF THE TALK 1 Anomalies 2 Strategy for computing global anomalies 3 The M5-brane geometry 4 Anomaly inflow 5 The self-dual field global anomaly 6 The fermion global anomaly 7 Anomaly cancellation A NOMALIES Consider a Euclidean quantum field theory defined on a compact oriented manifold M. Its gravitational anomalies can be detected as follows: The partition function Z can be seen as a function over the space of Riemannian metrics M on M. The group G of diffeomorphisms acts on M. A gravitational anomaly is present if the partition function is not invariant under this action. Indeed, the Ward identity relates the variation of the partition function under a diffeomorphism to the divergence of the stress-energy tensor. A NOMALIES Checking the invariance of Z directly is hard in general, but one can proceed as follows: Z defines a hermitian line bundle A over M/G. (More precisely, a G-equivariant bundle over M.) Z is the pull-back of a section of A . If there exists a hermitian connection on A with vanishing curvature and holonomies, then Z is G-invariant. In all cases at hand, A admits a natural connection ∇A . The curvature of ∇A is called the local anomaly and its set of holonomies is the global anomaly. If the local and global anomalies are trivial, the theory is anomaly-free. S TRATEGY FOR COMPUTING GLOBAL ANOMALIES Pick a path c ⊂ M with c(1) = hc(0), h ∈ G. Construct the mapping torus over c: Mc = (M × I)/[(x, 0) ∼ (h.x, 1)]. Endow it with a metric gc(t) × gc / . If there exists W such that ∂W = Mc , one can compute the holonomy of ∇A from geometrical data on W, in the limit → 0. In the case of a chiral fermionic theory, A is the determinant bundle of the chiral Dirac operator, and we have 1 1 ln holf (c) = index(DW ) − lim →0 2πi 2 [Witten - 1985], [Bismut-Freed - 1986] IDW W T HE M5- BRANE GEOMETRY We consider a smooth compact oriented 6-manifold M embedded in a smooth oriented spin 11-manifold X. Let N be the normal bundle of M in X. We require: w1 (N ) = 0 , w2 (N ) = w2 (M) , w5 (N ) = 0 . The boundary of a tubular neighborhood of M in X defines a 4-sphere π ˜ → bundle M M. ˜ (Induce a Metric: The metric on TX|M defines a natural metric on M. ˜ as the unit sphere bundle in N .) metric on TN and embed M T HE M5- BRANE GEOMETRY C-field: Recall that the M-theory C-field on X is best described by a shifted differential character of degree 4, where the shift is given by 1 2 p1 (X). ˜ of the field strength G on Practically, this means that the restriction G ˜ takes the form M ˜ = −1˜ G p1 + f + π ∗ (F) 4 ˜ f is a global angular form where ˜p1 is the 1st Pontryagin form of M, ˜ and F ∈ Ω4 (M). on M ❩ The term f is present because the five-brane sources G in X. It can be chosen canonically, as the half the Euler form of the vertical tangent ˜ bundle of M. T HE M5- BRANE GEOMETRY We consider as well this setup for 4-sphere bundles over π 7-dimensional mapping tori M˜c → Mc and over 8-dimensional π ˜ → manifolds W W bounded by mapping tori. π∗ (f 2 ) + 12 p˜1 is a lift of the degree 4 Wu class to integral cohomology. [Witten - hep-th/9912086] Rationally π∗ (f 2 ) = 0. [Bott-Cattaneo - dg-ga/9710001] ⇒ λ = 21 ˜p1 is a characteristic element for the wedge product pairing on 8-manifolds W without boundary: x∧x= W x∧λ W Enters the definition of the self-dual field. mod 2 . S ETUP FOR ANOMALY CANCELLATION ˜ be the space of metrics on M ˜ obtainable from a metric on X Let M according to the procedure above. ˜ are parameterized by Ω3 (M). The space of gauge fields on M ˜ The space of background fields is therefore B = Ω3 (M) × M. The group of local transformations is G = Ω3❩ (X) Write F = B/G. ˜ DiffX (M). S ETUP FOR ANOMALY CANCELLATION We consider a loop c in F. ˜ of a mapping c determines a metric and a C-field on the fibers ( M) ˜ c with base c. They can be converted to a metric and a C-field torus M ˜ c. on M ˜ over a We assume that there exists an oriented 4-sphere bundle W ˜ c on the boundary, and manifold W with boundary, restricting to M satisfying the same restrictions on the Stiefel-Whitney class of the associated rank 5 vector bundle. Non-trivial and probably hard cobordism problem! A NOMALY INFLOW M-theory at low energy contains a Chern-Simons term CS11 = 2πi X 1˜ ˜ ˜ ˜ C ∧ G ∧ G − C ∧ I8 6 , I8 = 1 48 ˜p2 + ˜p1 2 Let N be the tubular neighborhood of the five-brane. CS11 is not gauge invariant on X\N. Its variation under gauge transformations and diffeomorphisms is the anomaly inflow. The loop c is associated to a mapping torus (X\N)c with fiber X\N ˜ c . The variation of CS11 along the loop is given by and boundary M 1 ∆c (CS11 ) = 2πi (X\N)c 1˜ ˜ ˜ ˜ G ∧ G ∧ G − G ∧ I8 6 2 A NOMALY INFLOW Thanks to its integrality properties on closed manifolds, ∆c (CS11 ) can be computed by 1 ∆c (CS11 ) = 2πi ˜ W 1˜ ˜ ˜ ˜ G ∧ G ∧ G − G ∧ I8 6 ˜ and integrate over the 4-sphere One can use the explicit form of G ˜ to obtain fibers of W 1 ∆c (CS11 ) = 2πi using 61 π∗ (f 3 ) = W 1 1 1 F (F − λ) + I8 + p˜21 + p2 2 32 24 1 24 p2 . [Bott-Cattaneo - dg-ga/9710001] T HE SELF - DUAL FIELD GLOBAL ANOMALY The global anomaly of the self-dual field can be expressed as an integral over a manifold W bounded by the mapping torus Mc : 1 ln holSD (c) = lim 2πi = lim W W 1 1 L − G2 8 2 1 1 (L − λ2 ) − F(F − λ) 8 2 This formula has not yet been proven completely rigorously. It could in principle be off by a sign on certain loops, but this would imply that cohomological type IIB supergravity is anomalous. T HE SELF - DUAL FIELD GLOBAL ANOMALY A consistency check is to compute the global gauge anomaly. Mc = M × S1 with a constant vertical metric. In this case the purely gravitational terms vanish. The self-dual field gauge anomaly 1 ln holSD (c) = − 2πi W 1 F(F − λ) 2 cancels with the gauge anomaly inflow: 1 ∆c (CS11 ) = 2πi W 1 F (F − λ) 2 T HE FERMION GLOBAL ANOMALY Let S+TM be the chiral spinor bundle of M and S N be the spinor bundle constructed from N . In general these bundles do not exist globally on M, but S+TM ⊗ S N does. The fermions on the five-brane are real sections of S+TM ⊗ S N . T HE FERMION GLOBAL ANOMALY The associated global anomaly is computed by [Witten - 1985], [Bismut-Freed - 1986] 1 1 ln holf = 2πi 2 index(D) − lim ID W where D is the Dirac operator on S+TW ⊗ S N . and N ) is the index density of D. ˆ ID = A(TW)ch(S S+TW ⊗ S N admits a quaternionic structure, so index(D) is even and does not contribute to the global anomaly. A NOMALY CANCELLATION Let us check the cancellation of global anomalies. We have 1 ln holM5 = lim 2πi 1 1 1 1 (ID + L − λ2 + I8 + p˜21 + p2 ) 8 8 32 24 W But λ = 12 ˜p1 , so the 3rd and 5th terms cancel. Moreover, an explicit computation of the degree 8 components of the index densities show that 1 1 ID + L + I8 + p2 = 0 8 24 [Witten - hep-th/9610234], [Freed, Harvey, Minasian, Moore - hep-th/9803205]. C ONCLUSION We conclude that that the global gravitational anomalies of the M5-brane cancel, when the inflow from the bulk is taken into account. ˜ → bounded by M ˜ c → Mc This is true only if the 4-sphere bundle W exists. The relevant cobordism group seems hard to compute. The same computation with W a closed manifold also gives a straightforward proof of the cancellation of local anomalies. This result trivially generalizes to the other chiral five-branes, namely the type IIA and heterotic E8 × E8 five-branes, as they both can be lifted straightforwardly to M5-branes in an M-theory background.
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