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Global Anomalies in M-Theory

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Global Anomalies in M-Theory CERN-TH/97-277 hep-th/9710126 Global Anomalies in M -theory Mans Henningson Theory Division, CERN CH-1211 Geneva 23, Switzerland [email protected] Abstract We rst consider M -theory formulated on an op en eleven-dimensional spin-manifold. There is then a p otential anomaly under gauge transforma- tions on the E bundle that is de ned over the b oundary and also under 8 di eomorphisms of the b oundary. We then consider M -theory con gura- tions that include a ve-brane. In this case, di eomorphisms of the eleven- manifold induce di eomorphisms of the ve-brane world-volume and gauge transformations on its normal bundle. These transformations are also po- tentially anomalous. In b oth of these cases, it has previously b een shown that the p erturbative anomalies, i.e. the anomalies under transformations that can be continuously connected to the identity, cancel. We extend this analysis to global anomalies, i.e. anomalies under transformations in other comp onents of the group of gauge transformations and di eomorphisms. These anomalies are given by certain top ological invariants, that we explic- itly construct. CERN-TH/97-277 Octob er 1997 1. Intro duction The consistency of a theory with gauge- elds or dynamical gravity requires that the e ective action is invariant under gauge transformations and space-time di eomorphisms, usually referred to as cancelation of gauge and gravitational anomalies. The rst step towards establishing that a given theory is anomaly free is to consider transformations that are continuously connected to the identity. The cancelation of the corresp onding anomalies, often called p erturbative anomalies, imp oses some constraints on the chiral eld content of the theory. Given that the p erturbative anomalies cancel, it makes sense to investigate transformations in other comp onents of the group of gauge transformations and di eomorphisms. An anomaly under such a transformation is usually referred to as a global anomaly. A general formula for the quantum contribution of chiral elds to global anomalies was given in [1]. Provided that the p erturbative anomalies cancel, the global anomaly is a top ological invariant, i.e. it it invariant under smo oth deformations of the data, und thus only dep ends on the top ological classes of for example the space- time manifold and the gauge-bundle. The requirement that the anomaly vanishes for an arbitrary transformation imp oses some restrictions on these ob jects. In string theory, the requirements of sup ersymmetry and vanishing anomalies are par- ticularly constraining, b ecause of the high dimensionality of space-time. For the purp ose of computing anomalies, it is enough to know the low-energy e ective sup ergravity the- ory. The non-chiral typ e IIA sup ergravity theory obviously has no anomalies. The chiral typ e IIB sup ergravity theory has a p otential p erturbative gravitational anomaly, but the contributions from the various chiral elds `miraculously' cancel against each other [2]. Typ e I sup ergravity coupled to some sup er-Yang-Mills theory always has a non-vanishing p erturbative quantum anomaly. It can however be cancelled by a Green-Schwarz mech- anism [3] involving an anomalous transformation law at tree-level for the two-form eld, provided that the gauge group is SO(32) or E E . This discovery actually preceeded 8 8 the construction of the SO(32) and E E heterotic string theories. Global anomalies in 8 8 string theory was rst considered in [1]. The result is that the known string theories are free from global anomalies when formulated in ten-dimensional Minkowski space. How- ever, there are interesting non-trivial restrictions on consistent compacti ctions to lower dimensions. There is by now mounting evidence that the di erent string theories should be seen as particular limits of a conjectured theory called M -theory. In the long wave-length limit, 1 this theory should reduce to eleven-dimensional sup ergravity. It should also admit certain typ es of top ological defects, in particular space-time b oundaries and ve-brane solitons. Much has b een learned ab out M -theory by studying the mechanism for cancellation of p erturbative anomalies for various con gurations including such defects. In this way,itwas discovered that when M -theory is de ned on an op en eleven-manifold, there is an E E 8 8 sup er-Yang-Mills multiplet propagating on the b oundary. The cancellation of p erturbative gauge and gravitational anomalies involves a subtle interplay between contributions from this multiplet, the bulk degrees of freedom and a generalized Green-Schwarz mechanism [4][5]. The situation is even more interesting for M -theory con gurations involving a ve-brane. The p erturbative gauge and gravitational anomalies from the ve-brane world- volume theory can b e partially cancelled by an anomaly in ow from the surrounding eleven- dimensional space [6][7]. There is however a remaining part, whose cancellation seems to require additional world-volume interactions and also imp oses a certain top ological restriction on the ve-brane con guration [7][8]. Given our present incomplete understanding of M -theory, it seems that any further information ab out this theory would be valuable. The purp ose of the present pap er is to carry the analysis describ ed ab ove one step further byinvestigating also global anomalies. In section two, we consider the case of M -theory de ned on an op en eleven-manifold. In section three, we instead consider M -theory con gurations including a ve-brane. In b oth of these cases, we derive an explicit formula for the top ological invariants describing the global anomalies. 2. Anomalies on op en eleven-manifolds We consider M -theory on an eleven-dimensional op en spin-manifold Y . The massless degrees of freedom propagating in the bulk of Y are those of the eleven-dimensional su- p ergravity multiplet, i.e. a metric G , a three-form p otential C and a fermionic MN MNP Rarita-Schwinger eld . Atlow energies, the dynamics of these elds is governed by the M eleven-dimensional sup ergravity action [9]. The bulk action p ossesses a classical invariance under di eomorphisms of Y , and since we are in an o dd numb er of space-time dimensions, this symmetry is obviously not sp oiled by any chiral anomaly. However, there is p otentially an anomaly, often called the parity anomaly, in the bulk of Y , which is asso ciated with the Rarita-Schwinger eld [10]. The op erator in M the kinetic term of this eld is Hermitian in eleven dimensions, but has in nitely many 2 p ositive and negative eigenvalues, leading to a p otential sign problem in the de nition of the fermionic path-integral measure [2]. This will show up as a change of the bulk e ective bulk action under a di eomorphism of Y . To describ e this anomaly, it bulk 1 is convenient to intro duce a twelve-dimensional manifold (Y S ) , called the mapping torus, as follows: We start with the cylinder Y I , where I is an interval, and equip it with a metric that smo othly interp olates between the original metric on Y at one of the b oundaries of I and the metric obtained from it by the transformation at the other 1 b oundary. Finally,we glue together the two b oundaries of I to form (Y S ) . The bulk anomaly is then given by [11] 3 1 1 1 Index (RS ) Index (D ) ; (2:1) = i 0 bulk (Y S ) (Y S ) 2 2 1 1 where Index (RS ) and Index (D ) denote the indices of the Rarita-Schwinger 0 (Y S ) (Y S ) 1 and Dirac op erators on (Y S ) . It follows from charge conjugation symmetry that these indices are even in twelve dimensions, so is a multiple of i, corresp onding to the bulk sign ambiguity in the fermionic path integral measure. Precisely in twelve dimensions, 1 the combination of indices that app ears in (2.1) is related to the signature of (Y S ) 1 (Y S ) as 1 1 1 1 = Index (RS ) 3 Index (D ): (2:2) 0 (Y S ) (Y S ) (Y S ) 8 1 It follows that is a multiple of 16 and that the bulk anomaly can be written as (Y S ) i 1 : (2:3) = bulk (Y S ) 16 The massless degrees of freedom on the b oundary M of Y include a left-handed Rarita- Schwinger eld and a right-handed spinor eld originating from the Rarita-Schwinger eld of the bulk theory. There is also a set of left-handed spinor elds in the M adjoint representation of E that together with a set of gauge elds A make up a sup er 8 Yang-Mills multiplet propagating on the b oundary. These elds give rise to an anomaly under di eomorphisms of Y that induce di eomorphisms of M , and also under gauge transformations of the E bundle V over M . To describ e such a transformation , we 8 1 consider the mapping torus (M S ) and the E bundle V over it. These ob jects are 8 1 constructed in analogy with (Y S ) by identifying the b oundaries of the cylinder M I 1 1 after a twist by . We note that (Y S ) is b ounded by(MS ) . The anomalous change 3 under the transformation of the e ective action obtained byintegrating out ef f ef f the fermionic elds is then given by a general formula derived in [1] as i = ; (2:4) ef f 2 1 where denotes a certain -invariant on (M S ) .
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