Corners in M-theory Hisham Sati ∗ Department of Mathematics University of Maryland College Park,MD 20742 Abstract 1 1 0 M-theorycanbedefinedonclosedmanifoldsaswellasonmanifoldswithboundary. Asanextension, 2 weshowthatmanifoldswithcornersappearnaturallyinM-theory. Weillustratethiswithfoursituations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic n a string theory in relation to twelve dimensions, and the two M-branes within M-theory in the presence J of a boundary. The M2-brane is taken with (or as) a boundary and the worldvolume of the M5-brane 4 is viewed as a tubular neighborhood. We then concentrate on (variant) of the heterotic theory as a 1 corner and explore analytical and geometric consequences. In particular, we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner(s). The ] analysisinvolvesconsideringM-theoryondisconnectedmanifolds,andmakesuseoftheextensionofthe h Atiyah-Patodi-Singerindex theorem to manifolds with corners and theb-calculus of Melrose. t - p e h [ 1 v 3 9 7 2 . 1 0 1 1 : v i X r a ∗e-mail: [email protected] Contents 1 Introduction 1 2 Manifolds with corners and their relevance in M-theory 2 2.1 Basic definitions and relevant tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Occurrence in M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 M-theory on AdS spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 The M5-brane as a tubular neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 The M2-brane and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 The heterotic theory as a corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Analytical and geometric aspects of M-theory with corners 8 3.1 Formulation using Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.1 The non-supersymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 The supersymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Formulation using the signature operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 The phase of the partition function for manifolds with corners via the signature. . . . 14 3.2.2 Orientation-reversinginvolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1 Introduction The boundary of a manifold X cannot itself be a boundary, but there are situations in physics (and mathe- matics) which demand that sense be made out of some variationon the notion of boundaries of boundaries. Forexample,weknowthatheteroticstring theoryshouldbe aboundaryofM-theory[20][21]. Thelatter in turn is best described globally using a bounding twelve-dimensional theory [48] [8]. A naive boundary of a boundarydoesnotexistase.g. theboundaryoperatorinhomologyisnilpotent. However,thishasanatural setting within manifolds with corners. With this we then can view the heterotic theory as codimension-two corner of the twelve-dimensionaltheory. We recall the basics of manifolds with corners in section 2.1. Closed manifolds and manifolds with boundary are special cases of manifolds with corners. Also products of two manifolds with boundary form an interesting class of examples within manifolds with corners of codimenison-two We consider situations in M-theory, in addition to the heterotic boundary, where manifolds with corners are needed in order to describe the physical system. This is discussed in section 2.2 and includes the following. AdS/CFT correspondence. Eleven-dimensional supergravity admits solutions with an Anti-de Sitter space as a factor, most famously AdS S7 [16] and AdS S4 [38]. These spaces appear as the near 4 7 × × horizon limits of the M2-brane and the M5-brane, respectively, via the AdS/CFT correspondence which relatesquantum supergavityonthe AdS factorto the conformalfield theory onthe boundary ∂AdS [30]. In section2.2.1 we show that the lift ofthese solutions to twelve dimensions leads naturally to a manifold with corners as the product of two manifolds with boundaries. The M5-brane. TheM5-branewithworldvolumeW6 canbedescribedintermsoftubularneighborhoods in the target 11-dimensional manifold Y11. When Y11 has no boundary the resulting manifold arising from the sphere bundle of the normal bundle to the embedding W6 ֒ Y11 has a boundary. Now M-theory → itself on a manifold with boundary certainly makes sense [21] [7] [45] and so we ask what happens to the descriptionoftheM5-braneinthatcase. Whilethemainthemein[7][45]wasforwhenY11 hasaboundary, the description of the M5-brane was given only for the case when Y11 is closed (in [7] the M5-brane was related to M-theory with boundary only via anomalies involving torsion). In section 2.2.2 we show how the resulting manifold will be a manifold with corners of codimension-two. This uses the general result of [23] 1 thattheremovalofatubularneighborhoodofanysubmanifoldcreatesamanifoldofonecodimensionhigher. Therefore, including an M5-brane on an eleven-dimensional manifold with a boundary leads naturally to a manifold with corners. The M2-brane. The M2-brane in M-theory can have boundaries on the M5-brane [46]. When M-theory is considered on a manifold with boundary, then the two-dimensional boundaries of the M2-brane can end onthe ten-dimensionalboundaryofM-theory,inthe Horava-Wittenset-up[21]. Therefore,wetakethe M2- brane within M-theory with boundary, that is on a twelve manifold which is a product of a three-manifold with boundary with an eight manifold. Then we wrap the M2-brane on the former; the ten-manifold which is the product of the membrane boundary with the eight-manifold will be a corner for the twelve-manifold. In fact, this will be essentially the heterotic corner. We describe this in section 2.2.3. Heterotic M-theory. Heterotic string theory is essentially a boundary of M-theory [20] [21]. The M- theorypartitionfunctiononaSpineleven-manifoldwithboundarywasconsideredin[7]withanemphasison eleven rather than on twelve-dimensions. For topologicaland global (e.g. index theory) purposes, M-theory in turn is considered as a boundary of the bounding twelve-dimensional theory on Z12 [48] [8]. Hence, in the connectiontoheterotic stringtheory,the bounding twelve-dimensionaltheory requireshavingseemingly a ‘boundary of a boundary’. In section 3 we consider the effect of studying this from the point of view of theboundingtwelve-dimensionaltheory. Weprovidetwoformulations,oneusingDiracoperators(insection 3.1)andanotherinvolvingthesignatureoperators(insection3.2), makinguseoftheemergenceofthelatter in [45]. As in [14], we consider the Horava-Witten theory on a ten-dimensional Spin manifold M10 from two points of view. First, via the product with the interval [0,1] M10 (the “upstairs” formulation), which for × us nicely connects to manifolds with corners by taking a further Cartesian product with another interval. We do this for most of section 3. Second, via S1/Z M10 (the “downstairs” formulation) with Z acting 2 2 × as an orientation-reversinginvolution. We consider the eleven-manifold as a boundary of a twelve-manifold in the presence of an orientation-reversinginvolutionand study the effect onthe signature operatorand the correspondingeta-invariantsinsection3.2.2. Thisallowsustoformulatethephaseofthepartitionfunction. We studyanalyticalandgeometricaspectsofthe theoryinthis setting usingmainlythe constructionsin [3] [17] and [36] and the survey [28]. In particular we consider the global reduction to ten dimensions of the phaseofthepartitionfunction,usingb-eta-invariantswithintheb-calculus[33]. Thisallowsformoregeneral boundary conditions than those of Atiyah-Patodi-Singer (APS) [2] used in [8]. The discussion requires considering M-theory on disconnected eleven-dimensional spaces. We also consider the case of multiple ten-dimensional (heterotic) components in the setting of manifolds with corners. While this is mostly a physicspaper,we havechosento identify main(physical)results andobservations by recording them as propositions and lemmas, mainly as a way of keeping track of the main statements. 2 Manifolds with corners and their relevance in M-theory We first recall in section 2.1 the basics of manifolds with corners and then we provide our applications to M-theory in section 2.2. 2.1 Basic definitions and relevant tools We now give the basic definitions and some of the properties that we need in the applications to M-theory, which we discuss starting in the following section. 2 The basic definitions. A differentiable manifold with corners is a topological space covered by charts which are locally open subsets of Rn = [0, )n [9] [5]. Adding information about faces leads to manifold + ∞ with faces. Imposing conditions on how the faces piece globally together leads to s restrictive class called n -manifolds [23]. This is a manifold with faces together with an ordered n-tuple (∂ X,∂ X, ,∂ X) 0 1 n−1 h i ··· of faces of X which satisfy the following conditions: (1) The boundary is formed of n disconnected components ∂X =∂ X ∂ X; 0 n−1 ∪···∪ (2) The intersection ∂ X ∂ X is a face of ∂ X and of ∂ X for all i=j. i j i j ∩ 6 The number n is called the codimension of X. We will be mainly interested in the case n=2. Products and codimension. Theproductofan m -manifoldwithan n -manifold m+n -manifold. A h i h i h i 0 -manifoldis a manifold without a boundarywhile a 1 -manifoldis a manifold with boundary. So we can h i h i create many manifolds with boundary by multiplying manifolds of these two different types. Furthermore, we can create n -manifolds from products of 0 -manifolds with n -manifolds, 1 -manifolds with n 1 - h i h i h i h i h − i manifolds, and so on. In the main case of interest, which is 12-manifolds with corners of codimension-two, we can construct many such spaces by taking a product of a k-dimensional i -manifold with a (12 k)- h i − dimensional 2 i -manifoldwithi=0,1. Moreexplicitly,wecantaketheproductofaclosedmanifoldwith h − i amanifoldwithcornersaswellastheproductoftwomanifoldswithboundary,thesumofwhosedimensions is 12. We consider two simple examples of manifolds with corners of codimension-two. Example 1. The positive quadrant. Let R2 denote the closed positive quadrant of R2, that is R2 = + + (x1,x2) R2 : x1 0,x2 0 . The boundary of R2 in R2 is the set of points at which one or both { ∈ ≥ ≥ } + coordinates vanish. The points in R2 at which both coordinates vanish are called its corner points. The + boundaryofa smoothmanifoldwithcornersisingeneralnotasmoothmanifoldwithcorners. Forexample, the boundary of R2 is the union ∂R2 = H H , where H = (x1,x2) R2 : xi = 0 , i = 1,2, is a + + 1 ∪ 2 i { ∈ + } one-dimensional smooth manifold with boundary. Example 2. Lie groups with action of maximal torus. Let G be SU(2) or SO(4), the Lie groups of rank 2, and let T2 be the corresponding maximal torus. Then T2 acts on the product (D2)2 of 2 disks D2 by complex multiplication. The resulting associated fiber bundle G T2 (D2)2 is a 2 -manifold. For SU(2) × h i this is five-dimensional, while for SO(4) this is eight-dimensional. For more on such examples see [25]. We will be interested in integrating forms on manifolds with corners. Integration over the boundary amounts to integrating over the boundary components. We illustrate this with an example. Example 3. The square in R2. The square is a manifold with corners of codimension-two. Its edges are boundary hypersurfaces and its corners are codimension-two faces. Let I I =[0,1] [0,1] be the unit squareinR2,andsupposeω isasmooth1-formontheboundary∂(I I). Co×nsiderthem×apsF :I I I i × → × given by F (t)=(t,0), F (t)=(1,t), F (t)=(1 t,1), F (t)=(0,1 t) . (2.1) 1 2 3 4 − − The four curve segments in the sequence traverse the boundary of I I in the counterclockwise direction. × Then Stokes’ theorem for a manifold with corners gives [26] ω = ω+ ω+ ω+ ω. Such ∂(I×I) F1 F2 F3 F4 integration over rectangles should be familiar from electromagnetism, although it is usually not cast in this R R R R R language. One of the main advantages of using manifolds with corners is that, for example, the cube which is not a smooth manifold would be smooth as a manifold with corners. We will also need to study differential forms and cohomology on manifolds with corners. 3 L2-cohomology. A manifold with corners can be viewed as a manifold with singularities. De Rham cohomologydoesnotcapturetheinformationatthesingularitiesorcorners. Tomakeupforthis,onerestricts to the subcomplex of square-integrable differential forms, which leads to L2-cohomology. Let (Y,g ) be a Y RiemannianmanifoldandletΩp =Ωp(Y)bethespaceofsmoothp-formsandL2 =L2(Y)theL2completion of Ωp with respect to the L2-metric. The differential d is defined to be the exterior differential with the domain dom(d) = ω Ωp : dω L2(Y) , where Ωp = Ωp(Y) L2(Y) is the space of square-integrable { ∈ (2) ∈ } (2) ∩ smooth p-forms. The L2-cohomology is then the cohomology of the cochain complex 0 Ω0 (Y) d Ω1 (Y) d Ω2 (Y) d Ω3 (Y) d , (2.2) −→ (2) −→ (2) −→ (2) −→ (2) −→··· thatis,Hp (Y)=kerd /Imd . ThenaturalmapHp (Y) Hp(Y;R)viatheusualdeRhamcohomology (2) i i−1 (2) → isanisomorphismforY acompactmanifoldwithcornersbecausetheL2 conditionisautomaticallysatisfied for all smooth forms. For a nice exposition on this see [6]. Hodge theory for a manifold with corners is discussed in [39] [34]. Smoothing corners. Manifolds with corners are smooth in the sense of having charts locally as open subsets of Rn. However, they look like they should be singular at the corner. What is the explanation to + this? One thing one could do is smooth out the corner via a diffeomorphism, which is not an isometry. For instance, if the corner is that of a quadrant then one can replace rectangular coordinates with polar coordinatesandprovideasmoothingofthecornerbyconsideringonlynonzerovalueoftheradialcoordinate. This is called total boundary blow-up [32] (see also [28]). In our context will be interested in manifolds of the form Z ∼= [0,1)s1 ×[0,1)s2 ×M, where s1 and s2 are Cartesian coordinates on the two intervals. Near thecornerM,introducepolarcoordinatesvias =rcosθ ands =rsinθ sothatthetotallyblown-upspace 1 2 is Ztb ∼=[0,ε)r×[0,π/2]θ×M, forε>0. We havediffeomorphisminsteadof isometrybecause intersections of hypersurfaces at M do not have to occur at right angles, but any angle in the plane can be related by a diffeomorphism to the standard upper right quadrant. We will have this blow-up implicitly in mind in dealing with manifolds with corners. Tostudythe phaseofthepartitionfunctionweneedtoconsiderDiracoperatorsandtheircorresponding eta-invariants. Continuous spectrum and the b-trace. A Dirac operatoron a manifold with corners has a continuous spectrum, and hence trying to define the eta invariantwill involve infinite traces. The way aroundthis is to usethe b-tracewithintheb-calculus[33]. Forourpurposes,the mainideacanbe summarizedasfollows(see alsothe nextexample). The b-traceis definedintermsofthe b-integralforanoperatorO (schematically)as b b Tr(O):= tr(O). (2.3) ZY Then the corresponding eta-invariant will be defined using this trace as [33] bη(D)= 1 ∞ 1 bTr(De−tD2)dt. (2.4) √π √t Z0 Let us illustrate this with an example, which will be useful for us later. Example 4. The interval over a manifold. Consider [0,1] , the unit intervalwith coordinate s, which s willbe fiberedoveramanifoldM. The functionds/s isnotintegrableover[0,1] , sothatthe corresponding s heat operator is not trace class. However, the function sz, Rez > 0, is integrable with respect to ds/s over [0,1] . This suggests using an integral which corresponds to the usual integral when z is zero. As s nicely illustrated in [28], let f C∞(Y) be a smooth function on a manifold Y. Then for all complex ∈ numbers z with Rez > 0, the integral F(z) = szfdg exists and it extends from Rez > 0 to define a Y R 4 meromorphic function on all of C. Note that sz = ezlogs is an entire function of z for s > 0. Thus, f can be assumed to be supported on the collar [0,1] M of Y. Then F(z) is well-defined for Rez > 0 since s × szf(s,m) is integrable with respect to the measure (ds/s)dh as long as Rez > 0. Here m is a point in M and dh is a measure on M. Now expand f(s,m) in Taylor series at s = 0: f(s,m) ∞ skf (m). Since the integral sz+kf (m)dsdh is equal to 1 f (m)dh then the function F∼(z) ekx=t0endskfrom [0,1]×m k s z+k M k P Rez > 0 to be a meromorphic function on C with only simple poles at z = 0, 1, 2, with residue R R { − − ···} at z = 0 given by f (m)dh = f(0,m)dh. The b-integral of f is the regular value of F(z) at z = 0, M 0 M b fdg =Reg F(z), such that the residue of F(z) at z =0 is given by Res F(z)= f(0,m)dh. Y z=0 R R z=0 Y R We will also be interested in considering the kernels of Dirac operators on manifolds wRith corners. Infinite-dimensional kernels. The dimensions of the kernels are generically infinite so that the Dirac operator is not Fredholm in general. We will consider the effect of this in section 3.1. We now illustrate this in the simple example of the square. Consider the Cauchy-Riemann operator ∂ = ∂ +i∂ on the z x y square [0,1] [0,1] . The manifold and hence the operator are of product type. Then the kernel ker∂ is x y z × infinite-dimenisonal since this kernel consists of all holomorphic functions on the square. Compactification of manifolds with cylindrical ends to manifold with corners. In order to deal with the non-Fredholmpropertyofthe Dirac operatoronthe manifold with cornersZ, one has to introduce anothermanifoldZ ofthesamedimensionwhichisformedbyattachinginfinite cylinderstothecollarsofZ. This will be used in section 3.1. The manifold Z can be compactified by introducing the change ofvariables x1 =es1 and x2 =bes2. As si , xi 0 and so this change of variables compactifies Z to be the interior →∞ → of a compact manifold with corners of codimenbsion-two Z. The metric then transforms to the b-metric as b dx 2 dx 2 gZ =ds2+ds2+gM ; bgZ = 1 + 1 +gM . (2.5) 1 2 x x (cid:18) 1 (cid:19) (cid:18) 1 (cid:19) The Maslov index. Whenthetwelve-manifoldZ12 hasnoboundary,thereareniceadditivityproperties, for example the Novikov additivity of the signature [37]. The Atiyah-Patodi-Singer index theorem [2] gives the index ofthe Dirac andsignature operatorson manifolds with boundary in terms ofthe A-genusand the L-genus,respectively,andthedefectsgivenbythecorrespondingetainvariantsontheboundary. Inthecase of corners, the signature is no longer additive, but there is a correction term in Wall’s nonbadditivity [47]. The signature defect is the Maslov index of certain Lagrangian subspaces related to the cohomology of the boundaryY11 andthe corner. This, andthe correspondinggeneralizationusing[3][17][36] and[28],will be discussed in section 3.1 for the Dirac operator and in section 3.2 for the signature operator. 2.2 Occurrence in M-theory In this section we consider four situations, three of which are related to M-branes in M-theory, and one related to heterotic string theory, where manifolds with corners appear naturally. The first one is M-theory onAdSspaces,whichareconfigurationsthatoccurasnearhorizonlimitsofM-branes. Thesecondonearises byconsideringtheM5-braneasatubularneighborhoodineleven-dimensionalspacetimewithboundary. The third arises when considering boundaries in relation to the M2-brane. This includes the M2-brane having a boundary (ending on the M5-brane) or the M2-brane itself being considered as a boundary for instance when studying its partition function. The fourth views (a variant of) heterotic string theory as a corner in the twelve-dimensional bounding theory. 2.2.1 M-theory on AdS spaces Anti-deSitterspaceisaLorentzianspacewithboundaryatspatialinfinity. TheEuclideanversionisgivenby ahyperbolicspace,withveryinterestingboundarystructureatinfinity. Thereforeconsideringthe boundary 5 ofAdSspaceamounts,inanappropriatesense,tolookingattheboundaryofM-theoryas∂AdS S11−i for i × i=4,7. Compactifying the Euclidean boundary gives a product of spheres. In particular, for the M5-brane this gives S3 S7. Now the internal spaces S7 and S4 are boundaries of the 8-disk D8 and the 5-disk D5, × respectively. M-theory itself can be viewed as a boundary in twelve dimensions, so that from the point of view of this bounding theory we have spaces of the form AdS D12−i for i = 4,7. We notice that both i × factorsintheproductaremanifoldswithboundaries,andhencethe productitselfisamanifoldwithcorners of codimension-two, i.e. is a 2 -manifold. h i Theinternalspheresinthe products withAdSspacescanalsobe replacedby homogeneousspacesG/H, where G and H are Lie groups, with analogous near horizon structures [4]. In fact, general Einstein spaces M11−i, for i=4,7, with Killing spinors – and hence are Spin – can be used as well (see [12] and references therein). Thus,inordertodetectcorners,wewouldliketoaskwhetherthespacesAdS M11−icanbelifted i × to twelve dimensions. This reduces to checking whether M11−i can be boundaries. For M7 this is always the case since the relevant bordism group is trivial ΩSpin = 0; that is the Spin manifold M7 is always the 7 boundary of some eight-manifold, say W8. However, for M4 this is not the case since the bordism group is nottrivial,ΩSpin =Z. ByRohlin’stheorem,aclosedorientedSpin4-manifoldM4 isnullcobordantinΩSpin, 4 4 i.e. is the boundaryofa compactorientedSpinsmooth5-manifoldW5 ifandonly if the signatureσ(M4)of M4 vanishes. Thus the signature is a complete cobordisminvariant. The isomorphismΩSpin ∼= Z sends any cobordism class [M4] to σ(M4)/16. In particular, the Kummer surface K = z4+z4+4 z4→+z4 CP3, 4 { 1 2 3 4} ⊂ whose signature is σ(K )= 16, provides a generator for ΩSpin. We have 4 − 4 Proposition 1 (i) The near horizon limit of the M2-brane can always be described as corner for the twelve- dimenisonal bounding theory. (ii) The near horizon limit of the M5-brane can can be described as a corner for the twelve-dimensional bounding theory provided that the internal four-manifold is an Einstein space with zero signature. We will consider the M5-brane and the M2-brane themselves in section 2.2.2 and section 2.2.3, respec- tively. Examples of Spin 4-manifolds with zero signature include the 4-sphere S4, the projective space RP4 and their quotients by finite groups. Dimensional reductions of the latter type are considered e.g. in [13]. Classes of examples include ones for which the A-genus vanishes, since in four dimensions the A-genus and the Hirzebruch L-genus are related by a simple numerical factor. By the result of Atiyah-Hirzebruch the A-genus vanishes if the manifold admits a smoobth (isometric) circle action [1]. Interestingly,beven in the non-Spin case (say for us Spinc), such a result still holds [18]. The resulting theory on the orbitof the circle abctionisten-dimensionaltypeIIAstringtheory. TheM5-branewillgiverisetoatypeIIANS5-brane,which is of the same dimension, so that the dimension of the transverse space is reduced by one. Therefore, we have the nice compatibility result Proposition 2 The near horizon limit of the M5-brane can be described as a corner when M-theory is taken with a circle action, that is when the theory is related to type IIA string theory. For example, for M4 = S3 S1 this leads to type IIA string theory on AdS S3, studied e.g. in [41]. 7 × × On the other hand, for S7 the circle action gives a supersymmetric backgroundin type IIA string theory of theformAdS CP3 firstconsideredin[10]. Inthesecasestheten-dimensionalcornersare∂AdS S3 S1 4 7 × × × and ∂AdS S7, respectively. 4 × 2.2.2 The M5-brane as a tubular neighborhood Here we consider the extension of the description of the M5-brane as a tubular neighborhood to the case when Y11 has a boundary. This results, upon removing of a tubular neighborhood, in a manifold with corners of codimension-two. 6 ConsideranM5-branewithworldvolumeW6,consideredasa(closed)submanifoldinsideaclosedeleven- manifold Y11. Removing a tubular neighborhood of the M5-brane leads to a manifold with a boundary, as illustrated in [7] and used in [45]. While both of these references are concerned mainly with the case when Y11 has a boundary, that was restrictedto a closedY11 when dealing with tubular neighborhoods. Now we provide a description of the case when ∂Y11 = using the formalism in [5] [9] [23]. 6 ∅ Let ι : W6 ֒ Y11 be the embedding of the M5-brane in spacetime with normal bundle N11 W6, → → viewedasatubularneighborhoodofW6 inY11. Theunitspherebundle ofradiusr is theassociatedbundle S4 10 W6, and the corresponding disk bundle of radius r is D5 11 W6. Removing this disk → S → → D → bundle leads to an eleven-manifold Y11 =Y11 11 with boundary ∂Y11 = 10, the sphere bundle. r −D r S IfY11 isamanifoldwithboundarythentheremovalofatubularneighborhoodoftheM5-branefromY11 will result in a manifold with corners of codimension-two. Then, assuming that Y11 has multiple boundary components ∂ Y11 (i=1, ,n), W6 is a manifold with faces and becomes a manifold with boundary if we i ··· identify ∂ W6 = W6 ∂ Y11. We can interpret this as the boundary of M5-brane on the M9-brane, or the i i ∩ M5-brane in heterotic M-theory. Let us now consider the relation to type IIA string theory. For that, we assume that Y11 admits a differentiable circle action as in [8] [31] [42], and assume that the boundary ∂ Y11 is invariant under this i circle action. We would like to identify the corner in this case. The set of nonzero normal vectors v is N11 W6. ThisisacteduponbythepositivereallineR∗ = α R α>0 viamultiplicationbyapo{sit}ive scala−r: v αv. Thespherebundle 10 canbe identified+with{th∈e qu|otient(}N W6)/R∗. Weextendto the cylinder b7→undle over the sphere bunSdle 10 R= 11, which is a trivial line b−undle the+fiber at R∗v being Rv, and identify 10 with the zero sectioSn o×f 11. LCet 11 11 be the non-negativehalf of 11; an+element rv of the fiber ofS 11 over R∗v is in 11 if rC 0. If UC+is⊂anCopen neighborhood of W6 in CN11, denote by C + C+ ≥ U the inverse image of U under the canonical map 11 N11. C+ C+ → The S1-manifold with corners of codimension-two will be, as a set, the disjoint union (Y11 W6) 10. − ∪S Define a tubular neighborhood map, that is an S1-equivariant diffeomorphism T of an open S1-invariant neighborhood U of W6 in N11 onto an open neighborhood U′ of W6 in Y11 with the properties that T W6 | is the inclusion map W6 Y11 and the induced map T : N11 N11 of the normal bundle of W6 in ∗ ⊂ → U into the normal bundle of W6 in U′ is the identity map. The tubular map (see [9]) induces a map T′ : U (Y11 W6) 10, with respect to which T′ is a diffeomorphism onto a neighborhood of 10 + C → − ∪S S and which induces the given structure on (Y11 W6). Now let T be a second tubular map, thus defining 1 − a second structure on (Y11 W6) 10. Then the identity map on (Y11 W6) 10 is an isomorphism of − ∪S − ∪S the these two structures, so that (Y11 W6) 10 becomes a well-defined manifold with corners. − ∪S Let p : (Y11 W6) 10 Y11 be the natural projection, which is the identity on (Y11 W6) and − ∪S → − bundle projection on 10. Define the boundary to be ∂ ((Y11 W6) 10)=p−1(∂ Y11) for i=0,1 i i S − ∪S (Y11 W6) 10 oo ?_ ∂ (Y11 W6) 10 (2.6) i − ∪S − ∪S id×pr (cid:0) p (cid:1) (cid:15)(cid:15) (cid:15)(cid:15) Y11 oo ?_ ∂iY11 and∂ ((Y11 W6) 10)= 10. This showsthat((Y11 W6) 10) becomesanS1-manifoldwith corners 2 − ∪S S − ∪S of codimension-two. We summarize Proposition 3 The M5-brane worldvolume in an eleven-dimensional manifold with boundary is a manifold with corners, described above. 7 2.2.3 The M2-brane and boundaries Consider M-theory ona Spin eleven-manifoldY11 whichis a product of two Spin manifolds X3 M8. Take × X3 to be a three-manifoldwith a boundary ∂X3 =Σ , a Riemann surface,andM8 a closedeight-manifold. g Now take an M2-brane with boundary to wrap around X3 and identify the boundary of the M2-brane with the boundary of X3. Then we try to lift to twelve dimensions by making M8 into a boundary of a nine- dimensional manifold N9, with ∂N9 = M8. However, we cannot always perform these steps because the Spin cobordismgroupineight dimensions is notzero. Infact, ΩSpin =Z Z, generatedby the quaternionic projective plane HP2 and a generator which is one-fourth the sq8uare∼of t⊕he Kummer surface 1(K3)2. If we 4 weretoalwaysfindaSpinboundarythenwewouldconsiderΣ M8 isthecornerofthetwelve-dimensional g × manifold Z12. The latter is the product of two manifolds with boundary,namely X3 and N9, and so indeed it is a manifold with corners of codimension-two. We could also try to take Z12 to be just oriented and not necessarily Spin, and the same for M8. Then in trying to lift from M8 to N9 (again just oriented) we have to check that the obstruction in the oriented cobordism group in dimension eight is zero. In general this is not the case since Ω = Z Z, generated by the projective spaces CP4 and CP2 CP2. We then have 8 ∼ ⊕ × Proposition 4 TheM2-branewithaboundarygivesrisetoaten-dimensionalcornerinthetwelve-dimensional theory, provided an eight-dimensional zero bordism is used for the transverse space. 2.2.4 The heterotic theory as a corner ThetopologicalandglobalanalyticaspectsofM-theoryarebestdescribedusingaliftfromelevendimensions totwelvedimensions,wherethetheoryonY11 isconsideredfromthepointofviewofthetheoryonatwelve- dimensional Spin manifold Z12 bounding Y11, that is ∂Z12 =Y11 [48]. On the other hand, heterotic string theory can be considered on M-theory with a boundary, that is when Y11 itself has a boundary. The naive boundaryofaboundarydoesnotexist. However,manifoldswithcornerscometotherescue,sothatheterotic string theory can be viewed as a corner of the twelve-dimensional theory and the picture is consistent. The rest of the paper will concerned with expanding around this interpretation. In the following section we will consider analytical and geometric consequences of viewing the heterotic theory as a corner. 3 Analytical and geometric aspects of M-theory with corners The goalof this sectionis to exploreanalyticalconsequences oftaking the heterotic theory to be a cornerin the twelve-dimensionaltheory. Our discussion will mostly focus on the Dirac and signature operators,their eta-invariants, and the corresponding phase of the partition function. 3.1 Formulation using Dirac operators In this section we consider M-theory on two disconnected components, both of which form the boundary of the twelve-dimensional bounding theory, and which intersect on one corner, representing the heterotic theory. This is the opposite to the usual situation, where M-theory is taken on one component and the heterotic theory is taken on two disconnected components. The analytical constructions we apply here are very nicely surveyed in [28], to which we refer heavily throughout this section. Mass regularizations and perturbations will play an important role. The fields in heterotic string theory. Let S± denote the Spin bundles on M10. The fermionic fields in heterotic string theory consist of a gravitino ψ, which is a section of T∗M10 S+, a dilatino λ, which is a e -valued section of S+, and a gaugino χ, which is a section of S−. Here e⊗is the Lie algebra of the 8 8 Lie group E . We will work with general twisted spinors, that is with sections of S± E, where the vector 8 ⊗ 8 bundle E can be taken as the E bundle or the tangent bundle (minus appropriate number of trivial line 8 bundles) according to the context. The case with no corners. For comparison, let us briefly recall the case with no corners [14]. Consider Y11 = [0,1] M10 with the product metric, where M10 is a closed ten-dimensional Spin manifold. Then × ∂Y11 = M M , where M = M10 and M = M10 (that is, M10 with the opposite orientation). Let P± be the 0lo∪calb1oundary co1nd∼itions forthe D0ir∼ac−operatorD correspondingto spinorsin S± being zero, Y ∂Y imposed respectively on M and M . Then [14] 0 1 index(D ,P±)=index(D ), (3.1) Y M where D is the Dirac operator on M10. As explained in [14], this is the case for Horava-Witten theory M [20] [21], which we consider in more detail at the end of this section. The heterotic theory can also be viewed from the point of view of reduction of M-theory on S1/Z , where Z is an orientation-reversing 2 2 involution. More generally, let Y11 be an eleven-manifold with an orientation-reversingisometric involution τ : Y11 Y11 and with a lift τ : SY11 SY11 to the Spin bundle which anticommutes with the Dirac → → operator D and satisfies τ2 =1. Then D : S±Y11 S∓Y11, where S±Y11 are the -eigenspaces of τ. Y Y 7−→ ± When Y11 = S1 M10 with τ a reflection on the circle S1 and M10 is a compact Spin ten-manifold, then e × the same formula (3.1) holds [14]. e e The case with corners. Now consider Z12 as a compact oriented Riemannian 12-manifold with corners of codimension-two and metric gZ. Assume that Z12 has exactly two boundary hypersurfaces Y11 and Y11 1 2 that intersect in exactly one codimension-two face M10. The two hypersurfaces correspond to M-theory on two disconnected spaces. Near each hypersurface Y11, we assume that Z12 has a collar neighborhood i Z12 ∼= [0,1)si ×Yi11 where the metric is a product gZ = ds2i +giY, with giY the metric on Yi11. Then the product decomposition near each Yi11 can be taken to be Z12 ∼= [0,1)s1 ×[0,1)s2 ×M10 near the corner wherethemetricisaproductgZ =ds2+ds2+gM,withgM ametriconM10. Heres ands are,asbefore, 1 2 1 2 the coordinates on the ‘square’ over M10. In what follows we apply some of the results (surveyed) in [28]. The resulting Dirac operators on Y11 and on M10 starting from one on Z12. Let E and F be i Hermitian vector bundles over Z12 whose restrictions to Y11 are E and F , i = 1,2, respectively. The i i i restrictions to M10 are denoted E and F . What we have in mind are Spin bundles, possibly twisted by 0 0 vectorbundles,suchasanE vectorbundleorthetangentbundle. StartingwithaSpinbundleS =S+ S− on Z12, this reduces to S =8 S+ or S =S− on Y11. The choice depends on the boundary coZnditioZn⊕s. IZn Y Z Y Z turn, the restriction of SY to the ten-dimensional boundary will be SY|M10 ∼= SM ∼= SM+ ⊕SM−. The two splittings lead to local boundary conditions for the Dirac operators on Z12 and on Y11. LetD :C∞(Z12,E) C∞(Z12,F)beaDiracoperatoronZ12 whichisofproducttypeD =Γ (∂ +D ) → i si i near each hypersurface on the collar Z12 ∼= [0,1)si × Yi11, where Γi is a Dirac matrix, i.e. a unitary isomorphism from E into F and where D : C∞(Y11,E ) C∞(Y11,E ) is a (formally) self-adjoint i i i i i → i i Dirac operator on the 11-dimensional manifold with boundary Y11. Furthermore, assume that on the i product decomposition near the corner, the Dirac operator takes the form D = Γ ∂ +Γ ∂ +B where 1 s1 2 s2 B : C∞(M10,E ) C∞(M10,F ) is a Dirac operator on the ten-dimensional manifold without boundary 0 0 → M10. On the collar Z12 ∼=[0,1)s1 ×[0,1)s2 ×M10 we have Γi(∂si +Di)=Γi∂si +B, (i=1,2), so that D =Γ−1Γ ∂ +Γ−1B and D =Γ−1Γ ∂ +Γ−1B (3.2) 1 1 2 s2 1 2 2 1 s1 2 The fact that each D is (formally) self-adjoint, D∗ = D is compatible with the Clifford algebra identity i i i Γ−1Γ +Γ−1Γ = 2δ and gives the condition B∗Γ = Γ−1B( here Γ−1 = Γ∗). Then we relate the 11- i j j i ij i i dimensional operator to the 10-dimensional operator via D = Γ(∂ +D ), where Γ = Γ−1Γ and D = 1 s2 M 1 2 M Γ−1B. the operatorD is the DiracoperatoronM10 induced by D . The Dirac operatorinduced fromD 2 M 1 2 has a simple expression in relation to D and hence can be considered equivalent: D = Γ(∂ +D ), M 2 − s1 M with D =ΓD . M M e e 9