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Anomaly Cancellations on Lower-Dimensional Hypersurfaces by Inflow from the Bulk

Lower-dimensional (hyper)surfaces that can carry gauge or gauge/gravitational anomalies occur in many areas of physics: one-plus-one-dimensional boundaries or two-dimensional defect surfaces in condensed matter systems, four-dimensional brane-worlds in higher-dimensional cosmologies or various branes and orbifold planes in string or M-theory. In all cases we may have (quantum) anomalies localized on these hypersurfaces that are only cancelled by ``anomaly inflow'' from certain topological interactions in the bulk. Proper cancellation between these anomaly contributions of different origin requires a careful treatment of factors and signs. We review in some detail how these contributions occur and discuss applications in condensed matter (Quantum Hall Effect) and M-theory (five-branes and orbifold planes)

This work is dedicated to the memory of my friend and collaborator Ian Kogan. His interest in physics spanned almost all of theoretical physics. I hope the present contribution goes a little bit in this direction.

Introduction
Quantum field theories involving chiral fields coupled to gauge fields and/or gravity may have anomalies. These anomalies are a breakdown of gauge, or local Lorentz or diffeomorphism invariance respectively, at the one-loop level. More specifically, the (one-loop) quantum effective action lacks these invariances necessary for renormalizability and unitarity. In a consistent theory all these anomalies must cancel. 1 Absence of anomalies has been much used as a criterion for models in high energy physics, from the prediction of the charmed quark [1] to the choice of gauge group of the superstring [2].
Pure gravitational anomalies can only occur in 2, 6, 10, . . . dimensions while gauge or mixed gauge-gravitational anomalies are possible in all even dimensions [3]. On the other hand, quantum field theories in odd dimensions cannot be anomalous. Nevertheless, there are many interesting odd-dimensional theories that possess even-dimensional hypersurfaces with chiral matter localized on these surfaces. Typically the chirality originates either from an orbifold-like projection, or it is the property of a given solution (gravity background) with the corresponding "anti-solution" having opposite chirality.
Standard examples in eleven-dimensional M-theory [4] are the ten-dimensional orbifold planes arising from the Z 2 -projection in the Horava-Witten realization of the E 8 × E 8 heterotic string [5] and the (six-dimensional) five-brane carrying a chiral tensor multiplet, while the anti five-brane carries the same multiplet but of opposite chirality [6]. One could also mention the G 2 compactifications with conical singularities treated as boundaries of the elevendimensional space-time [7]. Other examples are 3 + 1-dimensional brane-world cosmologies with chiral matter in a 5-dimensional supergravity theory. A well-known example in condensed matter is the treatment of the 1 + 1-dimensional chiral edge currents in the 2 + 1-dimensional Quantum Hall Effect [8]. One might also consider chiral vortices, again in 2 + 1 dimensions, or two-dimensional defect surfaces in three-dimensional (Euclidean) systems.
Typically, these even-dimensional chiral "subsystems" possess one-loop anomalies. This does not contradict the fact that the original odd-dimensional theory is anomaly-free. Consider for example the effective action of eleven-dimensional M-theory. When computing the functional integral one has to sum the contributions of five-branes and of anti-five-branes and, of course, the anomalous contributions, being opposite, cancel. However, we rather like to think of M-theory within a given background with some five-branes in certain places and anti five-branes in others, maybe far away, and require local anomaly cancellation, i.e. on each (anti) five-brane separately, rather than just global cancellation. Remarkably, such local cancellation is indeed achieved by a so-called "anomaly inflow" from the bulk; M-theory has eleven-dimensional Chern-Simons like terms that are invariant in the bulk but have a anomalous variations on five-branes or on boundaries, precisely cancelling the one-loop anomalies locally [9,10].
In this paper we will explain in some generality such anomaly inflow from the bulk and how it can and does cancel the gauge and gravitational anomalies on the even-dimensional hypersurfaces. We will discuss why anomaly inflow always originates from topological terms (in odd dimensions). Usually, when discussing anomaly cancellation between different chiral fields one need not be very careful about overall common factors. Here, however, we want to consider cancellations between anomaly contributions of very different origin and special attention has to be paid to all factors and signs (see [4]). To this end, we also discuss in some detail the continuation between Euclidean and Minkowski signature, which again sheds some new light on why it must be the topological terms that lead to anomaly inflow.
In the next section, we begin by a general discussion of anomalies and anomaly inflow from the bulk, spending some time and space on the subtle continuation between Euclidean and Minkowski signature. Part of this section is just a recollection of standard results on oneloop anomalies [11] with special attention to conventions and signs. We explain how anomaly inflow uses the descent equations and why it necessarily originates from a manifold of higher dimension than the one on which the anomalous theory lives. Section 3 describes an elementary application to the (integer) Quantum Hall Effect where the effective bulk theory is a Chern-Simons theory and the boundary degrees of freedom are the chiral edge currents; anomaly cancellation by inflow from the bulk correctly explains the quantized Hall conductance. In Section 4, we describe two examples of anomaly cancellation by inflow in M-theory in quite some detail: on five-branes and on the Z 2 -orbifold planes. For the five-branes, in order to get all signs and coefficients consistent, we rederive everything from scratch: the solution itself, the modified Bianchi identity, the zero-modes and their chirality, the one-loop anomaly and the FHMM [10] mechanism for the inflow. For the orbifold planes we insist on the correct normalization of the Bianchi identity and describe the modification of the Chern-Simons term obtained in [4] necessary to get the correct inflow. Finally, in Section 5, we briefly mention analogous cancellations in brane-world scenarios.
2 One-Loop Anomalies and Anomaly Inflow

Conventions
We begin by carefully defining our conventions. They are the same as in [4]. In Minkowskian space we always use signature (−, +, . . . , +) and label the coordinates x µ , µ = 0, . . . D − 1. We always choose a right-handed coordinate system such that (With x 0 being time and for even D, this is a non-trivial statement. In particular, for even D, if we relabelled time as x 0 → x D then x 1 , . . . x D would be a left-handed coordinate system!) We define the ǫ-tensor as A p-form ω and its components are related as and its dual is We have * ( * ω) = (−) p(D−p)+1 ω and Finally we note that the components of the (p + 1)-form ξ = dω are given by (where the brackets denote anti-symmetrization with total weight one) and that the divergence We define the curvature 2-form R ab = 1 2 R ab νσ dx ν ∧ dx σ in terms of the spin-connection ω ab as R ab = dω ab + ω a c ∧ ω cb . Here a, b, c = 0, . . . D − 1 are "flat" indices, related to the "curved" ones by the D-bein e a µ . The torsion is T a = de a + ω a b ∧ e b . The Riemann tensor R µρ νσ is related to the curvature 2-form via R ab νσ = e a µ e b ρ R µρ νσ , and the Ricci tensor is R µ ν = R µρ νρ while the Ricci scalar R is given by R = R µ µ . With this sign convention, (space-like) spheres have R > 0.
For gauge theory, the gauge fields, field strength and gauge variation are given by Thus F is anti-hermitian and differs by an i from a hermitian field strength used by certain authors. 2 For gravity, one considers the spin connection ω a b as an SO(2n)-matrix valued 1-form. Similarly, the parameters ǫ a b of local Lorentz transformations (with ǫ ab = −ǫ ba ) are considered as an SO(2n)-matrix. Then (2.10) For spin-1 2 fermions the relevant Dirac operator is (E µ a is the inverse 2n-bein) (2.11)

Continuation Between Minkowski and Euclidean Signature
We now turn to the continuation to Euclidean signature. While the Minkowskian functional integral contains e iS M , the Euclidean one contains e −S E . This implies However, for a Euclidean manifold M E it is natural to index the coordinates from 1 to D, not from 0 to D − 1. One could, of course, simply write ix 0 = x 0 E ≡ x D E . The problem then is for even D = 2n that dx 0 . . x 2n−1 ) was a right-handed coordinate system then (x 1 , . . . x 2n E ) is a left-handed one. This problem is solved by shifting the indices of the coordinates as This is equivalent to a specific choice of an orientation on the Euclidean manifold M E . In Then, of course, for any tensor we similarly shift the indices, e.g.
as usual, and for a p-form In particular, we have for p = D The dual of a p-form ω E is defined as in (2.5) but using ǫ E . It then follows that * ( * ω E ) = (−) p(D−p) ω E (with an additional minus sign with respect to the Minkowski relation) and, as in It follows from the preceding discussion that the Euclidean action is not always real, not even its bosonic part. The original (real) Minkowskian action can contain two types of (locally) Lorentz invariant terms, terms that involve the metric like such as and topological (Chern-Simons type) terms that do not involve the metric such as (if D is odd) Hence the imaginary part of the Euclidean bosonic action is given by the topological terms.
Note that from now on we will not write the wedge products explicitly, but tr AF 2 will be short-hand for tr A ∧ F ∧ F , etc.
There is a further subtlety that needs to be settled when discussing the relation between the Minkowskian and Euclidean forms of the anomalies. One has to know how the chirality matrix γ is continued from the Euclidean to the Minkowskian and vice versa. This will be relevant for the 2n-dimensional submanifolds. The continuation of the γ-matrices is dictated by the continuation of the coordinates we have adopted (cf (2.13)): In accordance with Ref. [11] we define the Minkowskian and Euclidean chirality matrices γ M and γ E in 2n dimensions as Both γ M and γ E are hermitian. Taking into account (2.22) this leads to i.e. what we call positive chirality in Minkowski space is called negative chirality in Euclidean space and vice versa. This relative minus sign is somewhat unfortunate, but it is necessary to define self-dual n-forms from a pair of positive chirality spinors, both in Minkowskian space (with our convention for the ǫ-tensor) and in Euclidean space (with the conventions of [11]). 3 Indeed, as is well-known, in 2n = 4k + 2 dimensions, from a pair of spinors of the same chirality one can always construct the components of an n-form H by sandwiching n (different) 3 Since we will take [11] as the standard reference for computing anomalies in Euclidean space, we certainly want to use the same convention for γ E . On the other hand, we have somewhat more freedom to choose a sign convention for γ M . The definition (2.23) of γ M has the further advantage that in D = 10, γ M = γ 0 M . . . γ 9 M which is the usual convention used in string theory [12]. Our γ M also agrees with the definition of [13] in D = 2, 6 and 10 (but differs from it by a sign in D = 4 and 8). 2)) and it is obtained from 2 spinors ψ I (I = 1, 2) satisfying γ M ψ I = +ψ I .
In Euclidean space H E is called self-dual if (cf [11]) (with ǫ E given by (2.17)) and it is obtained from 2 spinors χ I (I = 1, 2) satisfying γ E χ I = +χ I . With these conventions a self-dual n-form in Minkowski space continues to an antiself-dual n-form in Euclidean space, and vice versa, consistent with the fact that positive chirality in Minkowski space continues to negative chirality in Euclidean space. The situation is summarized in Table 1 where each of the four entries corresponds to any of the 3 others.

Minkowskian
Euclidean spinors positive chirality negative chirality n-form self-dual anti-self-dual Table 1: Correspondences between the (anti-) self-duality of n-forms in 2n = 4k + 2 dimensions and the chirality of the corresponding pair of spinors are given, as well as their Euclidean, resp. Minkowskian continuations.
As we will recall below, the anomalies are given by topological terms with the chiralities corresponding as discussed above. One also has to remember that the continuation of the effective action Γ includes an extra factor i according to Eq. (2.12). In conclusion, the anomaly of a positive chirality spinor (or a self-dual n-form) in Minkowski space is given by δΓ M = M 2n MÎ 1 2n if in Euclidean space the anomaly of a negative chirality spinor (or an anti-self-dual n-form) is given by 2n . This will be discussed in more detail in the next subsection.

The One-Loop Anomalies
This subsection is a summary of the results of [11] where the anomalies for various chiral fields in Euclidean space were related to index theory. This whole subsection will be in Euclidean space of even dimension 2n. We first give the different relevant indices. The simplest index is that of a positive chirality spin-1 2 field. Here positive chirality means positive Euclidean chirality as defined above. For spin-1 2 fermions the relevant Euclidean Dirac operator is (cf. (2.11)) Then by the Atiyah-Singer index theorem where ch(F ) = tr exp i 2π F is the Chern character andÂ(M 2n ) is the Dirac genus of the manifold, given below. The subscript 2n indicates to pick only the part which is a 2n-form.
Note that if the gauge group is k G k , then ch(F ) is replaced by k ch(F k ).
Another important index is that of a positive chirality spin- 3 2 field. Such a field is obtained from a positive chirality spin- 1 2 field with an extra vector index by subtracting the spin- 1 2 part. An extra vector index leads to an additional factor for the index density, tr exp i 2π (2.31) The third type of field which leads to anomalies is a self-dual or anti-self-dual n-form H in 2n = 4k + 2 dimensions. Such antisymmetric tensor fields carry no charge w.r.t. the gauge group. As discussed above, a self-dual tensor can be constructed from a pair of positive chirality spinors. Correspondingly, the index isÂ(M 2n ) multiplied by tr exp i 2π 1 2 R ab T ab , where T ab = 1 2 γ ab as appropriate for the spin-1 2 representation. Note that the trace over the spinor representation gives a factor 2 n in 2n dimensions. There is also an additional factor 1 2 from the chirality projector of this second spinor and another factor 1 2 from a reality constraint (H is real), (2.32) L(M) is called the Hirzebruch polynomial, and the subscript on D A stands for "antisymmetric tensor". (Note that, whileÂ(M 2n ) tr exp i 2π 1 4 R ab γ ab carries an overall factor 2 n , L(M 2n ) has a factor 2 k in front of each 2k-form part. It is only for k = n that they coincide.) Of course, the index of a negative chirality (anti-self-dual) field is minus that of the corresponding positive chirality (self-dual) field. Explicitly one has: parameter v of the form Local Lorentz anomalies are treated analogously. Note that To avoid these complications, we will simply refer to the anomalous variation of the effective action, δ v Γ E [A] as the anomaly. So our anomaly is the negative integrated divergence of the quantum current (multiplied with the variation parameter v).
A most important result of [11] is the precise relation between the anomaly in 2n dimensions and index theorems in 2n+2 dimensions, which for the pure gauge anomaly of a positive chirality The standard descent equations dQ 1 2n = δ v Q 2n+1 and dQ 2n+1 = tr F n+1 relate Q 1 2n to the invariant polynomial tr F n+1 . Comparing with (2.33) we see that the pure gauge anomaly is thus given by . This is immediately generalized to include all gauge and local Lorentz anomalies due to all three types of chiral fields where I 2n+2 equals −2πi times the relevant index density appearing in the index theorem in 2n + 2 dimensions (corrected by a factor of − 1 2 in the case of the antisymmetric tensor field, see below). This shows that the Euclidean anomaly is purely imaginary. It is thus convenient to introduceÎ as I = −iÎ so that Explicitly we have (always for positive Euclidean chirality, respectively Euclidean self-dual Note that if A = A α λ α , B = B β λ β and tr λ α λ β = −δ αβ (the λ α are anti-hermitian) then e.g. tr AB = −A α B α and δ δA α tr AB = −B α . Hence one must define δ δA = −λ α δ δA α so that δ δA tr AB = B. Another way to see this minus sign in δ δA is to note that A α = − tr λ α A. (2.47) The last equation contains an extra factor − 1 2 with respect to the index (2.32). The minus sign takes into account the Bose rather than Fermi statistics, and the 1 2 corrects the 2 n+1 to 2 n which is the appropriate dimension of the spinor representation on M 2n while the index is computed in 2n + 2 dimensions. Note that in the cases of interest, the spin- 3 2 gravitino is not charged under the gauge group and in (2.46) the factor of ch(F ) simply equals 1. To facilitate comparison with references [12] (GSW) and [14] (FLO) we note that The flip of sign between I FLO andÎ 2n+2 is such that I 1 FLO directly gives the variation of the Minkowskian effective action for positive chirality spinors in Minkowskian space (with our definition of γ M ).
Before we go on, it is perhaps useful to look at an explicit example in four dimensions.
Consider the simple case of a spin-1 2 fermion of negative Minkowskian chirality coupled to SU(N) gauge fields. In the Euclidean, this corresponds to positive chirality and hence the anomalous variation of the Minkowskian effective action is δΓ M = Î 1 4 , whereÎ 1 4 is related via the descent equations toÎ 6 which is obtained from (2.45) aŝ (2.50) Note that this is real since by (2.9) tr F 3 is purely imaginary. Also, there is no mixed gaugegravitational anomaly since the relevant term ∼ tr R 2 tr F vanishes for SU(N) gauge fields.
It is only for U(1) gauge fields that one can get a mixed gauge-gravitational anomaly in four dimensions. Using the descent equations one explicitly gets It is important to note that we are only discussing the so-called consistent anomaly. Indeed, since our anomaly is defined as the variation of the effective action it automatically satisfies the Wess-Zumino consistency condition [15] and hence is the consistent anomaly. There is also another manifestation of the anomaly, the so-called covariant anomaly (which in the present tr vF 2 ). The latter is not relevant to us here and we will not discuss it further (see however Ref. [16]).
Finally, it is worth mentioning that the anomalies are "quantized" in the following sense: once we have normalized the gauge and gravitational fields in the usual way (so that F = dA+A 2 and R = dω + ω 2 ) the anomalies have no explicit dependence on the gauge or gravitational coupling constants. In a given theory, the total anomaly is a sum of the fixed anomalieŝ I spin 1 2 ,Î spin 3 2 andÎ A with coefficients that count the multiplicities of the corresponding fields, i.e. are integers. Of course, this came about from the relation with index theory, and there was just no place where any coupling constants could show up. Another way to see this is to recall that the anomalies are one-loop contributions to the effective action coming from exponentiating determinants. In the loop expansion of the effective action only the one-loop term is independent of the coupling constants.

Anomaly Cancellation by Inflow
We have seen that the anomalous variation of the one-loop quantum effective action is To appreciate the role of the higher-dimensional embedding, let us first remark that a (consistent) anomaly in 2n dimensions cannot be cancelled by adding to the classical invariant action a local non-invariant 2n-dimensional "counterterm" Γ with the sameÎ 2n+2 as before; the invariant polynomial is insensitive to the addition of a local counterterm.
While addition of a local counterterm cannot eliminate the anomaly, it can be used to shift between two different expressions of the "same" anomaly. Consider as an example the mixed U(1) gauge-gravitational anomaly for a negative chirality spin-1 2 fermion in four Minkowskian dimensions characterized by the invariant 6-form (2.54) Addition of the counterterm allows interpolation between the two expressions of the anomaly since . The preceding discussion shows that the anomaly cannot be cancelled by adding local terms defined on the same 2n-dimensional manifold on which live the chiral fields responsible for the anomaly. Instead, we will consider local terms defined on a higher-dimensional manifold which contains the 2n-dimensional one as a submanifold.
The simplest example is a 3-dimensional manifold M 3 whose boundary is a 2-dimensional manifold M 2 = ∂M 3 . In practice, one has to pay attention to the orientations of ∂M 3 and M 2 and be careful whether what one calls M 2 is ∂M 3 or −∂M 3 , i.e. ∂M 3 with opposite orientation.
Suppose that on M 2 lives a chiral spin- 1 2 field coupled to a gauge field A. The gauge anomaly is (for positive Euclidean chirality) where Q CS 3 is the usual Chern-Simons 3-form, obviously obeying Now suppose that the 3-dimensional Euclidean action contains a Chern-Simons term As discussed in Section 2.1, this topological term needs to be purely imaginary in order to correspond to a real term in the Minkowskian action. On the other hand, being imaginary in the Euclidean case is exactly what is needed to match the anomalous part of the effective action, as we now proceed to show. Under a gauge variation, the Chern-Simons term transforms as which would vanish if M 3 had no boundary. By Stoke's theorem we have Thus the non-invariance of the Chern-Simons term is localized on the 2-dimensional boundary manifold M 2 and, with the coefficient chosen as above, it exactly cancels the one-loop anomaly.
This is called anomaly cancellation by anomaly inflow from the bulk. This example is particularly simple as the Chern-Simons term is nothing but S CS = i Î 3 and the anomaly inflow is governed directly by the descent equations δÎ 3 = dÎ 1 2 . As an example of a somewhat different type, consider a 5-dimensional Minkowskian theory involving a U(1)-gauge field and gravity and suppose it admits solutions that are analogous to magnetic monopoles in 4 dimensions. In 5 dimensions these are magnetically charged string or vortex like solutions. Their world-volume is a 2-dimensional manifold W 2 . In the presence of such a solution, the Bianchi identity dF = 0 is modified as where α is some coefficient measuring the magnetic charge density on the string and δ W 2 is a Dirac distribution 3-form with support on the 2-dimensional world-volume W 2 . It has the Typically, on W 2 live some chiral fields. If we suppose that they carry no U(1)-charge and that there are n + positive and n − negative (Minkowskian) chirality spin-1 2 fields, there will only be a gravitational anomaly in two dimensions equal to This can again be cancelled by anomaly inflow from the bulk. Suppose there is a topological term in the 5-dimensional action involving F and the gravitational Chern-Simons 3-form, Its variation is (using again a descent relation analogous to (2.58)) and, if αβ = n − −n + 96π , this cancels the gravitational anomaly on the two-dimensional worldvolume. This second example is a very simplified version of the cancellation of the five-brane anomalies in M-theory, which will be discussed (with all its coefficients) in some detail below.
It is worthwhile to note a generic feature of anomaly inflow in the previous example. Suppose we decide to rescale the U(1)-gauge field by some factor η so that F →F = ηF . Then, the coefficient α in the Bianchi identity also gets rescaled as α →α = ηα so that it still reads dF =α δ (where nowÎ 1 2n is meant to contain all the contributions to the one-loop anomaly, with all the relevant signs and factors to take into account the different chiralities and multiplicities).
In any case, the condition for anomaly cancellation is the same in Euclidean and Minkowski signature,Î 1 2n + D (2n) = 0 . (2.68)

Anomaly Cancellation by Inflow in Condensed Matter: The Quantum Hall Effect
A most important example from condensed matter is the Quantum Hall Effect [8]. The relevant geometry of a Hall sample is two-dimensional with a one-dimensional boundary, e.g. an annulus.
Typically, the boundary has two disconnected pieces (edges) like the inner and outer boundary of the annulus. Adding time, the physics is on a 2 + 1 dimensional manifold with a 1+1 dimensional boundary.
A magnetic field B is applied perpendicular to the Hall sample and an electric field (E 1 , E 2 ) is present along the sample (usually perpendicular to the edges) resulting in a voltage drop. All this is again described by a 2+1 dimensional electromagnetic field A µ (vector potential) with field strength F µν = ∂ µ A ν − ∂ ν A µ such that (recall that our signature is (− + +)) When the filling factor (controlled by the ratio of the electron density and the magnetic field) takes values in certain intervals, one observes a vanishing longitudinal resistivity. The conductivity matrix being the inverse of the resistivity matrix, the longitudinal conductivity also vanishes and the current and electric field are related as  with an important role played by disorder (impurities) in the sample, leading to localization (see e.g. [8]). The fractional Quantum Hall Effect is more intriguing and has given rise to a large literature (which I will not cite). In both cases, effective field theories of the Chern-Simons type have played an important role, see e.g. Refs. [17,18,19,20,21,22].
Here, we will only consider a simple field theoretic model neglecting most of the subtleties discussed in the above-mentioned references, as well as in others. Consider an effective field theory given by a 2 + 1 dimensional Chern-Simons term of the electromagnetic vector potential A µ plus a coupling to the electromagnetic current j µ , The action (3.4) can be rewritten using forms (cf (2.3)) as It is well-known in the integer Quantum Hall Effect that there are chiral massless excitations on the boundaries (edge currents). They can be viewed as excitations of the incompressible two-dimensional electron gas or resulting from an interruption of the semiclassical cyclotron trajectories by the edges [8]. In any case, they are 1+1 dimensional chiral degrees of freedom.
In 1+1 dimensions it does not matter whether they are described as chiral bosons or as chiral fermions, both descriptions being related. Suppose there are n k species of them on the edge k (we label the two edges as k = 1, 2). Note that all species on a given edge have the same chirality. These chiral fermions being charged have a one-loop U(1) gauge anomaly. Recall that for U(1) gauge fields we replace A ≃ iqA and similarly for the field strength F ≃ iqF and for the gauge variation parameter v ≃ iqǫ, where q = −e is the (negative) electron charge. Then, tr v dA ≃ −e 2ǫ dA, and according to the general results of the previous section, the anomalous variation of the effective action on the k th edge is where the ± accounts for the (unspecified) chirality, 6 and we have used Eq. (2.57).
On the other hand, the bulk action S 2+1 is also anomalous due to the boundary and it gives an anomaly inflow The quantum anomalies (3.8) and the anomaly inflow (3.9) cancel if and only if σ H = (±n k ) e 2 2π . Since the anomaly should cancel on both edges k, this shows that ±n 1 = ±n 2 ≡ n and σ H = n e 2 2π , n ∈ Z (3.10) in agreement with Eq. (3.3). Anomaly cancellation by inflow from the bulk forces the Hall conductivity to be correctly quantized! As already noted, the fractional Quantum Hall Effect is more complicated and the edge excitations are described by more complicated quasiparticles involving exotic spins and statistics, so that our simple argument needs to be refined. Somewhat related arguments can be found in [20,21,22]. Other examples in 1+1 dimensional condensed matter where anomaly arguments play a role are quantum wires [22] and presumably vortices, as well as defect surfaces in 3-dimensional Euclidean statistical systems. Due to lack of competence, I will discuss none of them here. 6 One should also be careful about the orientations of M 2+1 and of the edges M k 1+1 to get the signs straight.

The low-energy effective action of M-Theory
M-theory has emerged from a web of dualities between superstring theories. In its elevendimensional uncompactified version it can be considered as the strong-coupling limit of type IIA superstring theory. This tells us that its low-energy effective action is that of elevendimensional supergravity first written by Cremmer, Julia and Scherk [23]. In Minkowski space its bosonic part reads (using our conventions as exposed in Section 2.1) where κ ≡ κ 11 is the 11-dimensional gravitational constant, R is the Ricci scalar, and G = dC.
The coefficients of the second and third term in this action can be changed by rescaling the C-field. Also, some authors use a different relation between G and dC. These issues have been extensively discussed in [4] where a table is given summarizing the conventions of various authors. Here, however, we will use the simple choice made in Eq. (4.1) which in the notation of Ref. [4] corresponds to α = β = 1. Note that the third term is a topological term, usually referred to as the Chern-Simons term.
The C-field equation of motion is or in components and the Einstein equations are Just as superstring theory possesses various D-branes, M-theory has two fundamental branes: membranes (2-branes) and 5-branes. Also, the low energy-effective action (4.1) certainly does receive higher-order corrections. Note that in eleven-dimensional supergravity there is no parameter besides the gravitational constant κ, and higher order necessarily means higher order in κ. The first such term is the famous Green-Schwarz term, initially inferred from considerations of anomaly cancellation on 5-branes by inflow [9,24]. It reads (in Minkowski space) where we assumed that one can freely integrate by parts (no boundaries or singularities), and where X 8 = dX 7 = 1 (2π) 3 Here T 2 is shorthand for and is interpreted as the membrane tension. The parameter ǫ can be fixed by various considerations of anomaly cancellation as we will show below. Since there have been some ambiguities in the literature we will keep ǫ as a parameter and show that all anomalies considered below cancel if and only if ǫ = +1 . The Euclidean continuations of the action (4.1) and the Green-Schwarz term are and (4.10)

The M-Theory Five-Brane
The 5-brane and anti-5-brane are solutions of 11-dimensional supergravity that preserve half of the 32 supersymmetries. The metric is a warped metric preserving Poincaré invariance on the (5 + 1)-dimensional world-volume (for flat 5-branes) and the 4-form G has a non-vanishing flux through any 4-sphere surrounding the world-volume. This is why the 5-branes are called "magnetic" sources. It will be enough for us to exhibit the bosonic fields only.
Although the original 11-dimensional supergravity is non-chiral, the 5-brane is a chiral solution; it carries a chiral (5 + 1)-dimensional supermultiplet which gives rise to anomalies.
Of course, the anti-5-brane carries the supermultiplet of opposite chirality. As a result, when computing an "M-theory functional integral" one has to sum over classical solutions of opposite chirality and the overall result is correctly non-chiral. However, we like to adopt a more modest view and consider M-theory in a given background with some number of 5-branes somewhere and some other number of anti-5-branes somewhere else. Then the anomalies cannot cancel between the different branes and anomaly cancellation must occur for each 5-brane or anti-5brane separately. This will be achieved by anomaly inflow from the two topological terms, the Chern-Simons and the Green-Schwarz term.
It is not too difficult to determine the nature of the chiral 6-dimensional supermultiplet living on the world-volume of a 5-brane [25]. What requires some more care is to correctly determine its chirality. We will see that the 5-brane acts as a "magnetic" source for the C-field leading to a modification of the Bianchi identity dG = 0. This is at the origin of anomaly inflow from the Green-Schwarz term [9] and similar to the mechanism outlined in Section 2.4 for the magnetic string. However, it was noticed [6] that there is a left-over "normal bundle" anomaly which is only canceled by further inflow from the (slightly modified) Chern-Simons term [10]. In principal, this should have fixed the coefficient ǫ of the Green-Schwarz term. In the literature one can find about as many times ǫ = +1 as ǫ = −1 (after eliminating the effect of using different conventions). This was the motivation in [4] to redo the whole computation from first principles. Here we will outline this computation again, with the result ǫ = +1.

The classical 5-brane solution
We work in Minkowski space and split the coordinates into longitudinal ones x α , α = 0, . .  [4], where one can also find a discussion of how things change under a rescaling of the C-field. The 4-form corresponding to (4.13) is ǫ mnpqs y s r 5 dy m ∧ dy n ∧ dy p ∧ dy q (4.14) and for any 4-sphere in the transverse space surrounding the world-volume we have the "magnetic charge" Hence, for the 5-brane the flux of G is positive and for the anti-5-brane it is negative.
The parameter r 0 sets the scale for the (anti-) 5-brane solution. One can compute the energy per 5-volume of the brane, i.e. the 5-brane tension T 5 as a function of r 0 . Using the Dirac quantization condition between membranes and 5-branes then relates the membrane tension T 2 and the 5-brane tension T 5 as T 2 T 5 = 2π 2κ 2 so that in the end 8π 2 r 3 0 = 2π T 2 , see [4] for details. (Recall from (4.7) that T 2 = (2π 2 /κ 2 ) 1/3 .) It follows that Eq. (4.15) can be rewritten This is equivalent to the modified Bianchi identity dG = ± 2π T 2 δ (5) where again the upper sign (+) applies for a 5-brane and the lower sign (−) for an anti-5brane. δ (5) W 6 is a 5-form Dirac distribution with support on the world-volume W 6 such that (6) . To summarize, the 5-brane and anti-5-brane solutions both have a metric given by (4.11).
The 4-form G is given by (4.14) and satisfies the Bianchi identity (4.17). The upper sign always corresponds to 5-branes and the lower sign to anti-5-branes.

The zero-modes
The (massless) fields that live on a five-brane are the zero-modes of the equations of motion in the background of the 5-brane solution. Hence, to determine them, we will consider the zero-modes of the bosonic equations of motion in this 5-brane background. The fermionic zeromodes then are simply inferred from the completion of the supermultiplet. The anti-5-brane background can be treated similarly (flipping signs in appropriate places).
Apart from fluctuations describing the position of the 5-brane, there are zero-modes of the C-field. A zero-mode is a square-integrable fluctuation δG = dδC around the 5-brane solution G 0 (given by (4.13) or (4.14) with the upper sign) such that G = G 0 + δG still is a solution of (4.3) or (4.2). Of course, G must also solve the Einstein equations to first order in δG. This will be the case with the same metric if the r.h.s. of (4.4) has no term linear in δG.
As mentioned above, the zero-modes must be square-integrable, The r-integral converges if and only if ζ > 0. Thus square-integrability selects ζ = +1 and, hence, H = dB is a self-dual 3-form on the world-volume.
To summarize, in Minkowski signature, on a 5-brane, there is a self-dual 3-form H (which continues to an anti-self-dual Euclidean 3-form H E ), while on an anti-5-brane the 3-form H is anti-self-dual (and continues to a self-dual Euclidean 3-form H E ). To complete the 6dimensional supermultiplets, we know that the self-dual 3-form is accompanied by two spinors of positive chirality, and the anti-self-dual 3-form by two spinors of negative chirality. We note that the same discussion can be equally well carried out entirely in the Euclidean case (see [4]), with the same result, of course.

The tangent and normal bundle anomalies
Now that we have determined the nature and chiralities of the fields living on the 5-brane world-volume, it is easy to determine the one-loop anomaly, using the results of Section 2.3.
For the Euclidean 5-brane we have an anti-self-dual 3-form and two negative chirality spinors.
While the 3-form cannot couple to gauge fields, the spinors couple to the "SO(5)-gauge" fields of the normal bundle. This coupling occurs via inherited from the eleven-dimensional spinor. Here a, b and i run from 1 to 6, while p, q = 7, . . . 11. Thus ω pq,i behaves as an SO(5)-gauge field A α i with generators λ α ∼ 1 2 γ pq . We see that the relevant SO(5) representation is the spin representation [6] and hence (R pq = dω pq + ω pr ω rq ≡ R ⊥ pq ) This trace appeared already in (2.32), except that there R ab was the curvature on the manifold (i.e. on the tangent bundle). One has ch(S(N)) = 4 1− 1 (4π) 2 The relevant anomaly polynomial includes an extra factor 1 2 from a chirality projector (as in (2.32)) as well as a minus sign for negative chirality. It is (R =R + R ⊥ ) The part not involving R ⊥ is just witĥ where X 8 is given in (4.6) (now with R →R) and The part −X 8 (R) is called the tangent bundle anomaly and −Î normal 8 the normal bundle anomaly.

Anomaly inflow from the Green-Schwarz and Chern-Simons terms
In this subsection we return to Minkowski space. As we have seen, the 5-brane has chiral zero-modes on its 6-dimensional world-volume with its Minkowski anomaly given by The tangent bundle anomaly −X 8 (R) is cancelled [9] through inflow from the Green-Schwarz term ∼ G ∧ X 7 (R). The latter, however, gives X 8 (R) = X 8 (R + R ⊥ ), not X 8 (R). The difference, as well as the normal bundle anomaly is cancelled through inflow from the Chern-Simons term as was shown in [6,10]. As a result, cancellation of the total 5-brane anomaly fixes both coefficients of the Green-Schwarz and Chern-Simons terms. In particular, it establishes a correlation between the two coefficients. Moreover, as we will see, cancellation can only occur if the sign of the anomaly due to the five-brane zero-modes is exactly as in (4.31), (4.29).
Let us first consider the simpler inflow from the Green-Schwarz term (4.5) in the form G ∧ X 7 . Using the Bianchi identity (4.17) we get where, as already noted, X 1 6 is X 1 6 (R). This corresponds via descent to an invariant polynomial Next, inflow from the Chern-Simons term is more subtle. We review the computation of [10], again paying particular attention to issues of signs and orientation. The two key points in [10] are: (i) the regularization δ (5) where ρ(r) rises monotonically from −1 at r = 0 to 0 at some finite distancer from the 5brane, and e 4 = de 3 is a certain angular form with S 4 e 4 2 = 1; and (ii) a modification of the Chern-Simons term close to the 5-brane, where G = dC.
The regularized Bianchi identity reads which is solved by (requiring regularity at r = 0 where e 4 is singular) Under a local Lorentz transformation, δe 3 = de 1 2 , and G is invariant if δC = 0 and δB = 1 2 e 1 2 . Note that [10] include the dρ ∧ B-term in C and hence get a non-trivial transformation for C.

If we let G = d C then the modified Chern-Simons term is
where M 11 \D ǫ W 6 is M 11 with a small "tubular" region of radius ǫ around the 5-brane worldvolume cut out. (Of course, this radius ǫ should not be confused with the ǫ which is the coefficient of the Green-Schwarz term.) Its boundary is where S ǫ W 6 is the 4-sphere bundle over W 6 . Note the minus sign that appears since the orientation of the boundary is opposite to that of the sphere bundle.
Under a local Lorentz transformation G and hence G are invariant and Inserting this variation into (4.37), and using d G = 0 one picks up a boundary contribution 8 In G = dC − π T 2 (dρ ∧ e 3 + ρ e 4 − 2dρ ∧ dB) the terms ∼ dρ cannot contribute to an integral over S ǫ W 6 . Also the contribution of the dC-terms vanishes in the limit ǫ → 0. Hence the only contribution comes from [10,26] SǫW 6 e 1 2 ∧ e 4 ∧ e 4 = 2 where p 2 (NW 6 ) 1 is related via descent to the second Pontrjagin class p 2 (NW 6 ) of the normal bundle given below. Using ρ(0) = −1 and (4.7) we arrive at  Using π 12 p 2 (NW 6 ) = 1 (2π) 3 we find that the total inflow corresponds tô Now it is easy to study anomaly cancellation. Invariance of the full quantum effective action requires that the sum of (4.29) and (4.45) vanishes. This gives four equations . It is quite amazing to see that anomaly cancellation requires four different terms to vanish, and they all do if the single coefficient ǫ is chosen as above. Note also that a rescaling of the C-field changes the coefficients of the Chern-Simons and Green-Schwarz terms, but cannot change the relative sign between them. The effect of such rescalings has been carefully traced through the computations in ref [4] where it can be seen that the resulting equations (4.46) are indeed invariant under these rescalings, as they should.
It is also interesting to note that the four conditions (4.46) for anomaly cancellation have enough structure to provide a check that we correctly computed the sign of the one-loop anomaly (if we believe that the anomaly must cancel). Suppose we replaced equation (4.29) bŷ

M -Theory on S 1 /Z 2 : the Strongly-Coupled Heterotic String
While compactification of M-theory on a circle S 1 leads to (strongly-coupled) type IIA superstring theory, compactification on an interval gives the strongly-coupled heterotic string [5].
There are two ways to view this latter compactification. On the one hand, one considers the compactification manifold as being ten-dimensional Minkowski space M 10 times the interval so that the 11-dimensional space-time has two boundaries, each of which is a copy of M 10 . This is called the "downstairs approach". On the other hand, the interval being S 1 /Z 2 , one may start with the 11-dimensional manifold being M 10 × S 1 and then perform the Z 2 orbifold projection.
In this case there are no boundaries, but two orbifold fixed-planes, each of which is again a copy of M 10 . This is called the "upstairs approach".
One may also consider more complicated compactifications on orbifolds like e.g. T 5 /Z 2 with many intersecting orbifold planes. The latter constructions have given rise to some model building, see e.g. [27].
Here we will work in the upstairs approach. As argued in [5] the orbifold projection eliminated half of the supersymmetry leaving only one chiral (ten-dimensional) gravitino on each of the ten-dimensional orbifold planes. This leads to a gravitational anomaly with an irreducible R 6 piece. The latter piece can be cancelled by adding E 8 gauge fields on each of the orbifold planes (interpreted as "twisted" matter). The total one-loop anomaly then no longer has this R 6 piece and, remarkably, has a factorized form on each of the planes, a necessary condition for anomaly cancellation by inflow from the Green-Schwarz and Chern-Simons terms. There has been a long series of papers discussing this cancellation that culminated with Ref. [28], each paper correcting some errors of the preceding ones. However, this was not the end of the story, since one of the authors of [28] realized that there was still an unnoticed numerical error, and to correctly obtain complete anomaly cancellation requires a slight modification of the Chern-Simons term in the vicinity of the orbifold planes, quite similar to what happened for the 5-brane as discussed above. This was reported in [4] and we will review these results in this subsection. The attitude taken in [4] was to show that anomaly cancellation in this case again determines the value of ǫ to be +1. Here, instead, we will consider that the coefficient of the Green-Schwarz-term is already fixed from the 5-brane anomaly cancellation and that with this value we correctly obtain anomaly cancellation also in the present case.

The one-loop anomalies on the orbifold 10-planes
As always in Minkowski signature, we label the coordinates as x µ , µ = 0, . . . 10. Here we will distinguish the circle coordinate x 10 ∈ [−πr 0 , πr 0 ] from the other xμ,μ = 0, . . . 9. The Z 2 -projection then acts as x 10 → −x 10 . As one can see from the Chern-Simons term, Cμνρ is Z 2 -odd and Cμν 10 is Z 2 -even (μ,ν,ρ = 0, . . . 9). The projection on Z 2 -even fields then implies e.g. that and all other components of C projected out. Also, this Z 2 -projection only leaves half of the components of the eleven-dimensional gravitino [5]. What remains is a ten-dimensional gravitino of positive chirality (in Minkowskian space), together with one negative chirality spin-1 2 field. Of course, in Euclidean space, this corresponds to one negative chirality spin-3 2 and a positive chirality spin-1 2 fermion. The 1-loop anomaly due to the eleven-dimensional gravitino on each 10-plane M A 10 , A = 1, 2 is thus given bŷ where one factor 1 2 is due to the Majorana condition and the other factor 1 2 due to the "splitting" of the anomaly between the two fixed planes [5]. R A denotes the curvature two-form on M A 10 which simply is the eleven-dimensional curvature R with its components tangent to S 1 suppressed. As is well known, such a polynomial has a tr R 6 -piece, and one must add an E 8 vector multiplet in the adjoint representation (Tr 1 = 248) with positive chirality (Minkowskian) Majorana spinors on each 10-plane. Then on each plane M A 10 one has a 1-loop anomaly corresponding toÎ thanks to the algebraic identity a 3 + b 3 = (a + b)(a 2 + b 2 − ab). Here X 8 is the relevant 8-form that appears in the anomaly-cancelling term of the heterotic string, (4.54)

Anomaly inflow and anomaly cancellation
To begin with, there is a slight subtlety concerning the coefficients of the Chern-Simons and Green-Schwarz terms in the upstairs formalism. To see this, we start in the downstairs formalism where S CS and S GS are given by integrals over an honest manifold with boundary which is M 10 times the interval I = S 1 /Z 2 . Then clearly the coefficients must be those given in the preceding subsections, Here κ is the eleven-dimensional κ as before. This can be rewritten in the upstairs formalism by replacing I . . . = 1 2 S 1 . . . and appropriately identifying the fields so that the integrand is Z 2 -even. This introduces an extra 1 2 in the coefficients. It is nevertheless customary to absorb this 1 2 in a redefinition of κ as Then one has and the Chern-Simons term looks conventionally normalized. However, due to the different dependence on κ, the Green-Schwarz term, when written in the upstairs formalism, has an extra factor of 2 −2/3 . This will be important later on.
The factorized form (4.51) of the anomaly on each ten-plane is a necessary condition to allow for local cancellation through inflow. Clearly, the I 4,A X 8 -term has the right form to be cancelled through inflow from the Green-Schwarz term, provided G satisfies a modified Bianchi identity dG ∼ A=1,2 δ A ∧ I 4,A , where δ A is a one-form Dirac distribution such that M 10 ×S 1 ξ (10) ∧ δ A = M A 10 ξ (10) for any 10-form ξ (10) . This is equivalent to prescribing a boundary value for G on the boundary planes in the down-stairs approach. Such a modified Bianchi identity is indeed necessary to maintain supersymmetry in the coupled 11-dimensional supergravity/10-dimensional super-Yang-Mills system [5]. In principle, this allows us to deduce the coefficient −ζ on the right-hand side of the Bianchi identity in the upstairs approach. It is given by −(4π) 2 κ 2 U λ 2 where λ is the (unknown) Yang-Mills coupling constant. Hence, we start with a Bianchi identity [5] (4.58) The variation of the Green-Schwarz term then is (recall δX 7 = dX 1 6 ) δS GS corresponds to an invariant polynomial As promised, this cancels the part of the anomaly (4.51) involving X 8 . Moreover, this cancellation is local, i.e. cancellation occurs on each plane separately. We see that anomaly cancellation fixes the value of ζ to be (4.60), thereby determining the value of the 10-dimensional Yang-Mills coupling λ in terms of the 11-dimensional gravitational coupling κ. Although this latter aspect has drawn some attention, one has to realize that the more interesting relation between λ and the 10-dimensional κ 10 involves the (unknown) radius r 0 of the circle, similarly to the relation between the type IIA string coupling constant and κ.
To study anomaly inflow from the Chern-Simons term we have to solve the Bianchi identity for G (as we did for the 5-brane). This involves several subtleties, discussed at length in [28].
One important point was to respect periodicity in the circle coordinate x 10 ∈ [−πr 0 , πr 0 ] which led to the introduction of two periodic Z 2 -odd "step" functions ǫ A (x 10 ) such that ǫ 1 (x 10 ) = sgn(x 10 ) − x 10 πr 0 and ǫ 2 (x 10 ) = ǫ 1 (x 10 ± πr 0 ). They satisfy Regularizing ǫ A (and hence δ A ) properly gives since their difference is a total derivative (ω 3,A is given in terms of the Chern-Simons forms on M A 10 and has no dx 10 component). This introduces an arbitrary real parameter b into the solution, Since G appears in the kinetic term ∼ G ∧ * G, as well as in the energy-momentum tensor, it must be gauge and local Lorentz invariant, δG = 0. This is achieved if [28] In [28] several arguments were given in favor of one particular value of b, namely b = 1, since only then G is globally well-defined. Furthermore, the higher Fourier modes of Cμν 10 are gauge invariant only for this value of b, which is a necessary condition for a safe truncation to the perturbative heterotic string. Last, but not least, it is only for b = 1 that G has no terms involving δ A which would lead to divergent pieces in the kinetic term G ∧ * G. Nevertheless, we will keep this parameter b for the time being and show in the end that anomaly cancellation also requires b = 1.
Note that, although G = dC, we still have G = d C as long as we stay away from the fixed planes. This motivates us to introduce a modified Chern-Simons term similar to what was done in Section 5 for the 5-brane or in [4] when discussing M-theory on singular G 2 -manifolds.
We take which away from the fixed planes is just ∼ C ∧ d C ∧ d C. Then (4.68) Note that we can freely integrate by parts (we assume that M 10 has no boundary). Furthermore, since both δ A and dC = dB ∧ dx 10 always contain a dx 10 , on the r.h.s of Eq. The modified Chern-Simons term contributes three terms ǫ ǫ δ. This factor of 3 was absent in [28] where inflow from the unmodified Chern-Simons term was computed. Also the result of [28] was obtained only after using S 1 dx 10 ǫ A ǫ B = πr 0 (δ AB − 1 3 ) which somewhat obscured the local character of anomaly cancellation. Now, however, due to the explicit δ A one-forms, the inflow from S CS is localized on the 10-planes M A 10 . Using (4.63) we find  As already mentioned there are many other arguments in favor of b = 1, but now we can conclude that also anomaly cancellation on S 1 /Z 2 requires b = 1, as argued in [28]. 10 Thus we have shown that all the anomalies are cancelled locally through inflow from the Green-Schwarz 11 and (modified) Chern-Simons terms with exactly the same coefficients as already selected from cancellation of the 5-brane anomalies.

Small radius limit and the heterotic anomaly cancelling term
Finally, it is easy to show that in the small radius limit (r 0 → 0) the sum S GS + S CS exactly reproduces the heterotic Green-Schwarz term. In this limit X 8 (R) and X 7 (R) are independent of x 10 and have no dx 10 components. From C =B ∧ dx 10 and δC given in (4.66) we identify the correctly normalized heterotic B-field as the zero mode ofB times (4π) 2 ζ 2πr 0 , where ω 1 2,Y M and ω 1 2,L are related to tr F 2 1 + tr F 2 2 and tr R 2 via descent. Next, using (4.65) and (4.60), the Green-Schwarz term (4.57) gives in the small radius limit (ω 3,Y M − ω 3,L ) ∧ X 7 . 10 In [28] inflow from the unmodified Chern-Simons term was computed. This is three times smaller than (4.70). Also the factor 2 2/3 in ζ was missing, so that the overall inflow δS CS appeared 12 times smaller. This discrepancy remained unnoticed since the anomaly cancellation condition was expressed as (4π) 5 κ 4 b 2  Mills theory with a global R-symmetry SU(4) which is anomalous. In this case, however, the non-invariance of the 5-dimensional Chern-Simons term does not provide an anomaly cancelling inflow, but explains the global SU(4) anomaly of the CFT (see e.g. [29]). The mathematics is the same, but its interpretation is different. In brane world scenarios, on the other hand, anomaly cancellation may be a valuable constraint. 12 In order to facilitate comparison with [12] we note thatX 8 = 1 (2π) 3 4! X GSW

8
, and S het as given in (4.78) exactly equals minus the expression given in [12]. The missing minus sign in [12] is due to a sign error related to the subtle issues of orientation, and is corrected e.g. when using the anomaly polynomials as given in [14].