Anomaly Inflow and p-Form Gauge Theories

Abstract

Chiral and non-chiral p-form gauge fields have gravitational anomalies and anomalies of Green-Schwarz type. This means that they are most naturally realized as the boundary modes of bulk topological phases in one higher dimensions. We give a systematic description of the total bulk-boundary system which is analogous to the realization of a chiral fermion on the boundary of a massive fermion. The anomaly of the boundary theory is given by the partition function of the bulk theory, which we explicitly compute in terms of the Atiyah–Patodi–Singer $\eta$-invariant. We use our formalism to determine the $\mathrm {SL}(2,{\mathbb {Z}})$ anomaly of the 4d Maxwell theory. We also apply it to study the worldvolume theories of a single D-brane and an M5-brane in the presence of orientifolds, orbifolds, and S-folds in string, M, and F theories. In an appendix we also describe a simple class of non-unitary invertible topological theories whose partition function is not a bordism invariant, illustrating the necessity of the unitarity condition in the cobordism classification of the invertible phases.

Appendices

Notations and Conventions

• ${\mathsf i}= \sqrt{-1}$ : the imaginary unit.

• $\mathrm{d}$: the exterior differential.

• X, Y, Z, M, N, ... : generic symbols for manifolds. (For general discussions, the dimensions are $\dim X =d$, $\dim Y = d+1$, $\dim Z = d+2$.)

• $\overline{X}$ : the orientation reversal of a manifold X.

• $H^p(X,{\mathbb {A}})$ : the cohomology on X with coefficients ${\mathbb {A}}(={\mathbb {Z}}, {\mathbb {R}}, {\mathbb {R}}/{\mathbb {Z}})$. Coefficients with a tilde, such as $\widetilde{{\mathbb {A}}}$, stand for twisted coefficient systems.

• $\Omega ^p(X)$ : differential forms of degree p, i.e. a p-form.

• $\Omega _\mathrm{closed}^p(X)$ : closed differential p-forms.

• Square bracket $[x ]_{\mathbb {A}}$ : the cohomology element corresponding to a cocycle x with coefficients ${\mathbb {A}}$. (The subscript ${\mathbb {A}}$ may be omitted if it is clear from context.)

• $\check{H}^p(X)$ : the differential cohomology group on X.

• A, B, C, a, b, c, ... : generic symbols for gauge fields as used usually by physicists.

• $\check{A}, \check{B},\check{C}, \check{a}, \check{b}, \check{c} ...$ : generic symbols for gauge fields as differential cohomology elements.

• $\check{A} = ({\mathsf {N}}_A, {\mathsf {A}}_A, {\mathsf {F}}_A)$ : the triplet representation of a differential cohomology element $\check{A}$.

• $\mathcal{{Z}}$ : the partition function.

• $\mathcal{{A}}\in {\mathbb {R}}/{\mathbb {Z}}$ : the phase $=\frac{1}{2\pi {\mathsf i}} \log \mathcal{{Z}}$ of the partition function $\mathcal{{Z}}$ of the bulk anomaly theory on closed manifolds. We simply call this $\mathcal{{A}}$ as the anomaly.

• $\mathcal{{Q}}\in {\mathbb {R}}/{\mathbb {Z}}$ : a quadratic refinement, possibly with $\mathcal{{Q}}(0)\ne 0$. We also use .

• : the APS $\eta$-invariant for a Dirac operator .

• : the index of a Dirac operator $\mathcal{{D}}$.

• $\text {spin-}G$: a tangential structure on a manifold X such that the tangent frame bundle $\mathrm {SO}(\dim X)$ is uplifted to $[\mathrm {Spin}(\dim X) \times G]/{\mathbb {Z}}_2$, for a group G. A choice of an injection ${\mathbb {Z}}_2 \rightarrow G$ is assumed.

• $\Gamma ^I$: gamma matrices. $\Gamma ^{I_1 I_2 \cdots I_m} = \frac{1}{m!} \sum _{\sigma } \mathop {\mathrm {sign}}(\sigma )\Gamma ^{I_{\sigma (1)}} \cdots \Gamma ^{I_{\sigma (m)}}$, where the sum is over all permutations $\sigma$.

• $\overline{\Gamma }$: the chirality operator or the ${\mathbb {Z}}_2$ grading on spin bundles (or Clifford modules).

We also have some remarks related to index theorems:

• In even spacetime dimensions 2m, the gamma matrices $\Gamma ^1, \cdots , \Gamma ^{2m}$ and the chirality operator $\overline{\Gamma }$ are related as $\overline{\Gamma }= {\mathsf i}^{-m} \overline{\Gamma }^1 \cdots \overline{\Gamma }^{2m}$. In odd dimensions $2m+1$, the gamma matrices are usually taken as ${\mathsf i}^{-m} \Gamma ^1 \cdots \Gamma ^{2m+1} = +1$. When the bundle with opposite representation ${\mathsf i}^{-m} \Gamma ^1 \cdots \Gamma ^{2m+1} = -1$ appears, we explicitly mention it.

• $\partial Y$, the boundary of a manifold Y, is taken with the orientation given by the following convention. If the neighborhood of the boundary has the form $(-\epsilon , 0] \times X \subset Y$ , then the oriented volume forms $\omega _Y$ and $\omega _X$ of Y and $X = \partial Y$ are related as $\omega _Y = \mathrm{d}\tau \wedge \omega _X$, where $\tau \in (-\epsilon , 0]$ and $\tau =0$ is the boundary. This is the standard orientation for Stokes’ theorem, but it is different from the usual convention for the APS index theorem in the literature. This leads to a sign change in the APS index theorem in front of the $\eta$-invariant. Namely, the APS index theorem on a manifold Z with boundary Y is of the form

  

  (A.1)

• In the conventions of the gamma matrices and the $\eta$-invariant used above, the anomaly of a chiral fermion with positive chirality $\overline{\Gamma }=+1$ is given by $\mathcal{{A}}= - \eta$ for a Dirac fermion (i.e. the bulk partition function is $\mathcal{{Z}}= \exp (-2\pi {\mathsf i}\eta )$), and $\mathcal{{A}}= - \frac{1}{2}\eta$ for a Majorana fermion (i.e. the bulk partition function is $\mathcal{{Z}}= \exp (-\pi {\mathsf i}\eta )$). For negative chirality fermions, the sign is reversed.

Some Sign Factors in M-Theory

The purpose of this appendix is to fix various sign factors which appear in M-theory. In particular, let $F^\mathrm{M}_4$ and $F^\mathrm{M}_7$ be M-theory 4-form and 7-form field strength. We will determine the sign s in the duality equation

$$\begin{aligned} *F^\mathrm{M}_4 = s {\mathsf i}F^\mathrm{M}_7. \end{aligned}$$

(B.1)

We want to determine whether it is $s=1$ or $-1$. After fixing some conventions of the Mp-branes and the fields $F^\mathrm{M}_{p+2}$, the value of s is not a convention, but is fixed.

In this appendix, we are only concerned with sign factors, and hence we neglect topology of $p+1$-form fields $C^\mathrm{M}_{p+1}$ and treat them as differential forms. In particular, $F^\mathrm{M}_{p+2} = \mathrm{d}C^\mathrm{M}_{p+1}$.

1.1 Convention of Mp-branes and $(p+1)$-form fields

We always use Euclidean signature for the metric unless otherwise stated. For gamma matrices, we take $\Gamma ^0$ to be imaginary antisymmetric and other $\Gamma ^I~(I=1,\cdots ,10)$ to be real symmetric. (In Lorentzian signature, all gamma matrices are real.) Then we see that the matrix

$$\begin{aligned} C={\mathsf i}\Gamma ^0 \end{aligned}$$

(B.2)

has the properties that

$$\begin{aligned} C \Gamma ^I C^{-1} = -(\Gamma ^I)^T = - (\Gamma ^I)^*~~~(I=0,1,\cdots , 10). \end{aligned}$$

(B.3)

Moreover, we take them to satisfy

$$\begin{aligned} {\mathsf i}^{-5} \Gamma ^0 \Gamma ^1 \cdots \Gamma ^{10}=1. \end{aligned}$$

(B.4)

We use these conventions for the gamma matrices.

For M5 and M2 branes, we use the following conventions. Let $Q^\mathrm{M}$ be the supercharge in 11-dimensions. An Mp-brane preserves the subgroups $\mathrm {SO}(p+1) \times \mathrm {SO}(10-p) \subset \mathrm {SO}(11)$ of the Lorentz symmetry and half of the supersymmetry. Then, if we put it on $x^{p+1}= \cdots x^{10}=0$ with the orientation determined by the volume form $\omega _{p+1}=\mathrm{d}x^ 0 \wedge \cdots \wedge \mathrm{d}x^p$, it is clear that the supercharges preserved by the Mp-brane should be given by

$$\begin{aligned} Q^{ \mathrm{M}p} = (1 \pm \Gamma ^{10} \Gamma ^{9} \cdots \Gamma ^{p+1} ) Q^\mathrm{M}. \end{aligned}$$

(B.5)

Here the ambiguity is only sign factors ± and not a general complex phase, because $Q^\mathrm{M}$ in Lorentz signature metric is real. The sign just specifies which we call as Mp-branes and which as anti-Mp-branes. We use the convention that Mp-branes (as opposed to anti-Mp-branes) with the worldvolume orientation

$$\begin{aligned} \omega _{p+1}=\mathrm{d}x^ 0 \wedge \cdots \wedge \mathrm{d}x^p \text { : positive volume form} \end{aligned}$$

(B.6)

are specified by the unbroken supercharges

$$\begin{aligned} Q^{ \mathrm{M}p} = (1 + \Gamma ^{10} \Gamma ^{9} \cdots \Gamma ^{p+1} ) Q^\mathrm{M}, \end{aligned}$$

(B.7)

or more explicitly

$$\begin{aligned} \mathrm{M}5&: Q^\mathrm{M5}=(1 + \Gamma ^{10}\Gamma ^{9}\Gamma ^{8}\Gamma ^{7}\Gamma ^{6})Q^\mathrm{M}, \end{aligned}$$

(B.8)

$$\begin{aligned} \mathrm{M}2&: Q^\mathrm{M2}= (1 + \Gamma ^{10} \Gamma ^{9} \Gamma ^{8}\Gamma ^{7}\Gamma ^{6} \Gamma ^{5} \Gamma ^{4} \Gamma ^{3}) Q^\mathrm{M}. \end{aligned}$$

(B.9)

These are just conventions.

Let us slightly rephrase the above conditions. The supersymmetry transformations are written as

$$\begin{aligned} \epsilon ^T C Q^\mathrm{M} \end{aligned}$$

(B.10)

where $\epsilon$ is the supersymmetry parameter, and $C (={\mathsf i}\Gamma _0)$ is the matrix defined above. The supersymmetry parameter for Mp-branes must satisfy,

$$\begin{aligned} \frac{1}{2}\epsilon ^T_p C (1+ \Gamma ^{10} \cdots \Gamma ^{p+1}) Q^\mathrm{M}= \epsilon ^T_p C Q^\mathrm{M} \end{aligned}$$

(B.11)

or

$$\begin{aligned} \Gamma ^{p+1} \cdots \Gamma ^{10} \epsilon _p = (-1)^{p}\epsilon _p. \end{aligned}$$

(B.12)

This is the supersymmetry parameter relevant for Mp-branes.

The sign convention of $(p+1)$-form fields $C_{p+1}^\mathrm{M}$ coupled to Mp-branes is determined by the following requirement. Let us consider the above Mp-brane with the orientation of the worldvolume $\omega _{p+1}$ given by (B.6). Let us also define

$$\begin{aligned} \delta _{10-p} (\mathbf {z}) := \delta (x^{p+1} ) \cdots \delta (x^{10} ) \mathrm{d}x^{p+1} \wedge \cdots \wedge \mathrm{d}x^{10}, \end{aligned}$$

(B.13)

where

$$\begin{aligned} \mathbf {z}= (x^{p+1}, \cdots , x^{10}). \end{aligned}$$

(B.14)

Then the coupling of $C_{p+1}^\mathrm{M}$ to the Mp-brane is given by

$$\begin{aligned} -S \supset 2\pi {\mathsf i}\int C_{p+1}^\mathrm{M} = 2\pi {\mathsf i}\int C_{p+1}^\mathrm{M} \wedge \delta _{10-p} (\mathbf {z}). \end{aligned}$$

(B.15)

The sign of $C_{p+1}^\mathrm{M}$ is defined by this coupling.

Including the kinetic term, the action contains

$$\begin{aligned} -S \supset - \frac{2\pi }{2} \int \mathrm{d}C_{p+1}^\mathrm{M} \wedge *\mathrm{d}C_{p+1}^\mathrm{M} + 2\pi {\mathsf i}\int C_{p+1}^\mathrm{M} \wedge \delta _{10-p} (\mathbf {z}), \end{aligned}$$

(B.16)

where we are using the Planck unit $2\pi \ell _\mathrm{M}=1$. The equation of motion is

$$\begin{aligned} (-1)^{p+1} \mathrm{d}*F_{p+2}^\mathrm{M} + {\mathsf i}\delta _{10-p} (\mathbf {z})=0. \end{aligned}$$

(B.17)

Now suppose that we have the duality equation of field strength

$$\begin{aligned} *F_{p+2}^\mathrm{M} = {\mathsf i}s_{p+2} F_{9-p}^\mathrm{M}, \end{aligned}$$

(B.18)

where $s_{p+2} = \pm 1$ are sign factors which we want to determine. Notice that we have already defined the sign convention for the fields $C_{p+1}$ and hence there is no freedom to modify this self-dual condition.

Then we get

$$\begin{aligned} \mathrm{d}F_{9-p}^\mathrm{M} = (-1)^p s_{p+2} \delta _{10-p} (\mathbf {z}). \end{aligned}$$

(B.19)

Because $*^2=1$ in odd dimensional Riemann manifold, we have $({\mathsf i}s_{p+2}) ( {\mathsf i}s_{9-p})=1$ or $s_{p+2} s_{9-p} = -1$. Let us set $s := s_4$. Then $s_7=-s$ and

$$\begin{aligned} \int _{S^7} F_{7}^\mathrm{M} = s, \qquad \int _{S^4} F_{4}^\mathrm{M} = s , \end{aligned}$$

(B.20)

where $S^{9-p}$ is the sphere surrounding the Mp-brane. We will see that the value of s is given by $s=+1$.

Before computing s, let us explain more about the structure of various signs and why they are important for the anomaly of M5-branes. The signs of $C_{p+1}$ are defined by (B.15), that is, $C_{p+1}$ and $- C_{p+1}$ are distinguished by the coupling to Mp-branes. The distinction between Mp-branes and anti-Mp-branes are defined by (B.8) and (B.9). They affect the computation of the anomaly in the following way.

First, the supercharge (B.8) determines the chirality of the worldvolume fields of the M5-brane. The chirality operator $\overline{\Gamma }^{{\mathrm {M}}5}$ on the M5-brane with the orientation $\omega _6= \mathrm{d}x^0 \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \wedge \mathrm{d}x^3 \wedge \mathrm{d}x^4 \wedge \mathrm{d}x^5$ is given as

$$\begin{aligned} \overline{\Gamma }^{{\mathrm {M}}5} = {\mathsf i}^{-3} \Gamma ^0 \Gamma ^1 \Gamma ^2 \Gamma ^3 \Gamma ^4 \Gamma ^5. \end{aligned}$$

(B.21)

By using (B.4), it can be written also as $\overline{\Gamma }^{{\mathrm {M}}5} = - \Gamma ^6 \Gamma ^7 \Gamma ^8 \Gamma ^9 \Gamma ^{10}$. Under this chirality operator, $Q^\mathrm{M5}$ has a definite chirality as

$$\begin{aligned} \overline{\Gamma }^{{\mathrm {M}}5} Q^\mathrm{M5} = - Q^\mathrm{M5} . \end{aligned}$$

(B.22)

From this, the worldvolumes fermions $\chi \sim [Q^\mathrm{M5} , \phi ]$ (where $\phi$ represent worldvolume scalars) has negative chirality $\overline{\Gamma }^{{\mathrm {M}}5} \chi = -\chi$ and so on. The chirality of the worldvolume fields affects the sign of the anomaly.

Next let us explain (B.9). We are interested in the M2-charge of the M-theory orbifold

$$\begin{aligned} {\mathbb {R}}^3 \times ({\mathbb {R}}^8/{\mathbb {Z}}_k). \end{aligned}$$

(B.23)

Here the orbifold action on the coordinate

$$\begin{aligned} \mathbf {z} = (z^1, z^2, z^3, z^4) = (x^3 + {\mathsf i}x^4, x^5 + {\mathsf i}x^6, x^7 + {\mathsf i}x^8, x^9 + {\mathsf i}x^{10}) \end{aligned}$$

(B.24)

is given by

$$\begin{aligned} \mathbf {z} \rightarrow e^{2\pi {\mathsf i}j /k} \mathbf {z} , \qquad (j =0,1,\cdots , k-1). \end{aligned}$$

(B.25)

The question is how to define the uplift of this action to spinors. We define the action in such a way that the supercharges preserved by this orbifold action is a subset of the supercharges (B.9). In other words, adding M2-branes to the orbifold does not break supersymmetry, while adding anti-M2-branes breaks it. The uplift of (B.25) on spinors $\Psi$ is either $\Psi \rightarrow + {\mathsf {R}}(j/k) \Psi$ or $\Psi \rightarrow -{\mathsf {R}}(j/k) \Psi$, where

$$\begin{aligned} {\mathsf {R}}(t) = \exp \left( - \pi t \left( \Gamma ^3\Gamma ^4 + \Gamma ^5\Gamma ^6 + \Gamma ^7\Gamma ^8 + \Gamma ^9\Gamma ^{10} \right) \right) . \end{aligned}$$

(B.26)

In fact, one can check

$$\begin{aligned} {\mathsf {R}}(t)^{-1} \left( \Gamma ^{1+2q} + {\mathsf i}\Gamma ^{2+2q}\right) {\mathsf {R}}(t) = e^{ 2\pi {\mathsf i}t} \left( \Gamma ^{1+2q} + {\mathsf i}\Gamma ^{2+2q}\right) \qquad (q=1,2,3,4). \end{aligned}$$

(B.27)

which corresponds to (B.24). The sign ambiguity in $\Psi \rightarrow \pm {\mathsf {R}}(j/m) \Psi$ is the standard one in going from $\mathrm {SO}$ to $\mathrm {Spin}$, and it determines the spin structure of ${\mathbb {R}}^8/{\mathbb {Z}}_k$. By requiring that ${\mathsf {R}}(j/k)$ preserves some of the charge $Q^{{\mathrm {M}}2}$, we conclude that the sign must be such that

$$\begin{aligned} \Psi \rightarrow +{\mathsf {R}}(j/k)\Psi . \end{aligned}$$

(B.28)

For example, ${\mathsf {R}}(1/2) = \Gamma ^3\Gamma ^4 \Gamma ^5\Gamma ^6 \Gamma ^7\Gamma ^8 \Gamma ^9\Gamma ^{10}$ and ${\mathsf {R}}(1/2)Q^{{\mathrm {M}}2} = Q^{{\mathrm {M}}2}$, and hence $+{\mathsf {R}}(1/2)$ preserves the same supercharges as M2-branes, while $-{\mathsf {R}}(1/2)$ preserves the same supercharges as anti-M2-branes. The choice (B.28) determines the spin structure of ${\mathbb {R}}^8/{\mathbb {Z}}_k$, and the spin structure affects the value of the $\eta$-invariant on $S^7/{\mathbb {Z}}_k$. In this way the choice (B.9) affects the anomaly of the orbifold.

1.2 Supergravity background and supersymmetry

Mp-branes are realized as extremal black p-brane solutions in supergravity. As we will see, the remaining supersymmetries (i.e. Killing spinors in the extremal black brane solutions) depend on the sign of the flux $F_{9-p}$. Our strategy is to relate the remaining supersymmetries (B.12) and the fluxes (B.20) and determine the sign factor s.

The Killing spinor equation has the following schematic form

(B.29)

where $\epsilon$ is the Killing spinor, $D_I$ is the covariant derivative, and

(B.30)

where $\Gamma ^{JKLM} =\Gamma ^{[J}\Gamma ^K \Gamma ^L \Gamma ^{M]}$ is the product of gamma matrices with the indices J, K, L, M antisymmetrized. The above form of the Killing spinor equation may be inferred just by simple considerations of Lorentz structure and a counting of mass dimensions if we recover the Planck scale $2\pi \ell _\mathrm{M}$. It is much more nontrivial to determine the coefficients $c_{1,2}$. According to the equation (13.13) of [133] , they are given by $|c_1| = 1/24$, $|c_2| = 1/8$ and $c_1c_2<0$. The overall sign of $c_1,c_2$ depends on the convention of $F^{{\mathrm {M}}}_4$.⁴⁴

Let us consider the Killing equation when the Lorentz index I is in the direction parallel to the Mp-brane, which we denote by the Greek letter $\mu$. Then by translational invariance, we have $\partial _\mu \epsilon = 0$. However, the covariant derivative $D_\mu$ is still nonzero. The metric of the extremal black p-brane solution is of the form

$$\begin{aligned} \mathrm{d}s^2 = E(r)^2 ( \mathrm{d}x_0^2 + \cdots + \mathrm{d}x_p^2) + F(r)^2(\mathrm{d}x_{p+1}^2+\cdots + \mathrm{d}x_{10}^2), \end{aligned}$$

(B.31)

where $r=|\mathbf {z}|$. In this metric, we can take the orthonormal frame $e_I^a$ as $e_\mu ^a = E(r) \delta ^a_\mu$ as long as $\mu$ is in the tangent direction. Then the spin connection $\omega _{\mu IJ}$ is

$$\begin{aligned} \frac{1}{4} \Gamma ^{IJ} \omega _{\hat{\mu }IJ} = \frac{1}{2} \Gamma _{\hat{\mu }} \Gamma ^{\hat{r}} F(r)^{-1} \partial _r \log E(r), \end{aligned}$$

(B.32)

where $\Gamma ^{\hat{\mu }}$ and $\Gamma ^{\hat{r}}$ are gamma matrices in the directions $x^\mu$ and r normalized in such a way that $(\Gamma ^{\hat{\mu }} )^2 = (\Gamma ^{\hat{r}} )^2=1$, and $\omega _{\hat{\mu }IJ} = E(r)^{-1} \omega _{ \mu IJ}$. Therefore, the Killing spinor equation is simplified to

(B.33)

In the M5 case ($p=5$), the ${F}^\mathrm{M}_4$ does not contain $\mu$ components and hence . On the other hand, in the M2 case ($p=2$), the term $*{F}^\mathrm{M}_4 \propto {F}^\mathrm{M}_7$ does not contain $\mu$ and r, and hence schematically ${F}^\mathrm{M}_4 \sim \mathrm{d}x^{\mu _1} \wedge \mathrm{d}x^{\mu _2} \wedge \mathrm{d}x^{\mu _3} \wedge \mathrm{d}r$. Thus we get . Therefore, we get

(B.34)

where $Z(r) = \frac{1}{2} F(r)^{-1} \partial _r \log E(r)$, and the sign in $\pm c_2$ depends on the sign in .

It is possible to rewrite in terms of which is defined in the similar way as in (B.30). Let us first notice that

$$\begin{aligned} \frac{1}{4!}\epsilon ^{I_1 \cdots I_{7} J_1 \cdots J_4}\Gamma _{J_1 \cdots J_4} = -{\mathsf i}\Gamma ^{I_1 \cdots I_7}, \end{aligned}$$

(B.35)

which follows from (B.4). Also, we have $F_4^\mathrm{M} = {\mathsf i}s *F_7^\mathrm{M}$ (because $*F_4^\mathrm{M} = {\mathsf i}s F_7^\mathrm{M}$ and $*^2=1$) which is explicitly written as

$$\begin{aligned} (F_4^\mathrm{M} )_{J_1 \cdots J_4} = ({\mathsf i}s) \cdot \frac{1}{7!}(F_7^\mathrm{M})^{I_1 \cdots I_7} \epsilon _{I_1 \cdots I_{7} J_1 \cdots J_4} . \end{aligned}$$

(B.36)

Therefore,

(B.37)

For the Mp-brane solution, the flux $F_{9-p}$ is such that

$$\begin{aligned} \int _{S^{9-p}}{F}^\mathrm{M}_{9-p}=s \end{aligned}$$

(B.38)

as we have seen in the previous subsection. Then we have

(B.39)

where the factor $| {F}^\mathrm{M}_{9-p}| \sim 1/r^{9-p}$ is a positive function of r. Therefore (B.34) is written for the Mp-brane solution as

$$\begin{aligned} \epsilon = \left( K_p Z(r)^{-1} | {F}^\mathrm{M}_{9-p}| \right) \Gamma ^{p+1} \cdots \Gamma ^{10} \epsilon , \end{aligned}$$

(B.40)

where $K_p$ is defined by

$$\begin{aligned} K_p = \left\{ \begin{array}{ll} - s (c_1+c_2), &{} (p=5), \\ -(c_1-c_2), &{} (p=2). \end{array} \right. \end{aligned}$$

(B.41)

Now let us notice that the sign of $Z(r) = \frac{1}{2} F(r)^{-1} \partial _r \log E(r)$ is independent of whether we consider branes or anti-branes, or whether we consider M2 or M5, because gravity is always an attractive force. (Its absolute value depends on whether we consider M2 or M5.) More explicitly, $\log E(r) \sim - 1 /r^{8-p}$ and hence $Z(r)>0$. Therefore, the sign of the right-hand-side of (B.40) is determined simply by the sign of $K_p$. By requiring that (B.40) coincides with (B.12), we get $Z(r) = |K_p | | {F}^\mathrm{M}_{9-p}|$ and

$$\begin{aligned} \mathop {\mathrm {sign}}(K_p) = (-1)^{p}. \end{aligned}$$

(B.42)

The actual values of $c_1$ and $c_2$ are such that $|c_2| > |c_1|$ and hence $\mathop {\mathrm {sign}}(c_2 \pm c_1) = \mathop {\mathrm {sign}}(c_2)$. Then (B.41) and (B.42) give $c_2>0$ and

$$\begin{aligned} s = +1, \end{aligned}$$

(B.43)

which is what we wanted to show.

Cohomology Pairing and $\eta$-Invariant Modulo 1 on Lens Spaces

In this appendix we compute the differential cohomology pairing on $S^7/{\mathbb {Z}}_k$ and the $\eta$-invariants mod 1 which are required in Sect. 7.2, and provide the values already quoted in (7.34). The strategy for the computation is to find a manifold Z whose boundary is $Y = S^7/{\mathbb {Z}}_k$. For our purposes, Z does not have to be spin; a $\text {spin}^c$ structure suffices. We also extend $\mathcal{{L}}_1$ and $\check{C}_1$ to Z, and use this Z to compute the quantities appearing in (7.34). Most of the discussions here can be generalized for $S^{2m-1}/{\mathbb {Z}}_k$ without much difficulty, so we take m to be general. Then we obtain a formula for the $\eta$-invariant mod 1 for lens spaces of general dimensions when the Dirac operator is coupled to general flat line bundles. We note that we basically follow the discussion in [112]. We also emphasize that the computation in this section only gives the $\eta$-invariants mod 1, and not the $\eta$-invariants themselves. This is enough for the purposes of Sect. 7.2, but may not be enough for some other purposes, such as using the $\eta$-invariant of the signature operator $\mathcal{{D}}^\mathrm{sig}$ which is multiplied by 1/8. A different computation of the $\eta$-invariants of lens spaces which gives their values as real numbers will be presented in Appendix D.

1.1 The geometry and the differential cohomology pairing

First, we consider $\mathbb {CP}^{m-1}$ and define a line bundle $\mathcal{{O}}(r)$ for an arbitrary integer $r \in {\mathbb {Z}}$ as

$$\begin{aligned}{}[ S^{2m-1} \times {\mathbb { C}}]/\mathrm {U}(1), \end{aligned}$$

(C.1)

where the $\mathrm {U}(1)$ acts as⁴⁵

$$\begin{aligned} S^{2m-1} \times {\mathbb { C}}\ni (\mathbf {z}, u) \mapsto ( e^{{\mathsf i}\alpha } \mathbf {z}, e^{ {\mathsf i}r \alpha } u) \qquad (\alpha \in {\mathbb {R}}). \end{aligned}$$

(C.2)

We denote the equivalence class of $(\mathbf {z},u)$ under the equivalence relation $(\mathbf {z}, u) \sim ( e^{{\mathsf i}\alpha } \mathbf {z}, e^{ {\mathsf i}r \alpha } u)$ as $[\mathbf {z}, u]$. Then $\mathcal{{O}}(r) =\{ [\mathbf {z}, u] \}$.

Now we take the total space of $\mathcal{{O}}( - k)$, and consider its subspace given by

$$\begin{aligned} Z&= \{ [\mathbf {z}, u ] \mid |u| \le 1\}, \end{aligned}$$

(C.3)

$$\begin{aligned} Y&= \{ [\mathbf {z}, u ] \mid |u| = 1\} . \end{aligned}$$

(C.4)

On Y, we can fix “the gauge symmetry” $(\mathbf {z}, u) \sim ( e^{{\mathsf i}\alpha } \mathbf {z}, e^{ - {\mathsf i}k \alpha } u)$ by taking $u=1$. Then the remaining gauge transformation is generated by $[\mathbf {z}, 1] = [e^{ 2\pi {\mathsf i}/k } \mathbf {z}, 1]$. Therefore, we conclude that $Y = S^{2m-1}/{\mathbb {Z}}_k$. The manifold Z has this lens space as the boundary, $Y =\partial Z$. One can also check that the orientation of Z as a complex manifold is compatible with the orientation of $S^{2m-1}/{\mathbb {Z}}_k$ induced from the standard orientation of $S^{2m-1}$.

On Z, we define a line bundle $\mathcal{{L}}_{s}$ ($s \in {\mathbb {Z}}$) by

$$\begin{aligned} \mathcal{{L}}_s = \{ [\mathbf {z}, u, v ] \} / (\mathbf {z}, u, v ) \sim (e^{{\mathsf i}\alpha }\mathbf {z}, e^{- {\mathsf i}k \alpha } u, e^{- {\mathsf i}s \alpha }v ) . \end{aligned}$$

(C.5)

Notice that $\mathcal{{L}}_s = \mathcal{{L}}_1^{\otimes s}$. The line bundle $\mathcal{{L}}_1$ extends the one defined in (7.30) from $Y =S^{2m-1}/{\mathbb {Z}}_k$ to Z. This is a pullback of $\mathcal{{O}}(-1)$ from $\mathbb {CP}^{m-1}$ to the total space of $\mathcal{{O}}(-k)$.

Let us consider a connection on $\mathcal{{L}}_1$ which becomes the flat connection in $Y = S^{2m-1}/{\mathbb {Z}}_k$. We represent the connection by using a differential cohomology element $\check{A} \in \check{H}^2(Z)$. Consider the holonomy $\exp (2\pi {\mathsf i}\int {\mathsf {A}}_A)$ of this connection around a loop

$$\begin{aligned}{}[e^{ 2\pi {\mathsf i}t/k }\mathbf {z}_0, u_0] \qquad ( 0 \le t \le 1) \end{aligned}$$

(C.6)

in $Y=S^{2m-1}/{\mathbb {Z}}_k$ for a fixed $(\mathbf {z}_0,u_0)$. The parallel transport of an element $[\mathbf {z}_0, u_0,v]$ of $\mathcal{{L}}_1$ is given by $[e^{ 2\pi {\mathsf i}t/k }\mathbf {z}_0, u_0,v]$. From the fact that

$$\begin{aligned} ( e^{2\pi {\mathsf i}/k}z_0,u_0, v) \sim (z_0, u_0, e^{2\pi {\mathsf i}/k}v), \end{aligned}$$

(C.7)

we can see that the holonomy of the flat connection around the loop is $e^{2\pi {\mathsf i}/k}$.

Next consider a two dimensional disk

$$\begin{aligned} D=\{ [\mathbf {z}_0, u] ; |u| \le 1 \} \subset Z \qquad (\mathbf {z}_0 \text {: fixed} ). \end{aligned}$$

(C.8)

Notice that the loop (C.6) is equal to $\partial D$ because $[e^{ {\mathsf i}t/k }\mathbf {z}_0, u_0] =[\mathbf {z}_0, e^{ {\mathsf i}t }u_0 ]$. From the above holonomy, we see that its curvature integral on the disk is given by⁴⁶

$$\begin{aligned} \int _D {\mathsf {F}}_A = \frac{1}{k} . \end{aligned}$$

(C.9)

In particular, $\mathcal{{L}}_1^{\otimes k}$ has a connection $k \check{A}$ which is trivial on Y and has the curvature integral given by $\int _D k {\mathsf {F}}_{A} =1$. This implies that the connection $k\check{A}$ can be continuously deformed (without changing the boundary values) to a connection which is trivial on Y and whose curvature is localized on

$$\begin{aligned} M= \{ [\mathbf {z}, u=0] \} \subset Z. \end{aligned}$$

(C.10)

This M is isomorphic to $\mathbb {CP}^{m-1}$. The localization of the curvature means $k {\mathsf {F}}_{A} \sim \delta (M)$ where $\delta (M)$ is the delta function localized on M.⁴⁷

We compute

$$\begin{aligned}&\int _Z ({\mathsf {F}}_{A})^m =\frac{1}{k} \int _Z ({\mathsf {F}}_{A})^{m-1} \delta (M) = \frac{1}{k} \int _M ({\mathsf {F}}_{A})^{m-1} \nonumber \\&=\frac{1}{k} \int _M (c_1(\mathcal{{L}}_1))^{m-1} = \frac{1}{k} \int _{\mathbb {CP}^{m-1} } c_1(\mathcal{{O}}(-1) )^{m-1} = \frac{(-1)^{m-1}}{k} . \end{aligned}$$

(C.11)

where we have used the fact that $\mathcal{{L}}_1$ restricted to $M \cong \mathbb {CP}^{m-1}$ is $\mathcal{{O}}(-1)$, and also used the standard fact that $\int _{\mathbb {CP}^{m-1} } c_1(\mathcal{{O}}(t) )^{m-1} = t^{m-1}$ for any $t \in {\mathbb {Z}}$.

Now we have done all the preparations to compute the pairing of $\check{C}_1$ on $Y = S^7/{\mathbb {Z}}_k$. In this paragraph we restrict our attention to the original case $m=4$. We define $\check{C}_1$ as $\check{C}_1 = \check{A} \star \check{A}$ which indeed gives $[{\mathsf {N}}_{C_1}] = c_1(\mathcal{{L}}_1)^2$ on Y as we have defined in (7.31). Then the pairing on Y is given by

$$\begin{aligned} (\check{C}_1, \check{C}_1) = \int _Z ({\mathsf {F}}_{C_1})^2 = \int _Z ({\mathsf {F}}_{A})^4 = - \frac{1}{k}. \end{aligned}$$

(C.12)

Thus we have obtained the first equation in (7.34). The pairing takes values in ${\mathbb {R}}/{\mathbb {Z}}$, so this equation is meaningful only mod 1.

1.2 The $\eta$-invariant

Next we want to go to the computation of the $\eta$-invariant. In this appendix we are interested not in $\eta$ itself, but only $\eta \mod 1$ for the purposes of computing anomalies. Thus we can use the APS index theorem to compute it by integrating the corresponding characteristic class on Z.

One point which we need to be careful is the following. We want to use the manifold (C.4). It is not always a spin manifold depending on the values of k (and m). However, the manifold Z (or any complex manifold) is a $\text {spin}^c$ manifold. For the purpose of computing the $\eta$-invariant, it is enough to have a $\text {spin}^c$ structure.

First let us discuss why Z is not necessarily $\text {spin}$. If we restrict to the submanifold $M \cong \mathbb {CP}^{m-1}$ defined in (C.10), the tangent bundle of Z splits as $TZ = T \mathbb {CP}^{m-1} \oplus \mathcal{{O}}(-k)$. The reason is that the normal bundle to M in Z is $\mathcal{{O}}(-k)$ which follows from the definition of Z as the total space of the $\mathcal{{O}}(-k)$ bundle over $\mathbb {CP}^{m-1}$. It is also well-known that if we add a trivial bundle $\underline{{\mathbb { C}}}$ to $T \mathbb {CP}^{m-1}$, we get a sum of m copies of $\mathcal{{O}}(1)$,

$$\begin{aligned} T \mathbb {CP}^{m-1} \oplus \underline{{\mathbb { C}}} \cong m \mathcal{{O}}(1). \end{aligned}$$

(C.13)

The bundle $m \mathcal{{O}}(1) \oplus \mathcal{{O}}(-k)$ has a spin lift only if $m+k$ is even. This is the obstruction to the existence of a spin structure on Z.

Instead of spin structure, we consider the following bundle on a complex manifold Z. The following discussion is generally true for any complex manifold and is well-known in algebraic geometry. Let ${T}^*Z$ be the complex cotangent bundle of the complex manifold Z, and let $\overline{T}^* Z$ be its complex conjugate bundle. (This $\overline{T}^* Z$ is isomorphic to the complex tangent bundle TZ by introducing an explicit hermitian metric on TZ. Using the bundle $\overline{T}^* Z$ may be more natural in the context of Dolbeault complex without an explicit hermitian metric.) We define

$$\begin{aligned} \mathcal{{S}}_c = \sum _{\ell =0}^{\dim _{\mathbb { C}}Z } \wedge ^\ell (\overline{T}^* Z), \end{aligned}$$

(C.14)

where $\wedge ^\ell E$ of a vector space E means the $\ell$-th antisymmetric product of E. This is $\text {spin}^c$ by the following reason. Let $\Gamma _I ~(I=1,\ldots , 2\dim _{\mathbb { C}}W)$ be the gamma matrices of $\dim _{\mathbb {R}}W = 2 \dim _{\mathbb { C}}W$ dimensional Clifford algebra. We take $a_i = (\Gamma _{2i -1} + {\mathsf i}\Gamma _{2i})/2$ and $a_i^\dagger = (\Gamma _{2i -1} - {\mathsf i}\Gamma _{2i})/2$. They may be regarded as creation and annihilation operators with $\{a_i, a_j^\dagger \} = \delta _{ij}$. Then we can find a representation of the Clifford algebra by first taking $|0\rangle$ such that $a_i |0\rangle =0$, and then consider $a_{i_1}^\dagger \ldots a_{i_\ell }^\dagger |0\rangle$. By this construction, we can see that $\mathcal{{S}}_c$ defined above is a $\text {spin}^c$ bundle (or more precisely an irreducible Clifford module) on which the Clifford algebra acts. More explicitly, on the bundle $\mathcal{{S}}_c$, the actions of $a_{i}$ and $a_i^\dagger$ are given by $a_{i}^\dagger = \mathrm{d}\bar{z}^i \wedge$ and $a_i = \iota _{\bar{\partial }_i}$, respectively, where $\mathrm{d}\bar{z}^i \wedge$ is an operator which acts on a differential form $\omega$ as $\omega \rightarrow \mathrm{d}\bar{z}^i \wedge \omega$, and $\iota _{\bar{\partial }_i}$ is its adjoint. The ‘chirality’, or equivalently the ${\mathbb {Z}}_2$ grading of the bundle, is determined by the degree $\ell$ of $\wedge ^\ell (\overline{T}^* W)$ mod 2.

The canonical line bundle of a complex manifold W is defined by

$$\begin{aligned} \mathcal{{K}}= \wedge ^{\dim _{\mathbb { C}}W} (T^*W). \end{aligned}$$

(C.15)

Then, if there exists a square root $\mathcal{{K}}^{1/2}$ of the canonical line bundle, we can define a spin bundle as

$$\begin{aligned} \mathcal{{S}}= \mathcal{{K}}^{1/2} \otimes \mathcal{{S}}_c. \end{aligned}$$

(C.16)

Therefore, $\mathcal{{K}}^{1/2}$ measures the difference between $\mathcal{{S}}$ and $\mathcal{{S}}_c$.

Let us return to the case of our manifold (C.4). Near the boundary, the manifold $Z = \{ [\mathbf {z},u] \}$ is embedded as a subspace of ${\mathbb { C}}^m/{\mathbb {Z}}_k$ by taking $\mathbf {w}=u^{1/k}\mathbf {z}$ as the complex coordinates of ${\mathbb { C}}^4/{\mathbb {Z}}_k$ as far as $\mathbf {w} \ne 0$. The equivalence relation is $\mathbf {w} \sim e^{2\pi {\mathsf i}/k}\mathbf {w}$.⁴⁸ The tangent bundle TZ is described by tangent vectors $\Delta \mathbf {w}$ with the equivalence relation $(\mathbf {w}, \Delta \mathbf {w}) \sim e^{2\pi i/k}(\mathbf {w}, \Delta \mathbf {w})$. From this, we see that TZ near the boundary is $m \mathcal{{L}}_{-1} = (\mathcal{{L}}_{1})^{-1} \oplus \cdots \oplus (\mathcal{{L}}_{1})^{-1}$, i.e. the sum of m copies of $\mathcal{{L}}_1^{-1}$. The canonical bundle near the boundary is $\mathcal{{K}}= \mathcal{{L}}_1^{\otimes m} = \mathcal{{L}}_m$. A square root of $\mathcal{{K}}$ exists near the boundary if m is even, and we take it to be $\mathcal{{K}}^{1/2} =\mathcal{{L}}_1^{\otimes (m/2)}$. Now we define

$$\begin{aligned} \mathcal{{S}}' = \mathcal{{L}}_{1}^{\otimes (m/2)} \otimes \mathcal{{S}}_c \end{aligned}$$

(C.17)

on Z. We soon explain the case of odd m.

When restricted to the boundary $Y = \partial Z$, the bundle $\mathcal{{S}}'$ gives a spin structure of $Y = S^{2m-1}/{\mathbb {Z}}_k$. The spin structure is not unique when k is even, because we can take a line bundle associated to a homomorphism ${\mathbb {Z}}_k \rightarrow {\mathbb {Z}}_2$ and modify the spin bundle by this line bundle. However, the spin structure of $\mathcal{{S}}'$ coincides with the one which is realized in the M-theory orbifold (which is given in (B.28) of Appendix B) as one can check by representing all bundles as a sum of powers of $\mathcal{{L}}_1$.

Inside Z, $\mathcal{{S}}'$ is not spin, but $\text {spin}^c$. However, that is not a problem in computing the $\eta$-invariant on $Y = S^{2m-1}/{\mathbb {Z}}_k$. We multiply the bundle by $\mathcal{{L}}_1^s$ to get

$$\begin{aligned} \mathcal{{S}}_s : = \mathcal{{L}}_1^{s} \otimes \mathcal{{S}}' = \mathcal{{L}}_1^{\otimes (s+m/2) } \otimes \mathcal{{S}}_c . \end{aligned}$$

(C.18)

This is a $\text {spin}^c$ bundle on Z. For the purpose of the computation of (7.34), we need to consider the $\eta$-invariant of the bundles with $s=0$ and $s=1$.

Although it is not directly relevant to (7.34), let us also comment on the case of odd m. In this case, $\mathcal{{S}}'$ is not well-defined. In fact, for odd m and even k, $S^{2m-1}/{\mathbb {Z}}_k$ is not a spin manifold as we explained before. However, the bundle $\mathcal{{S}}_s$ is well-defined if we take $s +m/2$ to be integer. The following computation is valid also for these cases.

Recall that Z is described as the total space of the $\mathcal{{O}}(-k)$ bundle on $\mathbb {CP}^{m-1}$, and $\mathcal{{L}}_1$ is the pullback of $\mathcal{{O}}(-1)$ to this total space. In particular, the canonical bundle $\mathcal{{K}}$ of Z is topologically given by $\mathcal{{K}}= \mathcal{{K}}_{\mathbb {CP}^{m-1}} \otimes \mathcal{{O}}(k) = \mathcal{{O}}(k-m)$. Here we have used the fact that the canonical bundle of $\mathbb {CP}^{m-1}$ is given by $\mathcal{{K}}_{\mathbb {CP}^{m-1}} = \mathcal{{O}}(-m)$, which follows from (C.13). The bundles $\mathcal{{O}}(s)$ here are understood as a pullback of $\mathcal{{O}}(s)$ from $\mathbb {CP}^{m-1}$ to Z. In fact, $\mathcal{{O}}(k) =\mathcal{{L}}_1^{\otimes (-k)}$ is trivial near the boundary $Y = S^{2m-1}/{\mathbb {Z}}_k$, so this canonical bundle reduces to what we have discussed above, that is, $\mathcal{{K}}= \mathcal{{L}}_1^{\otimes m}$ near the boundary. The expression $\mathcal{{K}}= \mathcal{{L}}_1^{\otimes (m -k)}$ is valid even inside Z.

Therefore, at the level of differential forms of the curvature (which is what is necessary for the computation of $\eta$ by using the higher dimensional manifold), we can split the bundle as

$$\begin{aligned} \mathcal{{S}}_s= \mathcal{{L}}_1^{\otimes (s+m/2) } \otimes \mathcal{{S}}_c = \mathcal{{L}}_1^{ \otimes (s + k/2)} \otimes \mathcal{{S}}, \end{aligned}$$

(C.19)

where we have formally set $\mathcal{{S}}= K^{1/2} \otimes \mathcal{{S}}_c$. Such a formal expression is valid when we consider the characteristic polynomial of the curvature in the index theorem. Let $\hat{A}(R)$ be the A-roof genus of Z defined explicitly in terms of the Riemann curvature tensor R. Let ${\mathsf {F}}_A$ be the curvature of the connection $\check{A}$ on $\mathcal{{L}}_1$ which was considered in the computation of (C.11). We can now compute the $\eta$-invariant on $Y = S^{2m-1}/{\mathbb {Z}}_k$ by using the APS index theorem. Let $\mathcal{{D}}_s$ be the Dirac operator acting on to the (positive chirality part of) $\mathcal{{S}}_s$ restricted to Y. From (C.19) we get

$$\begin{aligned} -\eta (\mathcal{{D}}_s) \equiv \int _Z \exp \left( (s+\frac{1}{2}k ){\mathsf {F}}_A \right) \hat{A}(R) \quad \mod 1. \end{aligned}$$

(C.20)

We can further simplify this expression as follows. The tangent bundle TZ is topologically the same as $T\mathbb {CP}^{m-1} \oplus \mathcal{{O}}(-k)$. The characteristic class does not change even if we add a trivial bundle $\underline{{\mathbb { C}}}$, and we have the splitting (C.13) Therefore, we get

$$\begin{aligned} TW \oplus \underline{{\mathbb { C}}} = \mathcal{{L}}_1^{k} \oplus \mathcal{{L}}_1^{-1} \oplus \cdots \oplus \mathcal{{L}}_1^{-1} : = \mathcal{{L}}_1^{\oplus (k, -1,\cdots ,-1)} , \end{aligned}$$

(C.21)

where there are m copies of $\mathcal{{L}}_1^{-1}$. Let $\hat{A}( \mathcal{{L}}_1^{\oplus (k, -1,\cdots ,-1)} )$ be the corresponding A-roof genus which is equal to $\hat{A}(R)$ (up to the continuous deformation of the connection from the Levi-Civita connection on TZ to the connection on $\mathcal{{L}}_1^{\oplus (k, -1,\cdots ,-1)}$ determined by the connection $\check{A}$ on $\mathcal{{L}}_1$). This can be represented as a polynomial of ${\mathsf {F}}_A$. Explicitly, it is given by

$$\begin{aligned} \hat{A}(\mathcal{{L}}_1^{\oplus (k, -1,\cdots ,-1)}) = \left( \frac{k{\mathsf {F}}_A/2}{\sinh (k{\mathsf {F}}_A/2)} \right) \left( \frac{{\mathsf {F}}_A/2}{\sinh ({\mathsf {F}}_A/2)} \right) ^m. \end{aligned}$$

(C.22)

The $\eta$ is now given by

$$\begin{aligned} -\eta (\mathcal{{D}}_s) \equiv \int _Z \exp \left( (s+ \frac{k}{2} ){\mathsf {F}}_A \right) \hat{A}(\mathcal{{L}}_1^{\oplus (k, -1,\cdots ,-1)}). \end{aligned}$$

(C.23)

The integrand is just a polynomial of ${\mathsf {F}}_A$. Moreover, we know from (C.11) that

$$\begin{aligned} \int _Z ({\mathsf {F}}_A)^m = \frac{(-1)^{m-1}}{k}. \end{aligned}$$

(C.24)

By using these results, we can compute the desired $\eta$-invariant.

The above result can be summarized as follows. Let us define power series of variables $y,x_1,\cdots ,x_{m+1}$ which we denote (by abuse of notation) as $\mathrm{ch}(y)$ and $\hat{A}(x_1, \cdots , x_{m+1})$, and $p(x_1,\cdots ,x_{m+1})$, by the following formulas:

$$\begin{aligned}&\mathrm{ch}(y) = \sum _{i} \mathrm{ch}_i(y) = e^y, \nonumber \\&\hat{A}(x_1,\cdots ,x_{m+1}) =\sum _i \hat{A}_i(x_1,\cdots ,x_{m+1})= \prod _{i=1}^{m+1} \left( \frac{ x_i /2}{\sinh (x_i /2)} \right) , \nonumber \\&p (x_1,\cdots ,x_{m+1}) = \sum _i p_i (x_1,\cdots ,x_{m+1}) = \prod _{i=1}^{m+1} (1+x_i^2), \end{aligned}$$

(C.25)

where $\mathrm{ch}_i$ is the degree i part of $\mathrm{ch}$, and $\hat{A}_i$ and $p_i$ are the degree 2i parts of $\hat{A}$ and p, respectively. $\hat{A}$ can be expanded by p as

$$\begin{aligned} \hat{A}_0=1, \qquad \hat{A}_1 = - \frac{1}{24}p_1, \qquad \hat{A}_2 = \frac{7p_1^2 - 4p_2}{5760}, \quad \ldots . \end{aligned}$$

(C.26)

By using these notations, the $\eta (\mathcal{{D}}_s)$ is obtained from (C.23) and (C.24) as

$$\begin{aligned} (-1)^m \eta (\mathcal{{D}}_s) \equiv \frac{1}{k} \sum _{i+2j = m} \mathrm{ch}_i \left( s+ k/2 \right) \hat{A}_j(k,1,\cdots ,1) \mod 1. \end{aligned}$$

(C.27)

This result was obtained by Gilkey [134] by a different method.

Now let us restrict to the case $m=4$. We get

$$\begin{aligned} p_1(k,1,1,1,1) = k^2+4, \qquad p_2(k,1,1,1,1) = 4k^2 + 6. \end{aligned}$$

(C.28)

Then we can compute the following, where all equalities are valid mod 1:

$$\begin{aligned} 12( \eta (\mathcal{{D}}_s) - \eta (\mathcal{{D}}_0) )&\equiv \frac{12}{k} \left( \mathrm{ch}_4( s+k/2) - \mathrm{ch}_4(k/2) \right) - \frac{1}{2k}p_1 \left( \mathrm{ch}_2( s+k/2) - \mathrm{ch}_2(k/2) \right) \nonumber \\&\equiv \frac{s^2( k^2- 2 + s^2 )}{2k} . \end{aligned}$$

(C.29)

By putting $s=1$, we get the second equation in (7.34). We can perform a consistency check of this result. For the background labeled by s, the gauge field $s \check{A}$ gives a 3-form background $\check{C} = (s \check{A}) \star (s \check{A}) = s^2 \check{C}_1$. Therefore, we have

$$\begin{aligned} 12( \eta (\mathcal{{D}}_s) - \eta (\mathcal{{D}}_0) ) = - \widetilde{\mathcal{{Q}}}( s^2 \check{C}_1) = - s^2 \widetilde{\mathcal{{Q}}}( \check{C}_1) - \frac{s^2(s^2-1)}{2} (\check{C}_1, \check{C}_1) . \end{aligned}$$

(C.30)

By using $- \widetilde{\mathcal{{Q}}}( \check{C}_1) = (k^2-1)/2k$ and $(\check{C}_1, \check{C}_1) = -1/k$, this equation gives the same result as (C.29) for general s.

We also compute

$$\begin{aligned} 30\eta (\mathcal{{D}}_0) - \frac{K_k(K_k-k)}{2k}&\equiv \frac{30}{k} \mathrm{ch}_4(k/2) - \frac{5}{4k}p_1 \mathrm{ch}_2(k/2) + \frac{7p_1^2 - 4p_2}{192k} - \frac{K_k(K_k-k)}{2k} \nonumber \\&= \frac{k^2-1}{24k} -\frac{k(k+1)(k-1)}{6} \nonumber \\&\equiv \frac{k^2-1}{24k}. \end{aligned}$$

(C.31)

This is the last equation in (7.34).

Computation of the $\eta$-Invariant on Lens Spaces

In this appendix we present methods for computing the values of the $\eta$-invariant on lens spaces $S^{2m-1}/{\mathbb {Z}}_k$ which are different from the method in Sect. C. We compute them as elements of ${\mathbb {R}}$ rather than ${\mathbb {R}}/{\mathbb {Z}}$.

The basic setup is as follows. Let

$$\begin{aligned} \mathbf {z} = (z^1, \cdots , z^m) \in {\mathbb { C}}^m. \end{aligned}$$

(D.1)

We consider the lens space defined by dividing the sphere $S^{2m-1} = \{ | \mathbf {z} | =1 \}$ by the ${\mathbb {Z}}_k$ action

$$\begin{aligned} \mathbf {z} \rightarrow e^{2\pi {\mathsf i}j /k} \mathbf {z} , \qquad (j =0,1,\cdots , k-1). \end{aligned}$$

(D.2)

We denote this space as $S^{2m-1}/{\mathbb {Z}}_k$.

On this space, we consider a $\text {spin}^c$ connection as follows. Let $\mathcal{{S}}_{{\mathbb { C}}^m}$ be the trivial $\text {spin}^c$ bundle on ${\mathbb { C}}^m$. We denote the coordinate of the fiber $\mathcal{{S}}_{{\mathbb { C}}^m}$ as $\Psi$. Then we define the action of ${\mathbb {Z}}_k$ by

$$\begin{aligned} \Psi \rightarrow e^{-2\pi {\mathsf i}j s /k} {\mathsf {R}}(j/k) \Psi \qquad (j =0,1,\cdots , k-1). \end{aligned}$$

(D.3)

where $s \in \frac{1}{2} {\mathbb {Z}}$ is a parameter which specifies the $\text {spin}^c$ connection, and

$$\begin{aligned} {\mathsf {R}}(\alpha ) = \exp \left( - \pi \alpha \left( \Gamma ^1\Gamma ^2 + \cdots + \Gamma ^{2m-1}\Gamma ^{2m} \right) \right) \qquad (0 \le \alpha \le 1). \end{aligned}$$

(D.4)

Here $\Gamma ^I ~(I=1,\cdots , 2m)$ are gamma matrices on ${\mathbb { C}}^m \cong {\mathbb {R}}^{2m}$. For this action to be a ${\mathbb {Z}}_k$ action, we must have

$$\begin{aligned} e^{-2\pi {\mathsf i}s } \exp \left( - \pi \left( \Gamma ^1\Gamma ^2 + \cdots + \Gamma ^{2m-1}\Gamma ^{2m} \right) \right) =1, \end{aligned}$$

(D.5)

or equivalently

$$\begin{aligned} s \in \frac{m}{2} + {\mathbb {Z}}. \end{aligned}$$

(D.6)

Then we define a bundle on $S^{2m-1}/{\mathbb {Z}}_k$ as,

$$\begin{aligned} \mathcal{{S}}_s = \{ (\mathbf {z}, \Psi ) \mid ~ |\mathbf {z} | =1, ~~ \overline{\Gamma }\Psi = + \Psi \} /{\mathbb {Z}}_k, \end{aligned}$$

(D.7)

where $\overline{\Gamma }$ is the chirality operator on $\mathcal{{S}}_{{\mathbb { C}}^m}$. This is a $\text {spin}^c$ bundle on $S^{2m-1}/{\mathbb {Z}}_k$.

We may use the notation that $\mathcal{{S}}_s = \mathcal{{S}}\otimes \mathcal{{L}}_s$, where $\mathcal{{S}}= \mathcal{{S}}_{s=0}$ is the spin bundle defined by the above construction with $s=0$, and $\mathcal{{L}}_s$ is a line bundle. This splitting is possible only for even m, but we may also formally use such splitting for odd m. Such a formal splitting is possible if we are only concerned with curvatures of the bundles.

We also consider bundles associated to bi-spinor fields $\Phi$. The relevant bundle on ${\mathbb { C}}^m$ for the bi-spinor is $\mathcal{{S}}_{{\mathbb { C}}^m} \otimes \overline{\mathcal{{S}}_{{\mathbb { C}}^m}} \otimes \mathcal{{L}}_t$, which means that $\Phi$ transforms as

$$\begin{aligned} \Phi \rightarrow e^{-2\pi {\mathsf i}j t /k} {\mathsf {R}}(j/k) \Phi {\mathsf {R}}(j/k)^{-1} \qquad (j =0,1,\cdots , k-1). \end{aligned}$$

(D.8)

Here $t \in {\mathbb {Z}}$ is an integer parameter. We construct the corresponding bundle on $S^{2m-1}/{\mathbb {Z}}_k$ by restricting the bi-spinor to $\overline{\Gamma }\Phi = \Phi$. We denote the bundle as $\mathcal{{S}}^\mathrm{sig}_t$; see Sect. 6.2 for the details of the signature index theorem.

Let $\mathcal{{D}}_s$ and $\mathcal{{D}}^\mathrm{sig}_t$ be the Dirac operator acting on sections of $\mathcal{{S}}_s$ and $\mathcal{{S}}^\mathrm{sig}_t$, respectively. We want to compute their $\eta$-invariant. We sometimes denote the $\eta$-invariant on a manifold Y as $\eta (Y)$ if the operator $\mathcal{{D}}$ is clear from the context, or if $\mathcal{{D}}$ is a general operator.

We present two methods: In Sect. D.1, we use ${\mathbb {Z}}_k$ orbifolds of the torus $T^{2m}$ to deduce the $\eta$-invariant. This method only uses the standard APS index theorem, but is only applicable to $k=2,3,4,6$. In Sect. D.2, we introduce and utilize the equivariant APS index theorem. This requires the reader to learn an additional mathematical machinery, but it allows the computation for arbitrary k. These two subsections can be read mostly independently.

1.1 APS index theorem on $T^{2m}/{\mathbb {Z}}_k$

Here we describe a method which only uses the APS index theorem, but is restricted to the cases $k=2,3,4,6$. It is done by computing the index on $T^{2m}/{\mathbb {Z}}_k$. The method here was used e.g. in [60, 65].

The (singular) manifold $T^{2m} /{\mathbb {Z}}_k$ is defined as follows. We first consider a torus $T^{2m}$ whose complex coordinate is denoted as $\mathbf {z} = (z^1, \cdots , z^m)$. It satisfies equivalence relations of the form $\mathbf {z} \sim \mathbf {z} + \mathbf {p}+\tau \mathbf {q}$, where $\mathbf {p}$ and $\mathbf {q}$ are vectors whose components are integers, and $\tau$ is the standard period matrix of the torus. For $k=2,3,4,6$ with specific $\tau$ for each k, we can act ${\mathbb {Z}}_k$ on the torus as in (D.2). Then we get $T^{2m}/{\mathbb {Z}}_k$. We can define $\text {spin}^c$ bundles in the same way as above.

The APS index theorem on $T^{2m}/{\mathbb {Z}}_k$ can be applied as follows. First, notice that the manifold $T^{2m}/{\mathbb {Z}}_k$ has orbifold singularities. Let $B_i$ be a small ball which is centered around a orbifold singularity labelled by i. The boundary is $\partial B_i = S^{2n-1}/{\mathbb {Z}}_{\ell _i}$ where $\ell _i$ is a divisor of k and depends on i. The index is defined as the APS index on a manifold which is obtained by subtracting $B_i$ from $T^{2m}/{\mathbb {Z}}_k$,

$$\begin{aligned} Z = T^{2m}/{\mathbb {Z}}_k - \bigsqcup _i B_i . \end{aligned}$$

(D.9)

This is nonsingular, and has as the boundary

$$\begin{aligned} \partial Z = \bigsqcup _i \overline{B_i}= \bigsqcup _i \overline{S^{2n-1}/{\mathbb {Z}}_{\ell _i}} \end{aligned}$$

(D.10)

where the overline means orientation reversal.

By the APS index theorem, we get

(D.11)

where we denote the $\eta$-invariant of the operator $\mathcal{{D}}_s$ or $\mathcal{{D}}^\mathrm{sig}_t$ on a manifold Y as $\eta (Y)$, and also similarly for the index. The minus sign is due to the fact that $\eta (\overline{Y}) = - \eta (Y)$.

More explicit form is determined by careful examination of the geometry. In the case $m=1$, one can see directly the following. $T^2/{\mathbb {Z}}_2$ has four orbifold points with $\ell _i=2$. $T^2/{\mathbb {Z}}_3$ has three orbifold points with $\ell _i=3$. $T^2/{\mathbb {Z}}_4$ has two orbifold points with $\ell _i=4$, and one orbifold point with $\ell _i=2$. $T^2/{\mathbb {Z}}_6$ has one orbifold point with $\ell _i=6$, one orbifold point with $\ell _i=3$, and one orbifold point with $\ell _i=2$.

Notice that if i is an orbifold point of $T^2/{\mathbb {Z}}_k$ with ${\mathbb {Z}}_{\ell _i}$ orbifold singularity, its pre-image in $T^2$ consists of $k/\ell _i$ points. Notice also that if i and j are two points in $T^2$ with orbifold singularity of type ${\mathbb {Z}}_{\ell _i}$ and ${\mathbb {Z}}_{\ell _j}$ respectively, then $\{i\} \times \{j\} \in T^2 \times T^2 =T^4$ is a fixed point with ${\mathbb {Z}}_\ell$ orbifold singularity, where $\ell$ is given by the greatest common divisor of $\ell _i$ and $\ell _j$. From these facts, one can determine the orbifold points of $T^{2m}/{\mathbb {Z}}_k$ from the knowledge of $T^2/{\mathbb {Z}}_k$.

The answer is given by

(D.12)

By solving these equations, we can determine the $\eta$-invariants as

(D.13)

The APS index theorem is valid under some boundary condition. In the present case of (D.9), the boundary condition is given by the following condition. We can shrink $B_i$ to zero size, and we extend the zero modes to the entire $T^{2m}/{\mathbb {Z}}_k$. The APS boundary condition is such that the extended zero modes remain finite at the orbifold points. This in turn implies that the zero modes come from the zero modes on $T^{2m}$ which are invariant under the ${\mathbb {Z}}_k$ action.

Therefore the index on $T^{2m}/{\mathbb {Z}}_k$ counts the net number of zero modes on $T^{2m}$ which are invariant under ${\mathbb {Z}}_k$. The zero modes on $T^{2m}$ are just constant modes $\partial _I \Psi =0$. The reason is that the Dirac operator is given by ${\mathsf i}\Gamma ^I \partial _I$, and $0=({\mathsf i}\Gamma ^I \partial _I)^2 \Psi = - \partial ^2\Psi$ which implies that $0 = \int _{T^{2m}}\Psi ^\dagger (- \partial ^2\Psi ) = \int _{T^{2m}}\partial _I \Psi ^\dagger \partial _I\Psi$. Therefore, we just need to count the number of components of the spinor field $\Psi$ which are invariant under ${\mathbb {Z}}_k$.

Let us consider the spinor field $\Psi$ which transforms like (D.3). We consider operators ${\mathsf i}^{-1} \Gamma ^{2i-1}\Gamma ^{2i}$ for $i=1,\cdots ,m$. We denote the eigenvalues of ${\mathsf i}^{-1} \Gamma ^{2i-1}\Gamma ^{2i}$ as $\sigma _i =\pm 1$. The chirality operator $\overline{\Gamma }$ is given by

$$\begin{aligned} \overline{\Gamma }= \prod _{i=1}^m \left( {\mathsf i}^{-1} \Gamma ^{2i-1}\Gamma ^{2i} \right) , \end{aligned}$$

(D.14)

and hence its eigenvalues are $\prod _{i=1}^m \sigma _i$. The condition that $\Psi$ is invariant under (D.3) is given by

$$\begin{aligned} s + \frac{1}{2} \sum _{i=1}^m \sigma _i \equiv 0 \mod k. \end{aligned}$$

(D.15)

We need to count the number of components of $\Psi$ satisfying this condition, and also determine the eigenvalues of $\overline{\Gamma }$ of these components. For this purpose, we rewrite the equation as

$$\begin{aligned} \sum _{i=1}^m \frac{ 1-\sigma _i }{2} \equiv s+\frac{m}{2} \mod k. \end{aligned}$$

(D.16)

Thus, the number of components which has $\sigma _i = -1$ is of the form $kj+s+\frac{m}{2}$ for $j \in {\mathbb {Z}}$. The chirality of such components is $\overline{\Gamma }= (-1)^{kj + s + \frac{m}{2}}$. Therefore, the index is given by using the binomial coefficient as

(D.17)

By putting this formula into (D.13), we get the values of the $\eta$-invariant.

The index of the operator acting on the bi-spinor field $\Phi$ transforming as (D.8) is more complicated. But the basic idea is the same. We just count the number of components which are invariant under (D.8).

We list some examples. For the operator $\mathcal{{D}}_s$, we get

(D.18)

For the operator $\mathcal{{D}}^\mathrm{sig}_t$, we get

(D.19)

Some of the results here were announced and used in [22].

1.2 Equivariant index theorem

The equivariant index theorem states the following. Let Z be a manifold with boundary $\partial Z = Y$. Suppose that a group G acts on the space Z (and any vector bundle on Z in which we are interested). We consider an element $g \in G$ such that the fixed points of the g action on Z are isolated points $p \in {\mathbb {Z}}$ which are not at the boundary, $p \notin Y$. In particular, g acts freely on Y.

Let $\mathcal{{D}}_Z$ be a Dirac operator which acts on sections of a bundle $\mathcal{{S}}_Z$ on Z. Let $\overline{\Gamma }$ be the chirality (or ${\mathbb {Z}}_2$ grading) operator $\{ \mathcal{{D}}_Z, \overline{\Gamma }\} =0$. Then the index can be defined as , where the trace is over the space spanned by the modes of $\mathcal{{D}}_Z$, and $\tau >0$ is an arbitrary positive constant. We can modify this definition by including $g \in G$ as

(D.20)

where on the right hand side g acts on the modes of $\mathcal{{D}}_Z$.

We also define the $\eta$-invariant twisted by g as follows. Consider the Dirac operator $\mathcal{{D}}_Y$ on Y which is constructed from $\mathcal{{D}}_Z$ as described in Sect. 4.2. Let $\psi _j$ be eigenmodes of $\mathcal{{D}}_Y$ which is in an irreducible representation $R_j(g)$ of G. Any mode in a single irreducible representation has the same eigenvalue $\lambda _j$. Then we define

$$\begin{aligned} \eta (\mathcal{{D}}_Y, g) = \frac{1}{2} \left( \sum _j \mathop {\mathrm {sign}}(\lambda _j) {{\,\mathrm{tr}\,}}R_j(g) \right) _\mathrm{reg}, \end{aligned}$$

(D.21)

where the subscript $\mathrm{reg}$ means an appropriate regularization.

If a point p is fixed by g, this element g acts on the fiber $(\mathcal{{S}}_Z)_{p}$ of the bundle $\mathcal{{S}}_Z$ by some matrix. We denote this matrix as $\rho _p(g)$. Also, g acts on the fiber $(TZ)_p$ of the tangent bundle TZ and we denote this matrix as $\tau _p(g)$.

Now we can state the equivariant index theorem [135]. The index for g satisfying the above conditions is given by

(D.22)

The sum is over the fixed points of g on Z. The trace ${{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho _p(g) )$ may also be called the supertrace of $\rho _p(g)$ under the ${\mathbb {Z}}_2$ grading $\overline{\Gamma }$.

The above equivariant index theorem may be understood as follows. We may try to prove the index theorem by the heat kernel method, which means that we use the expression (D.20) and take the limit $\tau \rightarrow +0$. Very roughly speaking, the boundary condition used in the APS index theorem is such that the boundary modes with $\mathop {\mathrm {sign}}(\lambda _i) = +1$ contributes to (D.20) with $\overline{\Gamma }=+1$, and the boundary modes with $\mathop {\mathrm {sign}}(\lambda _i) = -1$ contributes to (D.20) with $\overline{\Gamma }=-1$. This gives the boundary contribution $\eta (\mathcal{{D}}_Y, g)$ to the index. The bulk contribution is understood as follows. We can regard $H = \mathcal{{D}}_Z^2$ as a Hamiltonian of a quantum mechanical particle living on Z [136]. Then $e^{ - \tau \mathcal{{D}}_Z^2}$ is the Euclidean time evolution operator. Within a very short time $\tau \rightarrow +0$, it is very hard for a particle to go from a point $p \in Z$ to another point $g \cdot p \in Z$ unless these points are the same, $p = g \cdot p$. This implies that only the fixed points $p = g \cdot p$ contribute in the heat kernel method.⁴⁹ Near each fixed point p, we can approximate the manifold Z by a flat space ${\mathbb {R}}^D$ such that p corresponds to $0 \in {\mathbb {R}}^{D}$, where $D = \dim Z$. Then the trace ${{\,\mathrm{tr}\,}}( g \overline{\Gamma }e^{- \tau \mathcal{{D}}_Z^2})$ near the point p is given by

$$\begin{aligned}&{{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho _p(g) ) \int \frac{\mathrm{d}^D x\mathrm{d}^{D} k}{(2\pi )^{D} } e^{-{\mathsf i}k \cdot (\tau _p(g)x) } e^{- \tau k^2} e^{{\mathsf i}k \cdot x} \nonumber \\&\quad ={{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho _p(g) ) \int \mathrm{d}^{D}k\, e^{- \tau k^2} \delta \left( (1-\tau _p(g) )k \right) \nonumber \\&\quad = \frac{ {{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho _p(g) ) }{\det ( 1 -\tau _p(g) ) }. \end{aligned}$$

(D.23)

This is what appears in (D.22).

Now suppose that G is a finite group whose elements, except for the identity element $1 \in G$, satisfy the above conditions. We can use the equivariant index theorem to compute the $\eta$-invariant on a manifold Y/G which is smooth because of the assumption that G acts freely on Y. The $\eta$-invariant on this manifold is given by

$$\begin{aligned} \eta (\mathcal{{D}}_{Y/G} ) = \frac{1}{ |G|} \sum _{ g \in G} \eta (\mathcal{{D}}_Y, g) , \end{aligned}$$

(D.24)

where |G| is the number of elements of the finite group G. The $\eta (\mathcal{{D}}_Y, g)$ for $g \ne 1$ is given by (D.22), while for $g=1$ we simply use the ordinary APS index theorem

(D.25)

where F and R are gauge and Riemann curvatures on Z which are relevant to the index of the Dirac operator $\mathcal{{D}}_Z$. Therefore, we get

(D.26)

This is the general expression.

In a special case that there are no zero modes of $\mathcal{{D}}_Z$ and no curvature ($F=0$ and $R=0$), the formula becomes simple. This is the case for $Z = B^{2m} = \{ \mathbf {z} \in {\mathbb { C}}^{m} ; |\mathbf {z}| \le 1 \}$, $Y = S^{2m-1}$ and all the backgrounds are trivial.⁵⁰ We take $G = {\mathbb {Z}}_k$ which acts as (D.2). There is only a single fixed point $p =0 \in B^{2m}$. On this point we get

$$\begin{aligned} \det ( 1 -\tau (j) ) = |1 - e^{2\pi {\mathsf i}j /k}|^{2m}, \end{aligned}$$

(D.27)

where $j \in {\mathbb {Z}}_k$. The matrix $\rho (g)$ is determined from (D.3) or (D.8). In each case the trace ${{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho _p(g) )$ is given by

$$\begin{aligned} {{\,\mathrm{tr}\,}}(\overline{\Gamma }\rho (j) ) = \left\{ \begin{array}{ll} e^{-2\pi {\mathsf i}j s/k} ( e^{- \pi {\mathsf i}j /k} - e^{\pi {\mathsf i}j /k} )^m, &{} \text {(Dirac)}, \\ e^{-2\pi {\mathsf i}j s/k} ( e^{-\pi {\mathsf i}j /k} - e^{\pi {\mathsf i}j /k} )^m ( e^{-\pi {\mathsf i}j /k} + e^{\pi {\mathsf i}j /k} )^m,&{} \text {(signature)}. \end{array} \right. \end{aligned}$$

(D.28)

where we have used the fact that the chirality is given by $\overline{\Gamma }= \prod _{i=1}^m \left( {\mathsf i}^{-1} \Gamma ^{2i-1}\Gamma ^{2i} \right) .$ Therefore, the equivariant index theorem gives

$$\begin{aligned} \eta (\mathcal{{D}}_s)&= -\frac{{\mathsf i}^{-m}}{k} \sum _{j=1}^{k-1} \frac{e^{-2\pi {\mathsf i}j s/k} }{( 2 \sin ( \pi j/k) )^m } , \end{aligned}$$

(D.29)

$$\begin{aligned} \eta (\mathcal{{D}}^\mathrm{sig}_t)&= -\frac{{\mathsf i}^{-m}}{k} \sum _{j=1}^{k-1} \frac{e^{-2\pi {\mathsf i}j t/k} }{( \tan ( \pi j/k) )^m }. \end{aligned}$$

(D.30)

As examples, we list the values for the case of $m=4$, $s=0$ and $t=0$.

$$\begin{aligned} \begin{array}{c|ccccccccc} k &{} 2 &{} 3 &{} 4 &{} 5 &{} 6 &{} 7 &{} 8 &{} 9 &{} 10 \\ \hline \eta (\mathcal{{D}}_{s=0}, S^7/{\mathbb {Z}}_k) &{} -\frac{1}{32}&{} - \frac{2}{27}&{} - \frac{9}{64}&{} - \frac{6}{25}&{} - \frac{329}{864}&{} -\frac{4}{7}&{} - \frac{105}{128}&{} - \frac{92}{81}&{} -\frac{1221}{800}\\ \eta (\mathcal{{D}}^\mathrm{sig}_{t=0}, S^7/{\mathbb {Z}}_k ) &{} 0&{} - \frac{2}{27}&{} -\frac{1}{2}&{} -\frac{36}{25}&{} -\frac{82}{27}&{} -\frac{38}{7}&{} -\frac{35}{4}&{} -\frac{1064}{81}&{} -\frac{468}{25} \end{array}. \end{aligned}$$

(D.31)

Notice the agreement with the results in Sect. D.1 for $k=2,3,4,6$. The present method is valid for any k.

Non-unitary Counterexamples to Cobordism Classification

The recent understanding of the anomaly and the corresponding invertible phases states that unitary topological invertible phases are in bijection to the Pontryagin dual of the bordism classes [61, 62, 103, 137]. This statement is often called the cobordism classification of the invertible phases. In particular, the partition function of a unitary topological invertible phase is a bordism invariant.

Here we present a simple class of non-unitary invertible topological phases whose partition function is not a bordism invariant; in particular, its partition function on $S^D$ is $-1$.⁵¹ These examples illustrate the necessity of the unitarity condition in the cobordism classification. In this Appendix, D is the spacetime dimensions of the bulk invertible phase, which was denoted by $d+1$ in the main part of the text.

1.1 The simplest example

The simplest example is given by a “massive bc ghost system” in $D=1$ dimensions,

$$\begin{aligned} \mathcal{{L}}= b \left( {\mathsf i}\frac{\mathrm{d}}{\mathrm{d}t} - m \right) c = - b \left( \frac{\mathrm{d}}{\mathrm{d}\tau } +m \right) c , \end{aligned}$$

(E.1)

where t is the time coordinate, and $\tau = {\mathsf i}t$ is the Euclidean time coordinate. b and c obey Fermi-Dirac statistics, but they are not spinors; we will discuss their representations under Lorentz symmetry later for the case of general dimensions D, but for $D=1$ one can think of them just as scalars. We regularize this theory by Pauli-Villars regularization with Pauli-Villars mass M, and we take $m = - M$ and $M \rightarrow \infty$. Because $|m| \rightarrow \infty$, its Hilbert space is one-dimensional and spanned by the ground state. We need only $\mathrm {SO}(D)$ symmetry to define this theory, and no spin structure is necessary.

When $|M| = | m|$, the partition function on $S^1$ is given by

$$\begin{aligned} \mathcal{{Z}}(S^1) = \frac{\det ( \frac{\mathrm{d}}{\mathrm{d}\tau } +m) }{\det (\frac{\mathrm{d}}{\mathrm{d}\tau } +M)} = \frac{m}{M}. \end{aligned}$$

(E.2)

It is $\mathcal{{Z}}(S^1) = -1$ for $m=-M$. Obviously $S^1$ is trivial in the bordism group $\Omega ^1_{\mathrm {SO}}(\text {pt})$. This gives a counterexample to the cobordism classification.

The point is that b and c obey the periodic boundary conditions and the zero mode contribute the above factor m/M. (Nonzero modes do not contribute to the phase of the partition function.) If we instead consider a massive fermion, the circle $S^1_{\text {NS}}$ which is trivial in $\Omega ^1_{\text {spin}}(\text {pt})$ has the anti-periodic boundary condition, and in that case we get $\mathcal{{Z}}(S^1_{\text {NS}})=+1$. This is consistent with the cobordism classification. For the periodic boundary condition, we get $\mathcal{{Z}}(S^1_{\text {R}})=-1$ which is of course consistent because $S^1_{\text {R}}$ is the nontrivial element of $\Omega ^1_{\text {spin}}(\text {pt}) = {\mathbb {Z}}_2$.

1.2 Abstract description

The abstract description of the system which is closely related to the above one was discussed in [61]. Here we reproduce a sketch of the argument; we refer the reader to [61, 62] for more details on the axioms used below.

First we give general discussions. If we have two Hilbert spaces $\mathcal{{H}}_A$ and $\mathcal{{H}}_B$, then $\mathcal{{H}}_A \otimes \mathcal{{H}}_B$ and $\mathcal{{H}}_B\otimes \mathcal{{H}}_A$ are isomorphic. We want a way to identify their elements, so we want to have a map

$$\begin{aligned} \tau : \mathcal{{H}}_A \otimes \mathcal{{H}}_B \rightarrow \mathcal{{H}}_B \otimes \mathcal{{H}}_A. \end{aligned}$$

(E.3)

such that

$$\begin{aligned} \tau : |a\rangle \otimes |b\rangle \mapsto \epsilon _{a,b} |b\rangle \otimes |a\rangle , \end{aligned}$$

(E.4)

where $\epsilon _{a,b}$ is a sign factor. Mathematically, we need such $\tau$ to define the symmetric monoidal category of super vector spaces.

Physically, we might expect the states $|a\rangle$ and $|b\rangle$ to have statistics $\mathrm{deg} (a)$ and $\mathrm{deg} (b)$.⁵² Mathematically this is the ${\mathbb {Z}}_2$ grading in super vector spaces. $\mathrm{deg}$ is even for Bose-Einstein statistics and odd for Fermi-Dirac statistics. Then we expect

$$\begin{aligned} \epsilon _{a,b} = (-1)^{\mathrm{deg}(a )\mathrm{deg}(b ) }. \end{aligned}$$

(E.5)

Now we want to compute the partition function on $S^1$. For this purpose, we first consider an interval $I=[0,\beta ]$ and glue the two ends. The amplitude on the interval I is

$$\begin{aligned} e^{-\beta H} = \sum e^{-\beta E_a} |a\rangle \otimes \langle a| \in \mathcal{{H}}_A \otimes \mathcal{{H}}_A^* \end{aligned}$$

(E.6)

in the obvious notation. To glue the two ends we need to exchange $|a\rangle \otimes \langle a|$ to $\langle a| \otimes |a\rangle$ and then use a natural map (gluing) $\langle a| \otimes |b\rangle \rightarrow \langle a| b \rangle$. However, this exchange gives the sign

$$\begin{aligned} \tau : |a\rangle \otimes \langle a| \mapsto (-1)^{\mathrm{deg}(a) } \langle a| \otimes |a\rangle . \end{aligned}$$

(E.7)

Let us also consider another quantity, which we denote as $\mathrm{f}(a)$ and is defined by

$$\begin{aligned} (-1)^F |a\rangle = (-1)^{\mathrm{f}(a)} |a\rangle . \end{aligned}$$

(E.8)

Here $(-1)^F \in \mathrm {Spin}(D)$ is in the center. This quantity determines whether the state $|a\rangle$ is a spinor or not.

Now the partition function on $S^1$ is given by⁵³

$$\begin{aligned} \mathcal{{Z}}(S_\mathrm{{R}}^1)&= \sum e^{-\beta E_a} (-1)^{\mathrm{deg}(a) } , \end{aligned}$$

(E.9)

$$\begin{aligned} \mathcal{{Z}}(S_\mathrm{{NS}}^1)&= \sum e^{-\beta E_a} (-1)^{\mathrm{deg}(a) + \mathrm{f}(a)}, \end{aligned}$$

(E.10)

where we have used the fact that $S_\mathrm{{NS}}^1$ has an additional insertion of $(-1)^F \in \mathrm {Spin}(D)$. What makes the usual thermal partition function positive definite is the spin-statistics theorem $\mathrm{deg}(a) + \mathrm{f}(a)=0 \mod 2$.

The massive bc system considered above has $\mathrm{deg}(\Omega ) = 1$ but $\mathrm{f}(\Omega )=0$, where $|\Omega \rangle$ is the ground state, which is the only state in the limit of large mass gap. Therefore, we get $\mathcal{{Z}}(S^1)=-1$.

1.3 Generalizations to odd dimensions

We generalize the above system to theories in dimension $D=2n+1$. The following discussion is generally valid for $D = 4\ell +1$, but for $D=4\ell +3$ we will need to restrict to some specific dimensions as we discuss later.

Consider a manifold Y with $\dim Y = D=2n+1$. We take c and b to be sections of $\mathcal{{S}}\otimes \mathcal{{S}}^*$, where $\mathcal{{S}}$ is the spin bundle on Y and $\mathcal{{S}}^*$ is the dual to $\mathcal{{S}}$. For this theory itself, we do not need spin structure because of the relation

$$\begin{aligned} \mathcal{{S}}\otimes \mathcal{{S}}^* \cong \sum _{i=0}^n \wedge ^{2i} T^* Y. \end{aligned}$$

(E.11)

Notice that only even degrees appear. There is an isomorphism $\wedge ^{2i} T^* Y \cong \wedge ^{D-2i} T^* Y$, so we could have used odd degree instead. Sect. 6.2 for the details.

The Lagrangian of the theory is

(E.12)

Here gamma matrices $\Gamma ^I$ of only act on the first factor $\mathcal{{S}}$ in $\mathcal{{S}}\otimes \mathcal{{S}}^*$, while the covariant derivative $D_I$ acts on both factors. Thus is the Dirac operator relevant for the signature index theorem, which we discussed in detail in Sect. 6.2. However, we remark that $\mathcal{{D}}^\mathrm{sig}_Y$ in Sect. 6.2 acts on $\mathcal{{S}}\otimes (\mathcal{{S}}^* \oplus \mathcal{{S}}^*)$, so $\eta (\mathcal{{D}}^\mathrm{sig}_Y)$ of that section is twice the $\eta (\mathcal{{D}})$ of this appendix.

As before, we take $m = -M$ and $M \rightarrow \infty$. Then the partition function is

$$\begin{aligned} \mathcal{{Z}}(Y) = \exp ( -2 \pi {\mathsf i}\eta (\mathcal{{D}})). \end{aligned}$$

(E.13)

The zero modes of the operator $\mathcal{{D}}$ is given by the zero modes on $\sum _{i=0}^n \wedge ^{2i} T^* Y$. Let us define

$$\begin{aligned} b = \sum _{i=0}^{n} \dim H^{2i}(Y,{\mathbb {R}}), \end{aligned}$$

(E.14)

which is the sum of the Betti numbers of even dimensional cohomology. Then the number of zero modes is given by b, and

$$\begin{aligned} \eta (\mathcal{{D}}) = \frac{b}{2} + \frac{1}{2} \sum _{\lambda \ne 0} \frac{\lambda }{|\lambda |}, \end{aligned}$$

(E.15)

where the sum is over nonzero eigenvalues of $\mathcal{{D}}$. By the discussion in Sect. 6.2, this sum over nonzero eigenvalues is the same as of that section and hence we get .

Suppose that Y is a boundary of a $D+1$-manifold Z. By the signature index theorem (6.40) and the fact that , we have

$$\begin{aligned} \eta (\mathcal{{D}}) = \frac{1}{2} \left( b+\sigma (Z) - \int _Z L \right) . \end{aligned}$$

(E.16)

Here the signature is defined by the pairing on the relative cohomology $H^\bullet ( Z, \partial Z, {\mathbb {R}})$.

Let us consider the partition function on $Y = S^D$. It has $b=1$ coming from $H^0(S^D,{\mathbb {R}})$. We emphasize that $H^D(S^D,{\mathbb {R}})$ does not contribute because we summed only over even-degree cohomology. We can take the $(D+1)$-manifold as $Z = B^{D+1}$, which is the $(D+1)$-dimensional ball. The signature $\sigma (Z)$ and the Hirzebruch polynomial L (for a round sphere metric) are zero on the ball $B^{D+1}$. We conclude that

$$\begin{aligned} \eta (\mathcal{{D}}_{S^D}) = \frac{1}{2} \end{aligned}$$

(E.17)

and hence

$$\begin{aligned} \mathcal{{Z}}(S^D) = -1. \end{aligned}$$

(E.18)

Notice that $S^D = \partial B^{D+1}$ is trivial in the bordism groups of both $\mathrm {SO}$ and $\mathrm {Spin}$.

The above discussion was general for any $D=2n+1$. For $D=4\ell +1$, there is no perturbative gravitational anomaly in $d=4\ell$ and hence the invertible theory on $D=4\ell +1$ is topological. In fact, in these dimensions nonzero modes always cancel and we have

$$\begin{aligned} \mathcal{{Z}}(Y) = (-1)^{b}=(-1)^{\sum _i \dim H^{2i}(Y)}. \end{aligned}$$

(E.19)

It is interesting that the partition function is determined by Betti numbers.

In dimensions $D=7$ and $D =8\ell +3$, we can combine the above theory with the anomaly theories relevant to fermions and self-dual 2-form fields to cancel the perturbative anomaly. Then we get a topological theory. The anomaly theories for fermions and 2-form fields are unitary, and hence they do not change the above conclusion $\mathcal{{Z}}(S^D)=-1$.

1.4 Remark on the anomaly polynomial and the Euler characteristic class

Some theories we have discussed above, such as the ones in $D=4\ell +1$-dimensions, are topological theories in the sense that partition functions are topological invariants. But they can still be interpreted to have anomaly polynomials which are given by the Euler characteristic class. Let us explain this point.

To see the point clearly, and also to generalize the situation slightly, we consider a bundle P with the structure group $\mathrm {Spin}(D+1)$ which is not necessarily associated to the tangent bundle. More precisely, we consider a $(D+1)$-manifold Z with a bundle whose structure group is given by $[\mathrm {Spin}(D+1)_1 \times \mathrm {Spin}(D+1)_2]/{\mathbb {Z}}_2$, where the first $\mathrm {Spin}(D+1)_1$ is the Lorentz group and the second $\mathrm {Spin}(D+1)_2$ is the one we have introduced above. Below we consider as if the group is $\mathrm {Spin}(D+1)_1 \times \mathrm {Spin}(D+1)_2$ just for notational simplicity, but the discussions below make sense even if we divide the group by ${\mathbb {Z}}_2$.

Let $\mathcal{{T}}_{\pm }$ be the spin bundles with positive and negative chirality associated to P. We consider Dirac operators ${\mathcal {D}}_{Z,\mathcal{{T}}^*_{\pm }}$ coupled to $\mathcal{{T}}^*_{\pm }$. Namely, it acts on sections of $\mathcal{{S}}_{Z} \otimes \mathcal{{T}}^*_{\pm }$, where $\mathcal{{S}}_{Z} = \mathcal{{S}}_+ \oplus \mathcal{{S}}_-$ is the spin bundle on the manifold Z associated to the tangent bundle TZ. Suppose that the manifold Z has a boundary $\partial Z =Y$. We denote the relevant APS $\eta$-invariants as $\eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{\pm }} )$. Here the Dirac operator ${\mathcal {D}}_{Y,\mathcal{{T}}^*_{\pm }}$ acts on sections of $\mathcal{{S}}_Y \otimes \mathcal{{T}}^*_{\pm }$ where $\mathcal{{S}}_Y := \mathcal{{S}}_+|_{Y} = \mathcal{{S}}_-|_{Y}$. By using the APS index theorem, one can see that⁵⁴

$$\begin{aligned}&\mathrm{index}({\mathcal {D}}_{Z,\mathcal{{T}}^*_{+}}) - \mathrm{index}({\mathcal {D}}_{Z,\mathcal{{T}}^*_{-}}) = \int _Z E + \left( \eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{+}} ) -\eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{-}} ) \right) \end{aligned}$$

(E.20)

$$\begin{aligned}&\mathrm{index}({\mathcal {D}}_{Z,\mathcal{{T}}^*_{+}}) + \mathrm{index}({\mathcal {D}}_{Z,\mathcal{{T}}^*_{-}}) = \int _Z I(p, p') + \left( \eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{+}} ) + \eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{-}} ) \right) \end{aligned}$$

(E.21)

where E is the Euler characteristic class of P, and $I(p,p')$ is some polynomial of the Pontryagin classes of the tangent bundle (denoted as $p=\{p_1,p_2,\cdots \}$) and the Pontryagin classes of the bundle P (denoted as $p'=\{p'_1,p'_2,\cdots \}$). From the above formulas, we get

$$\begin{aligned} -\eta ({\mathcal {D}}_{Y,\mathcal{{T}}^*_{+}} ) = \frac{1}{2} \int _Z \left( E + I(p, p')\right) - \mathrm{index}({\mathcal {D}}_{Z,\mathcal{{T}}^*_{+}}) \end{aligned}$$

(E.22)

In particular, we see that $\int _Z \left( E + I(p, p')\right)$ is divisible by 2 if the boundary is empty, $\partial Z = \varnothing$.

Now let us consider the case that the bundle P is associated to the tangent bundle so that $\mathcal{{T}}_{\pm } = \mathcal{{S}}_{\pm }$. In this case, $\mathcal{{S}}:=\mathcal{{S}}_Y = \mathcal{{T}}_{+}|_{Y} = \mathcal{{T}}_{+}|_{Y}$. The above result implies that $\frac{1}{2} \left( E + I(p, p)\right)$ (where we have set $p'=p$) is the anomaly polynomial for the theory (E.13). Notice that if $D+1 = 4\ell +2$, the term $\int I(p,p')$ is zero. Also, for a hemisphere which has the standard metric, we have

$$\begin{aligned} \exp \left[ 2\pi {\mathsf i}\cdot \frac{1}{2} \int _\mathrm{hemisphere} \left( E + I(p, p') \right) \right] = -1, \end{aligned}$$

(E.23)

since the Euler number of the hemisphere is 1. More generally, the Euler number is a topological invariant even for manifolds with boundaries. (We usually need a term on the boundary related to extrinsic curvature, but the APS index theorem assumes that the region near the boundary is of cylindrical form, so the extrinsic curvature is zero.) Thus the theory (E.13) (for $D=4\ell +1$) is topological even though it has a nontrivial anomaly polynomial $\frac{1}{2} E$.

Let us remark that the theories considered here can still be related to bordism theory in a certain generalization. As we have seen above, the polynomial $\frac{1}{2} \left( E + I(p, p')\right)$ satisfies the integrality condition $\frac{1}{2} \int \left( E + I(p, p')\right) \in {\mathbb {Z}}$ on closed manifolds. This is valid for any P not necessarily associated to the tangent bundle. In particular, it is valid in the special case that the bundle $P \times _{\rho } {\mathbb {R}}^{D+1}$ (where $\rho$ is the vector representation of $\mathrm {Spin}(D+1)_2$) is only stably isomorphic to the tangent bundle TZ, that is, $P \times _{\rho } {\mathbb {R}}^{D+1} \oplus {\mathbb {R}}^K \simeq TZ \oplus {\mathbb {R}}^K$ for some K. By the result of [140], the theory in D-dimensions considered above give an element of the Anderson dual to a certain bordism theory, $(I\Omega ^G)^{D+1}(\text {pt})$. Here, we need to take the sequence of groups $G=\{G_k\}$ in that paper to be $G_k=\mathrm {SO}(k)$ for $k \le D+1$, and $G_k = \mathrm {SO}(D+1)$ for $k >D+1$ so that we can allow the Euler density E as a possible characteristic class.

Finally, we note that the choice of $G=\{G_k\}$ above does not satisfy the conditions⁵⁵ discussed by Freed and Hopkins [61], which are required to formulate reflection positivity. Therefore, there is no contradiction with the results of [61, 62], which relates unitary invertible theories with Anderson/Pontryagin duals of bordism groups for $G=\{G_k\}$ satisfying the conditions of Freed and Hopkins.