SciPost Phys. 7, 058 (2019)

Why are fractional charges of orientifolds
compatible with Dirac quantization?
Yuji Tachikawa1? and Kazuya Yonekura2
1 Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2 Faculty of Arts and Science, Kyushu University? yuji.tachikawa@ipmu.jp

Abstract
Orientifold p-planes with p ≤ 4 have fractional Dp-charges, and therefore appear inconsistent with Dirac quantization with respect to D(6−p)-branes. We explain in detail how
this issue is resolved by taking into account the anomaly of the worldvolume fermionsusing the η invariants. We also point out relationships to the classification of interacting
fermionic symmetry protected topological phases. In an appendix, we point out that theduality group of type IIB string theory is the pin+ version of the double cover of GL(2, Z).
Copyright Y. Tachikawa and K. Yonekura.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 21-05-2018
Accepted 10-10-2019
Check forupdatesPublished 01-11-2019
doi:10.21468/SciPostPhys.7.5.058

Contents
1 Introduction and summary

2

2 Careful analysis of anomalies
2.1 Conventions concerning index theorems
2.2 Conventions concerning string theory
2.3 O-planes and fermions
2.4 The η invariant and the anomaly cancellation

6
6
10
13
17

3 Relations to symmetry-protected topological phases
3.1 Extensions of the spin group
3.2 Properties of (s)pinors on RPn
3.3 O4 and RP4
3.4 O3 and RP5
3.5 O2 and RP6
3.6 O1 and RP7
3.7 O0 and RP8
3.8 Summary

19
20
21
22
22
23
23
25
25

A The duality group of type IIB string theory

25

References

27
1

SciPost Phys. 7, 058 (2019)1

Introduction and summary

The trouble: Dp-branes and Dq-branes are electromagnetic dual objects for p + q = 6, andtheir charges satisfy the Dirac quantization law with minimum possible value [1]. Now, it iswell known that an Op+ -plane has the Dp-brane charge +2 p−5 , where we define its charge byintegrating the flux around the angular direction RP8−p .1 In particular, its value is fractional
when p ≤ 4. Concretely, we have
O4+ O3+ O2+ O1+ O0+
Dp-charge

1
2

1
4

1
8

1
16

1
32

.

(1.1)

Naively, this contradicts the Dirac quantization law. The aim of this paper is to see how this contradiction can be resolved, by taking into account the anomaly of the worldvolume fermions of
Dq-branes. Our resolution is in part motivated by the recent development in the understandingof the classification of the interacting fermionic symmetry protected topological phases.
In this paper we only treat the Op+ -planes with the charge +2 p−5 , and leave the discussionof the Op− -planes with the charge −2 p−5 to another work.2 Whenever we say “Op-plane", thatmeans Op+ -plane in this paper.
The Dirac quantization law: Let us first recall why the Dirac quantization law is necessary.
Consider a Dq-brane wrapped on Y of spacetime dimension q + 1. It has a worldvolume
coupling
Z
Cq+1 ,

2πi

(1.2)

Y

where Cq+1 is the Ramond-Ramond (RR) potential sourced by Dp-branes. Cq+1 is not globallywell-defined when there is a nonzero flux, so the expression (1.2) is not entirely precise.
What we actually mean is that the RR couplings when the worldvolume is Y = Y1 andY = Y2 differ by the formulaZ
2πi

Fq+2 ,(1.3)

X

where X is a (q + 2)-dimensional manifold such that ∂ X = Y1 t Y2 and Fq+2 is the RR fieldstrength. For illustration, see Fig. 1. There, we depicted a case when Y1 is connected but Y2
consists of two disconnected components. This is in general necessary, since Dq-branes cansplit and merge as they interact.
The choice of such X is not unique. Therefore, we need to requireRR

e2πi X Fq+2 = e2πi X 0 Fq+2 ,(1.4)

for two (q + 2)-dimensional manifolds X and X 0 such that ∂ X = ∂ X 0 = Y1 t Y2 . Equivalently,
we need to requireZ
Fq+2 ∈ Z

for any ∂ X = 0.

(1.5)

X

The condition (1.5) is not satisfied for Op-plane when p ≤ 4. Therefore, the phase of thepartition function of Dq-branes in this background is not well-defined, if we only consider thephase coming from the coupling to the RR flux. For the string theory to be consistent, there
should be an additional source of the phase.
For reviews on orientifolds, see e.g [2–4]. A precise formulation of orientifolds in perturbative string theory
was given in [5], but in this paper we stick to a more traditional viewpoint.
2
In the case of the Op− -planes, the U(1) gauge fields on Dq-branes also contribute to the anomaly. See [6] forthe case of the O3− -plane.
1

2

SciPost Phys. 7, 058 (2019)

Y1

X

Y1

Y2

Xʹ

Y2

Figure 1: To compare the phases of the partition functions of D-branes wrappedaround Y1 and Y2 , we use a manifold X satisfying ∂ X = Y1 t Y2 . There can be
multiple such choices.

Previous analyses: The mildest case is when p = 4. Then, the violation of the Dirac quantization law is only a sign, since the D4-charge of an O4-plane is 1/2. In [7, 8], it was noticed
that the phase of the partition function of the world-volume fermionof D2-branes in thisRbackground has a global anomaly which also produces a sign (−1) X w4 , where w4 is the 4th
Stiefel-Whitney class of the spacetime manifold. This means that, instead of (1.5), we need
the shifted quantization condition
Z
Z
1
F4 =
w4 mod Z,
(1.6)
2 X
X
and this guarantees that the phase of the partition function of D2-branes are well-defined.
The main point of this paper is that this analysis can be generalized to all lower p in anatural manner.3 The essential ingredient is the realization that the most general form of
the anomaly of fermions is given in terms of the η invariant via Dai-Freed theorem [10–13],4
and this point of view also underlies the recent development of fermionic symmetry protected
topological (SPT) phases in terms of cobordisms [15].5 Let us quickly review this framework.
Fermion anomaly via η invariants: Suppose we would like to study a fermion system on(q + 1)-dimensional manifolds Y governed by the Dirac operator DY . We would like to characterize the phase difference of the partition function Zfermion (Y1 ) and Zfermion (Y2 ) on Y1 andY2 , connected by a (q + 2)-dimensional manifold X as before: ∂ X = Y1 t Y2 . We have a Dirac
operator DX on X naturally associated to DY . Then we havearg Zfermion (Y1 ) − arg Zfermion (Y2 ) = −2πη(DX ),(1.7)

where the right hand side is the η invariant associated to DX (with a particular boundary
condition at ∂ X ).
When X does not have a boundary ∂ X = 0 and X itself is a boundary of a (q + 3)dimensional manifold W , i.e. ∂ W = X , there is also a Dirac operator DW on W naturally
For a previous attempt for lower p, see [9].
R
In fact, it was already pointed out in [8] that the sign (−1) X w4 comes from the η
R invariant. We should alsomention that the string worldsheet has a similar issue in the Dirac quantization, where X H3 depends continuously
on X because d H 6= 0. Again this cancels agains the η invariant governing the anomaly of the worldsheet fermions[14].
5
We note that the relation between the global anomaly of fermions and the η invariant was already used inearly and important papers by Alvarez-Gaume, Della Pietra, Moore [16] and Witten [17, 18]. In [18], to show theabsence of the global anomaly in ten-dimensional E8 × E8 heterotic string theory, Witten even used the vanishing
spinof the certain cobordism group Ω11 (K(Z, 4)), which was proved in the appendix of that paper by R. Stong.
3
4

3

SciPost Phys. 7, 058 (2019)associated to DY and DX , and we have the Atiyah-Patodi-Singer index theorem [19–21] (seealso [22] for explanations for physicists)Z
η(DX ) = index(DW ) −

PW ,

(1.8)

W

where index(DW ) is the index of the Dirac operator DW and PW is a characteristic polynomialconstructed from the curvatures on W .
The combination of (1.7) and (1.8) can be thought of as a refinement of the anomaly
descent formalism often found in the standard quantum field theory textbooks
R which areavailable today. Namely, the equation (1.8) says that η(DX ) is more or less − X SX where
d SX = PW , and the equation (1.7) says that this Chern-Simons term controls the anomaly inthe phase of the fermion partition function. The point is that the η invariant contains additional information about the discrete constant part controlling the Rglobal anomaly, in addition
to the perturbative anomaly contained in the Chern-Simons term X SX .
Shifted quantization law via η invariants: After this quick review of the modern understanding of the fermion anomaly, let us come back to the question of the Dq-brane again.
The phase comes from the RR coupling and the fermion determinant. Therefore, the phasedifference of the partition function of a Dq-brane on Y1 and Y2 is given byZ
Fq+2 − 2πη(DX ),arg ZDq (Y1 ) − arg ZDq (Y2 ) = 2π

(1.9)

X

for an X = Y1 t Y2 . For this equation to be consistent independent of the choice of X , we needto haveZ
Fq+2 − η(DX ) = 0 mod Z,(1.10)

Xfor ∂ X = 0. The case p = 4 is reproduced since it is known that η(DX ) = 12 w4 mod Z in thiscase.
Let us now come back to our original question of the orientifolds. For the consistency of
the Dq-brane in the background of an Op-plane, we only have to show that
η(DRP8−p ) = ±2 p−5 mod Z,(1.11)

for the Dirac operator DRP8−p appropriate for a Dq-brane on the Op-plane background. The
precise sign will be determined later in Sec. 2.
The η invariant of real projective spaces: These η-invariants of real projective spaces werecomputed by Gilkey [23–25] using the equivariant version of the Atiyah-Patodi-Singer theorem
proved by Donnelly [26]. Let us sketch how the computation using the equivariant index
theorem goes.6 Suppose X = X̃ /G and X̃ = ∂ W̃ where G acts on W̃ such that the fixed pointsof the action of G are isolated points on the interior of W , see Fig. 2. We assume that there
are no curvature contributions from smooth parts. Then the equivariant version of (1.8) saysthatX
η(DX̃ , g) = index(DW̃ , g) −Pp ,(1.12)
pwhere η(DX̃ , g) and index(DW̃ , g) are defined by using the trace
P of g ∈ G instead of thedimension of the eigenspaces of the Dirac operator. The sum p is over the isolated fixed
In Sec. 2.4 we give another argument for the computation of the relevant η invariant without using the equivariant index theorem, following [11].
64

SciPost Phys. 7, 058 (2019)

̃
X

̃
W

p

G

̃ /G
X=X
Figure 2: The η invariant of a freely-acting quotient X = X̃ /G can often be computedby considering the equivariant APS index theorem on X̃ = ∂ W̃ such that the fixed
points p of G action on W̃ are isolated.

points p of the action of g ∈ G, and Pp is the local contribution at the fixed point p, given
solely in terms of how g acts there.
In our case, we take X = RPq+2 , X̃ = S q+2 , and W̃ = B q+3 , with G = Z2 acting on W̃ in thestandard manner. The only fixed point is at the origin. We haveη(DS q+2 , g = +1) = η+ (DS q+2 ) + η− (DS q+2 ),(1.13)η(DS q+2 , g = −1) = η+ (DS q+2 ) − η− (DS q+2 )(1.14)

andη(DS q+2 ) = η(DS q+2 , g = +1) = 0,

η(DRPq+2 ) = η+ (DS q+2 ).

(1.15)

Therefore we have1
Pp=0 .
(1.16)
2
What remains is the determination of the local contribution Pp=0 at the origin p = 0. This canbe done by a careful computation of Pp=0 since a general formula for it is known. But it canbe done more easily by a trick. We consider W̃ = T q+3 with the standard action of Z2 flipping
every direction. We have
η(DRPq+2 ) =0 = index(D T q+3 , g = −1) − 2q+3 P0 .

(1.17)

Now, the fermions on a Dq-brane arise from the dimensional reduction (or T-dual) of a
Majorana-Weyl fermion from ten dimensions. As such, the number of components of thefermions is real 16 or complex 8 in q + 1 dimensions. These 8 components are for chiral fermions, and hence the corresponding Dirac (instead of Weyl) fermion in q + 1 dimensions has 2 × 8 = 16 components. Then the Dirac fermion in two dimensions higher (i.e.
q + 3 dimensions) has 2 × 16 = 32 components. It turns out (see Sec. 2) that the Z2 action on the fermion components is given by the chirality operator Γ T q+3 on T q+3 up to asign as g : Ψ(~z ) 7→ ±Γ T q+3 Ψ(−~z ). The chirality operator Γ T q+3 in this transformation is required by the fact that the Dirac oeprator D T q+3 = iΓ I ∂ I on T q+3 is equivariant under g as
D T q+3 ◦g = g◦D T q+3 . Therefore, index(D T q+3 , g = −1) = ±32 = ±25 , and P0 = ±22−q = ±2 p−4 .
Plugging this into (1.16), we indeed get (1.11).
That said, we have been cavalier in the various signs and in particular the action of Z2 on
the fermions during the rough outline we gave above. We will spend Sec. 2 to carefully fix theconventions and the signs.

5

SciPost Phys. 7, 058 (2019)

Organization of the paper: The remainder of our paper consists of two more sections andan appendix, most of which can be read independently:• In Section 2, we carefully specify our conventions concerning index theorems and string
theory, determine the Z2 action relevant for the Op-plane, and carry out the strategy
outlined above. We confirm that all the fermions contribute to the anomaly by the same
sign, and we get 2 p−5 − 2 p−5 = 0 instead of 2 p−5 + 2 p−5 6= 0. This will be done in a
uniform computation independent of the dimensionality p.
• In Section 3, we interpret the global fermion anomaly we detected using the η invariants
in terms of cobordism invariants classifying the interacting fermionic SPT phases. We
see that it illustrates the dimensional hierarchy of SPTs with (anti)unitary Z2 symmetries
squaring to (±1) F discussed in Sec. 8 of [15].
• The appendix A, we discuss the precise nature of the symmetry group of the type IIB
string theory, which is usually called the SL(2, Z). We argue that it is a pin+ version of
the double cover of GL(2, Z).

2

Careful analysis of anomalies

2.1

Conventions concerning index theorems

First let us fix the conventions related to index theorems. We follow the conventions in [12];see in particular the Appendix A and B of that paper. We work entirely in Euclidean signature.
The spacetime dimension is denoted by d. We denote the spacetime indices as I, J, M , · · · .
These run over 0, 1, · · · , d − 1. We use I = 0 even in Euclidean signature.
We take the gamma matrices Γ I to be hermitian following the standard physics convention.
Also, we denote the chirality operator as Γ which gives the Z2 -grading relevant for the index
problem; This is the one which is usually denoted as γ5 in four dimensions. It must satisfy2
Γ = 1 and {Γ , Γ I } = 0. We call Γ = +1 as positive chirality and Γ = −1 as negative chirality.
We take the Dirac operator to be

D = iΓ I DI ,(2.1)

where DI is the covariant derivative.
On non-oriented manifolds, we need an additional gamma matrix which we denote by eΓ
because of the following reason. On a flat space, consider the reflection x J → −x J for some J.
We need a corresponding operation in the Pin± groups acting on a fermion Ψ. If we denote thisaction as Ψ(· · · , x J , · · · ) 7→ EJ Ψ(· · · , −x J , · · · ), then the matrix EJ must satisfy (i) Γ EJ = EJ Γ sothat the Z2 -grading is well-defined, and also satisfy (ii) Γ I EJ = EJ Γ I (I 6= J) and Γ J EJ = −EJ Γ J
for the Dirac operator to behave covariantly. These equations are satisfied if we define EJ as
EJ = αeΓ Γ J , where α = ±i for Pin+ and α = ±1 for Pin− . It is possible to realize Pin± (d) as a
subgroup of Spin(D) for some D > d, and then eΓ is given by the additional gamma matrices
in Spin(D).
APS index theorem: The index theorem is given as follows. In this paper we only need
the index theorems on even dimensional oriented manifolds or odd dimensional non-oriented
manifolds, so we restrict our attention to these cases. If we denote the number of zero modes
with positive chirality (Γ = +1) and negative chirality (Γ = −1) by n+ and n− respectively, we
have the APS index theorem of a Dirac operator DW on a manifold W with boundary ∂ W = X

6

SciPost Phys. 7, 058 (2019)asindex(DW ) = n+ − n− =ZÂ(R)ch(F ) + η(DX ),

(2.2)

W
i
where ch(F ) = tr exp( 2πF ) is the Chern character in the standard convention, Â(R) is thestandard Dirac genus, and DX is the boundary Dirac operator. The η invariant of a self-adjoint
operator D with the set of discrete spectrum of eigenvalues Spec(D) is defined as!
X
1
η(D) =
sign(λ) + dim(Ker D) ,(2.3)
2
λ∈Spec(D ), λ6=0with an appropriate regularization for the sum over all eigenvalues λ ∈ Spec(D). However,
the equation (2.2) needs more explanations about the orientation and the precise definitionof DX .
In even dimensions d = 2n, we define the totally anti-symmetric tensor by
ε I1 I2 ···I2n =


1
I1 I2
I2n
tr
Γ
Γ
Γ
·
·
·
Γ
.
i n 2n(2.4)

Then this totally anti-symmetric tensor ε I1 ···I d sets the orientation of the manifold. In the firstterm of (2.2), we have to use this orientation. This can be seen by e.g. carefully deriving
the index theorem by the standard method of Fujikawa, or by considering W = Σ × · · · × Σ
where Σ is a 2d Riemann surface and using the Riemann-Roch theorem. In odd dimensions,Â(R)ch(F )|d=odd is zero and hence the orientation does not matter.
The boundary Dirac operator DX is related to the bulk Dirac operator DW = iΓWI DI as
follows. Let x be a coordinate such that the boundary X is located at x = 0, and the interior
2of W is x < 0. The metric near the boundary is assumed to be of the form dsW= d x 2 + dsX2 .
Then by using the data near the boundary, we defineÒX ),DW = iΓWx (∂ x + D

ÒX = Γ x Γ I 0 DI 0 ,
D
W W

0

ÒX P + = iΓ I DI 0 ,DX = DX
W(2.5)

where we used
±
=
PW

1
(1 ± Γ W ),
2

0

0

+
.
ΓXI = −iΓWx ΓWI PW(2.6)

The index I 0 runs over the subset of {0, 1, · · · , d − 1} which are orthogonal to the direction+ÒX , and hence the eigenmodes of DÒX arespecified by x. Notice that PW
commutes with D+
preserved under the projection by PW . Thus DX is a self-adjoint operator.
Notice the following point. On non-oriented manifolds, there was no orientation and hence
there was no definite way to pick the sign of Γ W . If we change Γ W → −Γ W , the definition of positive and negative chirality is exchanged and hence the index becomesindex(DW ) → − index(DW ). Let us check how this is consistent with the index theorem. For
simplicity we assume that Ker DX is empty, although more general case can also be analyzed.
+−→ PW. Now, the vector spaces
If we change Γ W → −Γ W , the projection operator changes as PW+−
with PW = 1 and PW = 1 can be exchanged by the multiplication by ΓWx . This multiplication
ÒX → Γ x DÒ xÒ
by ΓWx also changes the boundary Dirac operator as DW X ΓW = −DX , and hence theη invariant is changed as η(DX ) → η(−DX ) = −η(DX ) where we used Ker DX = ∅. Therefore the index formula (2.2) behaves nicely under this change. This gives a quick motivation+
for the insertion of the projector PW
in the definition of DX . Without this projector, we haveÒX ) = 0.
η(DThe coefficient of the term η(DX ) in (2.2) in the present convention can be checked e.g. ina simple example of a 2d fermion coupled to a U(1) background field, by taking W = D2 ,7

SciPost Phys. 7, 058 (2019)

X = S 1 , and DW = i(σ1 D1 + σ2 D2 ). In this case, one can compute the eta invariant directlyfrom the definition, and check that the right-hand-side of (2.2) takes values in integers Z
for any background U(1) field. See also Appendix B of [12] for more discussions on generaloriented case.
For the APS index theorem to hold, we have to impose the APS boundary condition. For
the eigenmodes Ψ of DW , we require the boundary condition
¨
«
X
X
+ X
− X
Ψ|X ∈
cλ PW
Ψλ +
cλ0 PW
Ψλ ,
(2.7)
λ≥0

λ>0

ÒX , i.e. DÒX Ψ X = λΨ X . Namely, the restriction of Ψ on thewhere ΨλX are the eigenmodes of D
λλboundary X must be spanned by eigenmodes of non-negative eigenvalues for Γ W = +1 andpositive eigenvalues for Γ W = −1. This is imposed so that if the manifold W is extended by aÒX )Ψ = 0 in
cylinder region [0, ∞) × X , then the zero modes (which are solutions of (∂ x + D
the cylinder region) do not grow at infinity.
From this boundary condition, we can give a crude idea of why the boundary term η(DX )appears in the APS index theorem (2.2). Consider the heat kernel method for the index computation,2

index(DW ) = lim Tr(Γ W e−t(DW ) ).

(2.8)

t→+0

Near the boundary, the relevant modes are given by (2.7). The modes with Γ W = +1 andλ ≥ 0 contribute positively for this heat equation, while the modes with Γ W = −1 and λ > 0contributes negatively. However, the modes with Γ W = −1 and λ > 0 can be mapped to themodes with Γ W = +1 and λ < 0 by using ΓWx as mentioned above. Therefore, if we map
all the modes to Γ W = +1, the contribution of the mode with an eigenvalue λ of DX to theheat equation is proportional to sign(λ) with a positive proportionality constant. (Here we
define sign(λ = 0) = +1.) Their sum is the η invariant η(DX ) up to a positive proportionality
constant. It requires more work to show that the positive proportionality constant is just as in(2.2). See also [22] for more physical interpretation of the boundary η invariant.
Dai-Freed theorem: Now consider a manifold X with boundary ∂ X = Y . We assume that
there is a coordinate y such that Y is located at y = 0 and the interior of X is in the regiony < 0. The metric is assumed to be of the form dsX2 = d y 2 + dsY2
0
If we are given a Dirac operator DX = iΓXI DI 0 (which need not be Z2 -graded), we define
the boundary Dirac operator DY as
y

y

00

00

DY = ΓX ΓXI DI 00 = iΓYI DI 00 ,DX = iΓX (∂ y + DY ),

(2.9)

where
00

y

00

ΓYI = −iΓX ΓXI .

(2.10)We also define the boundary chirality operator Γ Y and the projection operators PY± asyΓ Y = ΓX ,

PY± =

1
(1 ± Γ Y ).
2

(2.11)In the above setup, the Dai-Freed theorem states the following. Let

DY−+ = PY− DY PY+
8(2.12)

SciPost Phys. 7, 058 (2019)be a chiral Dirac operator. Then the combination
det(DY−+ ) exp(−2πiη(DX ))(2.13)

takes values in C, although each of exp(−2πiη(DX )) and det(DY−+ ) takes values in onedimensional vector spaces called the determinant line and its inverse.
More physically, this means the following. The chiral Dirac operator DY−+ may suffer fromanomalies, and det(DY−+ ) may not be well-defined as a number in C. However, if we extend
Y to a manifold X such that ∂ X = Y , then we can make it well-defined by multiplying it by
the exponentiated η invariant exp(−2πiη(DX )) in the bulk. The η invariant itself suffers fromsome ambiguity on manifolds with boundary, since there is no preferred choice of boundary
conditions. The ambiguities of det(DY−+ ) and exp(−2πiη(DX )) cancel each other.
The Dai-Freed theorem has a very nice physical interpretation [11]. Consider a massive
fermion Ψ whose Euclidean action is given byZ−S E = −Ψ̄(−i DX + M )Ψ.

(2.14)

If we take the Pauli-Villars regulator mass as positive, M = +Λ and Λ → +∞, and the physicalfermion mass as negative and very large M = −Λ, the partition function is given by [16]Zbulk =det(−i DX − Λ)=det(−i DX + Λ)Y
λ∈Spec(DX )(−iλ − Λ)= e−2πiη(DX ) ,(−iλ + Λ)(2.15)

where we took the regularized version of sign(λ) appearing in the definition of the eta invariantas1(−iλ + Λ)2
sign(λ)reg = arg − 2,−1 < sign(λ)reg ≤ 1,
(2.16)πλ + Λ2
which gives the usual sign λ/|λ| for λ  Λ and which goes to zero, sign(λ)reg → 0 for |λ|  Λ.
Therefore, the partition function of the above massive fermion is given by Zbulk = e−2πiη(DX ) .
Now we take the fermion mass M to depend on the position as follows. Let ε > 0 be a verysmall length scale. We take M ( y) to be M ( y) = −Λ for y < −ε, and M ( y) = +Λ for y > +ε.
Namely, inside X the mass is negative as above, but outside X (i.e. y > 0) it is positive, andM (x) changes rapidly near the boundary Y . In this case, there is a localized chiral fermion on
Y which is given by the solutions of

Z y

y

(ΓX ∂ y + M (x))Ψ = 0,y

M ( y 0 )d y 0 .

Ψ ∝ exp −ΓX(2.17)

0y

For the solutions to be localized near Y , they must satisfy ΓX Ψ = +Ψ, and hence they have theypositive chirality under Γ Y = ΓX .
The total system is as follows. In the bulk x < 0, we have the partition function
Zbulk = e−2πiη(DX ) . On the boundary, the chiral fermion with positive chirality Γ Y = +1 gives
the partition function Zboundary = det(DY−+ ). The region x > 0 outside of X does not contributebecause of the cancellation between the physical fermion and the Pauli-Villars field. Therefore,
the total partition function of the system is
Z = Zboundary Zbulk = det(DY−+ ) exp(−2πiη(DX )).

(2.18)

This should be well-defined since the total system is a non-chiral theory and has a well-defined
UV regularization. This is one aspect of the physics of the Dai-Freed theorem [11]. See also[12] for another aspect.
9

SciPost Phys. 7, 058 (2019)

2.2

Conventions concerning string theory

Chirality in type IIB: In type IIB superstring theory, we require that the supercharges Q, Q̃
have the chirality Γ = −1,
Γ Q = −Q.

(2.19)

This is just a choice. Then, all other chiralities are automatically fixed.
For our purposes, what are important are the gaugino on D-branes and the duality equations for the field strengths of RR fields C. First, the chirality of gaugino is determined asfollows. Let us consider type I superstring theory. Its supercharge Q(type I) (= Q + Q̃) has thesame chirality as the one in type IIB. Moreover, the gaugino λ of the SO(32) vector multiplet
and the gauge field A I are roughly related as[Q(type I) , A I ] ∼ Γ I λ,{λ, (Q(type I) ) T C} ∼ (∂ I AJ − ∂J A I )Γ I Γ J ,(2.20)

where C is a charge conjugation matrix whose defining property is that CΓ I C−1 = −(Γ I ) T ,CΓ C−1 = −Γ , and (Ψ T C)Γ I Ψ 0 transforms as a Lorentz vector for two spinors Ψ and Ψ 0 of thesame chirality. This fixes the chirality of the gaugino of the “D9”-brane asΓ λ = +λ.

(2.21)

The fact that the gauginos have chirality Γ = +1 is valid for any Dp-brane under the followinginterpretation. The gamma matrices Γ 0 , · · · , Γ p are for the tangent bundle of the worldvolume,
while Γ p+1 , · · · , Γ 9 are for the normal bundle or equivalently the R-symmetry. One can simply
repeat the above argument, with the understanding that A I for I being normal directions arescalars. For the transformation (2.20) to be valid, we use the convention of T-duality which
will be discussed later.
Next, we determine the chirality of the 5-form field strength F5 = d C4 . Let Ψ I be thegravitino in type IIB. The gravitino has chirality Γ Ψ I = +Ψ I since we have the equation
I
I[Q, e M] ∼ Γ I Ψ M where e M
is the gravity field, and hence the chirality of the supercharge Q
and the gravitino Ψ I are opposite. The field strength F5 = d C4 appears in the transformationof Ψ I as [27]7{Ψ I , Q T C} ∼ (Γ I1 ···I5 F I1 ···I5 )Γ I .

(2.22)

From this transformation law and Γ Ψ I = +Ψ I , we getΓ Γ I1 ···I5 F I1 ···I5 = +Γ I1 ···I5 F I1 ···I5 .

(2.23)On the other hand, in the Euclidean signature,1Γ Γ I1 ···I p+2 = −i(−1) 2 (p+2)(p+3) ·1
ε I1 ···I p+2 J1 ···J8−p ΓJ1 ···J8−p ,(8 − p)!

(2.24)and hence the 5-form transforms as
1
F I ···I ε I1 ···I5 J1 ···J5 = −i F J1 ···J5 .
5! 1 5(2.25)In the notation of differential forms, it is simply written as?F5 = −i F5 .
The imaginary unit i appears because we are using the Euclidean signature.
7

See also page 393 [28] for helpful summary.

10(2.26)

SciPost Phys. 7, 058 (2019)

T-duality: Type II string theories have two Majorana-Weyl supercharges which we denote by
Q and Q̃. We always take Q to have chirality Γ Q = −Q. Then Q̃ has chirality Γ Q̃ = −Q̃ in type
IIB and Γ Q̃ = +Q̃ in type IIA.
Under the T-dual in the direction I, the supercharge Q is invariant, but the supercharge Q̃
before and after the T-dual are related asQ̃before = Γ I Q̃after .

(2.27)The sign is just a convention and it could as well be Q̃before = −Γ I Q̃after , but we use the above
convention for definiteness. In fact, there is a symmetry in type II string theories, usuallydenoted by (−1) F L , which acts on the supercharges as(−1) F L (Q) = Q,(−1) F L (Q̃) = −Q̃.

(2.28)

Then, the sign (2.27) in the T-dual can be changed by an additional operation of (−1) F L .
We remark that T-dual actions in two orthogonal directions I and J (I 6= J) do not commute. There is an extra sign factor (−1) F L in their commutation relation.
Let us start from type IIB and type I theories and take T-duals. We use the convention of
the relative sign of the two supercharges in type IIB such that the supercharge preserved in
type I string theory is the linear combination
Q + Q̃.

(2.29)

This means the following. In type IIB, there is a Z2 symmetry Ω, satisfying Ω2 = 1, which is
the orientation reversal in the string world sheet. In type I construction, we divide the theory
by this symmetry. If this Ω acts on supercharges asΩ(Q) = Q̃,Ω(Q̃) = Q,

(2.30)

then Q + Q̃ is the supercharge of type I. More discussions on the symmetries (−1) F L and Ω aregiven in Appendix A.
Consider a flat spacetime in which the directions p + 1, · · · , 9 are compactified on a torus
9−pT. By taking T-duals in the directions 9, 8, · · · , p + 1 in this order, the above supercharge intype I becomesQ(Dp) := Q + Γ 9 Γ 8 · · · Γ p+1Q̃.

(2.31)We use the convention that after the above T-duals, the D9-branes and O9-planes become
Dp-branes and Op-branes with the orientation d x 0 ∧ · · · ∧ d x p , rather than anti-Dp-branes
and anti-Op-planes. Then the above supercharge (2.31) is the linear combination which ispreserved by the Dp-branes and Op-branes with the above orientation.
We also define the RR fields such that the coupling to the Dp-brane is given byZ−S ⊃ 2πi

C p+1 ,

(2.32)

R
rather than −2πi C p+1 . This fixes the sign of the RR-fields.
In the above definitions of D-branes and RR-fields, the T-dual in the direction J acts on RR
fields as(Fk+1 ) I1 ···I k J = (Fk ) I1 ···I k .

(2.33)?Fk = β F10−k =⇒ ?Fk+1 = β(−1)k+1 F9−k .

(2.34)

From this, we get

11

SciPost Phys. 7, 058 (2019)

Combining this with (2.26), we get
1?Fk = +(−1) 2 k(k+1) i F10−k .

(2.35)

This is the self-duality condition of the RR-fields.
The sign factor in (2.35) has the following explanation. For simplicity, let us considera spacetime X without orientifold. RR-fluxes are classified by K-theory elements x ∈ K i (X )where i = 0 for Type IIA and i = 1 for Type IIB [29]. Let F (x) be RR fields corresponding
to thep

topological class x. In the de-Rham cohomology, its topological class is given by Âch(x). In[30], a complex structure J was introduced on the space of RR-fluxes given by J F (x) = ?F (x̄),
where x̄ is the complex conjugate of x. The Chern character ch(x) has the property that its
1
k-form part transforms as ch(x̄)|k = (−1) 2 k(k+1) ch(x)|k .8 Therefore, the complex structure1
J is explicitly given by J(F10−k ) = (−1) 2 k(k+1) ? Fk . The equation (2.35) just means thatJ = i. Therefore, in Euclidean signature manifolds, the space of RR-fluxes is holomorphic.
(See [29, 30] for more precise meaning of this statement.) This is the self-duality condition of
RR-fluxes in Euclidean signature spaces.
RR-flux and Dirac pairing:
the form

The Euclidean action involving RR fields C is schematically of

2π−S E ⊃ −2Z

d C p+1 ∧ ?d C p+1 + 2πi q pZ
Z (p+1)

C p+1 ,

(2.36)

where Z (p+1) is the world volume of a D-brane or O-plane, and q p is its charge which is q p = 1
for a single Dp-brane and q p = +2 p−5 for an Op+ -plane. This action is schematic since we havenot taken into account the self-dual equation in the sense of (2.35). However, the equation of
motion2π(−1) p+1 d(?F p+2 ) + 2πi q p δ(Z (p+1) ) = 0(2.37)

is valid, where δ(Z (p+1) ) is the delta function such thatZZ
Z (p+1)

C p+1 =

C p+1 ∧ δ(Z (p+1) ).

(2.38)

This δ(Z (p+1) ) is the Poincaré dual of Z (p+1) . By using (2.35), we get1d F8−p = +(−1) 2 (p+1)(p+2) q p δ(Z (p+1) ).

(2.39)

Notice that the imaginary unit i has disappeared.
Now, let X (8−p) be a closed submanifold with linking number+1 with Z (p+1) . Namely,
R
(9−p)
(9−p)
(8−p)
let W
be a manifold with ∂ W=X, and that W (9−p) δ(Z (p+1) ) = +1. From theabove, we getZ1

X (8−p)F8−p = +(−1) 2 (p+1)(p+2) q p .

(2.40)The phase ambiguity of a single Dq-brane with q = 6 − p wrapping a codimension-one submanifold Y (q+1) of X (q+2) (where q + 2 = 8 − p) is given as

Z



1
(2.41)
exp 2πiF8−p = exp +(−1) 2 (p+1)(p+2) 2πi q p .
X (8−p)For x ∈ K 1 (X ) = K −1 (X ), the Chern character is defined as follows. By definition, K −1 (X ) is given by elementsof K(X × S 1 ) whose restriction to X × {p} is trivial. Then we get ch(x) as an element of H even (X × S 1 ) which istrivial when restricted to X × {p}. We can push forward it to X by integrating over S 1 . By this process, we get odd
degree forms ch(x) ∈ H odd (X ). In particular, the (2n − 1)-form ch(x)|2n−1 transforms in the same way as 2n-form
ch(x)|2n under complex conjugation of x.
812

SciPost Phys. 7, 058 (2019)

2.3

O-planes and fermions

Let us consider the construction of the Op-plane which preserves the supercharge (2.31). The
local geometry near the Op-plane isR p+1 × (R9−p /Z2 ).

(2.42)In the construction, we divide the spacetime R p+1 × R9−p by the operation z~ → −~z , wherez~ = (x p+1 , · · · , x 9 ) are the coordinates orthogonal to the Op-plane. We also need to combine
it with the orientation reversal Ω of the string world sheet. More precisely, we need an uplift
to the group Spin in type IIB or Pin+ in type IIA. We denote this operation by R(9−p) Ω. In type
IIB, each of R(9−p) and Ω is well-defined independently, while in type IIA only the combinationR(9−p) Ω is well-defined for even p.
Explicitly, we take it to beR(9−p) Ω(Q) = Γ 9 Γ 8 · · · Γ p+1Q̃,R(9−p) Ω(Q̃) = Γ 9 Γ 8 · · · Γ p+1Q.

(2.43)

Applying this to (2.31) gives1

R(9−p) Ω(Q(Dp) ) = (−1) 2 p(p−1)Q + Γ 9 Γ 8 · · · Γ p+1Q̃,(2.44)

1

where we have used (Γ 9 Γ 8 · · · Γ p+1 )2 = (−1) 2 p(p−1) . The Dp-brane and the Op-plane must
preserve the same supercharge, so we have to combine R(9−p) Ω with (−1) F L in the constructionof the Op-plane;1R(9−p) Ω(−1) 2 p(p−1)F L (Q(Dp) ) = Q(Dp) .

(2.45)

Therefore, the Z2 action in the construction of the Op-plane is generated by
1

R(Op) := R(9−p) Ω(−1) 2 p(p−1)F L .

(2.46)

To compute the anomaly of the gaugino of the Dq-brane with q = 6 − p, it is important
to know the action of the above symmetry operation on the gaugino on the Dq-brane. Let us
place the Dq-brane at{x 0 = · · · = x p+2 = 0}.

(2.47)

We take the orientation of this Dq-brane asd x p+3 ∧ d x p+4 ∧ · · · ∧ d x 9 .

(2.48)

By the above convention of D-branes discussed around (2.31), we see that the superchargepreserved by this Dq-brane is given as
Q(Dq) := Q + (−1) p+1 Γ p+2 · · · Γ 1 Γ 0Q̃.

(2.49)

Here the sign factor (−1) p+1 appeared because(d x 0 ∧ · · · ∧ d x p+2 ) ∧ (d x p+3 ∧ · · · ∧ d x 9 ) =(−1) p+1 (d x p+3 ∧ · · · ∧ d x 9 ) ∧ (d x 0 ∧ · · · ∧ d x p+2 ). (2.50)
To make the notation simpler, we define
ΓR1 = Γ 9 Γ 8 · · · Γ p+1 ,ΓR2 = (−1) p+1 Γ p+2 · · · Γ 1 Γ 0 ,13(2.51)

SciPost Phys. 7, 058 (2019)

which satisfy
1(ΓR1 )2 = (−1) 2 p(p−1) ,1(ΓR2 )2 = (−1) p+1 (−1) 2 p(p−1) ,ΓR1 ΓR2 = (−1) p+1 ΓR2 ΓR1 .

(2.52)

By using them, we have Q(Dq) = Q + ΓR2Q̃ and1R(Op) (Q(Dq) ) =R(9−p) Ω Q + (−1) 2 p(p−1) ΓR2Q̃

1=ΓR1Q̃ + (−1) 2 p(p−1) ΓR2 ΓR1Q
1=(−1) 2 p(p−1) ΓR2 ΓR1 (Q + ΓR2Q̃),(2.53)

so we get
1

R(Op) (Q(Dq) ) = (−1) 2 p(p−1) ΓR2 ΓR1Q(Dq) .

(2.54)

By using this result, we want to see how the gaugino λ on the Dq-brane transforms under1
the transformation R(Op) = R(9−p) Ω(−1) 2 p(p−1)F L . Let us consider the supersymmetry transformation[Q(Dq) , φ I ] ∼ Γ I λ,(2.55)

where φ I are world volume scalar fields of the Dq-brane and the index I runs over the normal directions to the world volume. The supercharge Q(Dq) transforms as in (2.54). On theother hand, the scalars φ I are proportional to the positions of the Dq-brane, and hence theytransform as
−φ I I = (p + 1), (p + 2),(Op)(2.56)R (φ I ) =φI
I = 0, 1, · · · , p.
Therefore, the gaugino on the Dq-brane transforms as
1

R(Op) (λ) = −(−1) 2 p(p−1) ΓR2 ΓR1 λ
1= −(−1) 2 p(p−1) (Γ 0 · · · Γ p )(Γ p+1 Γ p+2 )2 (Γ p+3 · · · Γ 9 )λ1= −(−1) 2 p(p−1) iΓ p+1 Γ p+2 Γ λ,(2.57)

where Γ = i −5 Γ 0 · · · Γ 9 . This is the transformation of the gaugino in q + 1 dimensions placedat (2.47).
We are actually interested in the Dq-brane which is wrapped in a codimension-1 submanifold Y (q+1) of the (q + 2)-dimensional space X (q+2) given by~, z~) ∈ {0} × R9−p /Z2 ; |~X (q+2) = {(0z | = r},(2.58)

where z~ = (x p+1 , · · · , x 9 ) and r is an arbitrary positive constant. Before the division by Z2 ,the gaugino is a section of the spin bundle of R10 which contains the spin bundles of both thetangent and normal bundles of the worldvolume. We trivialize this total spin bundle by using
the flat metric of R10 , no matter how the worldvolume is curved. Then, the Z2 action relates
the gauginos at the positions z~ and −~z by
1R(Op) (λ(~z )) = −(−1) 2 p(p−1) iΓ x Γ y Γ λ(−~z ),(2.59)

where Γ x is the gamma matrix in the radial direction ~e x = z~/|~z |, Γ y is the gamma matrix in
(q+1)
(q+2)
the direction ~e y which is normal to Y
in X
. This equation follows from (2.57) by
14

SciPost Phys. 7, 058 (2019)slightly moving the Dq-brane from (x p+1 , x p+2 ) = (0, 0) to (x p+1 , x p+2 ) = (r, 0) and considering the point z~ = (r, 0, · · · , 0). In this case we have Γ x = Γ p+1 and Γ y = Γ p+2 .9 At the
point z~ = (−r, 0, · · · , 0) we have Γ x = −Γ p+1 and Γ y = −Γ p+2 , but the product is the same;(−Γ p+1 )(−Γ p+2 ) = Γ p+1 Γ p+2 . In the orientifold, we impose the identification under R(Op) as
1

λ(~z ) ∼ −(−1) 2 p(p−1) iΓ x Γ y Γ λ(−~z ).

(2.60)

Trivialization of the normal bundle: In the above computation, we must notice the following point. The gaugino is a section of the total spin bundle S(T Y (q+1) ⊕N Y (q+1) ) where T Y (q+1)
is the tangent bundle and N Y (q+1) is the normal bundle. The normal bundle is the bundle of
the R-symmetry from the point of view of the worldvolume theory. The sum T Y (q+1) ⊕ N Y (q+1)
is just the tangent bundle of the ten dimensional bulk restricted to the worldvolume Y (q+1) .
The result (2.60) is valid in the trivialization of the bundle which follows naturally from theflat space structure of the bulk spacetime R10 . However, to compare with the computation ofthe η-invariant discussed below, we want to trivialize the normal bundle N Y (q+1) rather than
the sum T Y (q+1) ⊕ N Y (q+1) so that the R-symmetry plays no role in the computation of theη-invariant.
The Dq-brane is moved within the plane x 0 = · · · = x p = 0, so these directions are always
trivial and we neglect them. Then the normal bundle N Y (q+1) is effectively spanned by the xdirection ~e x = z~/|~z | and the y-direction ~e y introduced above. We take the orthonormal frame(~e x , ~e y ) to trivialize N Y (q+1) .
For illustration, let us see the effect of trivialization of the normal bundle when the Dqbrane is at (x p+1 , x p+2 ) = (r cos θ , r sin θ ) and extending to other directions (x p+3 , · · · , x 9 ).
On the worldvolume, consider a point (x p+3 , · · · , x 9 ) = 0. At that point, the orthonormal
vectors in the x and the y directions are given by~e x = (cos θ , sin θ , 0 · · · , 0),~e y = (− sin θ , cos θ , 0 · · · , 0).

(2.61)

Then the gaugino λ0 after trivializing N Y (q+1) by the frame (~e x , ~e y ) is given byθ p+1 p+2
λ = exp
λ.
Γ
Γ
2
0



(2.62)

9
The reason that Γ y = Γ p+2 instead of Γ yR = −Γ p+2 is seen as follows. We have to recall the process of how tocompute the ambiguity of the coupling 2πi Cq+1 with the RR-field. It is simpler to consider the covering spaceX̃ (q+2) ∼= S (q+2) so that all (sub)manifolds are orientable.R Not only orientable, we actually need to orient them tocompute the coupling. Our computation of the RR-flux X (q+2) Fq+2 in Sec. 2.2 implicitly used the convention that
this sphere S (q+2) is oriented by a (q + 2)-form ω(q+2) such that d x ∧ ω(q+2) = d x p+1 ∧ · · · ∧ d x 9 where x = |~z |.
Moreover, the orientation of Y (q+1) must be given by a (q + 1)-form ω(q+1) such that d y ∧ ω(q+1) = ω(q+2) . ThisR
R
orientation is required by the Stokes theorem Y (q+1) Cq+1 = X 0(q+2) Fq+2 , where ∂ X 0(q+2) = Y (q+1) , and the factthat d y is in the outgoing direction as taken in Sec. 2.1. We also have to recall that the Dq-brane is oriented by(2.48). Therefore, at the point z~ = (r, 0, · · · , 0), we have ω(q+1) = d x p+3 ∧ d x p+4 ∧ · · · ∧ d x 9 . Combining this withd x = d x p+1 which gives ω(q+2) = d x p+2 ∧ d x p+3 ∧ · · · ∧ d x 9 at z~ = (r, 0, · · · , 0), we finally get d y = d x p+2 . This is
the reason that Γ y = Γ p+2 .
In the case of type IIA, the manifold X (q+2) = S (q+2) /Z2 = RP(q+2) is not orientable. However, the RR fieldchanges the sign under Z2 as Cq+1 → −Cq+1 in addition to the usual transformation under z~ → −~z . Therefore, theintegrations of Cq+1 and Fq+2 are well-defined without orientation. Their values are half of the integration of themin the oriented double cover such as S (q+2) .
The process of going from (q +1)-dimensional Y (q+1) to (q +2)-dimensional X (q+2) must be done in the same way
in the computation of both the RR-coupling and the η invariant. Therefore, the directions x and y determined
above from the consideration of the RR-coupling are exactly what should be used in the later computation of theη invariant.

15

SciPost Phys. 7, 058 (2019)

The points θ = 0 and θ = π are identified asπ
λ0 (θ = π) = exp
Γ p+1 Γ p+2 λ(θ = π)2 1p+1 p+2∼ΓΓ−(−1) 2 p(p−1) iΓ p+1 Γ p+2 Γ λ(θ = 0)1

= (−1) 2 p(p−1) iΓ λ0 (θ = 0),(2.63)(2.64)(2.65)

where we have used (2.60).
The Γ is the chirality operator of the gaugino. We have the chirality condition Γ λ = +λ as
discussed around (2.21).
Descent structure of fermions: To compute the anomaly by using the process described inSec. 2.1, we want to uplift the situation from Y (q+1) to~, z~) ∈ {0} × R9−p /Z2 ; |~W (q+3) = {(0z | ≤ r}.

(2.66)We remark that the “fermions" which appear in the following discussions are just mathematically introduced for the purpose of computation of the η-invariant, and not physical fields inthe string theory.
Let us consider a fermion on W (q+3) with gamma matrices ΓWI (I = p + 1, · · · , 9) and thechirality operator Γ W which transforms as
1

R(Op) (λ(q+3) (~z )) = +(−1) 2 p(p−1) Γ W λ(q+3) (−~z ).

(2.67)The insertion of Γ W in this equation needs explanation. On W (q+3) we use the Dirac operator
in flat space10

DW = i9X

ΓWI ∂ I .

(2.68)

I=p+1

This Dirac operator must transform covariantly under R(Op) . The spacetime coordinates transform as x I → −x I for I = p+1, · · · , 9, and hence the derivatives with respect to them transformas ∂ I → −∂ I . The Γ W , which anti-commutes with the gamma matrices ΓWI , was inserted so that
DW R(Op) (λ(q+3) ) = R(Op) (DW λ(q+3) ).

(2.69)

Thus the Dirac operator DW is equivariant under the Z2 operation R(Op) .
Following the process in Sec. 2.1, we first reduce it to~, z~) ∈ {0} × R9−p /Z2 ; |~X (q+2) = {(0z | = r}.

(2.70)

This is done by the projection
λ(q+2) =

1 + ΓW2λ(q+3) .

(2.71)

It transforms as
1

R(Op) (λ(q+2) (~z )) = +(−1) 2 p(p−1) λ(q+2) (−~z ),10X(2.72)

More precisely, we use some Weyl rescaling of it near the boundary X (q+2) so that the metric on W (q+3) nearis of the product form (−ε, 0] × X (q+2) .

(q+2)

16

SciPost Phys. 7, 058 (2019)because we have Γ W = +1 after the projection.
Next, we reduce the fermion λ(q+2) on X (q+2) to the chiral fermion λ(q+1) on Y (q+1) . Theyychirality operator is Γ Y = ΓX = −iΓWx ΓW as discussed in Sec. 2.1, and we identify it with thechirality operator Γ which is relevant for the worldvolume gaugino. The chiral fermion satisfies
Γ Y λ(q+1) = +λ(q+1) . The transformation is again
1

R(Op) (λ(q+1) (~z )) = +(−1) 2 p(p−1) λ(q+1) (−~z ),(2.73)

and we impose the identification R(Op) (λ(q+1) (~z )) ∼ λ(q+1) (~z ). However, we have to noticethat the above transformation is valid in the trivialization of the spin bundle which naturally follows from the flat space structure of W (q+3) . For the process of Sec. 2.1 to be valid,
we have to take the frame associated to (~e x , ~e y ). For example, by restricting attention to(x p+1 , x p+2 ) = (cos θ , sin θ ), we change the frame asθ p+1 p+2
θλ0(q+1) = exp
ΓW ΓWλ(q+1) = expiΓ Y λ(q+1)
2
2
 
θ
= exp
i λ(q+1) ,(2.74)
2
where we have used the chirality condition Γ Y λ = +λ. Then the identification is1

λ0(q+1) (θ = π) ∼ +i(−1) 2 p(p−1) λ0(q+1) (θ = 0).

(2.75)

This reproduces (2.65) after using Γ λ = +λ. Therefore, the fermion λ0(q+1) transforms in thesame way as the gaugino λ0 . We conclude that (2.67) is the correct transformation for theindex computation.

2.4

The η invariant and the anomaly cancellation

Setup:

Recall that we are interested in the following situation. The 10d manifold is
M (10) = R p+1 × (R9−p /Z2 ).

(2.76)

Z (p+1) = {x p+1 = · · · = x 9 = 0} = R p+1 × {0}.

(2.77)The Op-plane is located at

The flux of F8−p through~, z~) ∈ {0} × R9−p /Z2 ; |~X (8−p) = RP8−p = {(0z | = r}(2.78)

is given by q p = +2 p−5 for Op+ -plane. We also often use q = 6 − p. The η invariant on X (8−p)
is relevant for the Dq-brane wrapped on a codimension-1 submanifold Y (q+1) of X (q+2) .
Computation of the η invariant: Let us compute the η invariant following the strategy
discussed in the introduction. The part R p+1 in M (10) plays no role and hence we neglect it.
To compute the η invariant, we consider the manifold
T q+3 /Z2 ,(2.79)

where T q+3 is the (q + 3)-dimensional torus. We denote the fiber of the (s)pin bundle on T q+3
as S, and a (trivial) bundle on T q+3 as E. On the bundle whose total space is T q+3 × S ⊗ E, we
act Z2 by
1

R(Op) : (~z , s) 7→ (−~z , +(−1) 2 p(p−1) Γ W s),
17(2.80)

SciPost Phys. 7, 058 (2019)

where z~ = (x p+1 , x p+2 , · · · , x 9 ) are the coordinates of T q+3 and s is the coordinate of the fiber
S ⊗ E. The reason that we have introduced the bundle E is that the gauginos take values inthe bundle S ⊗ E for some E because of the symmetries Ω and (−1) F L as well as the normal
1bundle. The factor +(−1) 2 p(p−1) Γ W acting on s is the one which appears in (2.67).
We consider the Dirac operator DW = iΓ I ∂ I which acts on sections of the bundle(T q+3 × S ⊗ E)/Z2 ,(2.81)or in other words, DW is equivariant under the Z2 transformation. Near each fixed point of
T q+3 /Z2 , the local geometry is the same as Rq+3 /Z2 which appears in M (10) . Let
Btotal = B1 t B2 t · · · t B2q+3 be the disjoint union of small balls around each of the fixed
points of T q+3 /Z2 . Applying the APS index theorem to
W 0 = (T q+3 /Z2 ) \ Btotal ,(2.82)

we haveindex DW 0 = η(D∂ W 0 ) = η(D∂ B

total) = −η(D∂ Btotal ) = −2q+3 η(DX (q+2) ).

(2.83)

Here the overline on ∂ Btotal means the opposite spin/pin structure from that on ∂ Btotal whose
details are described in [31,32]. It is a generalization of the orientation flip, but this operation
is nontrivial even on non-orientable manifolds when there are pin structures. The equality
η(D∂ B ) = −η(D∂ Btotal ) is a simple consequence of the fact that we represent the Dirac optotal
erator DW near the boundary as ΓWx (∂ x + DX ) as discussed in Sec. 2.1, and the operator DX
changes the sign when we change the normal direction as x → x 0 = −x. In the last step, we
have also used the fact that each ∂ Bk (k = 1, · · · , 2q+3 ) is a copy of X (q+2) .
On the other hand, the index index DW 0 is simply computed from the number of solutions
1z ). In fact, the APS boundz ) = +(−1) 2 p(p−1) Γ W λ(q+3) (−~D T q+3 λ(q+3) = 0 which satisfy λ(q+3) (~ary condition is such that the solutions on W = (T q+3 /Z2 )\B are extended to T q+3 /Z2 without
any divergence at the fixed points.11 In other words, the index on W = (T q+3 /Z2 ) \ B is thesame as the index on T q+3 of the modes which are invariant under Z2 .
The Dirac equation on T q+3 is satisfied if and only if λ(q+3) is constant. Therefore, we
simply need to count the number of components satisfying1

λ(q+3) = +(−1) 2 p(p−1) Γ W λ(q+3) .

(2.84)

Then the index index DW is given by1

index DW = +(−1) 2 p(p−1) ×

1
1
× 4 × 8 = +(−1) 2 p(p−1) · 24 .
2

(2.85)

This formula is understood as follows. First, on the Dq-brane, the total number of complex
components of the gaugino is 8, since the 10d Majorana-Weyl fermion has 16 real or 8 complex
components. Next, in the process of going from dimension d = q + 1 for λ(q+1) to dimension
d +2 = q +3 for λ(q+3) , the number of components is increased by a factor of 4 = 2×2 becauseλ(q+1) is obtained from λ(q+3) by two projections 21 (1+Γ W ) and 12 (1+Γ Y ). Finally, the numberof fermions satisfying (2.84) is half of these components in (q + 3) dimensions, and all of them1have chirality Γ W = +(−1) 2 p(p−1) . Thus we obtain (2.85).
Therefore, from (2.83) we get1η(DX (q+2) ) = −(−1) 2 p(p−1) · 2 p−5 .
11(2.86)

In more detail, this argument requires some Weyl rescaling near the boundary. For the application of APS
index theorem, we take the metric near the boundary ∂ W as the product form (−ε, 0] × ∂ W . On the other hand,in the extension of the solution from (T q+3 /Z2 ) \ B to T q+3 /Z2 , the metric which follows from the flat metric on
T q+3 is more natural. These two metrics are related by Weyl transformation.

18

SciPost Phys. 7, 058 (2019)

Anomaly cancellation:as

By the above computation, we obtain the anomaly from the gaugino


1
exp(−2πiη(DX (q+2) )) = exp +(−1) 2 p(p−1) 2πi · 2 p−5 .
On the other hand, the RR coupling (2.41) gives

Z



1
exp 2πi
Fq+2 = exp +(−1) 2 (p+1)(p+2) 2πi · 2 p−5 ,(2.87)(2.88)X (q+2)where we have used q p
1
2 (p+1)(p+2)=+2 p−5 for Op+ .

By a simple computation we get

12 p(p−1)= −(−1). Therefore, the above two factors cancel each other to make(−1)
the partition function of the Dq-brane well-defined. That was what we wanted to show.

3

Relations to symmetry-protected topological phases

We used the η invariants in (q + 2) dimensions to capture global anomalies of fermions on
the worldvolume of a Dq-brane. According to [15, 31, 32], such anomalies are controlled
by cobordism invariants in (q + 2) dimensions, which also classify the interacting fermionicsymmetry protected phases in the said dimensions. We will study these cobordism invariants.
In this section we do not carefully keep the precise sign factors, since they have already beentreated in detail in the previous section.
For simplicity, we only consider spacetime manifolds of the form R8−q × X q+2 , and discuss
the (q+2)-dimensional cobordism invariants. This is not quite general in the context of string
theory, where we can also consider nontrivial topologies of the normal directions to X q+2 . In
fact, nontrivial normal bundles can produce anomalies of the worldvolume fermions which
are cancelled by a shift of the quantization condition of the RR fluxes [29].12
We include our discussions on the restricted set of worldvolumes with trivial normal bundles below, since it illustrates various variants of (s)pin bordism groups in various dimensions
which were discussed in Sec. 8 of [15] and will be reviewed in the next subsection. Namely,
we will see that we are considering a fermionic system in which the Lorentz group is upliftedto the symmetry group which is an extension of the spin group as follows:
string theory
IIA
IIB
IIA
IIB
4m
4m+1
4m+2
RP
RP
RP
RP4m+3projective space.
+
Z4−
structure group
pin
spin
pin
spin × Z2
uniform notation in Sec. 3.1 spin[3] spin[2] spin[1] spin[4](3.1)

Here SpinZ4 (d) = (Spin(d)×Z4 )/Z2 , where the generator of the Z2 is the diagonal combinationof the (−1) F ∈ Spin(d) and exp(πi) ∈ Z4 ⊂ U(1).
Before proceeding, we note that the relation between the fermionic symmetry protected
topological phases and the D-brane/O-plane systems was discussed from a different perspective by Ryu and Takayanagi [33,34]. Their aim was more about reproducing the periodic tableof free fermionic classifications.
Using the η invariant, the anomalies described in Sec. 4 of [29] can be seen as follows. Let us consider type
IIA string theory with a Dq-brane whose worldvolume is Y (q+1) . To consider the anomaly, we uplift the fermions to(q + 2)-dimensional space X (q+2) . On this manifold, there is also the normal bundle whose fiber is R(8−q) . When the
manifold X (q+2) is orientable and even-dimensional as in the case considered in [29], the η invariant of the Dirac
operator DX is given by exp(−2πiη(DX )) = (−1)index DX , where index DX here is the mod 2 index of DX definedby using the reality condition (i.e. Majorana condition) of the spinors including the normal as well as the tangent
bundles. When the normal bundle is nontrivial, this mod 2 index can have nontrivial values and hence we get a
nontrivial value for the anomaly (−1)index DX . In this particular case, the global anomalies are just sign factors.
12

19

SciPost Phys. 7, 058 (2019)

3.1

Extensions of the spin group

We can construct a class of extensions of the d-dimensional spin group Spin(d). Those extensions depend on an integer mod 4, which we denote as k. Let us denote that extended groupas Spin(d; k). They are constructed as follows.
We first let k be a positive integer. We then consider a subgroup Spin(d; k) ⊂ Spin(d + k)such that the elements of this subgroup commute with Spin(k) ⊂ Spin(d + k). In otherwords, if we act Spin(d; k) on Rd+k = Rd ⊕ Rk as the SO(d + k) matrix after the reduction
Spin(d + k) → SO(d + k), we demand that these elements should act on the part Rk as ±I k ,where I k is the unit k × k matrix. Because −I k is allowed, Spin(d; k) is twice as big as Spin(d),roughly speaking. By considering each case k = 1, 2, 3, 4 explicitly, one can check that we haveSpin(d; 1) = Pin− (d),+Spin(d; 3) = Pin (d),Spin(d; 2) = (Spin(d) × Z4 )/Z2 ,Spin(d; 4) = Spin(d) × Z2 .

(3.2)

One can also check that there are isomorphisms Spin(d; k+4) ∼= Spin(d; k), and hence Spin(d; k)only depends on k mod 4. Using this property, we generalize our definition of Spin(d; k) toarbitrary integral k.
There are natural homomorphisms
ρ1 : Spin(d; k) → O(d),ρ2 : Spin(d; k) → Z2 .

(3.3)

These homomorphisms are defined by first projecting from Spin(d + k) → SO(d + k) and thenconsidering the induced action on Rd ⊂ Rd+k and Rk ⊂ Rd+k respectively. These homomorphisms allow us to put G = Spin(d; k) into the following commutative diagram:ρ2

0 → Spin(d) → G −→ Z2 → 0↓↓ρ1
SO(d) → O(d),(3.4)

where the first line is an exact sequence. In fact Spin(d; k) for k = 1, 2, 3, 4 exhaust all such G
which fit in this diagram.
Let us call manifolds with the structure group Spin(•; k) as spin[k]-manifolds.13 In thespin[k]of d-dimensional manifolds with spin[k]-structure, there is a homobordism group Ωdmorphismspin[k]s : Ωdspin[k+1]→ Ωd−1,(3.5)

called the Smith homomorphism, defined as follows.14
On a spin[k]-manifold Md , we have a Z2 -bundle which is constructed from the Spin(d; k)bundle and the homomorphism ρ2 : Spin(d; k) → Z2 . In general, a Z2 -bundle is classified
by an element a ∈ H 1 (Md , Z2 ) representing the holonomy of the Z2 bundle. Moreover, anyelement a ∈ H 1 (Md , Z2 ) has a Poincaré dual submanifold Nd−1 ⊂ Md [35]. To construct thisPoincaré dual, we consider a real line bundle L on Md associated to the principal Z2 bundle.
We take a generic section f of this real line bundle L, and define Nd−1 = { f = 0} which issmooth if f is generic enough. One can see that the differential d f gives an isomorphism
L|Nd−1 ∼= N (Nd−1 ),

(3.6)

13
In general, given a real bundle ξ over BG, one can define the spin structure twisted by ξ on M to be a map
f : M → BG, i.e. a G-bundle, together with a spin structure of T M ⊕ f ∗ (ξ). There is a corresponding bordism
spin
group usually denoted by Ωd (BG, ξ) and called as the twisted spin bordism groups. See Chapter 4.4.3 of [25]for details on the twisted spin bordisms. Our spin[k] structure corresponds to taking G = Z2 and choosing ξ to be
L ⊕k where L is the real line bundle over BZ2 = RP∞ whose w1 is the generator of H 1 (RP∞ , Z2 ). In their notation
spin[k]spinΩd= Ωd (BZ2 , L ⊕k ).
14
For general discussion of Smith homomorphisms in various cobordism theories, see Chapter 3.3 of [25].

20SciPost Phys. 7, 058 (2019)

where N (Nd−1 ) is the normal bundle of Nd−1 in Md . Then we get
T Md |Nd−1 ∼= T Nd−1 ⊕ L|Nd−1 .

(3.7)

From the definition of the Spin(d; k) groups, it follows that the Spin(d; k) structure on
T Nd−1 ⊕ L|Nd−1 is the same as the Spin(d − 1; k + 1) structure on T Nd−1 . In fact, on a generalmanifold M , a spin[k] structure on T M is a spin structure on T M ⊕ L ⊕k . Therefore, we get amapspin[k]s : Ωdspin[k+1]

3 [Md ] 7→ [Nd−1 ] ∈ Ωd−1

.

(3.8)

We will use this homomorphism in later discussions.

3.2

Properties of (s)pinors on RPn

Now we consider (s)pin structures or its generalizations spin[k] structures on RPn . We will bebrief; for more mathematical details, the readers should consult [36] for a standard accountof pin structures in low dimensions, [37, 38] for those in general dimensions, and [25] for theη invariants of RPn and other fixed-point-free quotients of spheres.
First, let us recall that the total Stiefel-Whitney class of RPn is given byw(RPn ) = (1 + a)n+1 ,(3.9)

where a is the generator of H (RP , Z2 ). In particular, we have1n

w1 = (n + 1)a,w2 =(n + 1)n 2
a .
2(3.10)

Therefore RP2` is non-orientable while RP2`+1 is orientable.
For orientable manifolds, recall that the existence of the spin structure is equivalent to
w2 = 0, and the existence of the spinc structure is equivalent to the condition that w2 is a
mod-2 reduction of an integral class. For non-orientable manifolds, recall that the existenceof a pin+ structure or a pin− structure is equivalent to w2 = 0 and w2 + w21 = 0 respectively,
and that the existence of a pinc structure is equivalent to the condition that w2 is a mod-2
reduction of an integral class.
We note that a is the first Stiefel-Whitney class of the tautological real line bundle L overnRP , and a2 is the first Chern class of L ⊗ C. Therefore, every RP2` is pinc and RP2`+1 is spinc .
In more detail, RP4m+2 is pin− , RP4m+3 is spin, RP4m is pin+ , and RP4m+1 is spinc .
In fact, RP4m+1 is a spinZ4 manifold. Not only that, we can uniformly construct a spin[k]structure on RPn , for n + 1 + k ≡ 0 mod 4, as follows.
For this purpose, we first consider Rn+1 and uplift the structure group of the (trivial) tangent bundle T Rn+1 to Spin(n + 1; k). Then we take an element R ∈ Spin(n + 1; k) on Rn+1
given by

R := (Γ 0 Γ 1 · · · Γ n )(Γ n+1 · · · Γ n+k ).

(3.11)

When n + 1 + k ≡ 0 mod 4, the R defined above satisfies the following two conditions:

R2 = 1,R† Γ I R = −Γ I .

(3.12)

1
2 (n+k)(n+k+1)

For example, the first is satisfied since R2 = (−1). Then we can define a manifold
RPn = S n /{1, R} equipped with the Spin(n; k) structure which follows naturally from theSpin(n + 1; k) bundle on Rn+1 .
Note that this R is mapped toρ1 (R) = −I n+1 ∈ O(n + 1),

ρ2 (R) = −1 ∈ Z2 .

(3.13)

Because of the latter relation, we see that the generator a ∈ H 1 (RPn , Z2 ) is the holonomy of
the Z2 bundle.
21

SciPost Phys. 7, 058 (2019)

3.3

O4 and RP4

Let us first recall the case of O4-planes and D2-branes, which were discussed at length in [8] inthe context of M2-branes in non-orientable spacetime in M-theory. Here the angular directionpin+
RP4 has the pin+ = spin[3] structure, and therefore we need to consider Ω4 . This group is
known to be Z16 generated by RP4 , and the η invariant of a pin+ Majorana fermion of RP4
is known to give exp(2πi/16) [36, 39]. This Z16 famously classifies the interacting fermionic
SPTs in 3+1d [40–44], and also the anomaly of time-reversal invariant fermionic systems in
2+1d [11, 45], both with T2 = (−1) F .
That the gaugino on the D2-brane is a representation of pin+ can be seen as follows: the
IIA theory itself has the pin+ structure which naturally follows from the pin+ structure of Mtheory (see appendix A), and the supercharge preserved by D2-brane is QD2 = Q + Γ 9 · · · Γ 3Q̃
(2.31). Therefore, a spatial reflection along Γ I (I = 1, 2) on the D2-brane worldvolume actson the supercharge by Γ 9 · · · Γ 3 Γ I , which squares to +1.
The worldvolume theory of a D2-brane is N =8 supersymmetric. As such, the correspond4
ing η invariant in four dimensions is the eight times
R the generator, and assigns −1 to RP .
R
Since RP4 w4 = 1, this η invariant is given by (−1) M w4 for general 4-dimensional pin+ manifolds M .

3.4

O3 and RP5

In type IIB string theory, RP5 appears as the angular direction of an O3-plane. We already saw
above that RP5 is orientable, not spin, but spinc . That said, the natural structure in type IIB
string theory is not quite the spinc structure. Rather, we are using the center Z2 = {1, C } of theSL(2, Z) duality group. More precisely, what acts on the fermions is Z4 = {1, C , C 2 , C 3 }, since
C 2 = (−1) F (see appendix A). Indeed, C = Ω(−1) F and this combination appears in (2.46)
when p = 3. Then, the structure group SO(d) is lifted not to Spin(d) but to (Spin(d) × Z4 )/Z2
where the quotient identifies (−1) F ∈ Spin(d) and C 2 ∈ Z4 . This is an analogue of the spincstructure where we use Z4 instead of U(1); for the lack of a better name, let us call this a spinZ4
spinZ
structure. Therefore what interests us in this case is Ω5 4 . Notice that spinZ4 = spin[2] inthe notation of Sec. 3.1.
The basic feature of a spinZ4 = spin[2] manifold Md is as follows. The natural map
ρ2 : (Spin(d) × Z4 )/Z2 → Z4 /Z2 = Z2 defines a class a ∈ H 1 (Md , Z4 /Z2 ), defining a realline bundle L on Md . The spinZ4 = spin[2] structure on Md is a spin structure on T Md ⊕ L ⊕ L.
Therefore, we have the condition that w2 (Md ) = a2 . More physically, the bundle L ⊕ L may
be regarded as the tangent bundle of the F-theory torus T 2 on which C acts as a 180◦ rotation
as explained in appendix A.
The Poincaré dual of a can be realized as a submanifold Nd−1 ⊂ Md , as we already saw inSec. 3.1. Now we easily compute thatw2 (Nd−1 ) = w2 (Md ) − a2 = 0.

w1 (Nd−1 ) = a,(3.14)

This means that Nd−1 has a pin+ = spin[3] structure. This construction is clearly cobordism
invariant, and defines a homomorphism
spinZ4s : Ωdpin+→ Ωd−1 .

(3.15)

This is the Smith homomorphism given in (3.8).
When d = 5, the Smith homomorphism clearly sends our RP5 to RP4 , which we already
spinZ
pin+
pin+
recalled to be the generator of Ω4 = Z16 . In fact Ω5 4 ' Ω4 = Z16 .15
15

spinZ4To see this, one identifies Ωdspin' Ωd (BZ2 , ξ) as explained in footnote 13. We then apply the Atiyah-

22

SciPost Phys. 7, 058 (2019)In terms of the anomaly of fermions, consider a 4d Majorana fermion ψ on which Z4
acts as a multiplication by i. We can try to give a position-dependent mass term m(x)ψψ,
but m(x) is odd under Z4 /Z2 . Therefore, m needs to be zero along the Poincaré dual N3 ofa ∈ H 1 (M4 , Z4 /Z2 ). We now have a pin+ fermion, i.e. a time-reversal-invariant fermion with
T2 = (−1) F along N3 . This relates the anomaly of spinZ4 fermions in 4d and that of pin+
fermions in 3d, and the latter is famously characterized by Z16 .
The worldvolume theory of a D3-brane is N =4 supersymmetric. Therefore, it has four
copies of this minimal amount of spinZ4 fermion. The corresponding η invariant in 5d thenassigns four times the generator of Z16 , i.e. exp(±πi/2), to RP5 .
Remark on S-fold: The O3-plane has generalizations, called S-folds, which use more nontrivial elements of SL(2, Z) than C [48,49]. The anomaly cancellation in this case is much more
nontrivial than the O3-plane since the U(1) gauge field can contribute to the anomaly [50].
The relevant structure group is spinZ2k := (spin × Z2k )/Z2 , where Z2k is a spin cover of thesubgroup Zk ⊂ SL(2, Z) for k = 2, 3, 4, 6. (See appendix A for the reason why the spin coverof SL(2, Z) appears.) It would be interesting to work out the details.

3.5

O2 and RP6
pin−

Here RP6 is pin− = spin[1], and therefore we are interested in Ω6 . In type IIA string theory,pin− is realized as follows. Recall that M-theory has pin+ . Then, in type IIA, whenever we
act a spatial reflection in the direction I ≤ 9, we combine it also with the reflection along the
M-theory circle which we represent by I = 10. Because (Γ I Γ 10 )2 = −1 for I ≤ 9, it becomespin− . The line bundle L associated to the Z2 (orientation) bundle is physically interpreted as
the tangent bundle of the M-theory circle.
The reflection along the M-theory circle is realized by (−1) F L in the IIA theory, so we are
combining the pin+ reflections and (−1) F L to realize pin− . The factor (−1) F L is seen in (2.46)
when p = 2.
pin−
It is known that Ω6 ' Z16 . The η invariant of the minimal pin− fermion on RP6 is known
pin−

to be exp(±πi/8), and therefore RP6 generates Ω6 .
That the gaugino on the D4-brane is a representation of pin− can be seen just as we considered the pin+ -ness of the D2-brane above. Namely, the supercharge preserved by D4-brane
is QD4 = Q + Γ 9 · · · Γ 5Q̃ (2.31). Therefore, a spatial reflection along Γ I (I ≤ 4) on the D4brane worldvolume acts on the supercharge by Γ 9 · · · Γ 5 Γ I , which squares to −1. The worldvolume theory on a D4-brane is N =2 supersymmetric. Therefore, it has two copies of minimal
fermions, and assigns exp(±πi/4) to RP6 .

3.6

O1 and RP7
spin

In this case, RP7 is spin, but Ω7 = 0. Therefore, we should be seeing something else.
Recall that in 6d, a chiral fermion has a perturbative gravitational anomaly. Correspondingly, the η invariant of the smallest fermion in 7d is not a cobordism invariant. Instead, let us
consider a pair of a Weyl fermion with positive chirality and a Weyl fermion with negative chirality. We can further introduce a Z2 symmetry and declare that the positive-chirality fermion
is odd while the negative-chirality fermion is even. The perturbative gravitational anomaly is
spincanceled, and the corresponding η invariant in 7d gives a cobordism invariant for Ω7 (BZ2 ).
Hirzebruch spectral sequence (AHSS) for the twisted spin bordisms, constructed in [46, 47]. The E 2 page is in fact
the same as the AHSS for Ωspin (BZ2 ), and the difference is only in the differentials. From this one easily sees that
spinZ
spinZ
|Ω5 4 | ≤ 16. As we already saw that the homomorphism sends RP5 to RP4 and hence |Ω5 4 | ≥ 16, we are
done.

23

SciPost Phys. 7, 058 (2019)

The Z2 symmetry relevant for our string theory setup is generated by Ω as can be seen in (2.46)
with p = 1. Therefore the relevant structure is spin × Z2 = spin[4].
As discussed in Sec. 3.1, we can define a homomorphismspins : Ωdpin−(BZ2 ) → Ωd−1 .

(3.16)

spin

This was defined as follows: an element of Ωd (BZ2 ) comes from a pair of a spin manifold Md
and an element a ∈ H 1 (Md , Z2 ). The Poincaŕe dual of a can be represented by a submanifold
spin
spin
Nd−1 ⊂ Md , and Nd−1 is pin− . The kernel of s clearly includes Ωd ⊂ Ωd (BZ2 ), and therefore
we have a homomorphism
spin
pin−
s : Ω̃d (BZ2 ) → Ωd−1 ,(3.17)
spin

where Ω̃dspin

(BZ2 ) is the reduced bordism group Ωdspin(BZ2 ) = Ωd

pin−

spin

spin

⊕ Ω̃d

(BZ2 ).

A homomorphism in the inverse direction t : Ωd−1 → Ω̃d (BZ2 ) can be constructed asfollows. We consider the R2 -bundle L ⊕ R over Nd−1 , where L is the orientation line bundle
and R is a trivial real line bundle. We let Md0 be the unit circle bundle over Nd−1 constructedfrom this L ⊕ R. This Md0 has a natural spin structure, since the pin− structure on Nd−1 gives
the spin structure on T Nd−1 ⊕ L, which gives the spin structure on the total space of the bundle
L ⊕ R over Nd−1 , and the Md0 is the boundary of the unit disk bundle D ⊂ L ⊕ R over Nd−1 .
However, it is bordant to the empty manifold since ∂ D = Md0 . Now we introduce an additional
Z2 holonomy a ∈ H 1 (Md , Z2 ) which assigns −1 to the circle fiber. We denote this spin manifold
equipped with the Z2 holonomy as Md . This construction gives a map
pin−

spin

t : Ωd−1 → Ω̃d(BZ2 ).

(3.18)
spin

spin

Notice that the codomain of this map can be taken to be the subgroup Ω̃d (BZ2 ) ⊂ Ωd (BZ2 ),because if we neglect the Z2 holonomy, it is bordant to ∅ by the disk bundle as discussed
pin−
above. The two homomorphisms s and t satisfy s ◦ t = id. Therefore we see that Ωd−1 isspina direct summand of Ω̃d (BZ2 ). In fact, the homomorphism (3.17) is an isomorphism forgeneral d, known as the Smith isomorphism.16
pin−
We already saw that Ω6 = Z16 is generated by RP6 . Combining the Smith isomorphism
spin

spin

and Ω7 = 0, we see that Ω7 (BZ2 ) = Z16 generated by RP7 with a Z2 bundle whose w1 is
the generator of H 1 (RP7 , Z2 ).
The η invariant of a minimal 7d fermion on RP7 , with a standard choice of the metric,
is ±1/32, where the sign depends on the choice of the spin structure. The η invariant of aminimal 7d fermion depends continuously on the metric, however; a cobordism invariant isgiven by taking the difference, which assigns exp(±πi/8) to RP7 .
The worldvolume theory on a D5-brane is N =(1, 1) supersymmetric. Therefore, the perturbative gravitational anomaly cancels, and there is a Z2 symmetry Ω such that the massterm for the fermions is odd under it. This Z2 symmetry generated by Ω is a part of the duality
symmetry of type IIB superstring, under which g, C2 are even while C0 , B2 and C4 are odd.
The RP7 around an O1 has a nontrivial holonomy of this Z2 symmetry as shown in (2.46).
Therefore, the η invariant controlling the global anomaly of the worldvolume fermions on a
D5-brane is exp(±πi/8).
pin−

A proof goes as follows. We need to show that the kernel of s is trivial. So, let us assume N (d−1) is zero in Ωd−1 .
Then there is a pin− manifold W (d) such that ∂ W (d) = N (d−1) . Next we consider the orientation line bundle Z (d+1)
over W (d) . Z (d+1) is spin and has a canonical Z2 bundle on it. Z (d+1) is a manifold with corners. One boundary isof the form R × N (d−1) , which we paste to the tubular neighborhood of N (d−1) in M (d) × {1} in M (d) × [0, 1]. Then
M (d) × [0, 1] t Z (d+1) provides a bordism from M (d) to a certain X (d) . Now X (d) is spin, and the Z2 bundle on it istrivial since the Poincaré dual of the Z2 holonomy can be taken to be given by N (d−1) × [0, 1] t W (d) which doesspin
not touch X (d) . Therefore it is in Ωd , q.e.d. The authors thank R. Thorngren for providing this proof.
16

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SciPost Phys. 7, 058 (2019)

GSO projection in string theory: In passing, we mention that the same argument shows that
pin−
pin−
spin
Ω3 (BZ2 ) = Ω2 = Z8 . On the one hand, Ω2 = Z8 is generated by RP2 , whose Brown-Arf
spininvariant is 1/8. On the other hand, Ω3 (BZ2 ) = Z8 is generated by RP3 , and classifies theanomaly of 2d systems with equal numbers of left-moving and right-moving fermions, withan additional Z2 symmetry. The corresponding Z2 bundle can be thought of as measuring thedifference of the left-moving spin structure and the right-moving spin structure.
Consider N f non-chiral 2d Majorana fermions. The η invariant assigns exp(±N f πi/4) toRP3 . Therefore, only when N f is a multiple of 8, the system is non-anomalous and we canperform the sum over the left-moving and the right-moving spin structures independently, or
equivalently perform the chiral Gliozzi-Scherk-Olive projection [51]. Luckily, the superstring
worldsheet in the light-cone gauge has N f = 8. This is another mathematical accident underlying the consistency of superstring theory. This anomaly cancellation for N f = 8 was firstshown in [52].

3.7

O0 and RP8

Finally we briefly mention the case of O0 and RP8 . Here RP8 is pin+ . This leads us to examinepin+Ω8 . This is known to be ' Z2 × Z32 . The η invariant of the minimal pin+ fermion on RP8
is known to be exp(±πi/16), and is the generator of the Z32 part.
The worldvolume theory on a D6-brane is N =1 supersymmetric. Therefore, it assigns to
8
RP the phase exp(±πi/16).

3.8

Summary

We can summarize the discussions in this section as follows:orientifold
projective spacecobordism group· in our notation· explicitly
D-brane involved
susy on the D-brane±orientifold charge

O4
RP4

O3
RP5

O2
RP6

O1
RP7

Ω4
spin[3]
Ω4
Z16
D2
N =8
1
8 × 16= 21Ω5 4
spin[2]
Ω5
Z16
D3
N =4
1
4 × 16
= 14

spinZ

Ω6
spin[1]
Ω6
Z16
D4
N =2
1
2 × 16= 18

pin−Ω7 (BZ2 )spin[4]Ω7Z16
D5N =(1, 1)
1
1
1 × 16
= 16

pin+

spin

O0
RP8

pin+

Ω8
spin[3]Ω8.
Z32 × Z2
D6
N =1
1
1
1 × 32= 32

Acknowledgements
The authors would like to thank Y. Imamura for correspondence on supergravity. Y.T. is partially supported by JSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grant-in-Aid (Kiban-S), No.16H06335, and also by WPI Initiative, MEXT, Japan at IPMU,
the University of Tokyo. K.Y. is supported by JSPS KAKENHI Grant-in-Aid (Wakate-B),
No.17K14265.

A

The duality group of type IIB string theory

It is often said in the literature that the duality group of type IIB string theory is SL(2, Z). This
is not entirely precise: as we saw, the O3 background uses C ∈ SL(2, Z), but more precisely
C 2 = (−1) F . We also saw that the O1 background uses the Z2 symmetry Ω, which is noteven a part of SL(2, Z) but has nontrivial commutation relations with elements of SL(2, Z).
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SciPost Phys. 7, 058 (2019)

In this short appendix we specify the duality group more precisely. The basic statement is asfollows:17
The duality group of type IIB string theory is the pin+ version of the double cover of GL(2, Z).
Let us first recall the massless fields in type IIB supergravity. The bosons are the dilaton
e−φ , the metric g, the NS-NS B-field B2 , and the R-R fields C0 , C2 , C4 ; the fermions consist of
the dilatinos ψ, and the gravitinos Ψ I .
Non-perturbative symmetries: The action of the non-perturbative symmetry SL(2, Z) on
the bosonic fields is well-known. Schematically, we can say that g and C4 are invariant; two
two-forms (B2 , C2 ) transform as a doublet; and the combination τ := ie−φ + C0 is acted on by
the fractional linear transformation. The center of SL(2, Z) is Z2 = {1, C }. When we consider
the action on fermions, it is known that C in fact squares to (−1) F [55, 56]. This gives a
nontrivial extension of SL(2, Z) by Z2 = {1, (−1) F } known as Mp(2, Z).
Perturbative symmetries: From the perturbative worldsheet perspective, we see two Z2
symmetries, often denoted as Ω and (−1) F L . The former comes from the worldsheet orientation
reversal, and the latter is the left-moving fermion number. The parity under these two Z2
operations of bosonic fields are as follows:

R1 := Ω
R2 := (−1) F L

φ g C0
+ + −
+ + −

C4
−
−

B2
−
+

C2
+
− .

(A.1)

Here, we introduced another notations R1,2 for Ω and (−1) F L . This is because R1,2 extends
SL(2, Z) to GL(2, Z): Ra corresponds to the flip of the a-th coordinate, and sends τ → −τ̄. We
also see that R1 R2 = C , as it should be within GL(2, Z).
Now let us discuss the actions on the fermions. The worldsheet computation shows thatR1 2 = R2 2 = 1.

(A.2)

But they do not commute [3, Sec. 2.2]: Indeed, from the worldsheet point of view, we haveΩ(−1) F L = (−1) FR Ω = (−1) F (−1) F L Ω,

(A.3)

which in terms of R1,2 is given by

R1 R2 = (−1) F R2 R1 .

(A.4)

As an example, the action of them on the two supercharges Q and Q̃ of type IIB string theory
discussed in Sec. 2.2 is given as

 



 


Q
0 1
Q
Q
1 0
Q
R1
=
,
R2
=
.
(A.5)
Q̃
1 0
Q̃
Q̃
0 −1
Q̃
The relations (A.2) and (A.4) modulo (−1) F defines a Z2 × Z2 subgroup of O(2). The
element (−1) F then extends O(2) to Pin+ (2); R1 and R2 generate a subgroup isomorphic tothe dihedral group with eight elements [3, Sec. 2.2].
Combining the observations above, we see that, when the action on the fermions are taken
into account, GL(2, Z) is extended to a double cover of pin+ type.
17

It would be interesting to study the cancellation of the anomaly of this duality group in type IIB theory. Previous
analyses can be found e.g. in [53, 54].

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SciPost Phys. 7, 058 (2019)

F-theory interpretation: The structure we saw above has a natural F-theory interpretation.
We realize the F-theory as a limit of M-theory on a spacetime which is a T 2 -fibration. As iswell-known, the SL(2, Z) duality acts on T 2 as a change of basis of determinant one, and the
combination τ = ie−φ + C0 is the complex modulus of T 2 .
To see why SL(2, Z) is extended to Mp(2, Z), recall that the central element C ∈ SL(2, Z)
corresponds to a 180◦ rotation of the T 2 fiber. Clearly, C 2 then corresponds to a 360◦ rotation,
which acts by (−1) F . This means that C is in fact of order four, extending SL(2, Z) to Mp(2, Z).
To see why SL(2, Z) is extended to GL(2, Z), we realize that M-theory is in fact parity
symmetric. Therefore, we can consider the action of flipping only one coordinate of T 2 . When
we take the F-theory limit, this becomes a purely internal operation, which does not act on thespacetime of type IIB theory. They provide the operations R1 and R2 .
Finally, to see why the double cover of GL(2, Z) should be of type pin+ rather than pin− ,we recall that M-theory itself is pin+ rather than pin− . This is simply because the elevendimensional supercharge is a Majorana fermion, which is a representation of pin+ but not ofpin− .

References[1] J. Polchinski, Dirichlet branes and Ramond-Ramond charges, Phys. Rev. Lett. 75, 4724
(1995), doi:10.1103/PhysRevLett.75.4724, [arXiv:hep-th/9510017].
[2] S. Mukhi, Orientifolds: The unique personality of each spacetime dimension (1997),
arXiv:hep-th/9710004.
[3] A. Dabholkar, Lectures on orientifolds and duality (1998), arXiv:hep-th/9804208.
[4] C. Angelantonj and A. Sagnotti, Open Strings, Phys. Rept. 371, 1 (2002),
doi:10.1016/S0370-1573(02)00273-9, [Erratum: Phys. Rept. 376, 407 (2003),
doi:10.1016/S0370-1573(03)00006-1].
[5] J. Distler, D. S. Freed and G. W. Moore, Orientifold précis (2009), arXiv:0906.0795.
[6] C.-T. Hsieh, Y. Tachikawa and K. Yonekura, Anomaly of the electromagnetic duality of the Maxwell theory, Phys. Rev. Lett. 123, 161601 (2019),
doi:10.1103/PhysRevLett.123.161601, [arXiv:1905.08943].
[7] E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22, 1(1997), doi:10.1016/S0393-0440(96)00042-3, [arXiv:hep-th/9609122.].
[8] E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94, 195150(2016), doi:10.1103/PhysRevB.94.195150, [arXiv:1605.02391 [hep-th]].
[9] K. Dasgupta, D. P. Jatkar and S. Mukhi, Gravitational couplings and Z2 orientifolds,
Nucl. Phys. B 523, 465 (1998), doi:10.1016/S0550-3213(98)00155-2, [arXiv:hepth/9707224].
[10] X. Dai and D. S. Freed, η-invariants and determinant lines, J. Math. Phys. 35, 5155
(1994), doi:10.1063/1.530747, [arXiv:hep-th/9405012].
[11] E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88,
035001 (2016), doi:10.1103/RevModPhys.88.035001, [arXiv:1508.04715 [condmat.mes-hall]].

27

SciPost Phys. 7, 058 (2019)[12] K. Yonekura, Dai-Freed theorem and topological phases of matter, J. High Energ. Phys. 09,
022 (2016), doi:10.1007/JHEP09(2016)022, [arXiv:1607.01873 [hep-th]].
[13] E. Witten and K. Yonekura,
arXiv:1909.08775.

Anomaly Inflow and the η-Invariant (2019),[14] E. Witten, World-sheet corrections via D-instantons, J. High Energ. Phys. 02, 030 (2000),
doi:10.1088/1126-6708/2000/02/030, [arXiv:hep-th/9907041].
[15] A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, J. High Energ. Phys. 12, 001 (2015),
doi:10.1007/JHEP12(2015)052, [arXiv:1406.7329 [cond-mat.str-el]].
[16] L. Alvarez-Gaumé, S. Della Pietra and G. Moore, Anomalies and odd dimensions, Ann.
Phys. 163, 288 (1985), doi:10.1016/0003-4916(85)90383-5.
[17] E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100, 197 (1985),
doi:10.1007/BF01212448.
[18] E. Witten, Topological tools in 10-dimensional physics, Int. J. Mod. Phys. A 01, 39 (1986),
doi:10.1142/S0217751X86000034.
[19] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian Geometry.
I, Math. Proc. Camb. Phil. Soc. 77, 43 (1975), doi:10.1017/S0305004100049410.
[20] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry.
II, Math. Proc. Camb. Phil. Soc. 78, 405 (1975), doi:10.1017/S0305004100051872.
[21] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry.
III, Math. Proc. Camb. Phil. Soc. 79, 71 (1976), doi:10.1017/S0305004100052105.
[22] H. Fukaya, T. Onogi and S. Yamaguchi, Atiyah-Patodi-Singer index fromthe domain-wall fermion Dirac operator, Phys. Rev. D 96, 125004 (2017),
doi:10.1103/PhysRevD.96.125004, [arXiv:1710.03379 [hep-th]].
[23] P. B. Gilkey, The eta invariant and the K-theory of odd dimensional spherical space forms,
Invent. Math. 76, 421 (1984), doi:10.1007/BF01388468.
[24] P. B. Gilkey, The eta invariant for even dimensional PINc manifolds, Adv. Math. 58, 243(1985), doi:10.1016/0001-8708(85)90119-7.
[25] P. B. Gilkey, The geometry of spherical space form groups, World Sci., Series in Pure Mathematics 28, (2018), doi:10.1142/10467.
[26] H. Donnelly, Eta invariants for G-spaces, Indiana Univ. Math. J. 27, 889 (1978),doi:10.1512/iumj.1978.27.27060.
[27] J. H. Schwarz and P. C. West, Symmetries and transformations of chiral N = 2, D = 10
supergravity, Phys. Lett. B 126, 301 (1983), doi:10.1016/0370-2693(83)90168-5.
[28] Y. Imamura, Notes on Supergravity (in Japanese), http://www.th.phys.titech.ac.jp/~imamura/note/sugra.pdf.
[29] G. Moore and E. Witten, Self-duality, Ramond-Ramond fields, and K-theory, J. High Energ.
Phys. 05, 032 (2000), doi:10.1088/1126-6708/2000/05/032, [arXiv:hep-th/9912279].

28

SciPost Phys. 7, 058 (2019)[30] D.-E. Diaconescu, G. Moore and E. Witten, E8 gauge theory, and a derivation of K-theory from M -theory, Adv. Theor. Math. Phys. 6, 1031 (2002),
doi:10.4310/ATMP.2002.v6.n6.a2, [arXiv:hep-th/0005090].
[31] D. S. Freed and M. J. Hopkins, Reflection positivity and invertible topological phases(2016), arXiv:1604.06527.
[32] K. Yonekura, On the Cobordism classification of symmetry protected topologicalphases, Commun. Math. Phys. 368, 1121 (2019), doi:10.1007/s00220-019-03439-y,[arXiv:1803.10796 [hep-th]].
[33] S. Ryu and T. Takayanagi, Topological insulators and superconductors from D-brane, Phys.
Lett. B 693, 175 (2010), doi:10.1016/j.physletb.2010.08.019, [arXiv:1001.0763 [hepth]].
[34] S. Ryu and T. Takayanagi, Topological insulators and superconductors from string
theory, Phys. Rev. D 82, 086014 (2010), doi:10.1103/PhysRevD.82.086014,[arXiv:1007.4234 [hep-th]].
[35] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helvetici 28, 17 (1954), doi:10.1007/BF02566923.
[36] R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds, in Geometry of
Low-Dimensional Manifolds, vol. 2, Lond. Math. Soc. Lect. Note Ser. 151, 177 (1990).
[37] D. W. Anderson, E. H. Brown and F. P. Peterson, Pin cobordism and related topics, Commentarii Mathematici Helvetici 44, 462 (1969), doi:10.1007/BF02564545.
[38] R. C. Kirby and L. R. Taylor, A calculation of Pin+ bordism groups, Comment. Math. Helvetici 65, 434 (1990), doi:10.1007/BF02566617.
[39] S. Stolz, Exotic structures on 4-manifolds detected by spectral invariants, Invent. Math. 94,
147 (1988), doi:10.1007/BF01394348.
[40] L. Fidkowski, X. Chen and A. Vishwanath, Non-abelian topological order on the surface
of a 3D topological superconductor from an exactly solved model, Phys. Rev. X 3, 041016(2013), doi:10.1103/PhysRevX.3.041016, [arXiv:1305.5851 [cond-mat.str-el]].
[41] C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors inthree dimensions, Phys. Rev. B 89, 195124 (2014), doi:10.1103/PhysRevB.89.195124,[Erratum: Phys. Rev.B 91, 239902 (2015), doi:10.1103/PhysRevB.91.239902].
[42] M. A. Metlitski, L. Fidkowski, X. Chen and A. Vishwanath, Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold
way and fermionic Kramers doublets (2014), arXiv:1406.3032.
[43] A. Kitaev, Homotopy-theoretic approach to spt phases in action: Z16 classification of threedimensional superconductors, Lecture notes, http://www.ipam.ucla.edu/abstract/?tid=12389&pcode=STQ2015.
[44] T. Morimoto, A. Furusaki and C. Mudry, Breakdown of the topological classification Z for
gapped phases of noninteracting fermions by quartic interactions, Phys. Rev. B 92, 125104(2015), doi:10.1103/PhysRevB.92.125104, [arXiv:1505.06341 [cond-mat.str-el]].
[45] C.-T. Hsieh, G. Young Cho and S. Ryu, Global anomalies on the surface of fermionicsymmetry-protected topological phases in (3+1) dimensions, Phys. Rev. B 93, 075135(2016), doi:10.1103/PhysRevB.93.075135, [arXiv:1503.01411 [cond-mat.str-el]].
29

SciPost Phys. 7, 058 (2019)[46] P. Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann.
295, 745 (1993), doi:10.1007/BF01444915.
[47] A. V. Zhubr, Spin bordism of oriented manifolds and the Hauptvermutung for 6-manifolds,
in Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin’s Memorial, American
Mathematical Society Translations (2001), doi:10.1090/trans2/202/19.
[48] I. García-Etxebarria and D. Regalado, N = 3 four dimensional field theories, J. High Energ.
Phys. 03, 083 (2016), doi:10.1007/JHEP03(2016)083, [arXiv:1512.06434 [hep-th]].
[49] O. Aharony and Y. Tachikawa, S-folds and 4d N = 3 superconformal field theories, J. HighEnerg. Phys. 06, 044 (2016), doi:10.1007/JHEP06(2016)044, [arXiv:1602.08638 [hepth]].
[50] N. Seiberg, Y. Tachikawa and K. Yonekura, Anomalies of duality groups and extended conformal manifolds, Prog. Theor. Exp. Phys., 073B04 (2018) doi:10.1093/ptep/pty069,[arXiv:1803.07366 [hep-th]].
[51] F. Gliozzi, J. Scherk and D. Olive, Supersymmetry, supergravity theories and the dual spinor
model, Nucl. Phys. B 122, 253 (1977), doi:10.1016/0550-3213(77)90206-1.
[52] E. Witten, Global anomalies in string theory, in Symposium on Anomalies, Geometry, Topology, Argonne, Illinois, March 28-30 (1985), https://www.sns.ias.edu/witten/
symp-global-anomalies-stringtheory-1985.
[53] M. R. Gaberdiel and M. B. Green, An SL(2, Z) anomaly in IIB supergravity andits F-theory interpretation, J. High Energ. Phys. 11, 026 (1998), doi:10.1088/11266708/1998/11/026, [arXiv:hep-th/9810153].
[54] S. Mukhi, Dualities and the SL(2, Z) anomaly, J. High Energ. Phys. 12, 006 (1998),
doi:10.1088/1126-6708/1998/12/006, [arXiv:hep-th/9810213].
[55] T. Pantev and E. Sharpe, Duality group actions on fermions, J. High Energ. Phys. 11, 171(2016), doi:10.1007/JHEP11(2016)171, [arXiv:1609.00011 [hep-th]].
[56] D. R. Morrison. Private communication.

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