SciPost Phys. 9, 010 (2020)

Topological superconductors on superstring worldsheets
Justin Kaidi1 , Julio Parra-Martinez1,2 and Yuji Tachikawa3 ,with a mathematical appendix by Arun Debray 4
1 Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy,
University of California, Los Angeles, CA 90095, USA
2 Center of Mathematical Sciences and Applications,
Harvard University, Cambridge, MA 02138, USA
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
4 Department of Mathematics, University of Texas at Austin, TX 78712, USA

Abstract
We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible
phases, or equivalently topological superconductors, on the worldsheet. This allows us
to find that the unoriented Type 0 string theory with Ω2 = (−1)f admits different GSO
projections parameterized by n mod 8, depending on the number of Kitaev chains on
the worldsheet. The presence of n boundary Majorana fermions then leads to the classification of D-branes by K O n (X) ⊕ K O −n (X ) in these theories, which we also confirm by
the study of the D-brane boundary states. Finally, we show that there is no essentially
new GSO projection for the Type I worldsheet theory by studying the relevant bordism
group, which classifies corresponding invertible phases. In two appendixes the relevant
bordism group is computed in two ways.
This paper provides the details for the results announced in the letter [1].
Copyright J. Kaidi et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 18-02-2020
Accepted 16-07-2020
Check forupdatesPublished 21-07-2020
doi:10.21468/SciPostPhys.9.1.010

Contents
1 Introduction and summary
1.1 Generalities
1.2 GSO projections and K-theory classification of D-branes

11
3

2 The (1+1)d topological superconductors
2.1 Oriented invertible phases
2.2 Unoriented invertible phases

6
6
8

3 GSO projections
3.1 Oriented strings
3.2 Unoriented strings
3.3 Branes and K-theory

12
12
15
18

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SciPost Phys. 9, 010 (2020)

4 D-brane spectra via boundary fermions
4.1 (0+1)d Majorana fermions and their anomalies
4.2 D-branes and boundary fermions
4.3 K-theory classification of branes
4.4 Vacuum manifolds of tachyons as classifying spaces

20
20
23
25
28

5 D-brane spectra via boundary states
5.1 Matching non-torsion brane spectra
5.2 Matching torsion brane spectra
5.3 Pin+ Type 0 theories

29
30
30
32

6 No new Type I theories
6.1 ‘Spin structure’ on the Type I worldsheetd
6.2 The group fDPin(pt)
6.3 Invertible phases for DPin structure

33
34
35
36

A NSR formalism

37

B Boundary state formalism
B.1 Basics
B.2 Theta functions, partition functions and boundary state amplitudes
B.3 D-brane boundary states
B.4 O-plane boundary states

39
39
40
42
45

C Tadpole Cancellation

47

D Arf and ABK from index theory
D.1 η-invariants: generalities
D.2 η-invariants: examples
D.3 Quadratic forms and enhancements

50
50
51
54

d
E fDPin(pt) via the Atiyah-Hirzebruch spectral sequence

55

F

d
fDPin
(pt) via the Adams spectral sequence

References

1

Introduction and summary

1.1

Generalities

59
63

In perturbative formulations of superstring theories, one treats the 2d worldsheets of strings
as 2d quantum field theories with fermions. The most common treatment is the one due
to Neveu-Schwarz [2] and Ramond [3], often called the NSR formalism. There, one starts
with ten bosonic fields X µ=0,...,9 and ten left- and right-moving fermionic fields ψµ=0,...,9 andψ̃µ=0,...,9 . To remove the closed-string tachyon and at the same time obtain spacetime spinors,
one must perform a crucial step called the Gliozzi-Scherk-Olive (GSO) projection [4, 5]. The
two Type II superstring theories, Type IIA and Type IIB, arise due to a difference in the specifics

2

SciPost Phys. 9, 010 (2020)of this projection.
It was pointed out in [6, 7] that the GSO projection can be interpreted as a sum over the
possible spin structures on the worldsheet, with different consistent GSO projections corresponding to different ways of assigning complex phases to spin structures, in a manner consistent with cutting and gluing of the worldsheet. This point of view makes manifest the all-genus
consistency of known GSO projections, which is not evident in the one-loop analysis often presented to beginners of string theory e.g. in [8]. It does not, however, tell us whether we havefound all possible GSO projections. Indeed, possible consistent GSO projections for unoriented
superstring theories have not been studied systematically in the past.
One approach to the enumeration of all consistent GSO projections comes from a rather unexpected place, namely from the study of symmetry-protected topological (SPT) and invertible
phases of matter in condensed matter physics.1 For the purposes of this paper, an invertible
phase for a symmetry G can be defined as a system which has a one-dimensional gapped vacuum on any closed spatial manifold with a background field for G. By taking the infrared
limit, one can then isolate a quantum system whose entire Hilbert space on any closed spatial
manifold with any background field is one-dimensional and contains the vacuum only, with its
partition function simply a complex phase. We note that for an internal symmetry, the background field is simply a non-dynamical gauge field for the symmetry. On the other hand, forfermion number symmetry the background field is the spin structure, while for time-reversal
symmetry the background field is the “un-oriented-ness” of the spacetime.
An invertible phase provides a method for assigning complex phases to spin and otherstructures on a manifold, in a manner consistent with cutting and gluing. Conversely, anysuch assignment corresponds to an invertible phase. Therefore, if one can classify invertible
phases, one can classify all consistent GSO projections. Invertible phases with fermion number
symmetry and some additional discrete symmetries are usually called topological superconductors in the condensed matter literature. Therefore, the classification of GSO projections isequivalent to the classification of topological superconductors in (1 + 1) dimensions.
The classification of invertible phases has been an important topic of recent research intheoretical condensed matter physics. As will be reviewed below, a general answer in termsof bordism groups has been obtained, see e.g. [9–11]. The upshot is that with this result, we
can now carry out the classification of possible GSO projections on a given worldsheet, oncethe structure on said worldsheet is specified.
One important feature of a nontrivial invertible phase in d dimensions for a symmetry
G is that, if it is put on a spacetime with boundary, the (d − 1)-dimensional boundary theory
necessarily hosts nontrivial degrees of freedom. In particular, if G is unbroken on the boundary,
the G-symmetric boundary theory carries a corresponding G-anomaly. One familiar case isthat of the chiral anomaly of a fermion in spacetime dimension 2n, which is captured by a
Chern-Simons term in 2n + 1 dimensions. This is known as anomaly inflow [12, 13]. The
current understanding is that all anomalies2 in (d − 1) dimensions, both local and global, can1

In the literature of high-energy physics, the terms SPT and invertible phase are often used interchangeably.
In condensed matter physics they have subtly different connotations. In this paper we stick to the terminology of
invertible phases, which are more directly relevant for our purposes.
2
Some qualifications need to be added to this blanket statement. First, this framework is mostly about theanomalies of partition functions, and therefore does not immediately describe the conformal anomaly. Second,
anomalies of supersymmetry are less well understood, and it is not clear whether they can be described by bulk
invertible phases. That said, neither is conclusively outside of this framework. As for the first, the anomaly described by the Kitaev chain is about the impossibility of quantizing a single Majorana fermion, which is also notdirectly about the phase of the partition function. As for the second, the supersymmetry anomaly recently foundin [14,15], which is a superpartner of the anomaly in R-symmetry, was first found in the context of AdS/CFT [16].
We also note that the shortening anomaly of [17] is related to the fact that the scalar target space of the holographic
supergravity dual is often not Kähler. All this suggests that these anomalies might also be described in a suitable
generalization of the current framework. It would be interesting to work this out.

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SciPost Phys. 9, 010 (2020)

be characterized in terms of invertible phases in d dimensions.
Applying this observation to the worldsheets of superstrings, we conclude that different
invertible phases, i.e. different GSO projections, will require different boundary conditions on
the edges of worldsheets. Since the boundaries of worldsheets describe the D-branes to which
strings attach, this means that the properties of D-branes reflect the choice of GSO projection.
The discussions up to this point can be summarized schematically as follows:boundary anomaly:
bulk invertible phase∼ properties of D-branes : choice of GSO projection.

1.2

(1.1)

GSO projections and K-theory classification of D-branes

Let us now be more concrete. First, we recall the classification of invertible phases in terms ofbordism groups.3 Let X denote collectively the structure on the spacetime, appropriate for thesystems we would like to classify. For example, X consists of a spin structure and a G gauge
field for systems with fermion number symmetry and an internal symmetry G. We define the
X -bordism group ΩXd in dimension d to beΩXd := {d-dimensional manifolds with X structure}/ ∼,(1.2)

where the equivalence relation is introduced so that Md ∼ Md0 if and only if there exists Nd+1
with the structure X such that ∂ Nd+1 has Md as the incoming boundary and Md0 as the outgoingboundary. The group structure is given by the disjoint union. Then, the topological invertible
phases in spacetime dimension d are classified by [9–11]
fdX := Hom(ΩXd , U(1)).

(1.3)

This simply means that the topological invertible phase for an element α ∈ fdX assigns thepartition function α(Md ) ∈ U(1) in such a way that it only depends on the bordism class[Md ] ∈ ΩXd .4
In this paper we will encounter the following structures on the worldsheet: spin structure
for oriented Type 0 strings, Spin × Z2 structure for oriented Type II strings, Pin± structure for
two types of unoriented Type 0 strings, and “double pin” or DPin structure for Type I strings.
We remark here that originally the II, I, 0 in Type II, I, 0 strings referred to the number ofsupersymmetries in ten dimensions. Contrary to this usage, in this paper we refer to any NSR
strings with independent GSO projections on left- and right-moving spin structures as Type II,
and to any NSR strings with diagonal GSO projections as Type 0. Type I strings will then be
defined as Type II NSR strings on unoriented worldsheets.
The relevant dual bordism groups are listed in Table 1. There we have used a slightly moregeneral notation, with ΩXd (Y ) representing the bordism group of d-dimensional manifolds with
X structure equipped with a map to Y . Then for example a structure X 0 consisting of spin structure and a Z2 gauge field can equivalently be thought of as having (X , Y ) = (spin structure, BZ2 ),For other recent applications of bordism groups to high-energy theory, see [18–26].
More precisely, fdX as defined here classifies invertible phases whose partition functions do not depend continR
uously on the background fields. Therefore it includes e.g. the 2d theta term θ F /(2π), which only depends on
topological data, but it does not include e.g. the 3d gravitational Chern-Simons term, which does depend continuously on the metric. The latter is accounted for by considering (DΩX )d+1 instead, where D denotes the Anderson
dual. This group classifies the deformation classes of invertible phases which can depend continuously on the
background fields. Since we take the deformation classes, (DΩX )d+1 does not include the theta term, which can becontinuously varied. The torsion parts of both groups coincide: Tors Hom(ΩXd , U(1)) = Tors(DΩX )d+1 , since torsion
invertible phases are discrete and cannot depend continuously on the background data. Both Hom(ΩXd , U(1)) and(DΩX )d+1 are generalized cohomology theories, but their common torsion part is not. This unfortunately makes
the torsion part less mathematically natural. For all the cases of interest to us in this paper, all bordism groups are
finite, so these subtle differences can and will be ignored.
3

4

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SciPost Phys. 9, 010 (2020)

Table 1: Dual bordism groups relevant to our analysis. The first four columns areclassic [9, 27]. The last column is new.
d
2
3

d
fSpin
(pt)
Z2
0

d
fSpin
(BZ2 )
Z22
Z8

d
fPin
− (pt)
Z80dfPin+ (pt)
Z2
Z2

d
fDPin
(pt)
Z22
Z8

0

Spin

where BZ2 is the classifying space of Z2 gauge fields, and therefore ΩXd = Ωd (BZ2 ). Similarly, letting pt stands for a point, we have ΩXd = ΩXd (pt).
Let us begin by discussing the group f2Spin (pt) = Z2 . The nontrivial element is mathematically known as the Arf invariant, and assigns to a surface Σ with a choice of spin structure σ
a sign (−1)Arf(Σ, σ) . A spin structure is called even or odd depending on whether this sign is+1 or −1. There are many mathematical and physical definitions of the Arf invariant, one ofwhich is as the number modulo two of zero modes of the Dirac operator on the surface Σ with2
spin structure σ [28]. From this definition, we see easily that (−1)Arf(T , σ) is −1 if and only if
the spin structure σ is periodic along both cycles of the torus T 2 . There is also a combinatorial
definition [29, 30], which we will recall below.
In the continuum quantum field theory language, the Arf invariant may be written in termsof the partition function of a mass m Majorana fermion on (Σ, σ) as
Zferm (m  0; Σ, σ)/Zferm (m  0; Σ, σ). In general, the infinite-mass limit of a fermion partition function is known as an η-invariant in the mathematics literature, meaning that the
Arf invariant is an example of an η-invariant. There is also a discretized Hamiltonian version of this massive Majorana fermion defined on a spin chain — this is known as the Kitaev
chain [31]. In both of these descriptions, it is easy to argue that one needs a single Majorana
fermion on the (0 + 1)d boundary of the (1 + 1)d system hosting the Arf invariant theory.
A single Majorana fermion cannot be consistently quantized, since two Majorana fermionsact on a two-dimensional Hilbert space irreducibly. This means that, assuming that this Hilbert
space is the tensor product of two copies of the Hilbert p
space for a single fermion, the singlefermion Hilbert space would need to have dimension 2. This is one manifestation of ptheanomaly of the boundary theory, and will turn out to explain the difference by a factor of 2
between the tensions of D9-branes in Type IIA and Type IIB theories, originally found in [32].
Without the Arf invariant on the worldsheet, as will be the case for Type IIB strings, theendpoints of open strings will naturally couple to unitary bundles. Consideration of tachyon
condensation motivates one to introduce an equivalence relation on unitary bundles, leading
to the statement that stable D-branes on X are classified by complex K-theory K 0 (X ) [33]. On
the other hand, in the presence of the Arf invariant the boundary of the worldsheet needs
unitary bundles together with an additional Majorana fermion, or equivalently with an actionof the complex Clifford algebra Cl(1, C). Unitary bundles with an action of Cl(1, C), undera suitable equivalence relation implementing tachyon condensation, are classified by K 1 (X ),thus reproducing the known classification of the D-branes in the Type IIA theory [33, 34].
We next discuss the effects of including the topological superconductor corresponding to
the group f2Pin− (pt) = Z8 on worldsheets. The worldsheets can now be nonorientable and areequipped with a Pin− structure. This structure will be seen to be compatible with the Type 0
string, but not with the Type I string. The generator of the group Z8 of time-reversal invariant
topological superconductors is again the Kitaev chain, but now with the added assumption oftime-reversal invariance. The invertible phase corresponding to n modulo 8 is simply n copiesof the Kitaev chain, and has n time-reversal symmetric Majorana fermions on the boundary.
Physically, the reason that we need only consider n modulo 8 is that one can introduce a four5

SciPost Phys. 9, 010 (2020)

fermi interaction to the n = 8 theory which gives rise to a theory with unique ground state [35].
In continuum field theory language, the effective action describing the basic non-trivial phaseis the Arf-Brown-Kervaire (ABK) invariant, to be discussed below.
Roughly speaking, with n copies of the time-reversal-symmetric Kitaev chain on the worldsheet, open string endpoints can now couple to orthogonal bundles with an action of n timereversal invariant Majorana fermions, or equivalently with an action of the real Clifford algebraCl(n, R). Tachyon condensation then leads to the classification of D-branes by KO n (X ). As wewill see, a more careful analysis reveals that the classification is in fact by KO n (X ) ⊕ KO−n (X ).
In the case of Type I strings, the natural way to specify the worldsheet fermions is toconsider chiral fermions on the orientation double cover of the worldsheet. This leads to a
structure which we call “double pin” structure, since it will be shown to contain both Pin± as
subgroups. We will find by a standard algebraic topology computation that any invertible phase
one can add on the worldsheet is either the Arf invariant associated to the orientation double
cover, or a continuum version of the Haldane chain. We will find that the Arf invariant on
the double cover can be removed by performing a spacetime parity transformation along one
direction, meaning that it does not give rise to physically distinct theories. On the other hand,
the invertible phase corresponding to the low energy limit of the S = 1 Haldane chain [36,37],
whose partition function counts the number of RP2 modulo 2, gives rise to the difference
between O9± -orientifold planes, thus differentiating between Type I and Ĩ worldsheet theories.
We note in passing that Ryu and Takayanagi have pointed out in [38, 39] that the periodic
table [40,41] of free topological superconductors and topological insulators, based on K-theory
and KO-theory, can be naturally realized by considering D-branes in string theory. In thoseworks the topological superconductors were realized on brane worldvolumes, whereas here
we consider the topological superconductors on string worldsheets.

Organization
The aim of this paper is to give details on the results presented thus far in the Introduction.
The rest of this paper is organized as follows.
In Sec. 2, we begin by reviewing the necessary preliminary material concerning topologicalsuperconductors with several variants of spin structure. In Sec. 3, we study the effects ofworldsheet invertible phases on massless closed string and D-brane spectra. In the next two
sections, we give a more detailed analysis of the classification of D-branes. This is done fromtwo different perspectives: in Sec. 4 we study D-branes via boundary fermions and tachyon
condensation, while in Sec. 5 we utilize the boundary state formalism. The final section Sec. 6
is devoted to the algebraic-topological study of the possible invertible phases on the Type I
worldsheet.
Additional background information and details of calculations are given in the appendices.
In Appendix A, we briefly review the NSR formulation of superstring theory. In Appendix B,
we provide a short review of the boundary state formalism necessary for calculations in Sec. 5.
The results of this appendix are also utilized in Appendix C, in which we discuss the issue of
tadpole cancellation for some of the Type 0 theories discussed in this paper. The final threeappendices are more mathematical. In Appendix D we reobtain many of our results for Arf
and ABK invariants by means of index theory. Much of this appendix is due to E. Witten [42].
In Appendix E we provide the technical details of the algebraic-topological computation usedin Sec. 6, which uses the Atiyah-Hirzebruch spectral sequence. In Appendix F, written by Arun
Debray, we explain another computation of the same bordism group via the Adams spectral
sequence.
The results of this paper were already announced in a short letter [1] by the same authors.

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SciPost Phys. 9, 010 (2020)

The authors also note that when said work was nearing completion, they were informed5 that
E. Witten has an unpublished work with large overlap with theirs; two seminars he gave canbe found in [43, 44], and are highly recommended for their clarity in presentation.

2

The (1+1)d topological superconductors

The main topic of this paper is the addition of fermionic invertible phases to oriented andunoriented string worldsheets. In this section we begin by reviewing some basic facts about
fermions and (s)pin structures (see [6, 45] for more detailed reviews), as well as about the
known invertible phases for Spin, Spin × Z2 , Pin− , and Pin+ structures [9]. Also of importanceto us will be “double pin” or DPin structure, though we postpone a discussion of this to Section
6.
Many of the results that we obtain via combinatoric methods in this section can also be
obtained via index theory, i.e. by studying the properties of free fermions. For completeness,
we discuss this approach in Appendix D.

2.1

Oriented invertible phases

On an oriented d-manifold M the structure group of the tangent bundle T M is SO(d). In order
to consider fermions on M , we need to lift SO(d) to its double cover Spin(d) as specified by
the short exact sequence
0 → Z2 → Spin(d) → SO(d) → 0 .
(2.1)
There might be an obstruction to doing so, which is captured by the second Stiefel-Whitneyclass of T M , i.e. w2 (T M ) ∈ H 2 (M , Z2 ). If this class is trivial, we say that the manifold admits
a spin structure. Such a spin structure is generically not unique. Given a spin structure, we
can obtain another one by twisting by an element of H 1 (M , Z2 ).
In this paper, we focus on two-dimensional manifolds Σ, which we take to be the worldsheet of a string. Any orientable two-manifold admits a spin structure since w2 (T Σ) = w21 (T Σ)mod 2. For notational simplicity, from now on we write w i := w i (T Σ).
2.1.1

Invertible phase for Spin

Our primary interest is in fermionic invertible phases, i.e. phases which depend on a choiceof spin structure σ on Σ. In the absence of any symmetry besides fermion number (−1)f , thegroup capturing such phases is f2Spin (pt) = Z2 . The effective action for the corresponding
fermionic invertible phase can be written in terms of the Arf invariant [9],e2iπSeff (Σ, σ) = (−1)Arf(Σ, σ) ,(2.2)

where Arf(Σ, σ) is defined modulo 2. For simplicity, we will often leave the dependence on σ
implicit.
As discussed in the Introduction, this phase is a continuum version of the Kitaev chain [31].
In the continuum field theory language, this corresponds to the definition [45](−1)Arf(Σ, σ) :=Zferm (m  0; Σ, σ),Zferm (m  0; Σ, σ)(2.3)

where Zferm (m; Σ, σ) is the partition function of a free massive Majorana fermion of mass m.
To see that the right-hand side is ±1, we note that the non-zero eigenvalues E of the Dirac5

The authors thank Kantaro Ohmori and Matthew Heydeman for this information.

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SciPost Phys. 9, 010 (2020)

operator D comes in pairs ±E, since Γ := γ1 γ2 is globally well-defined on an oriented spin
surface and DΓ = −Γ D. Therefore,
Zferm (+m) Y i E + m Y (i E + m)(−i E + m)=
= (−1)index D .
(2.4)
Zferm (−m) E=0 i E − m E>0 (i E − m)(−i E − m)
The Arf invariant can also be defined combinatorially. To do so, given a spin structure σ onΣ, we define q̃(a) ∈ Z2 for each Z2 -valued 1-cocycle a on Σ by taking a non-self-intersecting
1-cycle A Poincaré dual to it and declaring¨
0 if the spin structure around A is NS,(2.5)
q̃(a) =
1 if the spin structure around A is R .
This function q̃(a) is known as a quadratic form and satisfiesZq̃(a + b) − q̃(a) − q̃(b) =a ∪ b.

(2.6)

Σ

There is a one-to-one correspondence between such quadratic forms and spin structures [30].
The Arf invariant can be defined in terms of this quadratic form as follows,(−1)Arf(Σ, σ) := p1

X(−1)q̃(a) .

|H 1 (Σ, Z2 )| a∈H 1 (Σ,Z )
2

(2.7)

To see that the right-hand side is ±1, we consider its square:
RHS2 ===

1|H 1 (Σ, Z2 )|
1
1|H (Σ, Z2 )|1|H 1 (Σ, Z2 )|X(−1)q̃(a)+q̃(b)a,b ∈H 1 (Σ,Z2 )X

R(−1)q̃(a+b)+ Σ a∪ba,b ∈H 1 (Σ,Z2 )X

R(−1)q̃(c)+ Σ a∪c ,(2.8)

a,c ∈H 1 (Σ,Z2 )R
where we have defined c = a + b ∈ H 1 (Σ, Z2 ) and used that Σ a ∪ a = 0 for an orientablemanifold. When c = 0, the summandR is 1 and the sum contributes a factor of |H 1 (Σ, Z2 )|.
When c 6= 0 there are equally many Σ a ∪ c = 0, 1 contributions by assumption of a nondegenerate intersection pairing, and hence these contributions cancel out. Thus we find thatRHS2 = 1.
We now focus on the torus T 2 , which admits four spin structures. We begin by listing all
elements of H 1 (T 2 , Z2 ), which is an order four group containing {0, a, b, a + b}. Here a and b
are mod 2 Poincaré duals of the A- and B-cycles of the torus, respectively. Then using formula(2.7), we have1
p 1 + e iπq̃(a) + e iπq̃(b) + e iπq̃(a+b)
4

1=
1 + e iπq̃(a) + e iπq̃(b) − e iπq̃(a) e iπq̃(b) ,
(2.9)
2
R
where we have made use of (2.6) and noted that a ∪ b = 1 for the two 1-cycles of the torus.
Depending on the spin structure, one has (q̃(a), q̃(b)) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}, for which
2(−1)Arf(T ) =8

SciPost Phys. 9, 010 (2020)2

we find that (−1)Arf(T ) assigns(0, 0) :7→ +1,(0, 1) :7→ +1,(2.10)(1, 0) :7→ +1,(1, 1) :7→ −1.

Here, we have represented the spin structure by lines on the torus — a grey dashed line
means that fermions are anti-periodic i.e. NS in the normal direction, whereas solid red lines
means that fermions are periodic i.e. R in the normal direction. From the perspective of
canonical quantization the red lines can be interpreted as insertions of (−1)f symmetry defects/operators.
On a manifold with boundary, the bulk invertible phase requires the presence of an odd
number of Majorana fermions on each boundary. As reviewed in Section 4.1, there is no
canonical way to quantize an odd-dimensional Clifford algebra. This can be thought of as ananomaly of the boundary system, which is compensated by the presence of the bulk invertible
phase.
2.1.2

Invertible phase for Spin × Z2

We will also need to consider Spin × Z2 -structures on Σ. These are given by a choice of spin
structure σ and a Z2 bundle with background gauge field a ∈ H 1 (Σ, Z2 ). As mentioned above,
H 1 (Σ, Z2 ) acts on the space of spin structures, so the choice of (σ, a) is equivalent to a choiceof two separate spin structures (σ L , σR ) := (σ, σ + a), where we take σ L and σR to be theleft- and right-moving spin structures. The corresponding invertible phases are classified by
f2Spin (BZ2 ) = Z22 , which is generated by the separate Arf invariants for σ L and σR :(−1)Arf(Σ, σ L ) ,(−1)Arf(Σ, σR ) .

(2.11)The discussion of each of these phases is identical to that in the previous section.

2.2

Unoriented invertible phases

We would now like to discuss invertible phases which can be formulated on unoriented manifolds. They can be thought of as phases protected by the action of time-reversal T.
2.2.1

Invertible phase for orientation

Before considering fermionic phases protected by T, let us discuss bosonic phases protected by
T. The structure group of the tangent bundle of an unoriented d-manifold M is O(d), whichcannot be reduced to SO(d). The obstruction to doing so is given by the first Stiefel-Whitneyclass of the tangent bundle w1 ∈ H 1 (M , Z2 ). For a 1-cycle C, we have¨I
0 if going around C preserves orientation,
w1 =(2.12)
1 if going around C reverses orientation.
C
Bosonic unoriented phases on the worldsheet are classified by f2O (pt) = Z2 , the generatorof which is the low-energy limit of the S = 1 Haldane chain [36, 37]. In the continuum field

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SciPost Phys. 9, 010 (2020)

theory language, the effective action for this phase is6R2

e2πiSeff (Σ) = (−1) Σ w1 .

(2.13)22
2πiSeff (RP )The generator of the bordism group ΩO= −1.
2 (pt) is the projective plane RP , i.e. e
One can easily calculate the value of the action on any other manifold by counting the numberof constituent RP2 appearing in its connected sum decomposition (mod 2). For instance, theMöbius strip M2 is a connected sum of the disc and the projective plane, i.e. M2 ∼= D2 #RP2 .
Hencee2πiSeff (M2 ) = −1.
(2.14)On the other hand the Klein Bottle K2 is a connected sum of two copies of the projective plane,i.e. K2 ∼= RP2 #RP2 , ande2πiSeff (K2 ) = 1.
(2.15)
When considered on a manifold with boundary, this phase captures the time-reversal anomalyof the (0 + 1)d boundary theory. This anomaly can be carried by a bosonic Kramers doublet
on the boundary, i.e. we have T2 = −1 instead of T2 = +1.
2.2.2

Pin structures

In order to describe fermionic invertible phases protected by T, we need to briefly review how
to put fermions on an unoriented manifold. There exist two different lifts of the O(d) bundle,
known as Pin± (d), which fit into the short exact sequence
0 → Z2 → Pin± (d) → O(d) → 0 .

(2.16)

Both Pin± (d) contain Spin(d) as their component connected to the identity, and their difference
lies in how time-reversal T and spatial reflection R lift,
Pin+ :

T2 = (−1)f ,R2 = 1 .

Pin− :

T2 = 1 ,R2 = (−1)f .

(2.17)

The corresponding obstruction classes are
Pin+ :

Pin− :

w2 ,

w2 + w21 .

(2.18)

Every two-manifold Σ has w2 + w21 = 0 mod 2, and hence admits a Pin− structure. However,the same is not true for Pin+ . For instance, the real projective plane RP2 has w2 6= 0, and so it
does not admit a Pin+ structure.
The action of Pin± on fermions Ψ can be given in terms of gamma matrices. In particular,reflection of the i-th coordinate acts on Ψ by the gamma matrix γi . The reflection squared istrivial in O(d), and therefore its lift when applied to a fermion is ±1. Then we have
{γi , γ j } = ±2ηi j ,(2.19)

for Pin± structure, respectively, where we temporarily use the Lorentzian signature and themetric ηi j is mostly plus. This explains why T squares to (−1)f when R squares to 1 and viceversa, as written in (2.17). The fermion fields Ψ transforming in this manner are sometimes
called “pinors.”
6This is also occasionally written as (−1)χ(Σ) , where χ(Σ) is the Euler characteristicof Σ. The reason for this isR
that w21 is equal to the mod-two reduction of the Euler class e, which satisfies Σ e = χ(Σ). We prefer writing thisphase in terms of Stiefel-Whitney classes in order to make bordism invariance manifest.

10

SciPost Phys. 9, 010 (2020)∼=

Figure 1: Möbius strip with an orientation-reversing line (in green).
In our study of unoriented string amplitudes, the behavior of pinors on the boundary of
the Möbius strip with Pin± structure will be particularly important. Recall that on a circle, one
can consistently define both anti-periodic and periodic fermions, i.e. fermions in the NS andR sectors. In contrast, the choice of NS or R on the boundary of the Möbius strip is fixed by
the choice of Pin± . We may see this as follows. Note that the Möbius strip can be constructedby taking a strip and gluing its ends together along an orientation-reversing line (Fig. 1) —upon crossing this line, we pick up an action of γi on fermions. Traversing the boundary of
the Möbius strip involves crossing this line twice, and this picks up an action of γ2i = ±1 on
fermions. Thus we conclude that boundary fermions on the Pin+ Möbius strip are in the R
sector, while those on the Pin− Möbius strip are in the NS sector.
2.2.3

Invertible phase for Pin− structure

We now study fermionic invertible phases protected by T such that T2 = +1. Such phases
are classified by f2Pin− (pt) = Z8 , which is generated by the Arf-Brown-Kervaire (ABK) invariant [46]. For recent work on this invertible phase, see e.g. [47–49]. The ABK invariant canbe thought of as the effective action of the Kitaev chain protected by time-reversal. In thecontinuum version, we have:
e2πiSeff (Σ, σ) = eπiABK(Σ, σ)/4 =Zferm (m  0; Σ, σ).
Zferm (m  0; Σ, σ)(2.20)

Here σ represents a choice of Pin− structure, which we will often omit from the argument of
ABK for brevity. Alternatively, the ABK invariant can be defined combinatorially aseπiABK(Σ)/4 = p

1

X|H 1 (Σ, Z2 )| a∈H 1 (Σ,Z )2eπiq(a)/2 ,(2.21)

where q is a quadratic enhancement q : H 1 (Σ, Z2 ) → Z4 satisfyingZq(a + b) − q(a) − q(b) = 2a∪b .

(2.22)

ΣOne may think of this q as a sort of doubling of the spin structure quadratic formq̃ : H 1 (Σ, Z2 ) → Z2 introduced earlier. Indeed, if the worldsheet is orientable thenq = 2 q̃ (mod 4). In that case one finds ABK(Σ) = 4 Arf(Σ) (mod 8), and (2.20) reduces to(2.2) as expected.
To see that the right-hand side of (2.21) is an eighth root of unity, again we consider its

11

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Figure 2: Projective plane with the single generator Z of its first homology class (inblue). To the right this is drawn as a sphere with a crosscap.
square:RHS2 ====

1
1|H (Σ, Z2 )|
1
1|H (Σ, Z2 )|
1
1|H (Σ, Z2 )|
1
1|H (Σ, Z2 )|X

i q(a)+q(b)a,b ∈H 1 (Σ,Z2 )X

R

i q(a+b) (−1) Σ a∪ba,b ∈H 1 (Σ,Z2 )X

R

i q(c) (−1) Σ (a∪c+a∪a)a,c ∈H 1 (Σ,Z2 )X

R

i q(c) (−1) Σ (a∪c+a∪w1 )a,c ∈H 1 (Σ,Z2 )q(w1 )= i,(2.23)
R
R
where we used a ∪ a = a ∪ w1 .
We will need some basic values for the ABK invariant. We begin by reviewing the case of22
RP , for which there is a single generator
R Z of H1 (RP , Z2 ), as depicted in Fig. 2. Its Poincaré
dual, z, is an unoriented cocycle with RP2 z ∪ z = 1. Then using (2.22) and q(0) = 0, weconclude that 2q(z) + 2 = 0 mod 4 and hence q(z) = 1 or 3. These label the two distinct Pin−structures on RP2 . The corresponding ABK invariants are easily calculated,
1
p 1 + e iπ/2 = e iπ/4 ,2
1
p 1 + e i3π/2 = e−iπ/4 .
2

2

1:

eπiABK(RP )/4 =3:eπiABK(RP )/4 =2(2.24)

We will also need the value of the ABK invariant on the Klein bottle. Because K2 is a connected sum of two copies of RP2 , it admits four choices of quadratic enhancement. Taking z1and z2 to be the basis naturally adapted to the connected sum gives(q(z1 ), q(z2 )) ∈ {(1, 1), (1, 3), (3, 1), (3, 3)}. However, it will behoove us to switch to the morefamiliar basis of H1 (K2 , Z2 ), for which a := z1 +z2 is Poincaré dual to the orientation-preserving
A-cycle while b := z2 is dual to the orientation-reversing B-cycle. The quadratic enhancements
are then (q(a), q(b)) ∈ {(0, 1), (0, 3), (2, 1), (2, 3)}. For each of these Pin− structures we compute the following values for the ABK invariant,(0, 1) :eπiABK(K2 )/4 =(0, 3) :eπiABK(K2 )/4 =(2, 1) :e

πiABK(K2 )/4(2, 3) :eπiABK(K2 )/4 =

=

12

1
(1 + 1 + i − i) = 1,2
1
(1 + 1 − i + i) = 1,2
1
(1 − 1 + i + i) = i,2
1
(1 − 1 − i − i) = −i.
2(2.25)

SciPost Phys. 9, 010 (2020)

It is useful to note thatR2eπi(4ABK(Σ))/4 = (−1) w1 ,(2.26)

namely that four copies of the ABK theory is the same as the nontrivial bosonic unoriented
phase discussed in Section 2.2.1.
The boundary theory of n copies of the ABK theory is detailed in Section 4.1. When n = 4,
we have a system of four Majorana fermions χi . We can then introduce a T-invariant interaction term χ1 χ2 χ3 χ4 . The ground state with this interaction is bosonic, two-fold degenerate,and has the projective anomaly T2 = −1, thus manifesting the equality (2.26) on the boundary. Similarly, when n = 8, we can consider a suitable quartic interaction under which theground state is unique [35].
2.2.4

Invertible phase for Pin+ structure

We now study fermionic invertible phases protected by T such that T2 = (−1)f . Such phases
are classified by f2Pin+ (pt) = Z2 , which is generated by the Arf invariant (2.7) on the orienta-tion double cover Σ̂ of the worldsheet, i.e. (−1)Arf(Σ̂) [9]. This gives ±1 on the Klein bottle,
depending on the spin structure of its double cover torus. Concretely, the Klein bottle admits
four Pin+ structures, two of which have NS sector fermions along the oriented A-cycle of K2 ,and two of which have R sector fermions along the A-cycle. For each of these Pin+ structures,
we must determine the corresponding spin structure on the orientation double cover, i.e. thetorus. To do so, let us consider for simplicity a rectangular torus with real coordinates x, ysatisfying x ∼ x +2π, y ∼ y +2L for some L. The original Klein bottle is obtained by imposing
the identification (x, y) ∼ (−x, y + L). The cycle in the x-direction is the oriented A-cycle onK2 , while the cycle in the y-direction is twice the unoriented B-cycle. From this we learn thatthe choice of Pin+ structure along the B-cycle is irrelevant on the double cover — the fermion
is always periodic along the cycle in the y-direction. Thus we conclude that the four Pin+structures are assigned phases (−1)Arf(Σ̂) = 1, 1, −1, −1.

3

GSO projections

In this section, we consider the addition of fermionic invertible phases to the worldsheet theories of various superstrings. This will allow us to enumerate all possible GSO projections. We
will focus on oriented Type II and Type 0 theories, as well as unoriented Type 0 theories. The
case of unoriented Type II (i.e. Type I) strings will be treated in Section 6. In our discussion
we will work in the NSR formalism, which is briefly reviewed in Appendix A. The propertiesof the low-lying states in this formalism are summarized in Table 2 for convenience.

3.1

Oriented strings

3.1.1

Type 0Let us begin by studying oriented Type 0 superstrings, which differ from Type II strings in that
there is a single spin structure for both left- and right-movers [6, 50]. The SPT phases which
can be consistently realized on the Type 0 worldsheets are classified by f2Spin (pt) = Z2 . As
we have seen above, the effective action for the non-trivial phase can be written in terms of
the Arf invariant (2.2). Depending on whether or not one allows for this non-trivial phaseon the worldsheet, one expects to arrive at two different Type 0 theories, with torus partition
functions
2
1X
Z (n) =(−1)n Arf(T , σ) Z[σ]Z[σ] ,n = 0, 1 ,
(3.1)
2 σ
13

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Table 2: Fermion parity and representations of the low-lying states of the closed NSR
superstring.

State

(−1)f

(−1)fL(−1)fR(NS−, NS−)

+1

−1

−1

1

(NS+, NS+)

+1

+1

+1

8 v ⊗ 8 v = 1 ⊕ 28 ⊕ 35

(R+, R+)

+1

+1

+1

8s ⊗ 8s = 1 ⊕ 28 ⊕ 35−(R−, R−)

+1

−1

−1

8c ⊗ 8c = 1 ⊕ 28 ⊕ 35+(R+, R−)

−1

+1

−1

8s ⊗ 8c = 8 v ⊕ 56

(R−, R+)

−1

−1

+1

8c ⊗ 8s = 8 v ⊕ 56

(NS+, R+)

+1

+1

+1

8 v ⊗ 8s = 8c ⊕ 56s(NS+, R−)

−1

+1

−1

8 v ⊗ 8c = 8s ⊕ 56c

Little group rep.

where Z[σ], Z[σ] are the standard left- and right-moving worldsheet torus partition functions
with spin structure σ. The two theories obtained in this way are Type 0B (n = 0) and Type0A (n = 1),
0A :
0B :


1
|Z[σ00 ]|2 + |Z[σ01 ]|2 + |Z[σ10 ]|2 − |Z[σ11 ]|2 ,
2

1
(0)
|Z[σ00 ]|2 + |Z[σ01 ]|2 + |Z[σ10 ]|2 + |Z[σ11 ]|2 ,Z =2Z (1) =

(3.2)

and their massless RR states are found to be
ȧ b
a ḃ
∈ (8s ⊗ 8c ) ⊕ (8c ⊗ 8s ) = (2 · 8 v ) ⊕ (2 · 56) ,, |0〉RR
0A : |0〉RRȧ ḃ
ab∈ (8s ⊗ 8s ) ⊕ (8c ⊗ 8c ) = (2 · 1) ⊕ (2 · 28) ⊕ 70 .
, |0〉RR
0B : |0〉RR(3.3)

The massless RR sector for Type 0A contains two 1-forms and two 3-form fields, while thatof Type 0B contains two scalars, two 2-forms, and a 4-form with no self-duality constraint.
This is exactly double the RR content of the corresponding Type II theories. This leads one to
expect a doubled brane spectrum, where for each p we have both Dp and Dp0 branes. This
will be discussed further in Section 3.3.
3.1.2

Type II

We now proceed to discuss the more familiar Type II superstrings. A characteristic feature ofthese strings is that they have separate spin structures for left- and right-movers, which meansthat the worldsheet is endowed with a Spin × Z2 structure. The anomalies and invertible
phases on such worldsheets are captured by f3Spin (BZ2 ) and f2Spin (BZ2 ), respectively. These
groups are listed in Table 1. As discussed in [51, 52], the fact that f3Spin (BZ2 ) = Z8 implies
that the number of physical pairs of left- and right-movers needs to be a multiple of eight tohave a non-anomalous GSO projection. This is indeed the case if the physical string lives inten dimensions.
On the other hand, the SPT phases on the worldsheet are classified by f2Spin (BZ2 ) = (Z2 )2 .
The two Z2 can be interpreted as separate left- and right-moving fermionic invertible phases,
as discussed in Section 2.1.2. In other words, the partition functions for these phases are givenby(−1)n L Arf(Σ, σ L )+nR Arf(Σ ,σR ) ,14

n L,R = 0, 1 ,(3.4)

SciPost Phys. 9, 010 (2020)

where σ L(R) is the left(right)-moving spin structure on the worldsheet Σ. The corresponding
torus partition functions for these theories are given by

 

X
1 X
(n L ,nR )n L Arf(T 2 , σ L )nR Arf(T 2 , σR )Z=(−1)
Z[σ L ] ×(−1)
Z[σR ] .
(3.5)
4 σ
σ
L

R

For instance, two cases are
1(Z[σ00 ] + Z[σ01 ] + Z[σ10 ] + Z[σ11 ]) Z[σ00 ] + Z[σ01 ] + Z[σ10 ] + Z[σ11 ] ,
4

1
(0,1)
Z
= (Z[σ00 ] + Z[σ01 ] + Z[σ10 ] + Z[σ11 ]) Z[σ00 ] + Z[σ01 ] + Z[σ10 ] − Z[σ11 ] .
4(3.6)
While there are seemingly four distinct SPT phases, there are in fact only two physicallydistinct Type II theories. To see this, recall the continuum definition of the Arf invariant:Z (0,0) =(−1)Arf(Σ, σ) =Zferm (m  0; Σ, σ),Zferm (m  0; Σ, σ)(3.7)

where Zferm (m; Σ, σ) is the partition function of a free massive Majorana fermion of mass m.
We note this formula holds at finite mass as well,(−1)Arf(Σ, σ) =Zferm (+m; Σ, σ),Zferm (−m; Σ, σ)(3.8)

which was already used in (2.4). In other words, upon flipping the sign of a mass term,m → −m, one generates a factor of (−1)Arf(Σ, σ) in the partition function. Note that such a flipof the mass term can be performed by (ψ, ψ̃) → (ψ, −ψ̃). Taking the limit m → 0, we findthat a Majorana-Weyl fermion ψ̃ has an anomaly under ψ̃ → −ψ̃, and generates (−1)Arf(Σ, σR ) .
This in particular means that the parity transformation along a single spacetime direction,say (ψµ=9 , ψ̃µ=9 ) → (−ψ9 , −ψ̃9 ), produces (−1)Arf(Σ, σ L )+Arf(Σ, σR ) , i.e. n L = nR = 1 in (3.4).
Therefore, there are only essentially two distinct Type II GSO projections, with the others being
related by spacetime parity transformation. The cases (n L , nR ) = (0, 0), (1, 1) are traditionallycalled Type IIB while the cases (n L , nR ) = (0, 1), (1, 0) are called Type IIA.
The reasoning above also explains why T-duality exchanges Type IIA/B. Recall that Tduality along a spacetime direction keeps (∂ X , ψ) fixed and implements (∂¯ X , ψ̃) → (−∂¯ X , −ψ̃).
Then by the previous paragraph, this generates (−1)Arf(Σ, σR ) , exchanging Type IIA/B. By thesame arguments, the two Type 0 theories are also exchanged by T-duality.
3.1.3

Comments on the two points of view on the effect of invertible phases

There are two ways of understanding the gauging of a global symmetry in the presence ofa non-trivial invertible phase. The point of view which we have taken so far is to take thetensor product of the original theory and the invertible phase, and to then gauge the relevant
symmetry. The Hilbert space of the invertible phase is one-dimensional, and therefore thischanges the way the global symmetry acts on the states of the original theory. For example, inthe case of Type 0 strings, we used the projectors
P0A =1(1 + (−1)f |0A ),
2

P0B =

1
(1 + (−1)f |0B )
2

(3.9)

and what produced the difference between the two was that on the RR sector, we had(−1)f |0A = −(−1)f |0B
15(3.10)

SciPost Phys. 9, 010 (2020)

due to the presence of the Arf theory.
More traditionally, the action of (−1)f was fixed once and for all, for example to be equalto (−1)f |0B , and different GSO projections were said to correspond to different projectors. For
example, in the RR sector, one would have written
RR
P0A
=

1
(1 − (−1)f ),
2

RR
P0B
=

1
(1 + (−1)f ).
2(3.11)

These two points of view clearly lead to the same results, and similar statements will be seen
to hold for unoriented strings. Though we will briefly discuss this traditional viewpoint when
we compare to the existing literature, we will mostly use the first point of view.

3.2

Unoriented strings

3.2.1

Orientation reversal on fermions and ground states

We now consider unoriented string theories. One way to obtain such theories is to gauge timereversal symmetry T on the worldsheet. T is an antiunitary symmetry that acts on worldsheet
fermions asTψ(t, σ)T−1 = ψ̃(−t, σ) ,Tψ̃(t, σ)T−1 = ψ(−t, σ) ,(3.12)

with T2 = 1. In string theory, it is often more common to describe this in terms of worldsheet
parity Ω, which is a unitary symmetry whose action is given byΩψ(t, σ)Ω−1 = −ψ̃(t, 2π − σ) ,Ωψ̃(t, σ)Ω−1 = ψ(t, 2π − σ) .

(3.13)

From this definition it is clear that Ω2 = (−1)f . The ability to choose between T or Ω is aconsequence of the CPT theorem. The fact that T2 = 1, or equivalently that Ω2 = (−1)f ,means that we are working with a Pin− structure on the worldsheet. In this case the action ofΩ on the ground states in the NSNS and RR sectors can be taken to be¨− |0̃〉R ⊗ |0〉R for (R±, R±),Ω |0〉NSNS = |0〉NSNS ,Ω |0〉R ⊗ |0̃〉R =(3.14)−i|0̃〉R ⊗ |0〉R for (R±, R∓).
One can also consider gauging Ω twisted by some Z2 symmetry.
Ωf := Ω(−1)fL . We find

Here we consider(Ωf )2 = Ω(−1)fL Ω(−1)fL = Ω(−1)fL +fR Ω = (−1)f Ω2 = 1(3.15)and hence gauging this operator gives Pin+ structure on worldsheets. In this case parity actson the NSNS and RR ground states as¨∓ |0̃〉R ⊗ |0〉R for (R±, R±),Ωf |0〉NSNS = −|0〉NSNS ,Ωf |0〉R ⊗ |0̃〉R =(3.16)∓i|0̃〉R ⊗ |0〉R for (R±, R∓).
Unlike for Type II strings where Ω is a symmetry of only Type IIB, for Type 0 strings Ω is asymmetry of both Type 0A and 0B. Hence we can obtain Pin− Type 0 theories by starting fromeither Type 0A or 0B and gauging Ω. Likewise, one might expect that we can obtain Pin+ Type0 theories by starting from either Type 0A or 0B and gauging Ωf . However, it turns out thatΩf cannot be consistently gauged in Type 0A, since it is incompatible with the Type 0A spin
structure projection [53–55].
In the rest of this section we study the consistent unoriented Type 0 strings in more detail.
We begin by analyzing the Pin− strings in Section 3.2.2, and then proceed to a discussion of thePin+ strings in Section 3.2.3. To the best of our knowledge, many of these theories have notbeen discussed in the literature — some preliminary works include [53–62]. For a condensed
matter perspective, see e.g. [63, 64].
16

SciPost Phys. 9, 010 (2020)

3.2.2

Pin− Type 0 Strings

Let us begin by discussing Pin− Type 0 strings. The group classifying the relevant invertible
phases is f2Pin− (pt) = Z8 , which is generated by the ABK invariant. We are thus led to predict
the existence of eight Pin− theories, each distinguished by the presence of n = 0, . . . , 7 copiesof ABK on the worldsheet.
In unoriented theories, the presence of a non-trivial invertible phase manifests itself in theaction of Ω on the different closed string ground states. In order to understand this action, we
make use of the values of the ABK invariant on the Klein bottle K2 obtained in Section 2.2.3.
In particular, it was found there that the Klein bottle admits four Pin− structures labelledby quadratic enhancements (q(a), q(b)) ∈ {(0, 1), (0, 3), (2, 1), (2, 3)} with respective values
e iπABK(K2 )/4 = 1, 1, i, −i. Recall that the first entry a corresponds to the orientation-preserving
cycle on K2 , while the second entry b corresponds to the orientation-reversing cycle.
We now want to interpret these results as the action of Ω and (−1)f on the closed string
Hilbert space. This can be done as follows. We begin by cutting K2 along the orientationpreserving cycle A to obtain a cylinder with an insertion of an Ω symmetry line. Since A isorientation-preserving, we know that q = 2 q̃ mod 4, where q̃(a) is the spin structure along
A. Consequently, the first and second Pin− structures correspond to NS structure along theA-cycle, while the third and fourth correspond to R structure along the A-cycle. We may theninterpret the partition function for each Pin− structure as the following traces on the torus.
We have:(0, 1) :↔


1
TrNSNS Ω e−2πlHcl
4(0, 3) :↔


1
TrNSNS Ω (−1)f e−2πlHcl
4(3.17)(2, 1) :↔


1
TrRR Ω e−2πlHcl
4(2, 3) :↔


1
TrRR Ω (−1)f e−2πlHcl
4where green lines represent orientation-reversal lines and red lines represent spin lines. In thisway, the value of the ABK invariant on K2 with Pin− structure labeled by (q(a), q(b)) can be
assigned to the action of Ω on the ground states with the appropriate cylinder spin structures.
In particular, we conclude that Ω acts trivially on NS ground states, whereas it acts with an
extra factor of i on R ground states. This implies that the presence of n copies of ABK changes
the action of Ω on the RR sector ground states by a factor of i n relative to (3.14), giving¨−i n |0̃〉R ⊗ |0〉R for (R±, R±),Ω|0〉NSNS = |0〉NSNS ,Ω|0〉R ⊗ |0̃〉R =(3.18)−i n+1 |0̃〉R ⊗ |0〉R for (R±, R∓).
Note that upon shifting n → n + 1, the additional factor of i changes the fermion-parityof the RR ground state, since Ω2 = (−1)f . Since the Type 0A/B theories differ by a projection onto states of worldsheet fermion number (−1)f = ±1, we see that theories with

17

SciPost Phys. 9, 010 (2020)

even n correspond to orientifolds of Type 0B, while theories with odd n correspond to orientifolds of Type 0A. This is also supported by recalling that on oriented manifolds Σ, theABK invariant reduces to ABK(Σ) = 4 Arf(Σ) (mod 8), and hence the partition function becomes e inπABK(Σ)/4 = (−1)n Arf(Σ) , which is precisely what distinguished the oriented Type
0A/B theories.
As far as the action of Ω on the vacuum (3.18) is concerned, theories differing by four copiesof ABK are indistinguishable. The reason for this is that only data about the Klein bottle K2 was−used to obtain (3.18). However, the manifold that generates the bordism group ΩPin(pt) is the22∼
projective plane RP , while K2 is a connected sum of two copies thereof, i.e. K2 = RP2 #RP2 .
Consequently, the partition function on K2 is insensitive to an additional sign that can ariseon manifolds whose decompositions contain an odd number of copies of RP2 . Indeed,
R 2four
4πiABK(Σ)/4
copies of the ABK theory is not trivial and gives partition function e= (−1) Σ w1 , asdiscussed in Section 2.2.3. As described in Section 2.2.1, unlike for the Klein bottle the Möbius
strip amplitude is sensitive to this sign. This sign turns out to give precisely the difference
between O9± orientifolds. More detail on this will be given in Section 5.1.
We may now study the closed string spectra of these theories. The action of Ω proposed in(3.18) does not project out the closed string tachyon in the NSNS sector, but has the following
implications for the spectra of RR fields. For n even, Ω projects out all (R±, R∓) states. The
cases n = 0 mod 4 and n = 2 mod 4 differ by a sign in the action of Ω, which projects out
the symmetric or antisymmetric combinations of (R±, R±) states. Then upon gauging Ω we
obtain the following RR spectra,[a b][ȧ ḃ]n = 0, 4 :|0〉RR ∈ 28 ⊂ 8s ⊗ 8s ,n = 2, 6 :|0〉RR ∈ 1 ⊕ 35− ⊂ 8s ⊗ 8s ,|0〉RR ∈ 28 ⊂ 8c ⊗ 8c ,(a b)(ȧ ḃ)|0〉RR ∈ 1 ⊕ 35+ ⊂ 8c ⊗ 8c .

(3.19)In the n = 0 mod 4 cases, only the two 2-forms survive the projection, while for n = 2 mod 4
only the two scalars and the 4-form survive. These spectra of RR fields are indeed a projectionof the Type 0B ones. For n odd, the extra factor of i in (3.18) projects out all the (R±, R±)
states, while the (R±, R∓) combinationsn = 1, 5 :n = 3, 7 :

1
ḃa
a ḃ
) ∈ 8 v ⊕ 56 ⊂ (8s ⊗ 8c ) ⊕ (8c ⊗ 8s ) ,+ |0〉RRp (|0〉RR
2
1
a ḃ
ḃa
) ∈ 8 v ⊕ 56 ⊂ (8s ⊗ 8c ) ⊕ (8c ⊗ 8s ) ,− |0〉RRp (|0〉RR
2(3.20)

survive the projection. This leaves a single set of 1- and 3-form fields. These states are part of
the Type 0A spectrum.
It is worth mentioning that because these theories possess neither spacetime fermions nor(anti-)self-dual form fields, they are all free of perturbative gravitational anomalies.
Finally, let us give a more traditional orientifold interpretation to the theories studied in thissection. In perturbative string theory we often refer not only to left/right-moving worldsheet
fermion number (−1)fL ,fR but also to left/right-moving spacetime fermion number (−1)FL ,FR .
We recall that (−1)F acts by +1 on the NS sector and by −1 on the R sector. We now considerΩF := Ω(−1)FL , which acts with an extra minus sign on the left-moving R sector. Above, we
saw that gauging Ω with two copies of the ABK theory gives the same minus sign. This suggests
the following identifications,n = 0, 4 :(0B, Ω)

n = 1, 5 :(0A, Ω)

n = 2, 6 :(0B, ΩF )n = 3, 7 :(0A, ΩF ) ,(3.21)

where the first element in parenthesis denotes the starting theory, and the second element
denotes the operator being gauged. The difference between theories differing by 4 copies of18

SciPost Phys. 9, 010 (2020)

ABK is the action of Ω or ΩF on Chan-Paton factors. This correspondence between the ABK
viewpoint and the orientifold viewpoint will be discussed further in Section 5.1.
3.2.3

Pin+ Type 0 Strings

We finally proceed to the case of Pin+ Type 0 strings, which were studied in [65,66]. The group
capturing potential invertible phases on the Pin+ worldsheet is f2Pin+ (pt) = Z2 . As reviewed
in Section 2.2.4, the effective action for this invertible phase is given by the Arf invariant of
the oriented double cover Σ̂ of the worldsheet, i.e. (−1)Arf(Σ̂) , whose generating manifold is
the Klein bottle. The Klein bottle was seen to admit four Pin+ structures, which we now label
as (0, 1), (0, 3), (2, 1), (2, 3) in analogy to the Pin− notation.7 By examining the spin structureon the double cover torus, these were assigned respective phases (−1)Arf(Σ̂) = 1, 1, −1, −1.
We now proceed as in the Pin− case above. First, we recast the Klein bottle partition
functions for the four Pin+ structures in terms of traces on the torus. This gives

1
TrNSNS Ωf e−2πl Hcl ,4

1(2, 1) ↔ TrRR Ωf e−2πl Hcl ,4(0, 1) ↔


1
TrNSNS Ωf (−1)f e−2πlHcl ,4

1(2, 3) ↔ TrRR Ωf (−1)f e−2πlHcl .
(3.22)
4(0, 3) ↔

The Arf(Σ̂) invertible phase assigns −1 to the Klein bottle with Pin+ structure (2, 1) and (2, 3),
and +1 to the other Pin+ structures. This means that the presence of the non-trivial invertible
phase changes the action of Ωf on R sector ground states by a sign relative to (3.16), but does
not change the action of (−1)f .
With this information, we may turn towards the analysis of the massless closed string
spectra of the theories. For the trivial phase, the orientifold projection keeps the symmetriccombinations of (R−,R−) and antisymmetric contributions of (R+,R+) in the Type 0B spectra.
In the non-trivial phase, one instead keeps the antisymmetric combinations of (R−,R−) andsymmetric contributions of (R+,R+),[a b](ȧ ḃ)n=0:|0〉RR ∈ 28 ⊂ 8s ⊗ 8s ,n=1:|0〉RR ∈ 1 ⊕ 35− ⊂ 8s ⊗ 8s ,|0〉RR ∈ 1 ⊕ 35+ ⊂ 8c ⊗ 8c ,(a b)[ȧ ḃ]|0〉RR ∈ 28 ⊂ 8c ⊗ 8c .

(3.23)

We note that these spectra are the same up to a spacetime parity transformation which
exchanges the self-dual and anti-self-dual 4-forms. This observation can also be explained
from the fact that the spacetime parity transformation generates (−1)Arf(Σ̂) on the worldsheet.
Indeed, in the Type II case, the same operation generated (−1)Arf(Σ,σ L )+Arf(Σ,σR ) as we saw
before, which is equal to (−1)Arf(Σ̂) when Σ is oriented.
We note that the RR spectra are equivalent to that of Type IIB, and the theory has a gravitational anomaly from the anti-self-dual 4-form, with no fermions to cancel it. As we discuss
briefly in Appendix C, consistency requires the theory to be coupled to fermionic open strings,
giving a U(32) gauge group [56, 60].

3.3

Branes and K-theory

In the above analysis we identified two oriented Type II strings, two oriented Type 0 strings,
and a number of unoriented Type 0 strings. In this subsection we discuss their spectra of stablebranes. To do so, we begin by briefly reviewing the well-known K-theory classification of stablebranes for oriented theories.
7

We do this for notational convenience only. There is no correspondence between quadratic enhancements andPin+ structures in general.

19

SciPost Phys. 9, 010 (2020)

Recall that oriented Type IIB on a spacetime X has stable D-branes which are classified by
the K-group K(X ) [33, 34]. This is the group of pairs of vector bundles (E, F ) over X subject
to an equivalence relation (E ⊕ H, F ⊕ H) ∼ (E, F ). More precisely, one should consider thee (X ), for which the bundles E and F are required to have the same rank.
reduced K-group K
Physically, the idea is to begin with a stack of equal numbers of D9- and D9-branes, and thento consider annihilation amongst these stacks. When the vector bundles over these stacks
are unequal this annihilation is not complete, and a residual brane of lower dimension is left
over [32, 33, 67]. It is expected that all branes can be obtained in this way.
e (X × S 1 ). One
For Type IIA, stable branes are classified by the higher K-group K 1 (X ) = Knemight entertain the possibility of allowing for even higher K-groups K (X ) for n > 1. However,
Bott periodicity states that for complex K-groups,e n (X ) = K
e n+2 (X ) .
K(3.24)

Thus the only distinct complex K-groups are those mentioned above, and both are realized bye n (S 9−p ), as listed in the firststring theories. The stable Dp-branes are captured by the groups K
two rows of Table 3.
As discussed in Section 3.1.1, the spectrum of massless RR fields in oriented Type 0A/B
theories is precisely double that of Type IIA/B. As such, one expects the spectrum of branes in
Type 0A/B to be doubled as well; the two branes of given worldvolume dimension (p + 1) are
typically denoted as Dp- and Dp0 -branes. It follows that the classification of stable branes is
via two copies of the complex K-groups just described. In other words, because there now exist
both D9- and D90 -branes, we must consider two separate pairs of vector bundles corresponding
to D9-D9 and D90 -D90 stacks. So the branes in the Type 0 theories are classified bye n (X ) ⊕ K
e n (X ) ∼e n (X ) ⊕ K
e −n (X ) .
K=K(3.25)The equality above follows from the mod 2 periodicity of complex K-theory. As we now discuss,
it is the latter form which generalizes to the unoriented case.
It has long been known that stable branes in unoriented Type I string theory are classifiedÝ ) [33,68]. Crucially, the reduced real K-groups have a mod 8 periodicity
by real K-theory KO(X[69],
Ý n (X ) ∼Ý n+8 (X ) .
KO= KO(3.26)

It is thus natural to guess that the eight Pin− Type 0 strings labeled by n mod 8 have stableÝ n (X ). More precisely, because one again expects a doubled spectrum
branes captured by KOfrom these Type 0 theories, the relevant group will be found to beÝ n (X ) ⊕ KO
Ý −n (X ) .
KO(3.27)In Sections 4 and 5, this group will be confirmed to classify the stable p-brane spectrum of thePin− Type 0 theory with n copies of ABK. Concretely, this spectrum is obtained by evaluating
KO±n (X ) on X = S 9−p , with the results listed in Table 3. The entries in this table can beÝ n (S k ) = KO n−k (pt), and then using the following values for realobtained by noting that KO
K-groups of points:
KO0 (pt) = Z, KO−1 (pt) = Z2 , KO−2 (pt) = Z2 , KO−3 (pt) = 0,
KO−4 (pt) = Z, KO−5 (pt) = 0,

KO−6 (pt) = 0,

KO−7 (pt) = 0.

(3.28)At this point we can check that the RR spectra we determined above agree with the non-torsion
Ý n (S 9−p )⊕ KO
Ý −n (S 9−p ). The aim of the next two sections is to establish the agreement
part of KO
including the torsion parts.
20

SciPost Phys. 9, 010 (2020)

For Pin+ Type 0 strings, we saw in Section 3.2.3 that these theories have the same RR
spectra as oriented Type IIB. As such, we expect to have the same classification via complex
K-theory as in that case. Since the Pin+ strings have less features not found previously thantheir Pin− counterparts, we will be very brief about them in what follows.
It is worth noting that whenever tadpole cancellation requires the addition of D9-branes,
the question of stability of Dp-branes must be revisited to account for the possibility of tachyonic modes of the strings stretched between the Dp- and D9-branes. In this case, the K-theory
classification outlined above may be modified, though we will not address these modifications.
Table 3: The ten K-groups capturing stable branes in the oriented Type II and(un)oriented Type 0 theories discussed above.

e
K
e
K1
0
Ý ⊕KO
Ý −0
KO
Ý −1
Ý 1 ⊕KOKOÝ 2 ⊕KO
Ý −2
KO
3
Ý ⊕KO
Ý −3
KO
Ý 4 ⊕KOÝ −4KO
Ý 5 ⊕KO
Ý −5
KO
6
Ý ⊕KO
Ý −6
KO
Ý 7 ⊕KO
Ý −7
KO

4

−1
Z
0
2Z2
Z2
2Z
0
0
0
2Z
Z2

0
0
Z
2Z2
Z⊕Z2
0
Z
0
Z
0
Z⊕Z2

1
Z
0
2Z
Z2
Z2
0
2Z
0
Z2
Z2

2
0
Z
0
Z
Z2
Z⊕Z2
0
Z⊕Z2
Z2
Z

3
Z
0
0
0
2Z
Z2
2Z2
Z2
2Z
0

4
0
Z
0
Z
0
Z⊕Z2
2Z2
Z⊕Z2
0
Z

5
Z
0
2Z
0
Z2
Z2
2Z
Z2
Z2
0

6
0
Z
0
Z⊕Z2
Z2
Z
0
Z
Z2
Z⊕Z2

7
Z
0
2Z2
Z2
2Z
0
0
0
2Z
Z2

8
0
Z
2Z2
Z⊕Z2
0
Z
0
Z
0
Z⊕Z2

9
Z
0
2Z
Z2
Z2
0
2Z
0
Z2
Z2

D-brane spectra via boundary fermions

In this section, we demonstrate the KO n ⊕ KO−n classification of stable branes for the Pin−
Type 0 theory with n copies of ABK. This is done by analyzing the Clifford modules carried byopen string endpoints. After doing so, we also study the spectra of non-stable branes in these
theories, including the gauge groups supported on their worldvolumes and the representationsof their open string tachyons.

4.1(0+1)d Majorana fermions and their anomalies

4.1.1

Fermions and Clifford algebras

We begin by considering systems of (0+1)d Majorana fermions, which appear on the boundaryof n copies of the ABK theory. Let us say we have r +s hermitian fermion operators ξa = (ξa )† ,a = 1, . . . , r + s, satisfying
ξ2a = +1
a = 1, . . . , r + s
(4.1)
and

¨

Tξa T−1 =+ξaa = 1, . . . , r,

−ξaa = r + 1, . . . , r + s.

21

(4.2)

SciPost Phys. 9, 010 (2020)

In the mathematics literature it is more common to consider operators invariant under T. Thiscan be achieved by defining¨
γa = ξa i = 1, . . . , r,(4.3)
γa = iξa i = r + 1, . . . , r + s.
The γa are no longer hermitian in general, but satisfy Tγa T−1 = γa . They generate the real Clifford algebra Cl(r, s). We often use the abbreviations Cl(+n) := Cl(n, 0) and Cl(−n) := Cl(0, n).8
The system of r + s fermions giving rise to Cl(r, s) can have anomalies in the realization of
T and (−1)f , which will be the topic of the next subsection. Before proceeding, we now give arough argument for why these anomalies depend only on r − s modulo 8. First we argue that
only r − s is relevant for the anomaly. The reason is that a pair of fermions with opposite T
transformations allow a T-invariant mass term, so they cannot be anomalous. For example,
when (r, s) = (1, 1), we can simply add a T-invariant mass term iξ1 ξ2 , which would trivialize
the vacuum, precluding any anomaly.
We next argue that only r − s mod 8 matters. This can be understood in two steps. As
the first step, we consider Cl(4). For this we can introduce a T-invariant quartic hermitian
interaction term
H = ξ 1 ξ2 ξ3 ξ 4(4.4)
to the system, for which the vacuum is two-dimensional, purely bosonic, and T2 = −1. This
realizes a Kramers doublet, on which
σ x := iξ1 ξ2 ,σ y := iξ1 ξ3 ,σz := iξ1 ξ4(4.5)

act as Pauli matrices.
As the second step, we combine two such Kramers doublets obtained from two copies ofCl(4), and make a single bosonic system with T2 = +1 and a unique vacuum. This can bedone by introducing a T-invariant term
H 0 := σ x ⊗ σ x + σ y ⊗ σ y + σz ⊗ σz ,(4.6)

which we note can be realized as a four-fermi operator using (4.5).
In other words, we can introduce to Cl(8) a four-fermion term ci jk` ξi ξ j ξk ξ` which is hermitian and T-invariant, such that its addition leads to a non-degenerate vacuum [35]. Therefore, eight Majorana fermions with the same time-reversal properties can be removed without
affecting the anomaly.
4.1.2

More general (0+1)d systems and anomalies

We do not necessarily have to couple n boundary Majorana fermions to n copies of the ABK
theory. We only have to couple a boundary system which has the same anomaly as n Majorana
fermions. We thus need to understand the possible anomalies concerning the realizations of
T and (−1)f . In the mathematical literature this analysis was first done in [71], in which thefollowing eight-fold classification in terms of three signs was given.
The first sign is the most subtle to define. We ask whether (−1)f can be realized in an
irreducible ungraded representation of the algebra. If this is possible, the representation is oftype +, and if not, it is of type −. As an example, consider Cl(+1). There are two ungraded
irreducible representations, which are real one-dimensional such that ξ1 = ±1. Clearly there
is no (−1)f operator that anticommutes with ξ1 , meaning that Cl(+1) is of type −.
8

Our convention is that Cl(−n) = Cn and Cl(+n) = Cn0 in the notation of Atiyah-Bott-Shapiro [70].

22SciPost Phys. 9, 010 (2020)

Table 4: The properties of T and (−1)f for the eight anomaly types n = 0, . . . , 7. The−
corresponding graded division algebras A = A0 ⊕ A1 , A1 = A+
1 ⊕ A1 are also given.

n
0
1
2
3
4
5
6
7

type
+
−
+
−
+
−
+
−

T and (−1)fT2commuteanticommute
anticommute
commutecommuteanticommute
anticommute
commute

R
R
R
H
H
H
H
R

n
0
1
2
3
4
5
6
7

A
A0 A+
A−
1
1
Cl(+0) =R
R
Cl(+1) = R ⊕ R R
R
Cl(+2) = R[2]C
C
Cl(+3) = C[2] H R3 R1
HH
Cl(−3) = H ⊕ H H R1 R3
Cl(−2) =H
CC
Cl(−1) =
C
R
R

p

dimR
1
p
2
2
p
2 2
2
p
2 2
2
p
2

In the following, for representations of type −, we adjoin (−1)f to the algebra and consider
the resulting irreducible graded representations. Again take Cl(+1) as an example. Then we
consider a representation given by




1 0
0 1
f
.
(4.7)
,
(−1) =
ξ1 =
0 −1
1 0
This is not irreducible as an ungraded representation but is irreducible as a graded representation.
The other two signs specifying the anomaly type are easier to define. The second sign isthe one appearing inT(−1)f = ±(−1)f T,(4.8)
while the third sign is the one appearing in(T)2 = ±1.

(4.9)

The eight types, labeled by n = 0, . . . , 7, are summarized in the left portion of Table 4. There,
we showed T2 = ±1 in terms of the corresponding division algebras R and H. In [71] it was
shown that the tensor product of a representation of type n and another of type n0 has the typen + n0 modulo 8. A very explicit analysis of Cl(+n) was given in Sec. 2.5.1 and 2.5.2 of [72],from which one can find that Cl(+n) is indeed of type n. Similarly, Cl(−n) is of type −n.
In [71] it was also shown that any graded irreducible representation of type n automaticallycontains an action of the minimal algebra A for that type. This information is shown in theright portion of Table 4. There, F[n] stands for the n×n matrix algebra over the field F, A0 andA1 are the bosonic and fermionic parts of the algebra, and A±
1 are the subspaces of A1 which
are hermitian and anti-hermitian, respectively.9 These eight graded algebras are known toexhaust the graded division algebras over R, i.e. graded algebras such that any homogeneous
element has an inverse.
4.1.3

On the boundary Hilbert space

Consider n copies of the Kitaev chain on a segment σ ∈ [0, π]. We call σ = 0 the left boundaryand σ = π the right boundary. The Majoranas ξ on the left and ξ0 on the right must have
opposite transformation properties under time-reversal [35,45]. This can be seen by imagining
9

Note that the hermitian conjugate on a real algebra is simply an involution satisfying (ab)∗ = b∗ a∗ .

23

SciPost Phys. 9, 010 (2020)

a process in which the endpoints of the segment join to give a closed circle. Once the endpoints
come together, the T-invariant mass term involving ξ and ξ0 should be able to gap the system,so ξ and ξ0 needs to have opposite T-transformation properties.
We work in the convention10 that we have n fermions ξi with Tξi T−1 = +ξi on the left andn fermions ξ0i with Tξ0i T−1 = −ξ0i on the right. They form Cl(n) and Cl(−n), respectively. The
Hilbert space associated to the open string segment, including the bulk and two boundaries,
can be identified with Cl(n) itself, on which Cl(n) acts from the left and Cl(−n) acts fromthe right. That the Hilbert space on the segment should naturally be equal to the boundary
algebra Cl(n) itself is clear from the state-operator correspondence. If we have other degrees
of freedom on the worldsheet, the Hilbert space on the segment is of the form

Hphys = Cl(n) ⊗ Hother dof .

(4.10)

Naively, one would like to say that this Hilbert
p space is the tensor product of the Hilbert
spaces on the two boundaries. The square root dimR A, listed in Table 4, is then what would
be taken as the dimension of the boundary Hilbert spaces. Note that this is not always an integer. It is sometimes useful to have a well-defined, non-anomalous boundary Hilbert spacewith integer dimension. This can be done by introducing n auxiliary boundary fermions Ξiwith TΞi T−1 = −Ξi on the left and n auxiliary boundary fermions Ξ0i with TΞ0i T−1 = +Ξ0i on
the right; this technique was used in e.g. [73] in the Type II setting. We note that auxiliaryboundary fermions have opposite T-transformation rules as compared to the physical boundary fermions. The fermions on the left boundary now form Cl(n, n) and can be quantizedwithout anomaly. This can be represented on a space V , thus providing Chan-Paton indices tothe boundary. We can do the same on the right. The Hilbert space on the segment, including
both the physical and auxiliary boundary fermions, is then of the form
V ⊗ V ∗ ⊗ Hother dof = Haux ⊗ Hphys ,(4.11)

where Haux = Cl(−n) and Hphys was defined in (4.10). Note that the elements of Hphys canbe found by finding operators (anti)commuting with all the auxiliary boundary fermions Ξ.
Below, when we say that “a boundary carries a representation of Cl(n),” we mean that
there are physical boundary fermions forming Cl(n), where n is taken modulo 8. If we use theauxiliary boundary fermions, they form Cl(−n).

4.2D-branes and boundary fermions

Let us now study the boundary fermions in the context of the worldsheet theory of Pin− Type0 strings, with n copies of the ABK theory. We first consider open strings ending on 9-branes.
We then have n boundary fermions on the left forming Cl(n) and n boundary fermions on
the right forming Cl(−n), as discussed above. The open-string Hilbert space, before GSO
projection, is of the form
Cl(n) ⊗ H0 ,(4.12)
where H0 is the open string Hilbert space of the massless worldsheet fields. Also as discussed
above, we can replace Cl(n) with any graded algebra having the same anomaly.
The restriction to 9-branes above meant that the only boundary fermions required were
those needed to cancel the anomaly of n copies of ABK. However, for branes of higher codimension there will be additional anomalies from bulk fermion zero-modes, which should beaccompanied by additional boundary fermions ζi . Let us set n = 0 mod 8 for the moment,
since the ABK boundary fermions can be easily reinstated later. In order to understand the
additional boundary Majoranas ζi , we must first understand when zero-modes can appear in
10

In Section 5, the choice of the convention here will correspond to a choice of definition of the O9-plane state.

24

SciPost Phys. 9, 010 (2020)

•

NS

•

•

Dp

Dp

NS

•

D9

D(9 − k)(a)•D(9 − k)0

R

•

∼
=

(c)

•

D(k + 1)

D9

•

Dp0

Dp

(b)

•

R

NS

•

D9

(d)

Figure 3: (a) A pair of Dp-branes with an NS sector string stretched between them;(b) an NS sector string stretched between a D(9 − k)- and D9-brane; (c) an R sector string stretched between a Dp- and Dp0 -brane; (d) an R sector string stretchedbetween a D(9 − k)0 - and D9-brane, which can be thought of as an NS sector string
stretched between a D(k + 1)- and D9-brane, as far as the behavior of the fermion
zero modes is concerned.
the bulk of the string. As reviewed in Appendix A, this depends on whether the ends of thestring have Dirichlet (D) or Neumann (N) boundary conditions. The NN, DD, and ND strings
satisfy the following boundary conditions,NN :ψµ (t, 0) = −η1 ψ̃µ (t, 0),ψµ (t, π) = +η2 ψ̃µ (t, π),DD :ψµ (t, 0) = +η1 ψ̃µ (t, 0),ψµ (t, π) = −η2 ψ̃µ (t, π),ND :ψµ (t, 0) = −η1 ψ̃µ (t, 0),ψµ (t, π) = −η2 ψ̃µ (t, π),(4.13)

where ψ, ψ̃ are left- and right-moving bulk fermions and η1,2 = ±1 specifies the boundary
conditions at σ = 0, π. Though only the relative sign in these conditions is ultimately important, our current conventions are such that η1,2 = +1 represents a Dp-brane at σ = 0, π, whileη1,2 = −1 represents a Dp0 -brane at σ = 0, π. This is described in more detail in Appendices
A and B.
The presence of zero-modes depends on the relative choice of η1,2 . This choice is in turn
related to the choice of NS or R sectors on the open string; we declare that the string is in the
NS sector for η1 = η2 and in the R sector for η1 = −η2 ; see Appendix A. From (4.13) we thenconclude that NS sector fermions can have zero-modes along ND and DN directions, whereas
R sector fermions can have zero-modes along NN and DD directions.
Consider now a coincident pair of 9-branes of type η (Fig. 3(a)). In light-cone gauge,
there are zero ND directions and a total of 8 NN + DD directions. Since open strings stretching
between branes of the same type are in the NS sector, any potential zero-modes would be in
ND directions, of which there are none. Hence there are no bulk zero-modes, and thus noboundary fermions. The same holds more generally for coincident pairs of p-branes of type η.
We now consider a 9-brane coincident with a (9 − k)-brane, both of type η (Fig. 3(b)).
We consider an open string between them, which is again in the NS sector since the branes
are still of the same type. However, there are now k DN directions, and hence k zero-modes.
We denote the restriction of these zero modes to the boundary at the (9 − k)-brane by ψ0i . Asin (3.12), the boundary time-reversal operator T acts on these by Tψ0i T−1 = ψ̃0i . By (4.13),
25

SciPost Phys. 9, 010 (2020)

if the (9 − k)-brane is at σ = 0 then we have ψ0i = ηψ̃0i , whereas if it is at σ = π we have
ψ0i = −ηψ̃0i . Together we conclude that¨

Tψ0i T−1 =+ηψ0i

σ = 0,−ηψ0i

σ = π.

(4.14)

When k 6= 0 mod 8, these zero-modes will lead to anomalies in T and (−1)f , and in orderto cancel them we must add extra boundary fermions. At σ = 0 there are two options: we may
add k boundary fermions ζi satisfying Tζi T−1 = −ηζi , or 8 − k boundary fermions satisfying
Tζi T−1 = +ηζi . These two choices are effectively identical, as they both behave as Cl(−ηk).
On the boundary at σ = π, one likewise requires a representation of Cl(+ηk).
For full consistency, we must check that the same additional boundary fermions also cancel
tentative anomalies in mixed strings stretching between Dp- and Dq0 -branes. To begin, consider coincident Dp- and Dp0 -branes (Fig. 3(c)). There are then zero ND directions and a totalof 8 NN+DD directions. Since the branes are now of opposite types, the open strings stretchedbetween them are in the R sector. Recall that in the R sector, there are zero-modes not from
ND directions, but rather from NN and DD directions. Hence in this case there are a total of 8
bulk zero-modes. But because there are eight of them, there is no time-reversal anomaly, andno boundary fermions are necessary. This is consistent with the fact that the strings stretchingfrom the Dp-branes to themselves, or from the Dp0 -branes to themselves, were shown to not
require any boundary fermions.
Another way to analyze this setup is to note that, insofar as counting zero-modes is concerned, switching η → −η is equivalent to switching Neumann and Dirichlet boundary conditions in all directions, as is evident from (4.13). Then the Dp0 -brane can be replaced bya D(10 − p)-brane, and we now have a setup with 8 ND directions and zero NN and DD directions (in light-cone gauge). Since the branes are of the same type now, the open string
stretched between them is in the NS sector, and we can again conclude that we have 8 zeromodes. This method of replacing the Dp0 -brane with a D(10 − p)-brane is useful in analyzing
e.g. the case of a D(9 − k)0 - and D9-brane, which can be replaced by a D(k + 1)- and D9brane (Fig. 3(d)). By our above analysis, this then has 8 − k bulk zero-modes which satisfy
Tψ0 T−1 = −(−1)ψ0 = +ψ0 at σ = 0. These can be cancelled by adding 8 − k boundary
fermions ζi satisfying Tζi T−1 = −ζi , or k boundary fermions satisfying Tζi T−1 = ζi , both ofwhich give a representation of Cl(k). This matches the results found before for a D(9 − k)0 brane. Using these techniques, one can easily check that the more general mixed strings between Dp- and Dq0 -branes have vanishing anomaly when one makes the above assignmentsof boundary fermions.
Finally, we may reintroduce non-zero n. As discussed above, this gives a representation ofCl(n) on the boundary at σ = 0 and of Cl(−n) on the boundary at σ = π. Let us now consider
an open string stretched between a (9 − k)-brane of type η at σ = 0 and a D(9 − k)-brane at
σ = π. In the theory with n copies of ABK, the endpoint at σ = 0 carries a representation ofCl(n − ηk), while the endpoint at σ = π carries a representation of Cl(−n + ηk).

4.3

K-theory classification of branes

We can now study the stability of D-branes in these theories. The analogous results in TypeI were obtained in [73–75]. We first note that the tachyon vertex operator ζ is a fermionic,
hermitian boundary operator such that the boundary interactionZµd t iψ0 ζDµ T (X )26

(4.15)

SciPost Phys. 9, 010 (2020)µ

is T-invariant. Here ψ0 is the restriction of the bulk fermion field to the boundary and T (X ) isµthe spacetime tachyon profile. In particular, the ψ0 appearing here should be those associatedto the Neumann directions, which have opposite T transformations as those given in (4.14).
Thus for the coupling to be T-invariant, we need ζ to have the same T transformations as
those given in (4.14),+ηζ
σ = 0,−1TζT =(4.16)−ηζσ = π.
Stable branes are those for which such ζ does not exist.
It is cumbersome to carry out the analysis below for the two cases η = ±1 separately.
This can be circumvented by analyzing the σ = 0 end when η = +1 and the σ = π end
when η = −1. This means that we work with the Clifford algebra Cl(n − ηk) for η = +1 andCl(−n + ηk) for η = −1, or in other words just Cl(ηn − k) for both cases. Therefore, below we
simply analyze Cl(ηn − k) and look for a tachyon vertex operator satisfying

TζT−1 = ζ,(4.17)

for both cases η = ±1. This means that the results to be obtained depend only onν := ηn − k.

(4.18)

When ν = 0 mod 8, the minimal choice of Chan-Paton algebra is simply R, as can be seenfrom Table 4. We can enlarge the Chan-Paton algebra by introducing N bosonic indices andN 0 fermionic indices. Then the Chan-Paton algebra has the bosonic part R[N ] ⊕ R[N 0 ] and00
the fermionic part RN ⊗ RN ⊕ RN ⊗ RN , where we recall that the algebra F[N ] stands for the0
N × N matrix algebra with entries in F. The T-invariant fermionic part is then RN ⊗ RN . Thismeans that the gauge group is O(N ) × O(N 0 ) and the tachyon field is in the bifundamental.
The stable branes are those such that the tachyon representation is empty, which occurs for
N ≥ 0 and N 0 = 0 or N = 0 and N 0 ≥ 0. These can be labeled by Z. The case ν = 4 is similar;
one simply replaces R by H and O by S p.
In the other six cases, the minimal choice of Chan-Paton algebra A already contains boththe bosonic and the fermionic part. We can introduce additional Chan-Paton indices i = 1, . . . , N ,thus enlarging the Chan-Paton algebra to A[N ] := A ⊗ R[N ]. From Table 4, we see that A[N ]has the following structure:
ν
1
2
3
5
6
7

A[N ]0A[N ]+
A[N ]−
1
1
R[N ] R[N ]symm. R[N ]anti.
C[N ] C[N ]symm. C[N ]anti.
H[N ] H[N ]symm. H[N ]anti.
H[N ]H[N ]anti. H[N ]symm.
C[N ]C[N ]anti. C[N ]symm.
R[N ]R[N ]anti. R[N ]symm.

(4.19)

where A[N ]0,1 denotes the bosonic and fermionic parts and A[N ]±1 denotes the hermitianand anti-hermitian parts. The symmetrization and antisymmetrization are defined as usual
for R and C, and as the symmetric and antisymmetric tensor square for the fundamental
representation of S p(N ) for H. It is instructive to check that the table above reduces to Table 4
when N = 1. From this it is easy to read off the gauge group and the representation of thetachyons for the D-branes, which is given in Table 5. One may check that by setting n = 0 in
Table 5 we get back the results of [73–75] for Type I.
The stable branes are those for which the tachyons are absent. For ν = 0, 4 we alreadydiscussed that they are classified by Z. For ν = 1, 2, 3, 5 the tachyon field is always present.
27

SciPost Phys. 9, 010 (2020)

Table 5: Gauge groups Gν and tachyon representations ρζ on the worldvolumeof N (9 − k)-branes of type η in the theory with n copies of ABK. Note thatν = ηn − k = ηn + p − 1 mod 8. “Bifund.” refers to the bifundamental representation, whereas “symm.” (“anti.”) refer to the symmetric (anti-symmetric) rank
2 tensor representations. We also listed the tensions of these branes, as well as their
K-theory classifications, which can be found by considering when the tachyon field isempty.
ν

0Gν

2

O(N ) × O(N ) O(N )ρζbifund.

λν1

KO (pt)

Z

ν

1
0

3

4
N
2

U(N ) S psymm. symm. symm.
p
p
2
2
2 2
0

0

0

Sp

N
2



× Sp

bifund.

5


N0
2



6
N
2

Sp

2

anti.
p
2 2

Z

0



7

U(N ) O(N )anti. anti.
p
2
2
Z2

Z2

Finally, for ν = 6, 7 the tachyon field is absent when N = 1 but appears when N = 2, andhence the classification is by Z2 . These results match with KOν (pt).
Let us reanalyze this setup using the auxiliary boundary fermions discussed in Section
4.1.3. Our physical boundary fermions form an algebra A whose anomaly is of type ν. We
introduce auxiliary boundary fermions Cl(−ν) to cancel the anomaly, so Cl(−ν) ⊗ A is nonanomalous, acting on the Chan-Paton vector space V . The tachyon field corresponds to ahermitian fermionic element ζ in A. This means that if such a tachyon field is present, theCl(−ν) action on V is extended to a Cl(1 − ν) action on V . In other words, on the D-branecharacterized by ν, a Chan-Paton space V with an action of Cl(−ν) is necessary, and it isunstable if and only if this Cl(−ν) action can be extended to an action of Cl(1 − ν). Using theresult in [70], this can be directly connected to KOν (pt). To see this, recall that in [70] Atiyah,
Bott, and Shapiro considered
M−n := free Z-modules generated by graded irreducible representations of Cl(−n)(4.20)and considered the natural map i ∗ : M−n−1 → M−n induced by i : Cl(−n) → Cl(−n − 1). It wasthen shown thatKO−n (pt) = M−n /i ∗ (M−n−1 ).
(4.21)
We note that by replacing Ξ by Ξ̂ = Ξ(−1)f , a graded representation of Cl(n) can be converted
to a graded representation of Cl(−n) and vice versa. Therefore we also haveKO−n (pt) = Mn / j ∗ (Mn+1 ) ,(4.22)

where j ∗ : Mn+1 → Mn is now induced by j : Cl(n) → Cl(n + 1). Then, Chan-Paton spaces with
Cl(−ν) action modulo those with Cl(−ν + 1) action are clearly classified by KOν (pt).
We note that this restatement also shows that the brane of type ν can naturally host an
orthogonal bundle with an additional action of Cl(−ν). It would be interesting to connect thisobservation to the definition of KO-theories in terms of Clifford bundles, which can be founde.g. in [76].
4.3.1Brane tension from boundary fermionsFinally, let us comment on the tensions of the various branes, both stable and unstable, identified thus far. In order to obtain the tension one computes a disc path integral, with the28

SciPost Phys. 9, 010 (2020)boundary of the disc anchored on the brane — this may be interpreted as the one-point function of the brane with a graviton in the closed string picture. If there are Majorana fermionspresent on the boundary of the disc, these will give a contribution to the path integral. In
particular, as mentioned in the Introduction
pthe contribution of a (0 + 1)d Majorana fermion
to the path integral is an overall factor of 2. This holds in all cases, including Type II andType 0 strings.
We begin by discussing the more familiar case of Type II, for which there are branes of onlya single type η = 1. As we have discussed, the Type IIB and IIA theories are distinguished by
the presence of respectively n = 0, 1 Majorana fermions
pon the boundary. For this reason, thenon-BPS D9-branes in Type IIA have a tension which is 2 times that of their BPS counterparts
in Type IIB. However, it is not the case that all branes in Type IIA have tensions which arelarger by this factor. Indeed, let us consider branes of codimension k. By considering open
strings stretched between such branes and D9-branes, we may conclude that strings ending
on such branes must have k bulk zero modes — this is argued for as above, by noting that achange in codimension leads to a change in the number of DN or ND directions for the open
string endpoints. It is then possible for the anomalies of the invertible phase and bulk zeromodes to cancel. In fact, since in this case the anomaly is only Z2 we can conclude that the
number of boundary fermionspis only |n − k| mod 2. As such, we conclude that Dp branes
in Type IIA/B have tensions ( 2)|n−k| mod 2 TpII , where TpII is the tension of the corresponding
stable non-torsion p-brane. In particular, this tells us that for Type IIB, branes with pp odd
have tensions TpII (essentially by definition) while for p even the branes have tensions 2TpII .
Likewise for Type IIA, branes with p even have tensions TpII , while branes with p odd haveptensions 2TpII .
We now generalize this to the case of Type 0 strings. As explained above, in this case thebulk theory carries the same anomaly as ν boundary Majorana fermions. It is natural to choose
the minimal realization of the anomaly in the boundary system, which is a system of Majorana
fermions in a representation of Cl(+ν) if ν < 4 and of Cl(−ν) if ν > 4. Then the tension of thecorresponding branes is weighted
p by the path integral over the system of fermions, yielding0
the result λν Tp where λν := dimR A as given in Table 4. For example, in the case of a D9p
or D90 -brane in the theory with 1 or 7 copies of ABK, we find that the tension is 2 times
that in the theory with no copies of ABK. The values λν are tabulated in Table 5, and will be
reproduced for stable branes via the boundary state formalism in Section 5.

4.4

Vacuum manifolds of tachyons as classifying spaces

We now reinterpret the results of this section in terms of tachyon condensation on the worldvolume of 9-branes. In the string theory literature, this is referred to as the Atiyah-Bott-Shapiro(ABS) construction [33, 34, 70].
We start from the 9-brane of type η. As before we use the trick of analyzing the σ = 0
boundary when η = +1 and the σ = π boundary when η = −1. The boundary then carries
an action of Cl(ηn). To construct a (9 − k)-brane located at X 1 = · · · = X k = 0, we choose a
set of time-reversal invariant Majorana fermion operators ζi=1,...,k in Cl(ηn) (for this purpose
one might need to replace ηn by ηn + 8m for some integer m) and use them to assemble thetachyon field

T (X ) = f (|X |)kX

X i ζi ,(4.23)

i=1

with f (|X |) some convergence factor chosen such that T (X ) obtains its vacuum value as
Pk|X | → ∞ and such that f (|X |)2 i=1 (X i )2 = 1. Upon condensation this tachyon gives rise
29

SciPost Phys. 9, 010 (2020)

to the desired (9 − k)-brane. As we used up k out of ηn Majorana fermions we originally hadon the 9-brane, we end up with the residual action of Cl(ηn−k), as argued before in a different
manner.
We now discuss the connection between tachyon profiles and KO-theory. Consider theusual 9-9 stack, supporting an O(N ) × O(N ) gauge group for n = 0 mod 8. We now follow
Sen’s construction and consider a domain wall configuration of the tachyon such that it condenses to produce a D8-brane [32]. This involves assignment of a vacuum value of ζ(X ) toeach point on the “sphere” S 0 at infinity. Thus such domain wall configurations are classified
by π0 (V9 ), where V9 is the vacuum manifold for the tachyon on D9-brane. The tachyon potential should be such that it breaks O(N ) × O(N ) to the O(N ) supported on the resulting D8.
A minimal assumption is thenV9 =O(∞) × O(∞) ∼= O(∞) ,O(∞)(4.24)

where we took the formal limit in which the number of original D9-branes is infinite. We canrepeat the argument for lower Dp-branes, and find that they would similarly be classified by
π8−p (V9 ).
The Dp-brane can also be obtained from tachyon condensation on Dq-branes with p < q < 9.
The vacuum manifold Vq for the tachyon field on the Dq-brane can be determined from Table 5:
S p(∞)S p(∞) × S p(∞)
O(∞)
U(∞)
×Z ,
V3 =
, V4 =, V5 =,
O(∞) × O(∞)
O(∞)
U(∞)
S p(∞)S p(∞)U(∞)O(∞)
O(∞) × O(∞)
V6 =× Z , V7 =, V8 =, V9 =,S p(∞) × S p(∞)
S p(∞)
U(∞)O(∞)(4.25)
where the subscript is defined modulo 8. One then expects that Dp-branes can be classified by
πq−p−1 (Vq ).
Ý 0 (S 9−p )All of this is compatible with the statement that the Dp-branes are classified by KO
thanks to the mathematical fact [76] that KOn := Vn+2 is the classifying space of KO-theory,in the sense that
Ý n (X ) = [X , KOn ].
KO(4.26)V2 =The KOn form an Ω-spectrum, which entails the relation KOn ' ΩKOn+1 . We then have
Ý 0 (S 9−p ).
πq−p−1 (Vq ) = [S q−p−1 , KOq−2 ] = [pt, Ωq−p−1 KOq−2 ] = [pt, KOp−1 ] = KO

5

(4.27)

D-brane spectra via boundary states

In the previous section, we were able to verify the K-theory classification of Pin− Type 0 strings
from the open string perspective. In this section, we rederive these results from a closed string
perspective. In addition to gaining more intuition about the behavior of invertible phases
on worldsheets, we will use the closed string perspective to verify our earlier results on thetensions of stable branes appearing in each theory. These tools will also be used to address theissue of tadpole cancellation in Appendix C.
From the closed string point of view, D-branes correspond to states in the closed string
Hilbert space. The basic idea is to define these states by imposing the open string boundary
conditions (4.13) as gluing conditions. Upon appropriate rotation of the Euclidean worldsheet,
one finds that the correct conditions to impose on boundary states |Bp, η〉 are
µ

µ

µ

µ

N : (αµn − α̃−n )|Bp, η〉 = (ψµr − iηψ̃−r )|Bp, η〉 = 0 ,D:(αµn + α̃−n )|Bp, η〉 = (ψµr + iηψ̃−r )|Bp, η〉 = 0 ,
30

(5.1)

SciPost Phys. 9, 010 (2020)

as reviewed in Appendix B. Here r is integer/half-integer for the R/NS sector. The solution tothese equations ends up being the coherent state defined in (B.4). Furthermore, in Appendix
B it is shown that the physical D-brane states in Type 0 theories are in fact linear combinationsof these |Bp, η〉, of the form
|Dp, η〉 =1 1
p (η|Bp, η〉NSNS + |Bp, η〉RR) ,NBp 2

η = ±1 ,

(5.2)
5

p−4

where |Dp, +〉 represents a Dp-brane, |Dp, −〉 represents a Dp0 -brane, and NBp = 2 2 (4π2 α0 ) 2
is a normalization factor obtained in Appendix B.3. This result will be the starting point for
our analysis in this section.

5.1

Matching non-torsion brane spectra

In this subsection we begin by rederiving the stable non-torsion brane spectra of the eight Pin−
Type 0 theories via the boundary state formalism. The analysis of the torsion brane spectra
will be carried out in Section 5.2.
To understand the spectrum of branes in the theory with n copies of ABK, we need tounderstand the action of the worldsheet parity operator Ω on D-brane states in the presenceof the invertible phase. We begin by understanding the action of Ω on the constituent boundary
states |Bp, η〉. In Section 3.2.2, we described the action of Ω on the ground states in the theory
with n copies of ABK. From this and (B.9) it easily follows that the action of Ω on the boundary
states is given byΩ|Bp, η〉RR = i ην |Bp, η〉RR ,Ω|Bp, η〉NSNS = |Bp, η〉NSNS ,(5.3)

with the parameter ν = ηn − k defined in Section 4.
The branes which survive the orientifolding are those which are left invariant by Ω; i.e. thosefor which ν is zero modulo 4. Therefore, the non-torsion Dp-branes are in one-to-one correÝ n (S k ), while the non-torsion Dp0 -branes are in
spondence with non-torsion elements of KOÝ −n (S k ), as listed in Table 3.
one-to-one correspondence with non-torsion elements of KO
Although ν is a mod 8 parameter, the presence of non-torsion branes depends on ν only
mod 4. Indeed, this is to be expected since (5.3) was obtained by considering only the valuesof ABK on the Klein bottle K2 . However, as discussed in Section 3.2.2, K2 is not the generating
manifold for f2Pin− (pt) = Z8 . Rather, the generating manifold is RP2 , and since K2 ∼= RP2 #RP2
we currently have access to only a Z4 of the full Z8 .
In contrast, the open string states do depend on ν mod 8. This is because the action of Ωon open string ground states comes from the Möbius strip amplitude. The Möbius strip M22contains a single RP2 , whose amplitude is e iπABK(RP )/4 = e±iπ/4 , as we saw in Section 2.2.3.
Then we have the schematic actionΩ|0; i j〉 ∼ e±inπ/4 |0; ji〉 ,(5.4)

where i, j are Chan-Paton indices. Therefore shifting n → n + 4 changes the action of Ω on the
Chan-Paton factors by a minus sign, between symmetric and anti-symmetric. This is precisely
as expected when going between O9− - and O9+ -planes, and in agreement with the fact thatÝ n+4 (X ) ∼KO
KS p n (X ).
=ß

5.2

Matching torsion brane spectra

We now use the boundary state formalism to check that the torsion brane spectra predicted byÝ n (X ) ⊕ KO
Ý −n (X ) are reproduced by the theory with n copies of ABK. The study of torsion
KO
31

SciPost Phys. 9, 010 (2020)

branes [32, 77, 78] in the boundary state formalism for unoriented theories was first donein [79–81]. Here we will adapt these methods to the Pin− Type 0 theories.
In contrast to the generic Type 0 non-torsion brane (5.2), the generic torsion branes ofType 0 consists of only the NSNS portion,
Ý|Dp, η〉 =λν η
p |Bp, η〉NSNS ,NBp 2

η = ±1 ,(5.5)

where λν > 0 is a normalization factor related to the tension of the torsion brane, which will
be shown to depend only on ν. This state, though GSO invariant, is not stable in the oriented
Type 0 theory since the tachyon is not projected out. This can be detected by computing theoverlap of two boundary states in the closed string tree-channel, and then doing a modulartransformation to the open string loop-channel, where a tachyon appears. However, in theunoriented theory the gauging of Ω can project out the tachyon. This is seen at the level of theamplitude by cancellation of the tachyon piece of the cylinder amplitude with the analogous
piece of the Möbius strip amplitude.
To determine the spectrum of stable torsion branes, we begin by computing the closed
string cylinder diagram in tree-channel. Using the results of (B.22), we have
Z∞
Z∞
8
λ2ν
d` f3 (2i`)
−2πlH
cl |D
Ý
Ý
p, η〉 = 6 vp+1
AC2 =d l 〈Dp, η|e
.
(5.6)
9−p
8
2
` 2 f1 (2i`)
00
The functions f i (τ) are defined in (B.11) in terms of Jacobi theta functions and the Dedekind
eta function. We can translate the tree-channel amplitude to loop-channel using the transformation l = 1/2t and the modular S transformations in (B.12), which yields
Z∞8d t f3 (i t)
1 2
,
(5.7)
AC2 = λν vp+1
p+3
82(2t) 2 f1 (i t)0with the answer independent of η and n. Using the q-expansions of the f i (τ) given in (B.11),
we may isolate the tachyon contribution,
Z∞
1 2
dt
AC2 tachyon = λν vp+1
eπt .
(5.8)
p+3
2
2
(2t)
0
We must now check under which circumstances this can be cancelled by a contribution
from the Möbius strip. In order to calculate the Möbius strip amplitude, we use the orientifold
state |Op〉 introduced in (B.42) and calculate in the loop-channelZ∞ÝÝp, η〉 + 〈Dp, η|e−2π`Hcl |O9〉AM =d l 〈O9|e−2π`Hcl |D20= −λν vp+1



Z∞

 i

d` e

πnη
4

2(9−p)

f38 f4

p−1

f1

0

!



1
2i` +
2
2(9−p)
f48 f3p−1f1

πnη−e−i 4λ= − ν vp+1
2

Z∞


dt
p+3

0

(2t) 2e2(9−p)

f38 f4i π4 (9−p−nη)

p−1

f1
π

−e−i 4 (9−p−nη)

32

!



!




1 
2i` +
2

1
it +
2



2(9−p)

f48 f3

p−1

f1

!





1 
it +
.(5.9)
2

SciPost Phys. 9, 010 (2020)

Isolating the tachyon contribution yields
Z∞
hπ i
dt
A M2 tachyon = λν sinν vp+1
eπt .
p+3
4
2
(2t)
0

(5.10)

Combining (5.8) and (5.10), we haveZ∞h π i
1 
dt(AC2 + A M2 ) tachyon = λν λν + 2 sinν vp+1
eπt .
p+3
2
4
2
(2t)
0

(5.11)

Since λν is positive,the values of p for which the tachyon can be cancelled are those suchthat sin π4 ν < 0. The tension of the corresponding torsion brane is then λν Tp0 , where 
λν = −2 sin π4 ν and Tp0 is the tension of a stable non-torsion p-brane in Type 0. For example,
when n = 0 we conclude that cancellation is possible if p ∈ {−1, 0, 7, 8}, regardlessp 0 of η = ±1.
0
The corresponding branes have tension 2Tp for p = −1, 7 and tension 2Tp for p = 0, 8.
Ý n (S k ) ⊕ KO
Ý −n (S k ). The
These are in one-to-one correspondence with the torsion classes in KO
tensions determined here also match exactly with the ones listed in Table 5.
Note that in this example, it would naively seem that p = 6 yields an acceptable torsion
brane as well. However, this is not the case. To understand this, recall that before orientifolding these branes are unstable due to a tachyon in the p-p strings — these in particular are inthe (−1)f odd part of the spectrum, and are built out of a certain vacuum |0〉odd
pp . One thennotes that, without the presence of Chan-Paton factors [80],π−i 4 pΩ|0〉odd|0〉oddpp = epp .

(5.12)

Hence for p = 2, 6 we see that Ω2 = −1 on the vacuum. To compensate, we need to introduce
at least a two-dimensional Chan-Paton index i = 1, 2, in the doublet representation of S p(1).
Then the Ω projection keeps the antisymmetric combination [i j] of the tachyon, and thus thereÝ n (S k ) ⊕ KO
Ý −n (S k ).
is no stable torsion D6-brane. Indeed this class is absent from KO
The generalization of this condition for non-zero n is as follows. The goal is to check that2
Ω 6= −1 on the relevant ground state, lest 21 (1 + Ω) not be a valid projector for removing thetachyon. We know that non-zero n modifies the open string ground states such that the actionof Ω is modified by (5.4). The cases with n even descend from Type 0B, in which case we have
p even, whereas the cases with n odd descend from Type 0A, in which case we have p odd.
−i π2 ηn −i π2 pThus we conclude that Ω2 on the |0; n〉odde, and hence wepp ground state is given by emust exclude cases for which p + η n = ±2, 6, 10, 14, . . . .
To summarize, the conditions that must be satisfied by torsion (9 − k)-branes of type η inthe theory with n copies of ABK are the following,
hπ i
sin
ν <0,
9 + ν 6= ±2 mod 8 .
(5.13)
4Note that the combination of n, k appearing here, namely ν = ηn − k mod 8, is the same one
appearing in Section 4 and Section 5.1. We have already checked above that the analysis via
boundary states is in agreement with the previous analysis via boundary fermions when n = 0.
Since the results (5.13) only depend on ν, this agreement is simply extended to the general
case.

5.3

Pin+ Type 0 theories

We may now briefly turn to the analysis of the brane spectra in Pin+ Type 0 theories. Using
the action of Ω given in (5.3) (with ν → −k) and the action of (−1)fL given in (B.8), one findsΩf |Bp, η〉RR = i −ηk |Bp, −η〉RR .

Ωf |Bp, η〉NSNS = −|Bp, −η〉NSNS ,33(5.14)

SciPost Phys. 9, 010 (2020)in the trivial phase. In the non-trivial phase, one gets an extra sign in the action of Ωf on RR
ground states, so more generallyΩf |Bp, η〉NSNS = −|Bp, −η〉NSNS ,Ωf |Bp, η〉RR = i 2n−ηk |Bp, −η〉RR .

(5.15)

with n = 0, 1 labelling the trivial or non-trivial phase. The stable D-brane states which are
invariant under Ωf then take the form
|Dp〉 =

1 1(|Bp, +〉NSNS − |Bp, −〉NSNS + |Bp, +〉RR + i 2n−k |Bp, −〉RR) ,NBp 2(5.16)

reminiscent of the Type II states obtained in (B.27). In fact, these states are invariant underΩf for any value of k = 9 − p, both odd and even. On the other hand, the fully physical Dbrane states must also be invariant under (−1)f . It is easy to see that this requires 2n − k to
be even. Thus for both n = 0, 1, we keep all states with p odd, reproducing the full spectrumof non-torsion branes in Type IIB.
We may now ask about torsion branes. These would take the same form as in Type I, given
by
Ý
|D
p〉 =

λ 1
(|Bp, +〉NSNS − |Bp, −〉NSNS) ,NBp 2(5.17)

with λ > 0 a normalization factor. In order for such a brane to be stable, we require the open
string tachyon contributions from the cylinder and Möbius strip amplitudes to cancel. In orderto calculate the Möbius strip amplitude, one must make use of the appropriate orientifold
plane state, which is obtained in Section B.4 and shown in (B.47). Importantly, note that thePin+ structure forces this state to be entirely in the RR sector. This means that the Möbius strip
amplitude consists only of terms of the form NSNS〈Bp| . . . |Bp〉RR, which vanish. There is thus
no Möbius strip contribution at all, and hence we cannot expect any cancellation of tachyons,and so no stable torsion branes. In conclusion, the spectrum of stable branes in these theories
is precisely the same as for oriented Type IIB, and is classified by the complex K-group K(X ).

6

No new Type I theories

In this final section we classify unoriented Type II (i.e. Type I) strings. Unlike Type 0 strings
for which one may consider Pin± theories separately, the unoriented Type II strings possessa more complicated spin extension of O(d), which contains both Pin± (d) as subgroups. We
refer to this as DPin structure, since it is a “doubled” Pin structure, and we define it precisely
in Section 6.1. That such a thing is necessary is to be expected: as explained in Section 2.2,
for Pin− structure the boundary circle of the Möbius strip is automatically in the NS sector,
whereas for Pin+ structure the boundary circle is automatically in the R sector. In contrast, weknow that Type I strings allow both NS and R boundary conditions on the Möbius strip, so it
is clear that these worldsheets must incorporate both Pin± .
In order to understand possible anomalies and invertible phases on the worlsheet of undoriented Type II theories, it is necessary to calculate the groups fDPin(pt) for d = 2, 3. Thismay be done using the Atiyah-Hirzebruch spectral sequence for twisted spin bordism, as will
be recalled in Section 6.2. Details of the calculation are relegated to Appendix E. The final
result is that f2DPin (pt) = (Z2 )2 and f3DPin (pt) = Z8 . The latter implies that the unoriented
Type II string is anomaly-free in ten dimensions. The former would naively suggest a quartet of
string worldsheet theories, but as we discuss in Section 6.3 only two of these theories are physically distinct. The two distinct options are the traditional Type I and Ĩ strings, corresponding
to orientifoldings by orientifold O9∓ -planes.
34

SciPost Phys. 9, 010 (2020)

6.1‘Spin structure’ on the Type I worldsheet

The oriented Type II worldsheet has separate spin structures for left- and right-movers, necessitating a Spin × Z2 structure. We would now like to understand the unoriented lift of thisstructure. We note that the Z2 part of the Spin × Z2 structure acts on (ψ, ψ̃) by diag(+1, −1),and should therefore be mapped to diag(−1, +1) under orientation reversal. To formalize, wethen need an extension of O(d) by Z2 × Z2 ,0 → Z2 × Z2 → G → O(d) → 0(6.1)

such that the orientation reversal part of O(d) exchanges the two Z2 factors, and such thatwhen restricted to SO(d) the extension class is given by (w2 , w2 ). Another way of saying thisis that we would like a spin structure on the orientation double cover of the worldsheet. Notethat for the worldsheet we have d = 2, but we will work more generally for the moment.
We now consider the effect of the homomorphism s : Z2 × Z2 → Z2 , (a, b) 7→ a b in theextension. Over SO(d), the extension class of the image is w2 + w2 = 0. The image is also
invariant under the orientation reversal part of O(d). Therefore the extension
0 → s(Z2 × Z2 ) → s(G) → O(d) → 0(6.2)

is trivial and can be split: s(G) ' O(d) × Z2 . There are two natural splitting; once there is a
splitting, we can compose it with
O(d) × Z2 → O(d) × Z2
(g, c) 7→ (g, c det g)(6.3)

to get another. So G can also be put in the following sequence,
0 → Z2 → G → O(d) × Z2 → 0 .

(6.4)

Its extension class is a linear combination of w2 , w21 , aw1 , and a2 , where a is the generatorof H 1 (BZ2 , Z2 ). To determine which linear combination, we may argue as follows. First,
since Z2 is not extended to Z4 within G, the term a2 is not involved. Second, since the entire
group is not Pin± × Z2 , we need the term aw1 . Finally, since the extension is Spin(d) × Z2
over SO(d) × Z2 , the class w2 must be involved. This means that the extension class is either
w2 + w1 a or w2 + w21 + w1 a. These two are exchanged by the change of the splitting, since (6.3)
sends a 7→ a + w1 . Let us pick w2 + w21 + w1 a for definiteness, and write c : G → Z2 for the
projection to the Z2 factor. The kernel of c is Pin− (d). The kernel of det g : G → O(d) → Z2 isSpin(d) × Z2 , and the kernel of c det g is Pin+ (d). So G is an interesting mixture of all threegroups — as mentioned before, we refer to it as G = DPin(d).
For d = 2 we can construct the group DPin(2) more directly. We first let Spin(2) act on(ψ, ψ̃) via diag(e iθ /2 , e−iθ /2 ). Next we supplement this with a chiral Z2 acting via
Z = diag(+1, −1) ,(6.5)

and then further include orientation-reversing elements of Pin− (2), which can be chosen to beelements of the Clifford algebra Cl(−2),




0 i
0 −1
γ0 =
,
γ1 =
.
(6.6)
i 0
1 0
Note that we have
γ0 γ1 = diag(i, −i) ,
35

(6.7)

SciPost Phys. 9, 010 (2020)

which is a lift of a 180◦ rotation. This means that γ0 γ1 Z = i 1 is also a lift of a 180◦ rotation.
Then we see thatγ̃0 := iγ1 ,γ̃1 := −iγ0
(6.8)
are also in the group DPin(2). Together with Spin(2), these elements of Cl(+2) can be used
to form Pin+ (2). In this way, we see explicitly how DPin(2) contains all three of Spin(2) × Z2 ,Pin− (2), and Pin+ (2).
Now consider the Möbius strip M2 . As discussed above, the boundary circle of M2 is automatically in the NS sector for Pin− since γ2i = −1, whereas for Pin+ structure the boundary
circle of M2 is automatically in the NS sector since γ̃2i = 1. We may now contrast this withthe case of DPin structure. In that case, we can use either γi or γ̃i to construct M2 , and depending on this choice we can have NS or R on the boundary circle. From the boundary state
point of view, this means that the orientifold plane states should have both NS and R sector
contributions, which matches the well-known result (B.49).

6.2

d
The group fDPin(pt)

To understand the anomalies and invertible phases present on unoriented Type II worldsheets,d
we must calculate fDPin(pt) for d = 2, 3. These groups may be calculated by using the AtiyahHirzebruch spectral sequence (AHSS) for twisted spin bordism groups.
We first recall twisted spin structures. We consider a space X with a real vector bundle V
over it. Take a manifold M . A spin structure on M twisted by V is a pair
( f : M → X , spin structure on T M ⊕ f ∗ (V )).
Spin

We can then consider the corresponding bordism group Ωd
w1 (T M ) = f ∗ (w1 (V )),(6.9)(X ; V ). Note that we have

w2 (T M ) = f ∗ (w1 (V )2 + w2 (V ))(6.10)

since we have a spin structure on T M ⊕ f ∗ (V ).
For example, when when V is a zero-dimensional trivial bundle this reduces to an ordinary
spin bordism of X . As another set of examples, take L to be the real line bundle over BZ2 suchthat w1 (L) is the generator of H 1 (BZ2 , Z2 ) = Z2 . Then we have SpinΩd(BZ2 )
 Pin−Ωd(pt)SpinΩd (BZ2 ; L ⊕n ) =SpinZ4

(pt)
Ωd

 Pin
+
Ωd
(pt)

n = 0,
n = 1,
n = 2,
n = 3,(6.11)

where SpinZ4 = (Spin × Z4 )/Z2 .
The group DPin described above corresponds to taking X = BZ2 × BZ2 and using
V = (L1 ⊗ L2 )⊕ L2⊕3 . Here L1 and L2 are real line bundles such that w1 (L1 ) = w and w1 (L2 ) = a,where we denote the generators of H 1 (BZ2 × BZ2 , Z2 ) = Z2 × Z2 by w and a. Then
w1 (V ) = w,

w2 (V ) = wa.

(6.12)Let us now recall the basics of the AHSS. This review will not be comprehensive — formore thorough introductions to the AHSS, the reader can consult e.g. [18, 21, 25, 82]. The
basic ingredients in the AHSS are the E2 page and a set of differentials. For twisted spinp,qqqbordism, the E2 page consists of E2 = H p (X , fSpin (pt)). The underline in the group fSpin (∗)
denotes the fact that the coefficient system is twisted by w1 (V ) ∈ H 1 (X , Z2 ). The differentials
on the E2 page are as follows:
36

SciPost Phys. 9, 010 (2020)

p,2

p+2,1

• d22 : E2 = H p (X , Z2 ) → E2

= H p+2 (X , Z2 ) is given byd22 (x) = Sq2 x + w1 (V ) Sq1 x + w2 (V )x ,
p,1

p+2,0

• d21 : E2 = H p (X , Z2 ) → E2

(6.13)= H p+2 (X , U(1)) is given by
d21 (x) = ι Sq2 x + w1 (V ) Sq1 x + w2 (V )x = ι Sq2 x + w2 (V )x ,(6.14)

ι

where ι is the inclusion Z2 ,→ U(1).11
These differentials, together with d23 , were determined in [83]. They were also deduced
previously in [84, 85] when w1 (V ) is trivial. Alternatively, they can be deduced using theidentity
Spin
Spin(6.15)Ωd (X ; V ) = Ω̃d+dim V (Thom(V )) ,where Thom(V ) is the Thom space of V . Indeed, denoting the Thom class by U, this equality
impliesd22 (x)U = Sq2 (x U),
d21 (x)U = ι(Sq2 (x U)).
(6.16)We then simply use the Cartan formula for the action of the Steenrod square, and the definitionof the Stiefel-Whitney classes as Sqd U = w d (V )U.
Details on our calculation will be given in Appendix E, where we demonstrate how one
2,3
computes fX (pt) not only for X = DPin but also for X = Spin × Z2 and Pin± to illustrate the methods involved. The results of the computation are that f2DPin (pt) = (Z2 )2 andf3DPin (pt) = Z8 , precisely as for the oriented Type II strings.

6.3

Invertible phases for DPin structure

As in the case of oriented Type II strings, the fact that f3DPin (pt) = Z8 means that worldsheet
anomalies conveniently cancel in ten dimensions. Also as for oriented Type II strings, the result
f2DPin (pt) = (Z2 )2 implies four worldsheet theories, though two of these will bephysically inR
2

distinct from the others. The generators of f2DPin (pt) can be taken tobe {(−1) w1 , (−1)Arf(Σ̂) },R
2where Σ̂ is the orientation double cover of Σ. The generator (−1) w1 is a bosonic invertible
phase, which was discussed in detail in Section 2.2.1. The generator (−1)Arf(Σ̂) was discussed
in the context of Pin+ in Section 2.2.4.
The effects of these phases
R 2 on the Type I theory are easy to read off. We know that thepresence of the phase (−1) w1 , which has the effect of assigning −1 to Möbius strip amplitudes
and +1 to cylinder and Klein bottle amplitudes, corresponds to the choice of O9± -planes, i.e.
it distinguishes between Type I and Type Ĩ theories. On the other hand, adding (−1)Arf(Σ̂)
leads to physically indistinct theories. To see this, note that on an oriented worldsheet Σ, thepartition function contribution (−1)Arf(Σ̂, σ) can be interpreted as(−1)Arf(Σ, σ L ) × (−1)Arf(Σ, σR )(6.17)and can thus be absorbed for example by flipping ψ9 and ψ̃9 at the same time, i.e. by aspacetime parity flip in one direction, as explained in Section 3.1.2. So in fact the theory
obtained from the non-trivial invertible phase Arf(Σ̂) is not physically distinct from the one
with the trivial phase, and the two are instead related in the same way that Type IIA/B andType IIA/B0 were related.
The second term in d21 (x) can be dropped since the twisted Bockstein associated to 0 → Z2 → U(1) → U(1) → 0
is Sq1 +w(V ), which implies that ι ◦ (Sq1 +w(V )) = 0. Therefore ι(w(V ) Sq1 x) = ι(Sq1 Sq1 x) = 0. The authors
thank R. Thorngren for this point.
1137

SciPost Phys. 9, 010 (2020)

Acknowledgments
The authors thank the TASI 2019 summer school at University of Colorado, Boulder, where
JK and JPM were students and YT was a lecturer, for providing an ideal environment for collaboration; this paper grew out of a lunchtime conversation there. The authors are especiallygrateful to Edward Witten for many useful correspondences, and for providing much of thecontent of Appendix D. We also thank Arun Debray, Ryohei Kobayashi, and Ryan Thorngren
for their help in computing bordism groups, as well as Oren Bergman, Jake McNamara, Max
Metlitski, Matthew Heydeman, and Juven Wang for useful discussions. JPM thanks the Center
for Mathematical Sciences and Applications at Harvard University for hospitality while thiswork was being completed. YT is in part supported by WPI Initiative, MEXT, Japan at IPMU,
the University of Tokyo, and in part by JSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837
and JSPS KAKENHI Grant-in-Aid (Kiban-S), No.16H06335. JPM is supported by the US Department of State through a Fulbright scholarship. JK and JPM thank the Mani L. Bhaumik
Institute for generous support.

A

NSR formalism

Throughout this paper we work in the NSR formalism, where the worldsheet fields for theclosed string consist of scalars X µ and left- and right-moving Majorana-Weyl fermions ψµ , ψ̃µ ,all of which are vectors in the ten-dimensional target space [8, 86].
Closed strings: We choose our conventions such that the worldsheet spatial coordinateσ ∈ [0, 2π). For simplicity we work in light-cone gauge, with µ = 0, 1 being the light-conecoordinates. In this gauge the worldsheet action is
Z



1
1
µ
µ
µ
µ
d t dσ
∂
X
∂
X
−
∂
X
∂
X
+
iψ
(∂
+
∂
)ψ
+
i
ψ̃
(∂
−
∂
)
ψ̃
(A.1)
t
t µ
σ
σ µ
t
σ
µ
t
σ
µ ,
4π
α0
with µ = 2, · · · , 9. The oscillator expansions for these fields arev
t α0 X 1µµ0µαµn e−in(t−σ) + α̃µn e−in(t+σ) ,X (t, σ) = x + α t p + i
2 n n
X
X
ψ̃µr e−ir(t+σ) ,ψµ (t, σ) =
ψµr e−i r(t−σ) ,ψ̃µ (t, σ) =

(A.2)
(A.3)

r

r

where r ∈ Z or Z + 12 for R or NS boundary conditions, respectively.
We are mainly interested in the fermionic sector, and in particular in its zero-modes. There
are no zero-modes for NS boundary conditions, and hence the left- and right-moving groundstates, |0〉NS and |0̃〉NS, are unique in this sector. It is easy to write fermion
number opPµµerators in terms of the operators that count the number of modes, N = r,µ r ψ−r ψ r andPµµÑ = r,µ r ψ̃−r ψ̃ r , as (−1)fL = (−1)N−1 and (−1)fR = (−1)Ñ−1 . The ground states are odd,
i.e.
(−1)fL |0〉NS = −|0〉NS ,(−1)fR |0̃〉NS = −|0̃〉NS .
(A.4)
On the other hand, fermions with R boundary conditions do allow zero-modes, which
satisfy the anticommutation relationsµ{ψ0 , ψν0 } = δµν ,µ{ψ̃0 , ψ̃ν0 } = δµν ,38µ{ψ0 , ψ̃ν0 } = 0 .

(A.5)

SciPost Phys. 9, 010 (2020)

Left- and right-moving zero-modes then separately furnish representations of the Clifford algebra Cl(8). These representations act on the degenerate ground states, which can be spinors|0〉Ra , |0̃〉Ra ∈ 8s or conjugate spinors |0〉Rȧ , |0̃〉Rȧ ∈ 8c of the little group SO(8). The two irreducible
representations are distinguished by the eigenvalue of the left- and right-moving fermion numbers
Yp
Yp
f
f
µ
µ
(−1)0L =2 ψ0 ,(−1)0R =2 ψ̃0 .
(A.6)µµ

The full fermion number operators are obtained by combining these with the operators counting the number of massive modes,
ff(−1)fL = (−1)0L (−1)N ,(−1)fR = (−1)0R (−1)Ñ .

(A.7)

Worldsheet parity Ω acts on the worldsheet fields as in (3.13), from which it follows thatthe action on the oscillator modes in (A.2) and (A.3) isΩψ r Ω = e2πr ψ̃ r ,Ωαn Ω = α̃n ,Ωψ̃ r Ω = −e2πr ψ r .

(A.8)

Next we review the low-lying spectra of the closed superstring. We will use the usual
notation [8] and denote each of the low-lying states by (A±, B±), where A, B can be R or NS
and denote respectively the left- and right-moving sectors, while the signs indicate the fermion
parity. The NSNS sector ground state is(NS−, NS−) :|0〉NSNS = |0〉NS ⊗ |0̃〉NS ,(A.9)

and is a scalar tachyon. The massless NSNS spectrum is given by the level one states,µψ̃−1/2 ψ̃ν−1/2 |0〉NSNS ∈ 8 v ⊗ 8 v = 1 ⊕ 28 ⊕ 35 v

(NS+, NS+) :(A.10)and consist of a dilaton, 1, 2-form, 28, and graviton, 35 v . The RR sector ground states transform in the product of spinor representations of the little group SO(8),(R+, R+) :ab= |0〉Ra ⊗ |0̃〉Rb ∈ 8s ⊗ 8s = 1 ⊕ 28 ⊕ 35− ,|0〉RR(R−, R−) :

ȧ ḃ
|0〉RR
= |0〉Rȧ ⊗ |0̃〉Rḃ ∈ 8c ⊗ 8c = 1 ⊕ 28 ⊕ 35+ ,(R+, R−) :a ḃ
= |0〉Ra ⊗ |0̃〉Rḃ ∈ 8s ⊗ 8c = 8 v ⊕ 56,|0〉RR(R−, R+) :

ȧ b
|0〉RR
= |0〉Rȧ ⊗ |0̃〉Rb ∈ 8c ⊗ 8s = 8 v ⊕ 56 .

(A.11)All of these states are massless and can be identified as the RR scalar 1, vector 8 v , two-form
28, three-form 56, and (anti)self-dual four-form 35(−)+ . Finally, the NSR states areµ(NS+, R+) :ψ−1/2 |0〉NS ⊗ |0̃〉Ra ∈ 8 v ⊗ 8s = 8c ⊕ 56s ,(NS+, R−) :ψ−1/2 |0〉NS ⊗ |0̃〉Rȧ ∈ 8 v ⊗ 8c = 8s ⊕ 56c ,µ(A.12)

which include the dilatinos, 8s and 8c , and gravitinos, 56s and 56c . The RNS states are conjugate to these.
Open strings: When considering open strings, one can have Dirichlet (D) or Neumann (N)boundary conditions on each end, which relate the left- and right-moving oscillators. We define
the NN, DD, and ND directions of the open string as in (4.13). Though only the relative sign inthose conditions is important, the conventions shown there are such that η1,2 = +1 represents

39

SciPost Phys. 9, 010 (2020)

a Dp-brane at σ = 0, π, while η1,2 = −1 represents a Dp0 -brane at σ = 0, π.12 That this is so
will be discussed in the beginning of Appendix B.
Note that the relative choice of η1,2 is related to the choice of NS or R sectors on the open
string. In particular, the string is in the NS sector for η1 = η2 and in the R sector for η1 = −η2 .
This may be seen as follows. First, we may replace the left- and right-moving fermions on
σ ∈ [0, π] with a single chiral fermion on σ ∈ [0, 2π] by defining
ψ(t, σ) = η2 ψ̃(t, 2π − σ) ,π ≤ σ ≤ 2π .

(A.13)The question of NS vs. R is then a question about the (anti-)periodicity of this extended
fermion. In particular, say that ψ̃(t, 2π) = η3 ψ̃(t, 0) where η3 = +1 for R and −1 for NS.
Then we haveψ(t, 0) = η2 ψ̃(t, 2π) = η2 η3 ψ̃(t, 0) .

(A.14)

But from (4.13), we also have ψ(t, 0) = −η1 ψ̃(t, 0) and hence we conclude that η1 η2 = −η3 .
The anti-periodic NS case then corresponds to η1 = η2 , while the periodic R case corresponds
to η1 = −η2 . From (4.13) we then conclude that NS sector fermions can have zero modes
along ND and DN directions, whereas R sector fermions can have zero modes along NN and
DD directions. In constructing the open string fermion number operator (−1)f one must takethese zero modes into account, as was done above for the RR sector of the closed string.
As for the open string spectrum, let us just mention that the ground state in the NS sector,|0〉NS, is an open string tachyon with (−1)f |0〉NS = −|0〉NS. We will not review the massless
spectrum here.

B

Boundary state formalism

Here we review the boundary state formalism — for more details, the reader may consult[63, 86, 87].

B.1

BasicsFrom the closed string point of view, D-branes correspond to states in the closed string Hilbert
space. Beginning with the boundary conditions (4.13), one can transition to Euclidean signature t → −i t E , and then do a rotation of the worldsheet to interchange the t E and σ directions,
thereby going from the open string to the closed string picture. In particular, rotation by π2
takes (t E , σ) → (σ, −t E ). Under this transformation, the fermions transform as
ψµ (t E + iσ) → e iπ/4 ψµ (t E + iσ) ,ψ̃µ (t E − iσ) → e−iπ/4 ψ̃µ (t E − iσ) .

(B.1)

The boundary conditions at fixed σ then become conditions at fixed time — for example, on
the slice t E = 0 the initial conditions are
N : ψµ (0, σ) = +iη1 ψ̃µ (0, σ) ,D:ψµ (0, σ) = −iη1 ψ̃µ (0, σ) ,σ ∈ [0, 2π) .

(B.2)

The conditions at t E = π look the same, but with η1 replaced by η2 — note that we no longer
have the relative minus sign for the same condition on the two boundaries, c.f. footnote 12.
Note that the extra sign in (4.13) is needed to encode the same physical boundary condition at σ = π as at
σ = 0, as explained e.g. in footnote 69 of [63]. To summarize, because the boundaries at σ = 0, π have opposite
orientation, imposing the same boundary condition on the two boundaries involves a reflection of one of themt → −t, under which ψµ → e iπ/2 ψµ and ψ̃µ → e−iπ/2 ψ̃µ .
1240

SciPost Phys. 9, 010 (2020)

Let us now focus on the slice at t E = 0. We will denote η1 = η for simplicity. We define theboundary states |Bp, η〉 to be the operator statement of the boundary conditions (B.2) on theclosed string Hilbert space. Writing things in terms of Fourier modes and including bosonicconstraints as well, these boundary states are then defined by
µ

µ

µ

µ

N : (αµn − α̃−n )|Bp, η〉 = (ψµr − iηψ̃−r )|Bp, η〉 = 0 ,D:(αµn + α̃−n )|Bp, η〉 = (ψµr + iηψ̃−r )|Bp, η〉 = 0 ,(B.3)

where r is integer/half-integer for the R/NS sector. The general solution to these conditions
can be written as a coherent state,



p+2
∞
9
X
X
X
1
µ
µ
µ
µ
− 1
|Bp, η〉 ∝ exp
α−n α̃−n +α α̃ 
n µ=2n µ=p+3 −n −n
n=1


p+2
∞
9

X
X
X
µ
µ
µ
µ
−
+iη
ψ−r ψ̃−r +ψ−r ψ̃−r  |Bp, η〉(0)
(B.4)

r>0
µ=2
µ=p+3
up to a normalization factor which we discuss in Appendix B.3. Here |Bp, η〉(0) is the groundstate, which depends on the sector. In the NSNS sector the ground state is just the usual one(0)|Bp, η〉NSNS = |0〉NSNS .

(B.5)

In the RR sector there is an extra subtlety because we need to solve the gluing conditions for(0)
the zero-modes. This is easily done by noticing that |B7, η〉RR need only satisfy conditions of
the kind
µ
µ
(0)
(B.6)
(ψ0 − iηψ̃0 )|B7, η〉RR = 0 .
We can then build the rest of the boundary ground states as(0)|Bp, η〉RR =

9
Y
µ=p+3

µ

µ

(0)

(ψ0 + iηψ̃0 )|B7, η〉RR .

(B.7)

(0)

It is not entirely trivial to see the relation between |B7, η〉RR and the usual RR vacuum. This is
explained, for instance, in Appendix B of [87]. The key feature is that the relation involves aneven number of RR zero-mode operators. With this in mind it follows that(−1)fL |Bp, η〉NSNS = −|Bp, −η〉NSNS ,(−1)fL |Bp, η〉RR = (−1)7−p |Bp, −η〉RR ,(−1)fR |Bp, η〉NSNS = −|Bp, −η〉NSNS ,(−1)fR |Bp, η〉RR = |Bp, −η〉RR ,(B.8)

where (−1)fL and (−1)fR are the left- and right-moving worldsheet fermion numbers. Similarly
using (A.8) and (B.7) one can check thatΩ|Bp, η〉NSNS = |Bp, η〉NSNS ,Ω|Bp, η〉RR = −(−iη)7−p |Bp, η〉RR = i −ηk |Bp, η〉RR.

(B.9)

where k = 9 − p.

B.2

Theta functions, partition functions and boundary state amplitudes

In the remainder of this appendix, we will be calculating amplitudes for closed strings propagating between boundary states. In order to do so, some preliminary data will be needed. We
now review our conventions for the different theta functions that appear in one-loop string
41

SciPost Phys. 9, 010 (2020)

partition functions, and give a few useful formulas for partition functions and boundary state
amplitudes.
First, note that we will use the usual shorthand for theta functions with characteristic,
11
ϑ1 (z|τ) = ϑ(z|τ) ,

ϑ2 (z|τ) = ϑ 0(z|τ) ,
ϑ3 (z|τ) = ϑ0ϑ4 (z|τ) = ϑ

2
1
2

2

0
 
0
1
2

(z|τ) ,(z|τ) ,

(B.10)and the notation ϑi (τ) = ϑi (0|τ), where τ is the modular parameter of a torus. Recall that
ϑ1 is odd and vanishes at the origin, i.e. ϑ1 (τ) = 0. For convenience we define the following
combinations,
v
∞
∞
Y
Y
t ϑ (τ) p
2
1/12
2n
1/12
f1 (τ) = η(τ) = q(1 − q ) ,f2 (τ) == 2q(1 + q2n ) ,
η(τ)
n=1
n=1
v
v
u
∞
∞
Y
Y
t ϑ (τ)t ϑ4 (τ)
3
f3 (τ) == q−1/24(1 + q2n−1 ) ,f4 (τ) == q−1/24(1 − q2n−1 ) ,
η(τ)
η(τ)
n=1
n=1
(B.11)
1/3
where q = e iπτ and η(τ) = ϑ10 (0|τ)/2π
is the Dedekind eta function. It will be useful toknow the modular S transformations,pf1 (i/t) = t f1 (i t) , f2 (i/t) = f4 (i t) ,(B.12)
f3 (i/t) = f3 (i t) ,f4 (i/t) = f2 (i t) ,and the T transformations,ππf1 (i t + 1) = e i 12 f1 (i t) ,f2 (i t + 1) = e i 12 f2 (i t) ,π

1

(B.13)πf3 (i t + 1) = e−i 24 f4 (i t) ,f4 (i t + 1) = e−i 24 f3 (i t) ,1

as well as the more unfamiliar P = T 2 S T 2 S T 2 transformation,








p
i
1
1
i
1
1
f1
+
= 2t f1 i t +, f2+= f2 i t +
,
4t 2
2
4t 2
2








π
π
1
1
i
1
i
1
f3
+
= e i 8 f4 i t +,f4+= e−i 8 f3 i t +
,
4t 2
2
4t 2
2

(B.14)

where t ∈ R. The Jacobi “abstruse” and “triple product” identities
f2 (τ)8 − f3 (τ)8 + f4 (τ)8 = 0 ,f2 (τ) f3 (τ) f4 (τ) =p
2(B.15)

will be used to simplify results.
The functions f i (τ) introduced above are useful since the open and closed string sector
traces are written naturally in terms of them. Denote the boundary state amplitudes in thetree-channel in sector S as follows
Z̃SLR = S,L 〈B|e−2πlHcl |B〉S,R ,(B.16)

where for fermionic sectors |B〉 = |B, η〉 and L, R denote the boundary conditions — either
Neumann (N) or Dirichlet (D) — on each boundary. The amplitudes with opposite η on either

42

SciPost Phys. 9, 010 (2020)

side are given by exchanging N ↔ D on one boundary state, as can be seen in (B.3). In termsof the f i (τ), the bosonic contributions are found to be [86]
p
1
2
1
1
NN
ND
Z̃B =
,
Z̃B =
,
Z̃BDD = p,
(B.17)20
f1 (2il)f2 (2il)
4π α l f1 (2il)the fermionic contributions in the NSNS sector are
NN
Z̃NSNS= f3 (2il) ,ND
Z̃NSNS= f4 (2il) ,DD
Z̃NSNS= f3 (2il) ,(B.18)and in the RR sector
NN
Z̃RR= − f2 (2il) ,

ND
Z̃RR
= 0,

DD
Z̃RR
= − f2 (2il) .

(B.19)

Consider parallel Bp and Bq boundary states with q > p. Note that there are p−1 NN, 9−q
DD, and q − p ND directions. Also recall that changing η → −η is equivalent to exchanging
the boundary conditions N ↔ D. The amplitudes for exchanging closed strings between thesestates then take the simple formf3 (2il)8 f4 (2il)2(q−p),
9−qf1 (2il)8−q+p(4π2 α0 l) 2
Vp+1
f4 (2il)8 f3 (2i`)2(q−p)−2πl Hcl|Bq, −η〉NSNS =,NSNS 〈Bp, η|e9−qf1 (2il)8−q+p(4π2 α0 l) 2
Vp+1
f2 (2il)8−2πl Hcl
,
〈Bp,
η|e
|Bq,
η〉
=
−
RR
RR
9−q
8
(4π2 α0 l) 2 f1 (2il)−2πl Hcl|Bq, −η〉RR = 0 ,RR〈Bp, η|eVp+1−2πl Hcl|Bq, η〉NSNS =NSNS 〈Bp, η|e(B.20)

where Vp+1 is the regularized volume of the p-brane, which comes from the zero-modes of thescalars parallel to its worldvolume.
p Note that we have used the Jacobi triple product identity(B.15) to remove some factors of 2/ f2 (2il) from these results. The last amplitude vanishes
because η L = −ηR corresponds to Ramond boundary conditions in the direction orthogonal
to the boundary, for which there is a fermion zero-mode.

B.3

D-brane boundary states

B.3.1

Boundary state normalization

In order to calculate tensions or tadpole contributions, we will want to find the proper normalization for the boundary states. The way to do this is to impose matching of the tree- andloop-channel cylinder amplitudes. That is, we want to impose the following identities
1
2
NBp

1
2
NBp

1
2
NBp

1
2
NBp

Z∞

d l NSNS〈Bp, η|e−2πl Hcl|Bp, η〉NSNS

=

Z∞

0

0

Z∞

Z∞

d l NSNS〈Bp, η|e−2πl Hcl |Bp, −η〉NSNS

=

0

0

Z∞

Z∞

dl

RR〈Bp, η|e−2πl Hcl|Bp, η〉RR

=

0

0

Z∞

Z∞

dl

RR〈Bp, η|e−2πl Hcl=|Bp, −η〉RR

0

0

43



dt
TrNS e−2πt Hop ,
2t


dt
TrR e−2πt Hop ,
2t


dt
TrNS e−2πt Hop (−1)f ,
2t


dt
TrR e−2πt Hop (−1)f = 0 .
2t

SciPost Phys. 9, 010 (2020)

We may for instance focus on the first one, which in loop-channel gives the resultZ∞Z∞ −2πt H 
f3 (i t)8dt
dt
op = v
TrNS e
,
p+1
p+3
8
2t
(2t) 2 f1 (i t)00(B.21)

p+1

where vp+1 = Vp+1 /(4π2 α0 ) 2 is the regularized volume of the brane, in units of the string
length. On the other hand, in tree-channel we haveZ∞Z∞
(4π2 α0 ) p−4
1
d l f3 (2il)8−2πl Hcl
d
l
〈Bp,
η|e
|Bp,
η〉
=
v
. (B.22)
NSNS
NSNS
p+1
9−p
2
2
8
NBp
NBp
l 2 f1 (2il)001
We can translate the tree-channel amplitude to loop-channel using the transformation l = 2t
and the modular S transformations in (B.12), which yields
Z∞
f3 (i t)81(4π2 α0 ) p−4
dtv(B.23)
p+1
p−9
2
2t 2 (2t) 2 (t) 28 f1 (i t)8NBp0Z∞
f3 (i t)825 (4π2 α0 ) p−4
dtv=.
(B.24)
p+1
p+3
2
8
NBp
(2t) 2 f1 (i t)0Comparing (B.21) and (B.24), we find that the proper normalization for the boundary state is
p−4

5

NBp = 2 2 (4π2 α0 ) 2 .

(B.25)

One can check that imposing the other identities gives the same result.
B.3.2

Type II and I

The states |Bp, η〉 must be assembled into a D-brane state such that they give the right open
string amplitudes. Let us begin by finding the D-brane state in Type II. Since the open string
sector includes both NS and R strings we must haveZ∞d l 〈Dp|e−2πl Hcl |Dp〉

(B.26)

0

=

Z∞
0

dt
2t






−2πt Hop 1f−2πt Hop 1
f
TrNS e(1 + (−1) ) − TrR e(1 + (−1) ) .
2
2

A brief calculation then shows that the proper normalization for the D-brane state is|Dp〉 =

1 1(|Bp, +〉NSNS − |Bp, −〉NSNS + |Bp, +〉RR + |Bp, −〉RR) ,NBp 2(B.27)

with NBp as defined in (B.25). The choice of relative sign of the NSNS and RR contributions
differentiates branes and anti-branes.
Recall that for Type II theories we wish to gauge both (−1)fL,R , so we should keep only
states invariant under projection by
II
PNSNS
=



1
1 + (−1)fL 1 + (−1)fR ,
4

II
PRR
=



1
1 + (−1)fL 1 ± (−1)fR ,4(B.28)

with the two choices of sign corresponding to Type IIB (+) and Type IIA (−). Using (B.8) it iseasy to see that the boundary states in Eq. (B.27) are invariant when p is odd for Type IIB andwhen p is even for Type IIA.
44

SciPost Phys. 9, 010 (2020)

In the presence of multiple branes the boundary state acquires an extra overall group theory
factor G that accounts for the trace over the Chan-Paton space,¨Nfor U(N ) ,|Dp〉 → G|Dp〉 where G =.
(B.29)
2N for S p(N ), SO(2N )The latter implies that in Type I the D-brane states for even a single brane are normalized withan extra factor of 2.
B.3.3

Type 0

We now do the same analysis for D-branes in Type 0. In Type 0 the open strings stretching
between two branes of the same (different) type are in the NS (R) sector, so we must have
Z∞
Z∞


1
dt
−2πl Hcl
TrNS e−2πt Hop (1 + (−1)f ) ,(B.30)d l 〈Dp, η|e|Dp, η〉 =
2t
2
0
0
Z∞
Z∞


dt
−2πl Hcl−2πt Hop 1fd l 〈Dp, η|e|Dp, −η〉 = −TrR e(1 + (−1) ) .
(B.31)2t
20
0From this and the relations above it follows that the properly normalized Type 0 D-brane stateis|Dp, η〉 =1 1
p (η|Bp, η〉NSNS + |Bp, η〉RR) ,NBp 2

η = ±1 .

(B.32)

The factor of η in front of |Bp, η〉NSNS is needed so that the force between the branes is attractive. To see this, consider the NSNS contribution to the amplitude,
Z∞Z
vp+1 ∞
d l f4 (2il)8−2πl Hcl|Dp, −η〉NSNS = − 6
d l N SN S 〈Dp, η|e.
(B.33)
d l 9−p82
l 2 f1 (2il)00The contribution from the massless states can be extracted from the constant term in the
expansion
f4 (2il)81= − 8 + O(q1 ) ,8
f1 (2il)q(B.34)

where now q = e−2πl . The minus sign cancels with the overall sign in (B.33), yielding a
positive contribution and hence an attractive force.
Recall that for Type 0 strings we gauge only a diagonal spin structure (−1)fL +fR , and hence
we keep only states invariant under projection by
0
PNSNS
=


1
1 + (−1)fL +fR ,
2

0
PRR
=


1
1 ± (−1)fL +fR ,2(B.35)

with the two choices of sign corresponding to Type 0B (+) and Type 0A (−). Using (B.8), we
see that (B.32) are invariant for p odd in Type 0B and p even in Type 0A.
In contrast to (B.27) then, for each such p there are now two boundary states for the Type0 strings [87], which we will call Dp and Dp0 for |Dp, +〉 and |Dp, −〉 respectively. Note that
Dp0 -branes are not anti Dp-branes.
pFinally, note that the normalizations of Type 0 and Type II branes differ by a factor of 2.
On the other hand, the amplitude for exchanging closed strings in Type II receives an extra
contribution corresponding to R strings in the loop channel. This impliespthat the tensions of
the Type 0 branes are smaller than those of Type II, in particular Tp0 = TpII / 2 [88,89]. Finally,as stated before, when there are multiple branes the boundary state acquires an extra grouptheory factor (B.29).
45

SciPost Phys. 9, 010 (2020)

B.4

O-plane boundary states

B.4.1

Crosscap state normalization

In analogy to the discussion above, we can find the correct normalization of the crosscap statesthat correspond to O-planes by requiring that the tree-channel amplitude for exchanging aclosed string between a D-brane and a crosscap state matches the loop-channel Möbius strip
amplitude. We know that the crosscap states are related to the usual boundary state by a π/2translation in imaginary time, so we normalize them as|Cq, η〉 = −

nCq

NBq

i Hcl |Bq, η〉 ,(B.36)

where nCq is the normalization relative to the usual boundary state, and the minus sign isrequired to get negative tension. Then, the relations we must impose areZ∞1
d l NSNS〈Cq, η|e−2πl Hcl |Bp, η〉NSNS − NSNS〈Bp, η|e−2πlHcl |Cq, −η〉NSNS
NBp 0
Z∞


dt
=
TrNS e−2πt Hop Ω ,
2t
0
Z∞

1
d l NSNS〈Bp, η|e−2πl Hcl |Cq, η〉NSNS − NSNS〈Cq, −η|e−2πlHcl |Bp, η〉NSNS
NBp 0
Z∞


dt
TrNS e−2πt Hop (−1)f Ω ,
=
2t
0
Z∞

1
−2πl Hcl|Cq, η〉RR − RR〈Cq, −η|e−2πlHcl |Bp, η〉RR
dl
RR〈Bp, η|e
NBp 0
Z∞


dt
=
TrR e−2πt Hop Ω ,
2t
0
Z∞

1
−2πl Hcldl|Bp, η〉RR − RR〈Bp, η|e−2πlHcl |Cq, −η〉RR
RR〈Cq, η|e
NBp 0
Z∞


dt
=
TrR e−2πt Hop (−1)f Ω .
2t
0
As before, we can fix the normalization using any of these relations by first writing theloop-channel Möbius strip amplitude


Z∞
Z∞

8 2(q−p) 


f
f
dt
dt
1
−2πt Hopi π4 (q−p)  3 4
TrNS eΩ = −vp+1e
it +,(B.37)
p+3
8−q+p
2t
2
f
(2t) 200
1

and then calculating the corresponding tree-channel amplitude using the boundary statesZ∞1d l NSNS〈Cq, η|e−2πl Hcl |Bp, η〉NSNS − NSNS〈Bp, η|e−2πlHcl |Cq, −η〉NSNS
NBp 0


!
!
Z∞


8 2(q−p) 
8 2(q−p) 
nCq
f
f
f
f
dl  3 41
1 
4 3= − 5 vp+1
2il −
−
2il +
9−q
8−q+p
8−q+p
2
2
2
f1
f1
l 2
0


Z∞

8 2(q−p) nCqd l  f4 f3
 2il + 1 ,= 4 vp+1(B.38)
9−q
8−q+p
2
2
f1
l 2
0
46

SciPost Phys. 9, 010 (2020)

where in the second equality we used the modular T transformations in (B.13). Next, wetranslate the tree-channel amplitude to the loop channel using the transformation l = 1/8t
and the modular P transformations in (B.14) to getZ∞
8 i π4 (q−p) 2(q−p) 
nOq
e
−
f
f
dt
1
1
4
3


vp+1
it +
8−q+p
8−q+p
24
8t 2 (8t) q−922(2t) 2 f1
0


Z∞

8 2(q−p) 
nOq
f
f
1
dt
i π4 (q−p)  3 4
= − q−4 vp+1
it +.
(B.39)
e
p+3
8−q+p
2
2
f1
(2t) 20Comparing with the previous result, we find that the normalization of the |Cq, η〉 crosscap
state relative to the boundary state is
nCq = 2q−4 .
B.4.2(B.40)

Pin− Type 0

Finally, we must assemble the crosscap states into physical orientifold plane states. For thePin− theories, the Pin− structure on the worldsheet requires the boundary of the Möbius strip
to have NS boundary conditions. Thus we expect the Oq-plane state to be purely in the NSNS
sector. In addition, we know that the open strings on the orientifold are in the NS sector, so
the O-plane state must give the following loop channel resultZ∞d l (〈Dp, η|e−2πl Hcl |Oq〉 + 〈Oq|e−2πlHcl |Dp, η〉)

0

=

Z∞
0



dt
1
TrNS e−2πt Hop (1 + (−1)f )Ω .
2t
2

(B.41)

Thus the physical orientifold boundary state is
1|Oq〉 = p (|Cq, +〉NSNS − |Cq, −〉NSNS) .
2(B.42)

Importantly, the crosscap state carries crucial information about the presence of n copiesof ABK. To see this, it is easiest to consider the Klein bottle amplitude. Requiring that thetree-channel amplitudes for exchanging closed strings between two crosscaps match the loopchannel Klein bottle amplitudes gives for example
Z∞
Z∞


dt
−2πl Hcl
d l NSNS〈Cp, −η|e|Cp, η〉NSNS =TrRR e−2πt Hcl Ω ,(B.43)
2t
0
0
Z∞
Z∞


dt
−2πl Hcl
TrRR e−2πt Hcl (−1)f Ω .
(B.44)
d l NSNS〈Cp, η|e|Cp, −η〉NSNS =2t0
0
If we use these to calculate the normalization of the crosscap state as was done for the Möbius
strip, then for n = 0 mod 8 both (B.43) and (B.44) yield the same result (B.40).
For generic n, however, (B.43) and (B.44) are unequal complex conjugates and the resultin (B.40) needs to be modified. This may be seen seen as follows. In the presence of n copies of
ABK, we know that the action of Ω on the closed string RR Hilbert space is modified by a factorof i n when the Pin− structure is q(a, b) = (2, 1) or (2, 3), see (3.18). These Pin− structures areexactly the ones captured by the right-hand sides of (B.43) and (B.44), and hence for non-zeron the left-hand side must change by i n . In other words, we should redefine |Cp, η〉 by a phase
47

SciPost Phys. 9, 010 (2020)e iθ (n,η) such that e−iθ (n,−η) e iθ (n,η) = i n , a solution of which is θ (n, η) = π4 ηn mod 2π. The
correct crosscap states for the theory with n 6= 0 mod 8 can then be taken to be|Cq, η〉 = −2q−4 iπnη Hcl
e 4 i |Bq, η〉 .
NBq

(B.45)

This is what we must insert into (B.42) to obtain the physical orientifold plane state.
B.4.3

Pin+ Type 0

Similarly, for unoriented Pin+ Type 0 we know that the Oq-plane state must give the following
loop channel resultsZ∞
d l (〈Dp, η|e−2πl Hcl |Oq〉 + 〈Oq|e−2πlHcl |Dp, η〉)

0

=

Z∞
0



dt
−2πt Hop 1f
TrR e(1 + (−1) )Ω .
2t
2

(B.46)The physical orientifold state is then found to be
1|Oq〉 = − p (|Cq, +〉RR + |Cq, −〉RR) .
2(B.47)

The fact that this contains only RR sector contributions is the boundary state formulation of
the fact that fermions on the boundary of the Pin+ Möbius strip are automatically in the R
sector.
B.4.4

Type IFor completeness, we finally describe the physical orientifold plane states for Type I. These
are obtained by requiringZ∞(B.48)d l 〈Dp|e−2πl Hcl |Oq〉 + 〈Oq|e−2πlHcl |Dp〉
0

=

Z∞
0

dt
2t






−2πt Hop 1f−2πt Hop 1
f
TrNS e(1 + (−1) )Ω − TrR e(1 + (−1) )Ω .
2
2

The correct combination is found to be|Oq〉 =

1(|Cq, +1〉NSNS − |Cq, −1〉NSNS + |Cq, +1〉RR + |Cq, −1〉RR) .
2

The fact that this contains both NSNS and RR contributions means that DPin structure on theworldsheet must allow the boundary of the Möbius strip to be in the NS or R sectors, and thus
must contain both Pin± as subgroups.
As an aside, let us note that the normalization of the O9 state relative to a Type I D9 state
has an extra factor of 32, as expected by the usual Type I tadpole cancellation.

C

Tadpole Cancellation

In this appendix we discuss the issue of tadpole cancellation in the unoriented Type 0 theories.
We begin by considering the Pin− Type 0 theory with n copies of ABK. Before beginning any
calculations it is important to recall that in this case the orientifold state corresponding to the48

SciPost Phys. 9, 010 (2020)

= 0

+

+ 2×

Figure 4: The tadpole cancellation condition. The Möbius strip is represented by acylinder with one crosscap (the “X” at the end) and we must include separate contributions from crosscaps on the left and right ends. The Klein bottle is represented bya cylinder with two crosscaps.
O9-plane does not have an RR contribution; see (B.42). This means that the orientifold does
not carry RR charge, and hence we will only be encountering NSNS tadpoles. Such tadpoles
are not fatal since they can be cancelled by the Fischler-Susskind mechanism [90,91], but thisintroduces a spacetime dependent coupling. We thus ask in which cases these NSNS tadpoles
can be cancelled without resorting to this mechanism.
The goal is to calculate the cylinder, Möbius strip, and Klein bottle amplitudes and checkthat the tadpole contributions cancel amongst them (Fig. 4). Furthermore, since the putative
tadpoles are in the closed string sector we must focus on the amplitudes in the tree-channel.
For the moment we will focus on the cases with n even, which are orientifolds of Type 0B.
First we will consider the cylinder amplitude. We recall that in Type 0B we have twodifferent kinds of nine-branes, with corresponding boundary states|D9, η〉 =G 1
p (η |B9, η〉NSNS + |B9, η〉RR) ,
NB9 2

(C.1)

where G is a group theory factor which equals G = N for unitary gauge group and G = 2N for
orthogonal or symplectic gauge group; see (B.29). The corresponding antibranes are|D9, η〉 =G 1
p (η |B9, η〉NSNS − |B9, η〉RR) .
NB9 2

(C.2)

In order to avoid introducing RR tadpoles, we must only introduce brane-antibrane pairs, withboundary state|DD9, η〉 = |D9, η〉 + |D9, η〉 =G p
2 η |B9, η〉NSNS .
NB9

(C.3)

In terms of such boundary states the cylinder amplitude is given by

A C2 =

Z∞d l 〈DD9, η|e−2πlHcl |DD9, η〉 ,

(C.4)

0

which can be evaluated using the results collected in Appendix B to give
G2
v10
AC2 =
16

Z∞
dl
0

f38 (2il)
f18 (2il)

.

(C.5)

Worldsheet parity does not affect this amplitude so the result does not depend on n. We caneasily extract the massless NSNS tadpole contribution using the q-expansions in (B.11), giving
G2
AC2 tadpole =
v10
2

49

Z∞
dl .
0

(C.6)

SciPost Phys. 9, 010 (2020)

Next we calculate the Möbius strip amplitude. The O9-plane state was given in (B.42);
crucially, it was argued to be n-dependent. Using the result obtained there, the Möbius strip
amplitude can be evaluated to giveZ∞
A M2 =d l 〈DD9, η|e−2πlHcl |O9〉 + 〈O9|e−2πlHcl |DD9, η〉0= −2 G v10Z∞0 iη π n 8π
e 4 f3 − e−iη 4 n f48
1
2il +
dl
,
2
f18

(C.7)

with the tadpole being
5

A M2 tadpole = −2 G cos

h πηn i
4

Z∞
dl .

v10

(C.8)

0

Finally, the Klein bottle amplitude amplitude isZ∞

d l 〈O9|e−2πlHcl |O9〉

AK2 =

0

= 16 v10

Z∞


dl


π
π 
2 f38 − e−iη 2 n + e iη 2 n f48
f180(2il) ,

(C.9)

with tadpoleZ∞h πηn iAK2 tadpole = 16 1 + cos
v10
dl
2
0
Z∞
h πηn i29v10dl .
= 2 cos
4
0
2(C.10)

Putting all the contributions together we find that the total tadpole ish πηn i21(AC2 + A M2 + AK2 ) tadpole =G − 32 cos
v10
2
4

Z∞
dl .

(C.11)

0We may now read off the tadpole cancellation conditions. For n = 0 one can cancel the
NSNS tadpole by adding sixteen 9-9 pairs. If we choose these to consist of m D9-D9 pairs and0
16 − m D90 -D9 pairs, the resulting gauge group is [SO(2m) × SO(32 − 2m)]2 . The cases withm = 0, 16 are purely bosonic and have gauge group SO(32) × SO(32). For n = 4, we have thesymplectic version of n = 0 and the tadpole cannot be cancelled. For n = 2, 6 we have zero
tadpole contribution, and would seemingly not require addition of any nine-branes.
Next we discuss cases with n odd, which are orientifolds of Type 0A. In Type 0A there
do not exist any stable 9-branes, but there are unstable ones. These unstable branes do notcouple to RR fields, and are purely in the NSNS sector. Hence the corresponding states may
be written asÝ η〉 =|D9,G
η|B9, η〉NSNS ,NB9(C.12)

with NB9 the usual normalization factor accompanying |B9, η〉 andpG the corresponding grouptheory factor. This result differs from (C.3) only by a factor of 2. Then
calp by a similarπηn culation as above we conclude that tadpole cancellation requires G = 32 2 cos 4 . The
case of n = 1 allows the tadpole to be cancelled by the addition of sixteen 9-branes. If wechoose these to consist of m D9-branes and 16 − m D90 -branes, the resulting gauge group is
50

SciPost Phys. 9, 010 (2020)

SO(2m) × SO(32 − 2m). Similar statements hold for n = 7. The n = 5, 3 cases are the corresponding symplectic cases, for which the NSNS tadpole cannot be cancelled by the addition ofbranes.
Finally, we discuss the issue of tadpole cancellation for Pin+ strings. In contrast to the−
Pin theories studied above, for these theories the orientifold only has contributions from the
RR sector; see (B.47). Hence one has an RR tadpole which must be cancelled. A calculation
analogous to the one above shows that the tadpole can be cancelled by adding 32 D9 and 32
D90 -branes, giving total gauge group U(32). Though this introduces NSNS tadpoles [56, 60],
these do not render the theory inconsistent and can be removed via the Fischler-Susskind
mechanism [90, 91].
Note that it makes sense to talk about tadpoles in the Pin+ theories despite RP2 not admitting a Pin+ structure. The reason for this is that the tadpole is given by a one-point function
on RP2 , which corresponds to a punctured RP2 . The latter manifold is conformally equivalent
to the Möbius strip, which does in fact admit a Pin+ structure.
As a final note, whenever tadpole cancellation requires the addition of D9-branes, thequestion of stability of Dp-branes must be revisited to account for the possibility of tachyonicmodes of the strings stretched between the Dp- and D9-branes. In this case, the K-theory
classification outlined in Sections 4 and 5 will be modified, and branes which were previously
stable may become unstable.

D

Arf and ABK from index theory

In this appendix we rephrase many of the results on the Arf and ABK invariants given in Section
2 in terms of index theory. The majority of this appendix is due to E. Witten [42]. The authors
thank him for very kindly allowing them to reproduce the content here. Four-dimensional
analogs of many of these results can be found in Appendix C of [45].

D.1

η-invariants: generalities

Our normalization of the eta invariant isη(Σ, σ) =X

sgn(E) ,(D.1)

E

where E are the eigenvalues of the Dirac operator on Σ with spin structure σ, and the sum is
to be appropriately regularized. We work in the convention that sgn(0) = 1, so that η(Σ, σ)also counts zero-modes. Often we will omit Σ from the argument of η(Σ, σ).
Because the Arf and ABK invariants can be expressed as ratios of massive fermion path
integrals as in (2.3) and (2.20), they are examples of η-invariants. For example, for the Arf
invariant we haveπ
Zferm (m  0) Y i E + m(−1)Arf(Σ, σ) =(D.2)=
= e i 2 η(σ) .
Zferm (m  0)iE − mE
An analogous result holds for the ABK invariant.
The η-invariant is not necessarily a bordism invariant, but in the case of two-dimensional
theories the η-invariant modulo some integer is. This can be seen by appealing to the APS
index theorem, which states that the index of the Dirac operator on a manifold Yd+1 withboundary ∂ Yd+1 = X d is given in terms of the η-invariant as13
Z1
ind i DYd+1 = − η(X d , σ) +Â(R) ch(F ) .
(D.3)
2
Y
d+1

13

In the original notation of APS [92], what we are calling η is instead called 2ξ.

51

SciPost Phys. 9, 010 (2020)

Note that when d is even, the local term on the right-hand side vanishes, and as a result theη-invariant can be a bordism invariant. Combined with the fact that the left-hand side is an
integer, we see that the η-invariant modulo 2 is a bordism invariant. This can be refined
further.
Assume that the fermion system whose Dirac operator is used in the definition of the ηinvariant admits a mass term. This provides an invariant anti-symmetric bilinear form on the
eigenfunctions, and therefore introduce a quaternionic structure. Therefore the index is infact an even number, and η modulo 4 is a bordism invariant.
Let us now consider a spin 2-manifold. Then there exists a globally well-defined chirality
matrix Γ satisfying Γ 2 = 1 and {i D, Γ } = 0, and hence for any state of non-zero eigenvalue
E there is also a state with eigenvalue −E. Then the contributions to the η-invariant fromnonzero eigenvalues simply cancel out. Denoting the number of positive chirality zero-modes
with spin structure σ by ζ(σ), we haveη(σ) = 2ζ(σ) mod 4 ,(D.4)

where the factor of 2 arises because η(σ) counts both chiralities. This means thatπ(−1)Arf(Σ, σ) = e i 2 η(σ) = (−1)ζ(σ)(D.5)

generates at most a Z2 , as expected by our previous definitions of the Arf invariant.
We next consider the Pin− case. As argued above, the η-invariant is a mod 4 bordism
invariant. Let us now show that η takes half-integer values, and thus provides us with a mod
8 invariant. The half-integrality is proven as follows. Given a Pin− structure σ ∈ H 2 (Σ, Z2 ),there exists a “complementary” Pin− structure σ0 := σ + w1 which is obtained by twisting by
the orientation bundle. Then note that14
η(σ) + η(σ0 ) = 0 mod 4 .

(D.6)

4η(σ + a) − 4η(σ) = 0 mod 4(D.7)

We also have generally thatfor any a ∈ H 1 (Σ, Z2 ).15 In the case that a = w1 we have σ + a = σ0 , and thus combining(D.6) and (D.7) we conclude that η(σ) is generically half-integral. As a result, we have thatπe iπABK(Σ, σ) = e i 2 η(Σ, σ) generates at most Z8 , as expected.

D.2

η-invariants: examples

We now offer some explicit calculations of the Arf and ABK invariants in terms of their definitions in this appendix.
T 2 : A trivial example is that of the Arf invariant on T 2 with spin structure σ. In that casewe know that for the NSNS, RNS, and NSR spin structures we have ζ(σ) = 0, whereas for RR
we have ζ(σ) = 1. This together with (D.5) then reproduces the results of (2.10).
14
This equality is true because the left-hand side is the η-invariant of the spin structure on the oriented double
cover Σ̂. Note that Σ̂ is the boundary of the total space X of the unit disk bundle of the orientation line bundleof Σ. That Σ is Pin− is equivalent to X being spin. These facts together imply that Σ̂ is null-bordant, and so theright-hand side is 0 modulo 4.
15
To see this, note that 4η(σ) is the η-invariant of the Dirac operator with Pin− structure σ acting on a rank 4trivial real vector bundle V , whereas 4η(σ+ a) is the η-invariant of the Dirac operator with Pin− structure σ acting
on a rank 4 real vector bundle V 0 = A⊕4 , where A has the property that w1 (A) = a. Because the Stiefel-Whitneyclasses of V 0 all vanish, V 0 is trivial and has the same mod 4 η-invariant as V , thereby giving (D.7).

52

SciPost Phys. 9, 010 (2020)

RP2 : A less trivial result is to reproduce the values of ABK on RP2 . We compute it in two
ways. The first is to consider an orbifold of the three-torus T 3 /Z2 where Z2 acts as x i → −x i
for i = 1, 2, 3. The resulting space has eight fixed points at x i ∈ 21 Z. We may remove a smallball around each of these points to obtain a smooth manifold, with the boundary of this manifold being eight copies of RP2 . Then by the APS index theorem (D.3) for d = 2 we conclude
that η(RP2 ) = − 41 indi D, with i D the Dirac operator on the T 3 /Z2 with points removed. Using conformal invariance, it is possible to argue that the index of the Dirac operator on thismanifold is the same as that on the original T 3 /Z2 , so we need only compute this quantity.
Let us define H± to be the spaces of spinors on T 3 which satisfy ψ(−x) = ±ψ(−x). The
Dirac operator maps H± → H∓ , and the index of the Dirac operator on T 3 /Z2 is then justdefined to be the number of zero-modes in H+ minus those in H− . These numbers are easily
obtained: depending on the Pin− structure, the zero-modes are the 2-dimensional space ofconstant spinors in either H+ or H− , with no zero modes in the remaining space. Hence wehave indiD = ±2, and consequently η(RP2 ) = ± 21 . We may finally calculate the ABK invariantπ
2
to be e iπABK(RP ) = e±i 4 , matching the previous results in (2.24).
The second derivation of this result is a direct computation from the spectrum of the Dirac
operator. Instead of directly studying the Dirac equation on unoriented manifolds Σ we willconsider their orientable double covers Σ̂. These are equipped with an orientation-reversing
involution τ such that Σ = Σ̂/τ. We will make use of the following morphism
Pin− structures on Σ = Σ̂/τ−→τ-invariant spin structures on Σ̂ ,(D.8)

induced by the projection. This map is not injective, but rather two-to-one since given a Pin−structure σ, both σ and its twist by the orientation bundle σ0 lift to the same spin structureon the orientable double cover. It is not surjective either, since there are spin structures on Σ̂
which are not the lift of any Pin− structure. The τ-invariance of the spin structures on Σ̂ implies
that [τ, i D] = 0. Hence there is a basis of eigenspinors with a well-defined eigenvalue of τ.
The different eigenvalues of τ correspond to different Pin− structures σ and σ0 . In summary,we can extract the spectrum of the Dirac operator i D on Σ from that on the orientable double
cover Σ̂ by considering eigenspinors of i D on the latter with a fixed eigenvalue of τ.
Let us apply this strategy to RP2 . Its orientable double cover is a two-sphere S 2 , which hasa single spin structure. The spectrum of the Dirac operator on the two-sphere is well known
and is given by16
E = ±(n + 1) with multiplicity2(n + 1) and τ = ∓(−1)n ,(D.9)

with n ≥ 0. For convenience we will regularize the sum over eigenvalues (D.1) as follows
Xη = lim+sgn(E) e−ε|E| .
(D.10)ε→0E

From this information we can readily calculate the η-invariant of RP2 for either Pin− structure,


X
X
1
τ = ±1 : η = lim+ ∓2(n + 1) e−ε(n+1) ±2(n + 1) e−ε(n+1) = ∓(D.11)
ε→02n∈2Nn∈2N+1
reproducing our previous result.
16

The eigenspace decomposition is simply the spinor spherical harmonics. One way to quickly derive the eigenvalues is to use the operator-state correspondence of a free massless Dirac fermion in dimension d + 1. There, the(absolute value of the) eigenvalues of the Dirac operator on S d are the dilatation eigenvalues of the single-particle
operators of the form ∂ · · · ∂ ψ, which are therefore given by n + d/2.

53

SciPost Phys. 9, 010 (2020)

The Klein bottle: We now obtain the values for the η-invariant on the Klein bottle K2 . A trivial way to do so is to note that the η-invariant factorizes under connected sums,
η(Σ1 #Σ2 ) = η(Σ1 ) + η(Σ2 ). Then recalling that K2 = RP2 #RP2 , our previous results imply that η(K2 ) = 0, 0, ±1 depending on the choice of Pin− structure. This reproduces theresults of (2.25) for the ABK invariant.
A more fulfilling derivation of this result is to again consider the explicit Dirac spectrum.
The orientable double cover in this case is the torus T 2 , which we take to be rectangular withside lengths 1 and 2. That is, T 2 = R2 /Γ for the lattice Γ = Z⊕2Z. Taking x i = (x, y) to be thecoordinates on the torus, we have (x, y) = (x +1, y) = (x +1, y +2). The orientation-reversing
involution is τ(x, y) = (−x, y + 1). As was discussed in Section 2.2.4, of the four torus spin
structures only those which are periodic in the y-direction descend in the quotient.
We first consider the spin structure periodic in x. We begin by finding the eigenspinors of
the square of the Dirac operator, which is just the Laplacian, (i D)2 = −∆. These can be easily
constructed asu p (x i ) = f p (x i ) Ψ,if p (x i ) = e2πi x pi

with1
pi ∈ Γ ∗ = Z ⊕ Z ,2(D.12)

where f p (x i ) are the eigenfunctions of the Laplacian, with momenta taking values in the dual
lattice Γ ∗ , and Ψ a covariantly constant spinor. We can also construct eigenfunctions for the
spin structure antiperiodic in x by letting the momenta take values in Γ̃ ∗ = (Z + 12 ) ⊕ 12 Z. In
both cases it is easy to check that(i D)2 u(x i ) = 4π2 p2 u(x i ) .

(D.13)In terms of the u(x i ) we can construct the eigenspinors of the Dirac operator asv ± (x i ) = ±2π|p| u(x i ) + i D u(x i ) ,with

i D v ± (x i ) = ±2π|p| v ± (x i ) .

(D.14)

This spectrum is clearly symmetric, and hence if there are no zero-modes the η-invariant vanishes. The only case in which there are zero-modes is the case of periodic spin structure inboth directions, and then the multiplicity of the zero-mode is two so that η(T 2 , σRR) = 2, as
we know.
To get the corresponding results for the Klein bottle, we now keep the portion of the spectrum with fixed τ eigenvalue. To do so, it is useful to choose an explicit representation for
the gamma-matrices, say as γ1 = σ3 and γ2 = σ1 . Then in addition to acting on (x, y) in themanner shown above, the involution τ acts as σ1 on spinors. With this, it is easy to show thatthe eigenspinors with fixed eigenvalue under τ have p1 = 0, and hence require periodic spin
structure in the x-direction. Defining n := 2p2 , the remaining spectrum is
E = ±π|n|with multiplicity

1andτ = ∓(−1)n ,(D.15)

with the zero eigenvalue having multiplicity 2.
Two of the Pin− structures of K2 lift to antiperiodic spin structure in the x-direction, andconsequently have vanishing η-invariant. The remaining two Pin− structures lift to periodic
spin structure in the x-direction, and correspond to the two different eigenvalues for τ. The
resulting η-invariants areXXτ = ±1 :η(K2 ) = lim+ 2 ∓e−επ|n| ±e−επ|n| = ∓1 ,(D.16)
ε→0n∈2N

reproducing earlier results.

54n∈2N+1

SciPost Phys. 9, 010 (2020)

D.3

Quadratic forms and enhancements

Let us now make contact between index theory and the combinatoric definitions of Arf and
ABK given in (2.7) and (2.21). In order to do so, we first rewrite the quadratic form q̃(a) andenhancement q(a) in terms of indices.
We start with the oriented case. We considerq̃(a) := ζ(σ + a) − ζ(σ) mod 2(D.17)

for a given spin structure σ, where ζ(σ) is the number of zero modes of the positive-chirality
Dirac operator. We now verify that this is the quadratic refinement of the intersection form,
i.e. the relation (2.6) is satisfied. To do so, we must check thatZζ(σ + a + b) + ζ(σ + a) + ζ(σ + b) + ζ(σ) =a ∪ b mod 2(D.18)

holds. We note that the left-hand side is ζ(V ), the mod 2 index with spin structure σ for
the Dirac operator acting on a positive chirality spinor valued in a rank 4 real vector bundle
V = ε + A + B + AB, where ε is a trivial real line bundle and we have w1 (A) = a, w1 (B) = b,
and w1 (AB) = a + b. From this definition, it also follows that w1 (V ) = 0 and w2 (V ) = a ∪ b.
Therefore, V is topologically equivalent to H ⊕ L, the direct sum of a rank 2 real trivial bundle H
and a complex line bundle L with c1 (L) = w2 (V ) mod 2. This is because real vector bundles
on a Riemann surface are classified topologically by their rank and Stiefel-Whitney classes.
Clearly ζ(H) = 0 modulo 0, so we have ζ(V ) = ζ(L). Under the U(1) rotating L, the zeromodes of L have charge ±1, with respective numbers n± . We then have ζ(L) = n+ + n−
mod 2. By complex conjugation, we can replace a charge −1 mode of positive chirality with acharge +1 mode of negative chirality. Let m± denote the number of positive/negative chirality
modes of charge +1. Then we have n± = m± , and hence ζ(L) = m+ − m− mod 2. The righthandside Ris now the index
of the Dirac operator acting on L, which byRR
R the index theorem is
c1 (L) = w2 (V ) = a ∪ b mod 2. We then conclude that ζ(V ) = a ∪ b mod 2, thereby
confirming (D.18).
With the definition (D.17), it is now simply to check that our combinatorial definition (2.7)
is consistent with the definition (D.5). We haveX1(−1)Arf(Σ,σ) = p(−1)ζ(σ+a)−ζ(σ)|H 1 (Σ, Z2 )| a∈H 1 (Σ,Z )2!
X
1(D.19)= (−1)ζ(σ) p(−1)ζ(a) .
|H 1 (Σ, Z2 )| a∈H 1 (Σ,Z )2

The term in parenthesis can be shown to square to 1 using steps analogous to those used forthe combinatorial definition, and one can then fix the result to +1 by checking an explicit
example.
We now move on to the Pin− case. In that case we define the quadratic enhancement q(a)asq(a) = ζ(σ + a) − ζ(σ) mod 4 ,(D.20)

where σ is now a Pin− structure. We must check that (2.22) is satisfied by this definition. To
do so, let us first prove this in the special case of Σ = RP2 . There is then only one non-trivial
possibility for a and b, namely w1 . The identity (2.22) is trivially satisfied unless a = b = w1 ,so we focus on that case. Then noting that q(0) = 0, the identity we wish to prove isZ
q(w1 ) =w21 = 1 mod 2 .
55(D.21)

SciPost Phys. 9, 010 (2020)

RP2

RP2

Σ

Σ

Figure 5: A bordism between RP2 × Σ and itself, with a real vector bundle V 0 on it.
The blue line represents the Poincaré dual of w2 (V 0 ).
We now use the fact that q(w1 ) = η(σ0 ) − η(σ) mod 2. As we showed in the previous subsection, for RP2 one of the two η-invariants is + 12 , while the other is − 12 . Either way, we conclude
that q(w1 ) = 1 mod 2, thereby confirming the identity.
To prove the identity in generality, we now make use of bordism invariance. What wewould like to prove isZ
ζ(σ + a + b) − ζ(σ + a) − ζ(σ + b) + ζ(σ) = 2

a ∪ b mod 4 .

(D.22)

Equivalently, this is0

η(V ) − η(V ) = 2Z

w2 (V 0 ) mod 4 ,(D.23)

where V is a rank 8 trivial bundle and V 0 is a rank 8 bundle with w1 (V 0 ) = 0 and w2 (V 0 ) = a∪b.
Recall that in two dimensions w2 (V 0 ) is Poincaré dual to a point, while in three dimensions it
is dual to a curve. Then consider a bordism from RP2 × Σ to itself by means of a connected
sum, as shown in Fig. 5. In the figure, we have drawn a curve that starts at RP2 on the top left
and goes down to Σ on the bottom right. Consider a real vector bundle V 0 such that w2 (V 0 )is Poincaré dual to this curve. This gives a bordism between RP2 × Σ with w2 (V 0 ) = 1 on RP2
and 0 on Σ, and RP2 × Σ with w2 (V 0 ) = 0 on RP2 and 1 on Σ. Because (D.23) is unchangedby this change in V 0 , it must hold for any Σ, thus proving the claim.

E

d
fDPin
(pt) via the Atiyah-Hirzebruch spectral sequence
d=2,3

In this appendix we analyze the Atiyah-Hirzebruch spectral sequence (AHSS) for fX(pt)±
for X = Spin × Z2 , Pin , and DPin. Except for the last case X = DPin the outcome is wellknown; we include the computations here just to illustrate the method.
q
To write down the E2 page, we will need the groups H p (X , fspin (pt)). More concretely, weneed H ∗ (BZ2 × BZ2 , Z2 ) and H ∗ (BZ2 × BZ2 , U(1)), where the underline signifies that the firstZ2 acts on U(1) by complex conjugation and the second acts trivially. The first is standard: wehaveH ∗ (BZ2 × BZ2 , Z2 ) = Z2 [w, a] ,(E.1)
where w and a are the generators of H 1 (BZ2 × BZ2 , Z2 ) = H 1 (BZ2 , Z2 ) ⊕ H 1 (BZ2 , Z2 ). As forH ∗ (BZ2 × BZ2 , U(1)), they are determined as an abstract group in e.g. Appendix J.6 of [93];in particular all elements are annihilated by 2. For our purposes we will need more detailed
data. We note that the short exact sequenceι2·0 −→ Z2 −→ U(1) −→ U(1) → 0
56

(E.2)

SciPost Phys. 9, 010 (2020)

leads to the long exact sequenceβ

2·· · · −→ H d−1 (BZ2 × BZ2 , U(1)) −→ H d (BZ2 × BZ2 , Z2 )
ι

β

2·

−→ H d (BZ2 × BZ2 , U(1)) −→ H d (BZ2 × BZ2 , U(1)) −→ · · · (E.3)
Since
2·annihilates
everything,
we
see
that
d
d
H (BZ2 × BZ2 , U(1)) is a quotient of H (BZ2 × BZ2 , Z2 ) by the image of the twisted Bockstein
β = Sq1 +w. Therefore we findH 0 (BZ2 × BZ2 , U(1)) = U(1) ,(E.4)〈w, a〉,〈w〉〈w2 , wa, a2 〉H 2 (BZ2 × BZ2 , U(1)) = Z22 =,〈a(a + w)〉〈w3 , w2 a, wa2 , a3 〉,H 3 (BZ2 × BZ2 , U(1)) = Z22 =〈w3 , wa2 〉〈w4 , w3 a, w2 a2 , wa3 , a4 〉H 4 (BZ2 × BZ2 , U(1)) = Z32 = 2.
〈w a(a + w), a3 (a + w)〉
H 1 (BZ2 × BZ2 , U(1)) = Z2 =

(E.5)
(E.6)
(E.7)
(E.8)

This data can be checked e.g. by noticing that in this low degree range H d (BZ2 ×BG, U(1))
dwith T : Z2 ×G → Z2 given by T = w equals funoriented(BG). The generators of Ωunoriented(BZ2 )d1
can be taken to be e.g. S with nontrivial Z2 bundle for d = 1, RP2 with and without nontrivial
Z2 bundle for d = 2, (RP2 with and without nontrivial Z2 bundle) × (S 1 with nontrivial Z2
bundle) for d = 3, and RP4 with and without nontrivial Z2 bundle, and RP2 × RP2 with
nontrivial Z2 on the first factor for d = 4. We can then evaluate all elements of Z2 [w, a] on thegenerators with the identification that w is w1 of the manifold and a is w1 of the Z2 bundle.
With this information, we can now proceed to the calculation of the relevant groups. Beforeddcomputing fDPin(pt), we illustrate the technique in the known examples of fSpin(BZ2 ) and
1
1
d
fPin
± (pt). Below, the image of ι : Z2 ,→ U(1) is denoted by prefixing by 2 , since {0, 2 } ⊂ U(1).

fSpin (BZ2 )The E2 page needed for obtaining fSpin (BZ2 ) is
q
3
2

Z2

Z2

Z2

Z2

Z2

1

Z2

Z2

Z2

Z2

Z2

0

U(1)

1
2 Z2

0

1

(E.9)

1
2 Z2

2

57

3

4

p

SciPost Phys. 9, 010 (2020)

This can be found from the data given above by forgetting the pieces involving w. The differp,q
ential d2 starting from E2 with p + q ≤ 4 turns out to be zero. The E3 page is then
q
3
2

Z2

Z2

?

?

?

1

Z2

Z2

Z2

?

?

0

U(1)

1
2 Z2

0

1

(E.10)

1
2 Z2

2

0,2

3,0

3

4

p

The only possibly nontrivial d3 is d3 : E3 → E3 but a special property of untwisted bordism
says that every dn starting from E 0,q is zero. (This fact is explained below Theorem 9.10of [82].) Then this is also the E4 page, and E p,q with p + q ≤ 3 cannot change any further.
dFrom this we read off that fSpin(pt) for d = 1, 2, 3 contains 4, 4, and 8 elements, respectively. This agrees with known results.
fPin− (pt)
The E2 page in this case is
q
3
2

Z2

w

w2

w3

w4

1

Z2

w

w2

w3

w4

0

U(1)
0

1 2
2w

1

2

(E.11)

1 4
2w

3

4

p

This can be found from the data given above by forgetting the part involving a. For d2 startingfrom q = 2, one has d22 = Sq2 +w1 (V ) Sq1 +w2 (V ) = Sq2 +w Sq1 .
Then sinceSq2 (w2 ) = (Sq1 w)(Sq1 w) = w4 and Sq1 (w) = w2 , we findd22 (w) = w3 ,d22 (1) = d22 (w3 ) = 0 ,d22 (w2 ) = w4 .

(E.12)

1 4
w .
2(E.13)On the other hand we have d21 = 12 Sq2 , and henced21 (1) = d21 (w) = d21 (w3 ) = 0 ,d21 (w2 ) =Then the E3 page is
q
3
2

Z2

1

Z2

0

U(1)
0

?

?

(E.14)

?

w
1 2
2w

1

2

3 4

p

d
This predicts |fPin− (pt)| = 2, 8, 0 for d = 1, 2, 3, in agreement with known results.

58

SciPost Phys. 9, 010 (2020)

fPin+ (pt)
The E2 page in this case is
q
3
2

Z2

w

w2

w3

w4

1

Z2

w

w2

w3

w4

0

U(1)

1 2
2w

0

1

(E.15)

1 4
2w

2

3

4

p

This is obtained from the previous data by setting w
d22 = Sq2 +w1 Sq1 +w2 = Sq2 +w Sq1 +w2 , and sod22 (1) = w2 ,d22 (w) = d22 (w2 ) = 0 ,=We then havea.

d22 (w3 ) = w5 .

(E.16)On the other hand d21 = 12 Sq2 + 12 w2 and hence
1 2
w ,
2
Then the E3 page is

1 3
w ,
2

d21 (w) =d21 (1) =d21 (w2 ) = 0 ,d21 (w3 ) =1 5
w . (E.17)
2

q
3
2
1
0

w w2

?

?

w

3

?

w

1 4
2w

U(1)
0

(E.18)

1

2

3

4

p

d
This predicts |fPin+ (pt)| = 0, 2, 2 for d = 1, 2, 3, in agreement with known results.

fDPin (pt)
We finally arrive at the case of interest. The E2 page is
q
3
2

Z2

w, a

w2 , wa, a2

w3 , w2 a, wa2 , a3

w4 , . . .

1

Z2

w, a

w2 , wa, a2

w3 , w2 a, wa2 , a3

w4 , . . .

0

U(1)

1
2a

1 2 1
1 2
2 w , 2 wa = 2 a

1 2
1 3
2 w a, 2 a

1 4 1 31 2 2 1
1 4
3
2 w , 2 w a = 2 w a , 2 wa = 2 a

0

1

2

3

4

q
(E.19)We have d22 = Sq2 +w Sq1 +wa and d21 = 21 Sq2 + 21 wa. Then the E3 page is
q
3
2

a

?

?

?

1

a

a2

?

?

U(1)

1
2a

1 2
2w

1 3
2a

1 4
1 3
1 2 2 1
1 4
3
2 w = 2 w a = 2 w a , 2 wa = 2 a

0

1

2

3

4

0

59

(E.20)qSciPost Phys. 9, 010 (2020)

At this point we see that there can be at most four elements in f2DPin (pt) and eight elements
R2

in f3DPin (pt). We already know a subgroup Z2 × Z2 of f2DPin (pt), generated by (−1) w1 and(−1)Arf(Σ̂) , and thus we conclude that f2DPin (pt) = (Z2 )2 . We also know that the anomaly of
Majorana fermion on unoriented surfaces form Z8 , so we conclude that f3DPin (pt) = Z8 .
dfDPin(pt) via the Adams spectral sequenceFdIn this appendix17 , we compute fDPin(pt) for d ≤ 6 a different way, using the Adams spectral sequence. Though computations with the Adams spectral sequence are often difficult,the problem simplifies greatly when computing twisted spin bordism groups ΩXd , thanks toa technique that first appears in Davis’ thesis [94] and builds on work of Stong [95] andAnderson-Brown-Peterson [96].
We highly recommend Beaudry and Campbell’s paper [97] for a detailed introduction tothis method of computation and the ingredients that go into it, as well as several worked
examples. We assume familiarity with the definitions and notation they give.

∼∼
Theorem F.1. The low-degree dpin bordism groups are: ΩDPin= Z/2, ΩDPin= Z/2,
0
1
DPin ∼
DPin ∼
DPin ∼
DPin ∼
DPin ∼Ω2 = Z/2 ⊕ Z/2, Ω3 = Z/8, Ω4 = Z/2 ⊕ Z/2, Ω5 = 0, and Ω6 = Z/2 ⊕ Z/2.
For any finite abelian group A, there is a (noncanonical) isomorphism A ∼= Hom(A, U(1)),
dso this also computes fDPin (pt) for 0 ≤ d ≤ 6, and agrees with the calculations made in
Appendix E. Recall from Sec. 6.2 that a dpin structure is equivalent to a choice of two real line
bundles L1 , L2 → M and a spin structure on
T M ⊕ (L1 ⊗ L2 ) ⊕ (L2 )⊕3 .

(F.2)

One consequence is that if MTDPin denotes the Thom spectrum for dpin structures, so that
πk (MTDPin) ∼, then= ΩDPink
MTDPin ' MTSpin ∧ (BZ/2 × BZ/2) L1 L2 +3L2 −4 .

(F.3)

The second summand, (BZ/2 × BZ/2) L1 L2 +3L2 −4 , which we denote X to tame the notation, is
the Thom spectrum of the virtual vector bundle
V := (L1 ⊗ L2 ) ⊕ (L2 )⊕3 − R4 −→ BZ/2 × BZ/2.

(F.4)

∼
e
By (F.3), ΩDPin
=Ω
k
k

Spin

(X ).
Spin
e
We will compute Ωk (X ) for 0 ≤ k ≤ 6 for our X using the Adams spectral sequence,
employing a standard trick to work over A(1) := 〈Sq1 , Sq2 〉 rather than the entire Steenrod algebra. For details on how this works and many worked examples, see Beaudry-Campbell [97],who carefully explain and summarize how to use the Adams spectral sequence for these kindse ∗ (X ; F2 ) as an A(1)-module. Then, theof computations. The idea is that we must determine H
E2 -page of this Adams spectral sequence is
s,t

s,t

e ∗ (X ; F2 ), F2 ).
E2 = ExtA(1) (H(F.5)

f t−s (X ) ⊗ Ẑ2 ,(Definitions and notation are as in [97].) The spectral sequence converges to ko
where ko denotes connective real K-theory and Ẑ2 denotes the 2-adic integers. Furthermore,
f t−s (X ) is isomorphic to Ωe Spinwhen t − s ≤ 7, ko
t−s (X ) [96]. We will show, for our particular
17

This appendix was contributed by Arun Debray.

60

SciPost Phys. 9, 010 (2020)

e Spinchoice of X , Ω∗ (X ) lacks torsion for odd primes. Therefore tensoring it with Ẑ2 does not
lose any information. (In general, information can be lost when tensoring with Ẑ2 , but that
information can be computed by other means.) This allows us to use the spectral sequencee Spinabove to compute Ωt−s (X ) in the degrees of our interest.
e SpinProof of Theorem F.1. First we argue Ω∗ (X ) has no p-torsion for odd primes p. In fact, wee Spinwill show that if p is an odd prime, Ω∗ (X )⊗F p = 0. For any finitely generated abelian group
A, the p-torsion subgroup of A includes into the p-torsion subgroup of A ⊗ F p , so this suffices.
e k (MTSpin ∧ X ). Tensoring with F p , the map
e Spin (X ) ∼By definition, Ω=Hke k (MTSpin ∧ X ) ⊗ F p −→ H
e k (MTSpin ∧ X ; F p )
H

(F.6)

is injective, by the universal coefficient theorem.
The Künneth theorem computese∗ (MTSpin ∧ X ; F p ) as a sum of tensor products of the form H
e i (MTSpin; F p ) ⊗ H
e j (X ; F p ), so itHe j (X ; F p ) vanishes for all j. The twisted-coefficients Thom isomorphism tells
suffices to show He p on F p such thatus there is a (in this case nontrivial) Z[Z/2 × Z/2]-module structure F
e j (X ; F p ) ∼e p ).
H= H j (Z/2 × Z/2; F(F.7)

e p is pMaschke’s theorem implies that since #(Z/2 × Z/2) and p are coprime, and since Fthe p ) vanishes in degrees j > 0. Using that 0 group homology is thetorsion, H j (Z/2 × Z/2; Fe p ) = 0 as well. Thus
abelian group of coinvariants, one can check directly that H0 (Z/2×Z/2; FSpine ∗ (X ) has no p-torsion.
Ωe ∗ (X ; F2 ). As a graded abelian
On to the Adams spectral sequence. First we determine H
e 0 (X ; F2 ) denotes the Thom
group, this is characterized by the Thom isomorphism: if U ∈ Hclass, cup product with U is an isomorphism∼=e k (X ; F2 ).
(U·): H k (BZ/2 × BZ/2; F2 ) −→ H

(F.8)

There is no degree shift because the virtual vector bundle V → BZ/2 × BZ/2 (from (F.4)) has
rank zero. Let w := w1 (L1 ) and a := w1 (L2 ) in H 1 (BZ/2 × BZ/2; F2 ); thenH ∗ (BZ/2 × BZ/2; F2 ) ∼= F2 [w, a].

(F.9)

e ∗ (X ; F2 ) is determined by the following rules.
The A(1)-module structure on H
1. Sqi (U) = U w i (V ), where w i denotes the i th Stiefel-Whitney class. In this case, w1 (V ) = w
and w2 (V ) = wa.
2. The Cartan formula determines the Steenrod squares of a product. We only need Sq1and Sq2 , for which the Cartan formula specializes to
Sq1 (x y) = Sq1 (x) y + x Sq1 ( y)(F.10a)Sq2 (x y) = Sq2 (x) y + Sq1 (x) Sq1 ( y) + x Sq2 ( y).

(F.10b)

3. From the axiomatic properties of Steenrod squares, Sq1 (w) = w2 , Sq1 (a) = a2 , andSq2 (w) = Sq2 (a) = 0.
Using these three rules one can determine the action of Sq1 and Sq2 on any cohomology classof X , as it is a sum of products of U, w, and a. This is routine, and indeed we used a computer
program to make these calculations. The answer is displayed in Figure 6.
e ∗ (X ; F2 ) splits into several summands.
From this figure, we see that, as an A(1)-module, H
All summands pictured except the orange summand are isomorphic to shifts of A(1). The
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SciPost Phys. 9, 010 (2020)

12
U(w10 a + w8 a3 + w2 a9 + a11 )

11
10
9
8
7

Uβ
U w3 a 3

6

U w6

5
U w4

4
3

Uα
U w2

2
1
0

U a4

Ua
U

e ∗ (X ; F2 ) contains all elements of degree at mostFigure 6: This A(1)-submodule of H
7. Each dot represents an F2 summand, with its cohomological degree given byits height. The connecting lines, resp. curves, indicate an action by Sq1 , resp. Sq2 ,carrying the lower dot to the upper dot. This A(1)-module factors as several different
summands; we give each summand a different color. In the generators of the orange
summand, α := w2 a + a3 and β := w6 a + w4 a3 + w2 a5 + a7 .
orange summand, i.e. the one that contains U a, continues above what we draw in Figure 6
and is isomorphic to the mod 2 cohomology of the spectrum MO(1), the Thom spectrum of thetautological line bundle σ → BO(1) (see [97, Figure 4]); therefore we denote that summande ∗ (MO(1)).18 Specifically,by He ∗ (X ; F2 ) ∼e ∗ (MO(1)) ⊕ Σ2 A(1) ⊕ Σ4 A(1) ⊕ Σ4 A(1) ⊕ Σ6 A(1) ⊕ Σ6 A(1) ⊕ P, (F.11)
H= A(1) ⊕ H
where P has no elements of degree less than 8. Hence, below degree t − s = 8, the E2 -pageof the Adams spectral sequence (F.5) is the direct sum of the E2 -pages of the summands otherthan P, and these have all been calculated. For Σk A(1), there is a single F2 summand ine ∗ (MO(1)), see [98, Example 6.3]. Putting these together, thebidegree s = 0, t = k; for H
E2 -page for this spectral sequence is3
2
1(F.12)

0
0

1

2

3

4

5

6

In this diagram, the x-axis is t − s and the y-axis is s. Therefore a differential d r moves one
degree to the left and r degrees upwards. Each dot represents an F2 summand of the E2 e ∗ (X ; F2 ) are responsible for which
page; the different colors indicate which summands of H18e ∗ (MO(1))
Strictly speaking, we have only calculated this summand up to degree 12, and it could differ from H
in larger degrees. This would only affect the E2 -page in degrees larger than we use and display in (F.12), so thecalculation is the same in either case.

62

SciPost Phys. 9, 010 (2020)∗,∗

data on the E2 -page. The E2 -page carries an action by ExtA(1) (F2 , F2 ). The vertical lines
1,1

indicate action by an element h0 ∈ ExtA(1) (F2 , F2 ), and the diagonal lines indicate action by
2,1

h1 ∈ ExtA(1) (F2 , F2 ); see [97, Example 4.1.1] for more on h0 and h1 . All differentials are h0 and h1 -linear, i.e. d r (hi x) = hi d r (x) (i = 0, 1). In this example, the only differential within
the range displayed in (F.12) that could be nonzero is the d2 from bidegree (4, 0) to bidegree(3, 2). Often, h0 - and h1 -linearity allow one to deduce that differentials vanish, but this does
not provide any information about this d2 , so we have to do something different.
There will also be a question of extension problems: the line t − s = k is the associatede Spin (X ). This in particular introduces an amgraded of a filtration, possibly nontrivial, on Ωke Spin (X ): it could either be Z/2 ⊕ Z/2 or Z/4. Fortunately, when we show this d2
biguity in Ω
2vanishes in Corollary F.16, we will also be able to resolve this ambiguity.
Recall that a dpin structure on M is data of two line bundles L1 , L2 → M and a spin structure on T M ⊕(L1 ⊗L2 )⊕(L2 )⊕3 . Computing w1 and w2 of this bundle with the Whitney sum formula shows that if (M , L1 , L2 ) has a dpin structure, w1 (M ) = w1 (L1 ) andw2 (M ) = w1 (M )(w1 (M ) + w1 (L2 )).
Lemma F.13. The assignment from (M , L1 , L2 ) to a smooth representative of the Poincaré dual−e Spin (X ) → ΩPin .
of w1 (L2 ) ∈ H 1 (M ; Z/2) induces a map DL2 : Ωd−1dProof. These kinds of arguments are standard in bordism theory (e.g. [24, 25, 27]), so wewill be succinct. Let i : N ,→ M be a smooth representative for the Poincaré dual of w1 (L2 )and ν → N be the normal bundle; then w(ν) = 1 + w1 (L2 ). Using the short exact sequence
0 → T N → ν → T M |N → 0 and the Whitney sum formula, we get
w1 (N ) = i ∗ (w1 (M ) + w1 (L2 ))(F.14a)∗w2 (N ) = i (w2 (M )) + w1 (N )w1 (ν)
∗

2

∗

2

= i (w1 (M ) + w1 (M )w1 (L2 ) + (w1 (M ) + w1 (L2 ))w1 (L2 ))
2= i (w1 (M ) + w1 (L2 ) )2= w1 (N ) ,(F.14b)(F.14c)(F.14d)(F.14e)so N admits a pin− structure; a choice of pin− structure amounts to the additional data of anullhomotopy of the map w2 +w21 : N → K(Z/2, 2). A choice of dpin structure on (M , L1 , L2 ) includes (up to a contractible choice) data of nullhomotopies of the maps
w1 (M ) + w1 (L1 ): M → K(Z/2, 1) and w2 (M ) + w1 (M )(w1 (M ) + w1 (L2 )): M → K(Z/2, 2);
via(F.14a),this
inducesanullhomotopy
of
the
map
w1 (N ) + i ∗ (w1 (M ) + w1 (L2 )): N → K(Z/2, 1), which then induces a nullhomotopy of
w2 + w21 : N → K(Z/2, 2) via the rest of (F.14). The proof of bordism invariance of this construction is as usual.
DL2 is an example of a Smith homomorphism. For a general discussion of Smith homomorphisms, see e.g. [25, §4].
−−e Spin (X ) → ΩPin ∼Lemma F.15. The image of DL2 : Ω.
= Z/8 contains a generator of ΩPin
2
2
3

Proof. Let a ∈ H 1 (RP3 ; F2 ) be the generator. Let L1 → RP3 be trivial and L2 → RP3 be thetautological bundle, so w1 (L1 ) = 0 and w1 (L2 ) = a. Since w(RP3 ) = (1 + a)4 = 1 + a4 = 0,
w1 (RP3 ) = 0 = w1 (L1 ) and w2 (RP3 ) = 0 = w1 (RP3 )(w1 (RP3 )w1 (L2 )). Hence (RP3 , L1 , L2 )admits a dpin structure; choose one.
The standard embedding RP2 ,→ RP3 represents the homology class Poincaré dual to a,so DL2 (RP3 , L1 , L2 ) is the pin− bordism class of RP2 with one of its two pin− structures. KirbyTaylor [27, §3] describe how to show that RP2 with either choice of pin− structure generates
−
ΩPin
.
2
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SciPost Phys. 9, 010 (2020)

Corollary F.16.
Spin

e
1. Ω
3

(X ) ∼= Z/8, so the d2 noted above vanishes.

2. The extension
0

/ Z/2

/Ω
e Spin (X )
2

/ Z/2

/ 0,

(F.17)e Spin (X ), splits.
which comes from the Adams filtration on Ω
2
Proof. For (1), let x, y be elements on the E∞ -page, i.e. elements of the associated graded of
the Adams filtration. It is a general fact about the Adams spectral sequence that if h0 x = y,e Spinthen there are preimages x, y ∈ Ω∗ (X ) of x, resp. y, such that 2x = y. For example,
supposing the d2 of interest were nonzero, the line t −s = 3 on the E∞ -page (i.e. the associatede Spin (X )) would contain exactly two Z/2 summands linked by an h0 ; hence there
graded of Ω2e Spin (X ) ∼e Spin (X ) with x 1 = 2x 2 , so Ω
would be nonzero x 1 , x 2 ∈ Ω= Z/4. On the other hand,3
3
if d2 = 0, there would be three Z/2 summands linked by h0 s, so there would be nonzero
e Spin (X ) ise Spin (X ) would be Z/8. That is, Ωe Spin (X ) with x 1 = 4x 2 , and hence Ω
x1, x2 ∈ Ω
3
33
isomorphic to either Z/8, if the d2 in question vanishes, or Z/4, if that d2 does not vanish.
e Spin (X ) admits a surjective map to Z/8, so Z/4 does not work.
Lemma F.15 says Ω3On to (2). Like in the above case with h0 , it is a general fact about the Adams spectral
e Spinsequence that if h1 x = y, then one can choose preimages x and y in Ω
∗ (X ) such that∼
η · x = y, where η is the generator of π1 S = Z/2. (Concretely, if x is the dpin bordism class ofsome manifold M , then η · x is the bordism class of S 1 × M , where S 1 has the dpin structure
induced from the nonbounding framing.)e Spin (X ) would be Z/4 rather than Z/2 ⊕ Z/2.
If the extension in (F.17) did not split, then Ω2However, we can rule this out: suppose it were Z/4, and let x be a generator. Then the imagee Spin (X ) (i.e. the t − s = 2 line of the Adams E∞ -page) isof x in the associated graded of Ω2the nontrivial element of the yellow Z/2 summand in bidegree (2, 0), and the image of 2x isthe nonzero element of the orange Z/2 summand in bidegree (2, 1). The h1 -action carries thisto the nonzero element of the orange Z/2 summand in bidegree (3, 2), so η · 2x 6= 0. Sincee Spin (X ) ∼2η = 0, however, this is a contradiction, forcing Ω= Z/2 ⊕ Z/2.
2
There can be no more nontrivial differentials or hidden extensions in the range shownin (F.12), so we are done.

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