WHAT BORDISM-THEORETIC ANOMALY CANCELLATION CAN DO FOR U ARUN DEBRAY AND MATTHEW YU

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WHAT BORDISM-THEORETIC ANOMALY CANCELLATION CAN DO
FOR U
ARUN DEBRAY AND MATTHEW YU
Abstract.
We perform a bordism computation to show that the
E7(7)
(
R
) U-duality
symmetry of 4d
N
= 8 supergravity could have an anomaly invisible to perturbative
methods; then we show that this anomaly is trivial. We compute the relevant bordism
group using the Adams and Atiyah-Hirzebruch spectral sequences, and we show the
anomaly vanishes by computing
η
-invariants on the Wu manifold, which generates the
bordism group.
Contents
1. Introduction 1
2. Placing the U-duality symmetry on manifolds 3
3. Anomalies, invertible field theories, and bordism 6
3.1. Generalities on anomalies and invertible field theories 6
3.2. Specializing to the U-duality symmetry type 7
4. Spectral sequence computation 9
4.1. Nothing interesting at odd primes 10
4.2. Computing the cohomology of B(Spin-SU8)10
4.3. The Adams Computation 15
4.4. Determining the Manifold Generator 18
5. Evaluating on the Anomaly 21
5.1. Evaluating on the Wu manifold 22
References 25
1. Introduction
One of the most surprising discoveries in the field of string theory is the existence of
duality symmetries. These symmetries show that the same theory can be described in
Date: October 12, 2022.
It is a pleasure to thank Diego Delmastro, Markus Dierigl, Dan Freed, Jonathan J. Heckman, Theo
Johnson-Freyd, Miguel Montero, and David Speyer for helpful comments. We also thank the Simons
Collaboration for Global Categorical Symmetries for hosting a summer school where some of this work
was conducted. Research at the Perimeter Institute is supported by the Government of Canada through
Industry Canada and by the Province of Ontario through the Ministry of Economic Development and
Innovation.
1
arXiv:2210.04911v1 [hep-th] 10 Oct 2022
2 ARUN DEBRAY AND MATTHEW YU
superficially different ways. In some cases, this can be seen via a transformation of the
parameters of the theory, or even the spacetime itself. One such symmetry is U-duality,
given by the group
En(n)
(
Z
). By starting with an 11-dimensional theory which encompasses
the type IIA string theory, and compactifying on an
n
-torus, we gain an
SLn
(
Z
) symmetry
from the mapping class group on the
n
-torus. We arrive at the same theory by compactifying
10d type IIB on a
n−
1-torus, and obtain an O
n−1,n−1
(
Z
) symmetry related to T-duality.
The group En(n)(Z) is then generated by the two aforementioned groups.
In the low energy regime of the 11d theory, which is 11d supergravity, we have an
embedding of
En(n)
(
Z
)
→En(n)
upon applying the torus compactification procedure. The
latter group is the U-duality of supergravity. One finds a maximally noncompact form of
En
after the compactification, and this is denoted
En(n)
(
R
). The maximally noncompact
form of a Lie group of rank
n
contains
n
more noncompact generators than compact
generators. For the purpose of this paper, we reduce 11-dimensional supergravity on a
7-dimensional torus. This gives a maximal supergravity theory, i.e. 4d
N
= 8 supergravity,
with an
E7
symmetry.
1
The noncompact form is
E7(7)
which is 133-dimensional and is
diffeomorphic, but not isomorphic, to SU8/{±1} × R70.
Because this is a symmetry of the theory, one can ask if it is anomalous, and in
particular if there are any global anomalies. Since 4d
N
= 8 supergravity arises as the
low energy effective theory of string theory, then a strong theorem of quantum gravity
saying that there are no global symmetries implies that the U-duality symmetry must be
gaugeable. Therefore, the existence of any global anomaly would require a mechanism
for its cancellation. It would therefore be an interesting question to consider if additional
topological terms need to be added to cancel the nonperturbative anomaly as in [
DDHM22
],
but we show that with the matter content of 4d maximal supergravity is sufficient to cancel
the anomaly on the nose.
The purpose of this paper is to answer:
Question 1.1.Can 4d
N
= 8 supergravity with an
E7(7)
symmetry have a nontrivial
anomaly topological field theory (TFT)? If it can, how do we show that the anomaly
vanishes?
We find that theories with this symmetry type can have a nontrivial anomaly, so we
have to check whether 4d N= 8 supergravity carries this nontrivial anomaly.
Theorem 1.2.
The group of deformation classes of 5d reflection-positive, invertible TFTs
on spin-
SU8
manifolds is isomorphic to
Z/
2. In this group, the anomaly field theory of 4d
N= 8 supergravity is trivial.
The order of the global anomaly is equal to the order of a bordism group in degree 5
that can be computed from the Adams spectral sequence. We find that the global anomaly
is
Z/
2 valued, but nonetheless is trivial when we take into account the matter content of
4d
N
= 8 supergravity. In order to see the cancellation we first find the manifold generator
1Dimensional reduction of IIB supergravity on an 6-dimensional torus also yields the same symmetry.
WHAT BORDISM-THEORETIC ANOMALY CANCELLATION CAN DO FOR U 3
of the bordism group, which happens to be the Wu manifold, and compute
η
-invariants on
it. Even if an anomaly is trivial, trivializing it is extra data, but our computation gives
us a unique trivialization for free; see Remark 3.6 for more. This bordism computation is
also mathematically intriguing because we find ourselves working over the entire Steenrod
algebra, however the specific properties of the problem we are interested in make this
tractable.
This work only focuses on U-duality as the group
E7(7)
rather than
E7(7)
(
Z
), because
the cohomology of the discrete group that arises in string theory is not known, and a
strategy we employ of taking the maximal compact subgroup will not work. But one could
imagine running a similar Adams computation for the group
E7
(
Z
) and checking that the
anomaly vanishes. There are also a plethora of dualities that arise from compactifying 11d
supergravity that one can also compute anomalies of, among them are the U-dualities that
arise from compactifying on lower dimensional tori. In upcoming work [
DY23
] we study
the anomalies of T-duality in a setup where the group is small enough to be computable,
but big enough to yield interesting anomalies.
The structure of the paper is as follows: in
§
2we present the symmetries and tangential
structure for the maximal 4d supergravity theory with U-duality symmetry and turn it
into a bordism computation. We also give details on the field content of the theory and
how it is compatible with the type of manifold we are considering. In
§
3we review the
possibility of global anomalies, and invertible field theories. In
§
4we perform the spectral
sequence computation and give the manifold generator for the bordism group in question.
In
§
5we show that the anomaly vanishes by considering the field content on the manifold
generator.
2. Placing the U-duality symmetry on manifolds
In this section, we review how the
E7(7)
U-duality symmetry acts on the fields of 4d
N
= 8 supergravity; then we discuss what kinds of manifolds are valid backgrounds
in the presence of this symmetry. We assume that we have already Wick-rotated into
Euclidean signature. We determine a Lie group
H4
with a map
ρ4:H4→
O
4
such that 4d
N
= 8 supergravity can be formulated on 4-manifolds
M
equipped with a metric and an
H4
-connection
P,
Θ
→M
, such that
ρ4
(Θ) is the Levi-Civita connection. As we review in
§
3, anomalies are classified in terms of bordism; once we know
H4
and
ρ4
, Freed-Hopkins’
work [FH21b] tells us what bordism groups to compute.
The field content of 4d
N
= 8 supergravity coincides with the spectrum of type IIB
closed string theory compactified on T6and consists of the following fields:
•70 scalar fields,
•56 gauginos (spin 1/2),
•28 vector bosons (spin 1),
•8 gravitinos (spin 3/2), and
•1 graviton (spin 2).
4 ARUN DEBRAY AND MATTHEW YU
Cremmer-Julia [
CJ79
] exhibited an
e7(7)
symmetry of this theory, meaning an action on
the fields for which the Lagrangian is invariant. Here,
e7(7)
is the Lie algebra of the real,
noncompact Lie group
E7(7)
, which is the split form of the complex Lie group
E7
(
C
).
Cartan [
Car14
,
§
VIII] constructed
E7(7)
explicitly as follows: the 56-dimensional vector
space
(2.1) V:= Λ2(R8)⊕Λ2((R8)∗)
has a canonical symplectic form coming from the duality pairing.
E7(7)
is defined to be
the subgroup of Sp(V) preserving the quartic form
(2.2)
q(xab, ycd) = xadybcxcdyda−1
4xabyabxcdycd+1
96 abc···hxabxcdxef xgh +abc···hyabycdyef ygh.
Thus, by construction,
E7(7)
comes with a 56-dimensional representation, which we denote
56.
E7(7)
is noncompact; its maximal compact is
SU8/{±
1
}
, giving us an embedding
su8⊂e7(7)
. Thus
π1
(
E7(7)
)
∼
=Z/
2; let
e
E7(7)
denote the universal cover, which is a
double cover.
There is an action of
e7(7)
on the fields of 4d
N
= 8 supergravity, but in this paper we
only need to know how
su8⊂e7(7)
acts: we will see in
§
3.2 that the anomaly calculation
factors through the maximal compact subgroup of
E7(7)
. For the full
e7(7)
story, see [
FM13
,
§2]; the e7(7)-action exponentiates to an e
E7(7)-action on the fields. The su8-action is:
(1)
The 70 scalar fields can be repackaged into a single field valued in
E7(7)/
(
SU8/{±
1
}
)
with trivial su8-action.
(2) The gauginos transform in the representation 56 := Λ3(C8).
(3)
The vector bosons transform in the 28-dimensional representation Λ
2
(
C8
), which
we call 28.
(4)
The gravitinos transform in the defining representation of
su8
, which we denote
8
.
(5) The graviton transforms in the trivial representation.
The presence of fermions (the gauginos and gravitinos) means that we must have data of
a spin structure, or something like it, to formulate this theory. In quantum physics, a strong
form of
G
-symmetry is to couple to a
G
-connection, suggesting that we should formulate
4d
N
= 8 supergravity on Riemannian spin 4-manifolds
M
together with an
e
E7(7)
-bundle
P→M
and a connection on
P
. The spin of each field tells us which representation of
Spin4
it transforms as, and we just learned how the fields transform under the
e
E7(7)
-symmetry,
so we can place this theory on manifolds
M
with a geometric
Spin4×e
E7(7)
-structure, i.e.
a metric and a principal
Spin4×e
E7(7)
-bundle
P→M
with connection whose induced
O
4
-connection is the Levi-Civita connection. The fields are sections of associated bundles
to
P
and the representations they transform in. The Lagrangian is invariant under the
Spin4×e
E7(7)
-symmetry, so defines a functional on the space of fields, and we can study
this field theory as usual.
WHAT BORDISM-THEORETIC ANOMALY CANCELLATION CAN DO FOR U 5
However, we can do better! We will see that the representations above factor through a
quotient
H4
of
Spin4×e
E7(7)
, which we define below in
(2.4)
, so the same procedure above
works with
H4
in place of
Spin4×e
E7(7)
. A lift of the structure group to
H4
is less data than
a lift all the way to
Spin4×e
E7(7)
, so we expect to be able to define 4d
N
= 8 supergravity
on more manifolds. This is similar to the way that the
SL2
(
Z
) duality symmetry in type
IIB string theory can be placed not just on manifolds with a
Spin10 ×Mp2
(
Z
)-structure,
2
but on the larger class of manifolds with a
Spin10 ×{±1}Mp2
(
Z
)-structure [
PS16
,
§
5],
or how certain
SU2
gauge theories can be defined on manifolds with a
Spinn×{±1}SU2
structure [WWW19].
Let
−
1
∈Spin4
be the nonidentity element of the kernel of
Spin4→SO4
and let
x
be
the nonidentity element of the kernel of
e
E7(7) →E7(7)
. The key fact allowing us to descend
to a quotient is that
−
1 acts nontrivially on the representations of
Spin4×e
E7(7)
above,
and
x
acts nontrivially, but on a given representation, these two elements both act by 1
or they both act by
−
1. We can check this even though we have not specified the entire
e7(7)
-action on the fields, because
−
1
∈e
E7(7)
is contained in the copy of
SU8
in
e
E7(7)
, and
we have specified the
su8
-action. Therefore the
Z/
2 subgroup of
Spin4×e
E7(7)
generated
by (−1, x) acts trivially, and we can form the quotient
(2.4) H4:= Spin4×{±1}e
E7(7) = (Spin4×e
E7(7))/h(−1, x)i.
The representations that the fields transform in all descend to representations of
H4
, so
following the procedure above, we can define 4d
N
= 8 supergravity on manifolds
M
with
ageometric
H4
-structure, i.e. a metric, an
H4
-bundle
P→M
, and a connection on
P
whose induced O4-connection is the Levi-Civita connection.
Remark 2.5.As a check to determine that we have the correct symmetry group, we can
compare with other string dualities. The U-duality group contains the S-duality group for
type IIB string theory, which comes geometrically from the fact that 4d
N
= 8 supergravity
can be constructed by compactifying type IIB string theory on
T6
. Therefore the ways in
which the duality groups mix with the spin structure must be compatible. As explained by
Pantev-Sharpe [
PS16
,
§
5], the
SL2
(
Z
) duality symmetry of type IIB string theory mixes
with the spin structure to form the group
Spin10 ×{±1}Mp2
(
Z
), where
Mp2
(
Z
) is the
metaplectic group from Footnote 2.
Therefore the way in which the U-duality group mixes with
{±
1
} ⊂ Spin4
must also
be nontrivial. Extensions of a group
G
by
{±
1
}
are classified by
H2
(
BG
;
{±
1
}
). If
G
is
connected,
BG
is simply connected and the Hurewicz and universal coefficient theorems
together provide a natural identification
(2.6) H2(BG;{±1})∼
=
−→ Hom(π2(BG),{±1}) = Hom(π1(G),{±1}).
2Here Mp2(Z) is the metaplectic group, a central extension of SL2(Z) of the form
(2.3) 1 {±1}Mp2(Z) SL2(Z) 1.
摘要：

WHATBORDISM-THEORETICANOMALYCANCELLATIONCANDOFORUARUNDEBRAYANDMATTHEWYUAbstract.WeperformabordismcomputationtoshowthattheE7(7)(R)U-dualitysymmetryof4dN=8supergravitycouldhaveananomalyinvisibletoperturbativemethods;thenweshowthatthisanomalyistrivial.WecomputetherelevantbordismgroupusingtheAdamsandAti...

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