Anomaly of the Electromagnetic Duality of Maxwell Theory
$
N

#
−

+

#
S

$

Chang-Tse Hsieh
Kavli IPMU & Institute for Solid States Physics, Univ of Tokyo

East Asian String Webinar
May 29, 2020

Collaboration
Yuji Tachikawa

Kazuya Yonekura(Kavli IPMU)(Tohoku Univ)

CTH-Tachikawa-Yonekura, arXiv: 1905.08943, PRL 123, 161601 (2019)
CTH-Tachikawa-Yonekura, arXiv: 2003.11550

Outline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions

2d
2. 4d
3. 6d
1.

• SummaryOutline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions

2d
2. 4d
3. 6d
1.

• SummaryMaxwell theory and EM duality
From Jackson, we know
2020 nowplays an important role in the consistency of strin
INTRODUCTION

Maxwell theory and EM duality
and

Z0 =

p

µ0 /✏0 (in SI units) or 1 (in Gaussian units)

Every physicist knows that the electromagnetic field isscribed classically by the Maxwell equation, and that iinvariant under the electromagnetic duality S : (E, B)(B, E). The properties of the electromagnetic duality inquantum theory might not be as well known to physicists
general, and in fact are not very well understood in the lite
ture. This is particularly true when going around a nontriv
path in the spacetime results in a duality transformation.1
this letter, we uncover a feature of the Maxwell theory aits duality symmetry in such a situation, namely that it haquantum anomaly.
We recall that a quantum theory in d + 1 dimensions wit
symmetry group G can have a quantum anomaly, in the sethat its partition function has a controllable phase ambi
ity. Our modern understanding is that such a theory is bethought of as living on the boundary of a symmetry protec
topological phase (SPT phase) in the (d + 1) + 1 dimensio

phase in 4+1
dimensions characterizing
the anomaly,
thedimensional
superconformal
field theory known
as the andE-strindimensional
superconformal
field theory ofknown
the E-strin
plays
an importantrole in the consistency
stringastheory.
plays an important role in the consistency of string theory.

Maxwell theory
and EM duality
INTRODUCTION

thou
INTRODUCTION
thou
topo
• Classical Maxwell theory: SO(2) duality symmetry
topo
bulk
(It can be extended to a larger symm, e.g. SL(2, ℝ ) [Gaillard-Zumino (81])
bulk
sion
p
Z0 = pµ0 /✏0 (in SI units) or 1 (in Gaussian units) (1)
sion
trody
• Quantum
knowelectric
andunits)
magnetic(1)Z0 =mechanically,µ0 /✏0 (in SIwe
units)or the1 (inGaussiantroda
hascharges must obey the Dirac quantization conditionhas
m of aqof aspec
qm = 2⇡n~, n 2 Z
(2)
spec
phas
qm = 2⇡n~, n 2 Z(2)or more generally, the Dirac-Zwanziger-Schwinger
phasW
quantization conditioningWt
q1 m2 q2 m1 = 2⇡n~, n 2 Z(3) (q ,m )(q1,m1)2 2ing2 +q1 m2 q2 m1 = 2⇡n~, n 2 Z
(3)
2 ++
(4n
Every physicist knows that the electromagnetic field is dedyons
(4n
theo
Every
physicist byknows
the electromagnetic
scribedclassicallythe that
Maxwellequation, and fieldthat isit deis

3

Departmenttheof spacetime
Physics,mTohoku
9
involves
the action ofSendai
the duali
r~0 University,

~ =
(3) W
field, Has well
as
its
SL(2,
Z)
generalizations.
03
4⇡µ0 rtheory in the situation whe
We consider the (3+1)-dimensional
Maxwellis 56 times that of a Weyl fermion. This resu
the spacetime involves the actionsymmetryof the dualitytransformationexchanging
protected
topological phasein 4+1~ thatfield, as well as its SL(2, Z) generalizations.
Wefindanomaly of thsidering the propertiesof athe(5+1)-dimensiona
qm
R~ em =
L
anomalythe4⇡
Maxwelltheoryplaysindependan (4)
impo
56 times
that of a(notWeylDirac’s
fermion.
ThisofresultisR derived
in: two
Aisheuristic
derivation
original
argument
in 1931)

Maxwell theory and EM duality

symmetry
protected topological phase in 4+1 dimensions characterizing th[Jackson,Ch. 6.12]

Zsidering the properties of a (5+1)-dimensional
superconformal field theory
1
angular momentum~ em~⇥~ in3theL= 2an INTRODUCTION~x ⇥ (EH)dx consistency(5) oanomaly of the Maxwell theory
plays
importantrolecof the EM field:

INTRODUCTION
q

~ =
E

q ~r
4⇡✏0 r3

R3

L=

1
~4 =
E

m Fµ⌫ F µ⌫q~r
4⇡✏0 r3(6)oMaxwellMaxwellI
II
MaxwellF µ⌫ ! F µ⌫ cos ⇠ + Gµ⌫ sin
⇠
(7)
III
0
~
Maxwellm r~
H = (1) 03
IV
Maxwell4⇡µ0 r
f
Weyl,
G
⇥
Z
2
Fµ⌫ ! Fµ⌫ cos ⇠ + ⇤ F µ⌫ sin ⇠ Weyl, Gf (8)
Z
1
3

N

INTRODUCTION
3
symmetry
protectedoftopologicalphase
in 4+1
dimensionsSendaicharacter
Department
Physics,mTohokuUniversity,
90
the spacetime
involves
theaction
of
the
duali
~
r
sidering the properties of(5+1)-dimensional superconformal(3)
field
~a =
1
H
field,
as
well
as
its
SL(2,
Z)
generalizations.
W
03
2⇡
4⇡µ
r
anomalyoftheMaxwelltheoryplays
animportant
role
in
the
consis0
We consider the (3+1)-dimensional Maxwell theory in the situation whe

is 56 times that of a Weyl fermion. This resu
q the~r duality
the spacetime involves the actionoftransformationexchanging
protected
topologicalphase
in 4+1~ = symmetryE(1)
3
4⇡✏0 r the properties~ thatfield, as well as its SL(2, INTRODUCTION
Z) generalizations.
Wefindanomaly of thsideringof athe(5+1)-dimensiona
qm
R~ em =
L
anomalythe4⇡
Maxwelltheoryplaysindependan (4)
impo
56 times
that of a(notWeylDirac’s
fermion.
ThisofresultisR derived
in: two
Aisheuristic
derivation
original
argument
in 1931)

Maxwell theory and EM duality

symmetry
protected topological phasein 4+1 dimensions characterizing th[Jackson,Ch. 6.12]~(2)Zsidering the properties of a (5+1)-dimensional superconformal field theoryq 1~r INTRODUCTION
angular momentum ~ L~~⇥~ in3the
= 2an
~
x
⇥
(
E
H)d
x(1)consistency
(5) o
E
=
anomaly of the Maxwell theoryem
plays
importantrole
3of the EM field:
4⇡✏0cr R3
m r~0
~
H=
(3)
03
4⇡µ0 r1
Nµ⌫
INTRODUCTION
m
q
~
r
q
~
L
=
F
F
(6)
R
(2)
µ⌫
~4 =
o
E
Maxwell4⇡✏0 r3~I
qm R
Maxwell
~
Lem =
(4)
II
4⇡ R
Maxwell
F µ⌫m! r~F0 µ⌫ cos ⇠ + Gµ⌫ sin
⇠
(7)
III
0
~
~
Maxwellq ~r H =m r (3)03~~Z
4⇡µ0 rE=
H = (1) 03
IV
13
Maxwell34⇡✏~ em~ ⇥ H)d~0 r=
L~x ⇥ (Ex 4⇡µ0 r (5)
2
f
c R3
Weyl,
G
⇥
Z
2
Z Fµ⌫ ! Fµ⌫ cos ⇠ + ⇤ F µ⌫ sin ⇠f (8)1Weyl, G~ ⇥ 1H)d~ Z 3xJ~em = 2~x ⇥ (E
(4)
R

3

3
symmetry
protectedoftopologicalphase
in 4+1
dimensionsSendaicharacter
Department
Physics,
TohokuUniversity,
90
thespacetime
involves
theaction
of
the
duali
~
INTRODUCTION m rsidering the properties of(5+1)-dimensional superconformal(3)
field
~a =
H
field, astheory
well4⇡µ
as itsranSL(2,Z) generalizations.
W
03
anomalyoftheMaxwell
plays
important
role
in
the
consis0
We consider the (3+1)-dimensional Maxwell theory in the situation whe

is 56 times that of a Weyl fermion. This resu
the spacetime involves the actionsymmetryof the dualitytransformationexchanging
protected
topological phasein 4+1q~r~= sidering3 the properties(1)anomaly of th~ thatfield, as well as its SL(2, INTRODUCTION
Z) E
generalizations.
We
findof athe(5+1)-dimensiona
qm
R
4⇡✏0 rL~ em =
1anomalythe4⇡
Maxwelltheoryplaysindependan (4)
impo
56 times
that of a(notWeylDirac’s
fermion.
ThisofresultisR derived
in: two
Aisheuristic
derivation
original
argument
in 1931)2⇡

Maxwell theory and EM duality

symmetry
protected topological phase in 4+1 dimensions characterizing th[Jackson,Ch. 6.12]~
R(2) field theoryZsidering the properties of a (5+1)-dimensional
superconformal

q 1~r INTRODUCTION
angular momentum ~ L~~⇥~ in3the
= 2an
~
x
⇥
(
E
H)d
x(1)consistency
(5) o
E
=
anomaly of the Maxwell theoryem
plays
importantrole
3of the EM field:
4⇡✏0cr R3
~0
m
r
~ =
H
4⇡µ0 r03

INTRODUCTION(q1,m1)~R(3)1 ) µ⌫
L = (q2~,mF2µ⌫
Fq
E
=
4

~r
4⇡✏0 r3

(2)

(6)
o

Maxwell
⇣q m
⌘R
I
~
m
q
1
2
1
2
Maxwell
~
Lem =
(4)
II
4⇡
4⇡
R
Maxwell
F µ⌫m! r~F0 µ⌫ cos ⇠ + Gµ⌫ sin
⇠
(7)
III
0
~
~
Maxwellq ~r H =m r (3)03~~
4⇡µ0 r ~ H = (1) 03
E=IV
3Maxwellqm
R
4⇡µ0 r (5)4⇡✏0 r L~ em =
f
4⇡ R
Weyl,
G
⇥
Z
2
Z Fµ⌫ ! Fµ⌫ cos ⇠ + ⇤ F µ⌫ sin ⇠f (8)1Weyl, G~ ⇥ 1H)d~ Z 3xJ~em = 2 Z ~x ⇥ (E
(4)
3

N

symmetry protected topological phase in 4+1 dimensions character
INTRODUCTION m r~0
sidering the properties of(5+1)-dimensional superconformal(3)
field
~a =
H
03
anomaly of the Maxwell
theory4⇡µplays
0 ran important role in the consis
INTRODUCTION

Maxwell theory andduality
q EM
~r
~
E=
~
INTRODUCTION3
qm
R
4⇡✏0 rL~ =(1)

A heuristic derivation (not Dirac’s originalemqargument~r4⇡ R in 1931) :~E=[Jackson, Ch. 6.12]
4⇡✏0 r3~Rangular momentumof the EM field:(q1,m1)~ em =
Lm r
4⇡µ0 r03~R21 ) µ⌫
L = (q2,mF2µ⌫
F
4 ~0
1 2

r03

(2)
(4)

4⇡µ0
4⇡
R
µ⌫
µ⌫
F m! r~F0 cos ⇠ + Gµ⌫ sin ⇠
~ =
H
(3)
03
4⇡µ0 r ~1)R̂qm= R n~, n 2 Z ✓ (5)But QM tells us (L~ emem=L
4⇡ R2
Z Fµ⌫ ! Fµ⌫ cos ⇠ + ⇤ F µ⌫ sin ⇠1~ ⇥ H)d~ 3x
J~em = 2 Z ~x ⇥ (E
(4)
4⇡

(5)

(2)

(3)

r
⇣ q mH~ =m m
~
q ⌘R
1

(1)

(2)

Z

~ em q= 1~r~ ⇥ H)d~ 3 x(1)
~ =
L
~
x
⇥
(
E
E
4⇡✏0cr23 R3
~
~0R
~ =
H

(4) 1

(6)

(3)
(7)

(4)
(8)

2⇡

Department of
INTRODUCTION
anomaly of the Maxwell theory plays
role in the✓ an important◆
INTRODUCTION
We consider
a bthe (3+1)-dimension
Z) of
T: 2
!SL(2,
ithey action
the spacetimec involvesdfield, as well as its SL(2, Z) general✓◆ INTRODUCTION
is 56 times that of a Weyl fermion.
✓ a b◆✓ ◆ (1)✓ topological◆ ✓ ◆phaa b 2 SL(2, Z)
symmetryqprotected
T
:
!
i
b y q
The quantum EM duality
preserve
the charge quan.
c d must
2 SL(2,Z)(1) acond.
!
sidering thepropertiesof a (5+1)-d
c d
m
c
d
m
✓
◆ anomaly of the Maxwell theory
play
q1 q2
✓ ◆n 2 ✓◆✓ ◆q1 m2 q2 m1 = det= 2⇡n~,Z (1)
q ! (2)✓ a b◆ q !
m2
T:
!im
y1
~ !
~
E
ac bd
E
T:
!i y
(2)
m
m
INTRODUCTION!

Maxwell theory and EM duality~

Hand thus is represented by an SL(2, ℤ) group, namelyq ◆ ~r
✓ ◆
✓ E
✓
~◆
= q
q
a
b
3
✓ ◆ !✓◆ ✓4⇡✏◆
0 r
qm
ac bd
q
m
!
m
c d
m
~ !
!
✓
◆ R
~!
~!
E
a
b
E
✓
◆
!
~
~
~
E
ac bd
E
H
H
!
~
~
c d
H
H
~0
m
r
~ =
H
03

!

c d

✓

~
H

◆
(2)
~
a b
E

~
E
!
(3)
✓H
~ ◆
~
c
d
H
a,
b,
c,
d
2
Z
a b (3)c d

2 SL(2, Z)(3)i.e. a, b, c, d 2 Z

ad(4) bc = 1
T : (4)! i yad✓bc =(4)◆1!

consider
the of(3+1)-dimensional
theoryin the rolesituation
the Wespacetime
involves
theMaxwell
action oftheory
the Maxwell
dualityexchanginanomalytheplaystransformation
animportantin thew
field,
as well as involves
its SL(2,theZ) action
generalizations.
We findthat the anomalyof tthe spacetimeof the dualitytransformationexchang
isfield,
56 timesthat
WeylZ)
fermion.
This result
derivedin twoindepenas wellas of
its aSL(2,generalizations.
Weisfindthat theanomalyofsymmetry
protected
phaseThis
in 4+1
dimensions
is 56 times
that of INTRODUCTION
atopological
Weyl fermion.
resultis derivedcharacterizingin two indepe
sidering
theprotected
propertiestopologicalof a (5+1)-dimensional
field theosymmetryphase in 4+1 superconformal
dimensions characterizing
anomaly
Maxwell
playsthean importantrole in cond.
the consistencysidering
thetheproperties
a (5+1)-dimensional
field theoThe quantum
EM ofduality
mustoftheory
preserve
chargesuperconformalquan.
anomaly of the Maxwell theory plays an important role in the consistenc

Maxwell theory and EM duality✓

q1 q2
qINTRODUCTION
m
=
det
2 1
m1 m2

◆

n 2 Z (B,(1)E).1 Th
1quantum
INTRODUCTION(B, E).theoTgeneraland,inquantumtheo
and thus is represented by an SL(2, ℤ) group, namely
q ~r
ture.
Thisand,is pi
general
✓
◆
~
E
=
(2)
0 1
3
path
the isspature.inThis
S: ✓(1)◆ 4⇡✏0 r
01 01
this
standard EM duality (1)(S-duality)
pathLetter,in thewe
sp
S:
1 0
its
duality
thisLetter,sym
w
generators~(3)
ititshasa quantu✓◆ R
dualitysy
1 1
Witten effect (2" shift of top.mine.
#−term)T : ✓(2)
it has a quant
01 11◆We recall th
mine.
T :
(2)
0 1
symmetry
grot
~0
m
r
We
recall~ =◆✓ H
that(4)its partiti
03
q1 m2

= 2⇡n~,

More on Maxwell theory
In ordinary Maxwell theory, there is no relation btw charges (q, m) andspin/statistics. We could have both neutral boson and fermion as the“origin” of the charge latticeOpen circles: primitive vectors,
representing single-particle
states (SL(2, ℤ) invariant)More on Maxwell theory
In ordinary Maxwell theory, there is no relation btw charges (q, m) andspin/statistics. We could have both neutral boson and fermion as the“origin” of the charge latticeOpen circles: primitive vectors,
representing single-particle
states (SL(2, ℤ) invariant)(q=1, m=0) and (q=0, m=1)
span the charge lattice

2

University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan3
Department of Physics, Tohoku University, Sendai 980-8578, Japan

More on Maxwell theory

We consider the (3+1)-dimensional Maxwell theory in the situation where going around nontrivial paths inthe spacetime involves the action of the duality transformation exchanging the electric field and the magnetic
field, as well as its SL(2, Z) generalizations. We find that the anomaly of this system in a particular formulation
is 56 times that of a Weyl fermion. This result is derived in two independent ways: one is by using the bulk
In ordinary
theory,there
is no characterizing
relation btw(q,them)andsymmetry
protectedMaxwell
topological phasein 4+1
dimensions
the charges
anomaly, andotheris by considering
the properties ofa (5+1)-dimensional
superconformal
field theory
as the E-string
spin/statistics.
We
could have bothneutral boson
andknown
fermionas thetheory. Thisanomaly of the Maxwell theory plays an important role in the consistency of string theory.

“origin” of the charge latticeS
ST
None charge conj. ZC
full EM duality SL(2, Z)
2 S-duality Z4 ST-symm. Z3
o
ll
0
0
0
Z9
Z9
✓
◆
IOpen circles: primitive vectors,
1
0
ll
0
Z2
C:
II
single-particle0
1
l representing
0
Z2
lIII
states0 (SL(2, ℤ)Z2invariant) Z2
Z2
Z4
Z9
Z36
lIV
✓
◆
0 1
⇥ Zf2 0
0
Z4
Z9 C
Z36
S:
Gf
0
Z16
Z32
Z9
Z288
1 0

(q=-1, m=0) and (q=0, m=-1)
the charge
latticeLE spanI. Partition
functionsand related data on S 5 /Zk .

T :INTRODUCTION

ST :

✓

✓

◆
1 1
0 1

0

1

◆

University of Tokyo, Kashiwa, Chiba 277-8583, Japan2 Institute for Solid State Ph
2
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581,3 Department
Japan
of P
3
Department of Physics, Tohoku University, Sendai 980-8578, Japan

More on Maxwell theory

We consider the (3+1)-dimensiona

We consider the (3+1)-dimensional Maxwell theory in the situation where going around nontrivial paths inthe spacetimeof t
the spacetime involves the action of the duality transformation exchanging
the electricinvolves
field andthe
the action
magnetic
as wellinasa its
SL(2, Z)
generaliz
field, as well as its SL(2, Z) generalizations. We find that the anomaly offield,
this system
particular
formulation
is 56 times
is 56 times that of a Weyl fermion. This result is derived in two independent
ways:that
oneofis abyWeyl
usingfermion.
the bulk T
In ordinary
theory,there
is no characterizing
relation btw(q,them)andsymmetry
protected
topologicalphase
symmetry
protectedMaxwell
topological phasein 4+1
dimensions
the charges
anomaly,andotheris by considering
the properties ofa (5+1)-dimensional
superconformal
field theory
the E-stringThissidering
theasproperties
oftheory.
a (5+1)-dim
spin/statistics.
We
could have bothneutral boson
andknown
fermionas theanomaly of the Maxwell theory plays an important role in the consistencyof stringoftheory.
anomaly
the Maxwell theory plays“origin” of the charge latticeS
ST
None charge conj. ZC
full EM duality SL(2, Z)
2 S-duality Z4 ST-symm. Z3o
INTRODUCTION
ll
0
0
0
Z9
Z9
✓
◆
IOpen circles: primitive vectors,
1
0
ll
0
Z2
C:
II
single-particle0
1
l representing
0
Z2
lIII
states0 (SL(2, ℤ)Z2invariant) ✓
◆
Z2
Z2
Z4
Z9
Z36
lIV
✓
◆
0
1
S: 0 1⇥ Zf2 0
0
Z4
Z9 S
Z36
S:
1 0
Gf
0
Z16
Z32
Z9
Z288
1 0

(q=0, m=-1) and (q=1, m=0)
the charge
latticeLE spanI. Partition
functionsand related data on S 5 /Zk .
INTRODUCTION

✓
◆
✓
◆
1 1
TT :: 0 1
0 1
ST :

✓

✓ 0

◆
1 ◆

University of Tokyo, Kashiwa, Chiba 277-8583, Japan2 Institute 3for Solid State Ph
Department of P
2
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581,3 Department
Japan
of P
3
Department of Physics, Tohoku University, Sendai We
980-8578,Japan
considerthe (3+1)-dimensiona

We consider the (3+1)-dimensiona

spacetime
involves
the action
We consider the (3+1)-dimensional Maxwell theory in the situationthe
wheregoing aroundnontrivial
pathsofin t
the spacetimeof tfield,
as wellasinvolves
its
SL(2,
Z)action
generaliza
the spacetime involves the action of the duality transformation exchanging
theelectric
field
andthethemagnetic
as wellthat
SL(2,generalizT
isfield,
56 timesofa WeylZ)fermion.
field, as well as its SL(2, Z) generalizations. We find that the anomaly of
thissysteminasa its
particular
formulation
is 56 times
that
Weyl
is 56 times that of a Weyl fermion. This result is derived in two independent
ways:
oneofis aby
usingfermion.
the phasebulk Tsymmetry
protected
topologicalIn ordinary
theory,there
is no characterizing
relation btw(q,them)symmetry
topologicalphase
symmetry
protectedMaxwell
topological phasein 4+1
dimensions
the charges
anomaly,andotheris by considering
theprotected
properties
ofanda (5+1)-dim
sidering
the properties ofa (5+1)-dimensional
superconformal
field theory
theE-string
theory.
Thissidering
theas
properties
oftheorya (5+1)-dim
spin/statistics.
We
could have bothneutral boson
andknown
fermionas theanomalyoftheMaxwellplays
anomaly of the Maxwell theory plays an important role in the consistencyof stringoftheory.
anomaly
the Maxwell theory plays

More on Maxwell theory“origin” of the charge latticeS
ST
None charge conj. ZC
full EM duality SL(2, Z)INTRODUCTION
2 S-duality Z4 ST-symm. Z3
INTRODUCTION
llo
0
0
0
Z9
Z9
✓
◆
IOpen circles: primitive vectors,
1
0
ll
0
Z2
C:
II
single-particle0
1
l representing
0
Z2
✓
◆
lIII
states0 (SL(2, ℤ)Z2invariant) 0
1
◆
S: ✓
Z2
Z2
Z4
Z9
Z36
lIV
✓
◆
0
1
S : 01 01⇥ Zf2 0
0
Z4
Z9 T
Z36
S:
1 0
Gf
0
Z16
Z32
Z9
Z288
1 0

(q=1, m=0) and (q=1, m=1)
the charge
latticeLE spanI. Partition
functionsand related data on S 5 /Zk .
INTRODUCTION

✓
◆
1
1
✓
◆
◆
T : ✓
1
1
0 1
TT :: 0 1
0 1

✓✓
00
✓
ST
:
ST :

◆
11 ◆
◆

More on Maxwell theory
Now let’s image a world where neutral particles are bosons. Then the
charge lattice can be associated w/ various “charge-spin relations”,
resulting extra four versions of Maxwell theory.
MaxwellI

MaxwellII

MaxwellIV

MaxwellIII

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

f

b

f

b

f

b

f

b

f

b

f

b

f

b

b

b

b

b

b

b

b

f

f

f

f

f

f

f

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

f

b

f

b

f

b

f

b

f

b

f

b

f

b

b

b

b

b

b

b

b

f

f

f

f

f

f

f

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

(1, 0): b(0, 1): b(1, 1): f

Figure 2. The charge lattice Q(1, 0): b(0, 1): f(1, 1): b

Figure 2. The charge lattice Q(1, 0): f(0, 1): b(1, 1): b

Figure 2. The charge lattice Q(1, 0): f(0, 1): f(1, 1): f

Figure 2. The charge lattice Q

formodular
these points
the2 (Z)equation q + ig = q0 (m⌧ + n). The action of the modular(Z) equation q + ig = q0 (m⌧ + n). The action of the modular
these points in the equation q + ig = q0 (m⌧ + n). The action of thegroupinSL
for thesegroup
pointsSL
in2 thefor thesegroup
pointsSLinthe equation q + ig = q0 (m⌧ + n). The action of the modular group S
2 (Z)preserves primitive vectors in the charge lattice.
serves primitive vectors in the charge lattice.
preserves primitive vectors in the charge lattice.
preserves primitive vectors in the charge lattice.

Givenof athespecificof thea charge
ven a specific theory, an important question is identifying the subset
chargetheory,latticean important question is identifying the subsetGiven
specificlatticetheory, an important question is identifying the subset Givenof the acharge
specificlatticetheory, an important question is identifying the subset of the chargethat states.
can be Hererealizedstates.
it is thet can be realized by single-particle states, as opposed to multi-particle
it isbythesingle-particle states, as opposed to multi-particlethat
can Herebe realizedby single-particle states, as opposed to multi-particle thatstates.
canHerebe realized
it is theby single-particle states, as opposed to multi-particle states. Here itprimitive vectors that represent single-particle states ([O2]).
mitive vectors that represent single-particle states ([O2]).
primitive vectors that represent single-particle states ([O2]).
primitive vectors that represent single-particle states ([O2]).

⇠⇠
We now
symmetries of the charge lattice. Firstly, the Weyl groupZ2 permutes= looknow look at symmetries of the charge lattice. Firstly, the Weyl group
W lookWe W
now⇠
at symmetries of the charge lattice. Firstly, the Weyl group
WeWnowat symmetries of the charge lattice. Firstly, the Weyl group W ⇠= Zat= Zlook= Z2 pe2 permutes
2 permutes
thecharges.
axes of thecharge thevector lattice, interchanging electric and magnetic the
charges.
Secondly,
the vector lattice, interchanging electric and magnetic charges.
axes of the charge vector lattice, interchanging electric and magnetic
Secondly,axes ofthe charge
the axesSecondly,of the charge
the vector lattice, interchanging electric and magnetic charges. Second
generalizationofdualitysymmetryofMaxwellisarotation
by
⇡/2
which
exchanges
electricallyeralization of duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges electricallygeneralization of duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges
generalizationelectricallyof duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges electcharged particles and magnetic monopoles. The ⇡/2-rotational symmetry

More on Maxwell theory
Now let’s image a world where neutral particles are bosons. Then the
charge lattice can be associated w/ various “charge-spin relations”,
resulting extra four versions of Maxwell theory.
MaxwellI

MaxwellII

MaxwellIV

MaxwellIII= all-fermion electrodynamics

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

f

b

f

b

f

b

f

b

f

b

f

b

f

b

b

b

b

b

b

b

b

f

f

f

f

f

f

f

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

f

b

f

b

f

b

f

b

f

b

f

b

f

b

b

b

b

b

b

b

b

f

f

f

f

f

f

f

b

b

b

b

b

b

b

b

b

b

b

b

b

b

f

b

f

b

f

b

f

f

b

f

b

f

b

f

(1, 0): b(0, 1): b(1, 1): f

Figure 2. The charge lattice Q

S
T(1, 0): b(0, 1): f(1, 1): b

Figure 2. The charge lattice Q

S(1, 0): f(0, 1): b(1, 1): b

T

Figure 2. The charge lattice Q(1, 0): f(0, 1): f(1, 1): f

Figure 2. The charge lattice Q

formodular
these points
the2 (Z)equation q + ig = q0 (m⌧ + n). The action of the modular(Z) equation q + ig = q0 (m⌧ + n). The action of the modular
these points in the equation q + ig = q0 (m⌧ + n). The action of thegroupinSL
for thesegroup
pointsSL
in2 thefor thesegroup
pointsSLinthe equation q + ig = q0 (m⌧ + n). The action of the modular group S
2 (Z)preserves primitive vectors in the charge lattice.
serves primitive vectors in the charge lattice.
preserves primitive vectors in the charge lattice.
preserves primitive vectors in the charge lattice.

Givenof athespecificof thea charge
ven a specific theory, an important question is identifying the subset
chargetheory,latticean important question is identifying the subsetGiven
specificlatticetheory, an important question is identifying the subset Givenof the acharge
specificlatticetheory, an important question is identifying the subset of the chargethat states.
can be Hererealizedstates.
it is thet can be realized by single-particle states, as opposed to multi-particle
it isbythesingle-particle states, as opposed to multi-particlethat
can Herebe realizedby single-particle states, as opposed to multi-particle thatstates.
canHerebe realized
it is theby single-particle states, as opposed to multi-particle states. Here itprimitive vectors that represent single-particle states ([O2]).
mitive vectors that represent single-particle states ([O2]).
primitive vectors that represent single-particle states ([O2]).
primitive vectors that represent single-particle states ([O2]).

permuted by SL(2, ℤ)

SL(2, ℤ) invariant⇠⇠
We now
symmetries of the charge lattice. Firstly, the Weyl groupZ2 permutes= looknow look at symmetries of the charge lattice. Firstly, the Weyl group
W lookWe W
now⇠
at symmetries of the charge lattice. Firstly, the Weyl group
WeWnowat symmetries of the charge lattice. Firstly, the Weyl group W ⇠= Zat= Zlook= Z2 pe2 permutes
2 permutes
thecharges.
axes of thecharge thevector lattice, interchanging electric and magnetic the
charges.
Secondly,
the vector lattice, interchanging electric and magnetic charges.
axes of the charge vector lattice, interchanging electric and magnetic
Secondly,axes ofthe charge
the axesSecondly,of the charge
the vector lattice, interchanging electric and magnetic charges. Second
generalizationofdualitysymmetryofMaxwellisarotation
by
⇡/2
which
exchanges
electricallyeralization of duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges electricallygeneralization of duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges
generalizationelectricallyof duality symmetry of Maxwell is a rotation by ⇡/2 which exchanges electcharged particles and magnetic monopoles. The ⇡/2-rotational symmetry

sidering the properties of a (5+1)-dimension
anomalyNoneof thecharge
Maxwelltheoryplays an impS
conj.
ZC
2 S-duality Z4

Maxwello000MaxwellI
0
ZC2
SZ4
None
chargeconj.
Z
S-dualityZII
2
4 ST-sy
Maxwell0Z2oMaxwell000IIIMaxwell0
Z2
I
IV
Maxwell
0
Z4
Z2 Z2
Z2
Z4
Maxwell• It seems natural to have these modified
II Maxwell theories
Maxwell
0Zf 0 Z2
Weyl,
G
⇥
0
Z4
2
III
IV
f
Z2 the• However, Maxwell = all-fermMaxwell
ED is
anomalous
Weyl,
G00 (while
Z16 otherZ32

All-fermion electrodynamics

IV
Z2 in purely
Z2 4d if
Z4
Maxwellthree are not), in the sense that it cannotf exist
Weyl, G
⇥TABLEZ2 0 I. Partition0(13);functions and Zrelated
data on4
microscopic DOF are only bosons[Wang-Potter-Senthil
Kravecf
Weyl,
G
0
Z16
Z32
McGreevy-Swingle (14); Thorngren (14); Wang-Wen-Witten(18)]

INTRODUCTIONTABLE I. Partition functions
and related data on S 5 /Z

Such a theory (in the absence of neutral fermions)Ø does not have a bosonic regulator (e.g. 4d U(1) lattice gauge theory)
INTRODUCTIONØ does not have a well-defined part. func. on some spacetime (e.g.CP
CP22)
Ø is the IR theory of some anomalous UV theory (e.g. ferm of isospin 4" + 3/2

w/ a refined SU(2) anomaly)
R
R
w
w w w
Ø must live on the boundary of a 5d bulk (w/ part. func. ( 1) (M5 1)2 M35) 2 3nnXHd represent the possible values of the “magnetic flux” of the
Universe
cosmic(WPI),
string excitations of the theory in 3 + I dimensions (orJapann 2 + I dimensions). In general, the magnetic fluxIPMU-19-0068
carried by a
Chiba 277-8581, Japan
TU-1088
elementoftheunbroken
group
H
that
encodestheresultofof Physics,
magnetic
dualityof thetheorydai
980-8578,
Japan pathalong
a closed
thatMaxwell
encloses thestring. To define themusta basepoint
goingspecify
around1 nontrivial
paths inx0 and a3 closed loop C that starts andTachikawa,
and Kazuya Yonekura
he electric field and the magnetic
irclesthestringexactly once.
(See fig. 1.) Then the flux is giventem
aSymmetries
particular formulation
isMaxwelland inMathematicsof the ofUniverse(WPI), theory might also be anomalous, and we’d
ilson
loop
ays: one
isChibaby operator
using
the bulk SPT
ashiwa,277-8583,
Japan

Symmetries of Maxwell theory

like to know which symm is anomaly-free and thus can be gauged.

nsidering
the propertiesof a Chiba
5+1
rsity
of Tokyo,
Kashiwa,
277-8581, Japans anomaly of the Maxwell theory

ology Group,
Department
Physics,
Ø One
exampleto consider
w/ extra dynamicalh(C,x0)=P exp of isdx~A1 . Maxwell theory(2.3)
University, Aoba-ku, Sendai 980-8578,
Japan(C, x~)

ifgauge fields, e.g. Alice electrodynamics [Schwarz (82)]ory in the situation where going around nontrivial paths in
ues ina “Alice
H(x0),
the subgrouptheandunderlying
string/loop”
with ofthe
ansformationexchanging
the electric
field
the magneticgroup G thatalong C of this“Cheshire[Wilczek
et al (90)]at theanomaly
in a particular
formulation isq, m! q, charge”
m system(11)
ved in two independent ways: one is by using the bulk SPT
aly, and the other is by considering the properties of a 5+1
Alice EMs the E-string
theory. This anomaly of the Maxwell theory
U(1) o Z2 = O(2)(12)
theory.

t knows that the electromagnetic field is dey by the Maxwell equation, and that it is“local” charge conjugation
he electromagnetic duality S : (E, B) 7!
along C
operties of the electromagnetic duality
the q, mq, m in!
(11)
might not be as well known to physicists in
Fig
from[Bucher-Lo-Preskill
ectC,arestarting
andwellendingat theknows
point
x0,
enclosesa 1oop of cosmic string.
Everyphysicistthatthe (92)]
electromagnetic
field is denot very
understoodin thelitera1)cularly scribedtrue whenclassicallygoing aroundnontrivial
bya theMaxwell equation, and that it isspecial case ofphase we find fo
We study the✓◆ing the relationa b
2 SL(2, Z)(6)
2 + 1d U(1)1 C
c d(4n we’d+ 1) + 1d s
Symmetries of Maxwell theory might also be anomalous, andlike to know which
gauged.studied✓ symm is◆anomaly-free and thus can betheory,
0 1
3 +et 1d(7) [BakØ Another example is to consider the Janus configuration
al. Maxwe
1 0(92); Giaotto-Witten (08)] where the spacetime has a duality duality
twist symmet
m

c d

m

Symmetries of Maxwell theoryE(x + L, y, z) = B(x, y, z)
B(x + L, y, z) =E(x, y, z)

(8)
(9)

1 One example is a

L, y, z) = B(xular setup was st[Ganor
et
al
(08
,10,
12,
14)]
Every physicist knows that the electromagnetic field is dehappens in a mo
Janus from wiki: God of beginnings,
ribed classically by the Maxwell equation, andthat it isknowledge. Ther
gates, transitions, time, duality,of the Maxwell tvariant under the electromagnetic duality S : doorways,(E, B)passages,
7! and ending
are related to theB, E). The properties of the electromagnetic duality in theAnomaly of symmetries• More precisely, we want to know the ’t Hooft anomaly of agiven symm, which obstructs the gauging of the symm.
• Such an anomaly manifests in a controlled manner and can beunderstood by anomaly inflow argument.
Ø Modern view: An n-dim anomalous theory is (most naturally) realized as aboundary mode of a (n+1)-dim symmetry protected topological (SPT)phase or invertible field theory (IFT) in (n+1)dHilbert space (on any closed manifold) is 1-dim(’t Hooft) Anomaly
on boundary theory
Bulk SPT phase
IFT w/ symm

Anomaly of symmetries☆ ☆ ☆ Fact :anomaly in n dim ó part. func. of (n+1)d bulk IFT on closed manifolds

Outline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions(1+1)d2. (3+1)d3. (5+1)d1.

• SummaryBefore discussing 4d Maxwell theory and its corresp 5d bulk
theory, let’s look at a simpler but related example:
2d chiral boson and 3d Chern-Simons theory

Outline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions

2d
2. 4d
3. 6d
1.

• Conclusion

anomaly in [? ]. Our result in this Letter can bes interpreted as a mixedgravitational
SL(2, Z)ved
twoofindependent
ways:
one
is by
using
SPT
, Z)inpart
WARM-UP:(1+1)-DIMENSIONALCHIRALaracterizing
theandANOMALY
other
isOF
by
considering
propertiesof a 5+1
considered
asthe
the
determination
ofthe
the bulk
puretheSL(2,Z) partof
WARM-UP: ANOMALY OF (1+? ]. Our result inthisanomaly,
Lettercan beBOSON
IN TERMS
OF(2+1)-DIMENSIONAL
U(1) theory BOSON IN TERMS OF (2+ly,andthepure
other
is
by
the
propertiesof aof5+1
anomaly.
field
theoryknown
theconsidering
theory.
Thisanomaly
the Maxwell
nationoftheSL(2,theZ)as
partofE-string
WARM-UP:ANOMALY
OF (1+1)-DIMENSIONALCHIRALCHERN-SI
CHERN-SIMONS
Our
computation
shows
that
the
anomalyoftheMaxwell
BOSON
TERMS
OF (2+1)-DIMENSIONAL
U(1)
hethe
Maxwellconsistency
string theory.
E-stringoftheory.
This anomaly
ofINtheMaxwelltheoryCHERN-SIMONS
theoryis 56 times that of a Weyl
fermion in a certain prescertainthat the preanomaly of theMaxwell
heory.
We start by recalling the w
cise formulationof the duality.
Where does this numbera Weyl fermion
pre- by recallingWe
start
the well-understoodcase of56 thenumber56 in a certain
come
WeWe
will
providean answerusing the property
start
by recallingthe well-understoodcase ofof the anomaly of the (1+1)-dimensio
ality. Where does thisnumberfrom?
56
anomalyof the (1+1)-dimensional chiral boson at the free fermion radius. This theory nproperty ofa (5+1)-dimensional
field theory
originallyanomaly ofsuperconformal
the (1+1)-dimensional
chiral bosonat the freede an answer using the property
offermionradius.
This
theorynaturally
lives
at
the
boundyerconformal
originallytheorynaturally
livestheat thebound- ary of the (2+1)-dimensional U
in [? ?fermion] and radius.
known Thisas theE-string
theory;
name
along
C
field theoryfoundoriginally
q,
m
!
q,
m
(11)
theoryarycomesof namethefrom(2+1)-dimensional
Chern-Simons
atat level
1 CS
y; as
thethenamek =1, forwhich the Eu
ary ofthe that(2+1)-dimensional
U(1) Chern-Simons
theoryU(1)thefact
it hasU(1)E8 global
symmetry.theoryThewn
E-string theory;
the
R
along
C
k two
= 1,branchesfor1 which
EuclideanactionSk=1 == ⇡i (A/2⇡)(F/2⇡) [? ? ? ].
level
k The
=theory
1, level
forRhaswhich
theEuclideanactionistheistensor
Sk=1
E-string
ofthevacua,
calledit has E8The
global symmetry.
metry.
R
q,
m
!
q,
m
(11)
⇡i Higgs(A/2⇡)(F/2⇡)? the? anomaly]. Higgs
The anomaly
is thethenterized by the partition function
ranchesof vacua,
calledthetensorand thebranchEcharac⇡i branch(A/2⇡)(F/2⇡)[? branch.
? ? ].[?OnThe
is thencharac8dthe tensor
Gaussianunits)(1)terizedby thepartitionfunction
theory on closed 3-dimensional manifo
nch. On the Higgs branch
theby
E8the
symmetry
is partition
HiggsedtoE
acts ofonthis28Chern-Simonsfermions theoryvia
7 , whichterizedfunctionof
thisChern-Simons
nch the E8
on closed
3-dimensional
manifolds
ME7 , which acts on 28 fermionsvia
self-dualchiral56-dimensional
fundamental
representation;
possi-boson Let us recall
that the acti
U(1)o3 .Zthis
O(2)
(12)
2 =is
)
onits
closed3-dimensional
manifolds
MR
ermionsvia
3.
mental representation; thispossi- a pseudo-real
Let us recall
that the action
at levelbleis since
representation
R withdim 2,
R =Sk=22k = 2⇡i (A/2⇡)(F/2⇡), is well-de
Rthat thehisispossiLetus
recallaction
at
level
2,
S
presentation R with dim RR = 2k
2⇡i (A/2⇡)(F/2⇡), is well-defined modulo 2⇡ik=2
when =
the
U(1)
o
Z
=
O(2)
(12)
2
m
2⇡i (2)(A/2⇡)(F/2⇡), is well-defined modulo 2⇡i when the
Z R = 2k

Anomaly of 2d chiral boson in terms of 3d U(1)Chern-Simons theory
Boundary theory of 3d U(1) CS theoey= 2d (self-dual) chiral boson

The (grav) anomaly of the chiral boson is characterized by part. func.
ZU(1)CSspin(M3 )3-manifold=of the U(1)1 CS a closed
M3:)

n 2 Z ZU(1)CS (M(3)3 ) =

)

E
B

◆

) ◆

Z

Z

[DA]top.trivial eR⇡i (A/2⇡)(F/2⇡)XeR⇡i (A/2⇡)(F/2⇡)#.

#
A:flatR
X
one-loop determinant ⇡i (A/2⇡)(F/2⇡)
classical saddle points (13)[DA]top.trivial e⇡i (A/2⇡)(F/2⇡) ⇥
e
.
R

"

⇥

"

A:flatEvery physicist knowsthat the electromagnetic field is described classically by the Maxwell equation, (13)and that it isinvariant under the electromagnetic duality S : (E, B) 7!
Every physicist knows that the electromagnetic field is de(B, E). The properties of the electromagnetic duality in the(4)symmetry acting
where c = c1 (FR) is the first
Chern class of the gauge bundle
We consider the (3+1)-dimensional Maxwell theory in the situation
going around ofnoitywheresymmetryothe
Z2duality= O(2)
the spacetime
action oftransformation exchanging (12)the electric field an)nd q(c) := e⇡i (A/2⇡)(F/2⇡). involves the U(1)the backgro
field, as well as its SL(2, Z) generalizations. We find that the anomalyduceof this system
in a particu
Anomalyof
2dchiral
boson
in
terms
of
3d
U(1)
is 56
timesothatZofa=Weylfermion. This level-1
result is derived
in two independent ways: one is by
We note that q(c) is simply
theclasU(1)O(2)(12)
2exponentiatedwhich means thZ(2)symmetry protected topological phase in 4+1 dimensions characterizing the anomaly, and theChern-Simons
theoryical action
evaluated atsideringa flatAs recalled
above, superconformal
defining field theory
theA.
propertiesof a (5+1)-dimensionalas thearou
E-strwhenknown
going
anomalyof (Mthe Maxwell
an important role in the consistency of string theory.
ZU(1)CS= theory plays3 ) orientedt) requires something morethananmanifold"and the#of the partition
f
Z
R
R
X
ntegration
on it.(3)Mathematically,qe⇡iisS-duality
known
as a ⇥
quadratic(A/2⇡)(F/2⇡)
n 2 Z ZU(1)CSchargeconj. ZZ ST-symm.
Z
fullduality SL(2, Z) .
[DA]
e⇡iEM(A/2⇡)(F/2⇡)(M3 ) = None
top.trivial
1
2
Maxwell0pairing 0on H "0 3 , Z). The
Z Arf
Z
#
efinementofthetorsion(M
inA:flat2⇡ ArgZ
R0
R
X
MaxwellZZ
T : No on-si(13)⇡i(A/2⇡)(F/2⇡)⇡i(A/2⇡)(F/2⇡))
variant
Arf(q)
isdefinedequationabove
and -is known . Maxwell
0by the Z
[DA]
e
⇥
e
top.trivial◆
where the eta in
Maxwell0ZE
A:flato take values inMaxwell
one eighth
ofphysicist
anZ integer.
We
end upZ with the Zfield is deEveryknows
Z
Z that the electromagnetic(4)
dif
1 (13)acting on the1
The phaseofGZ⇥ isgiven
by[Hsieh-Tachikawa-Yonekura(19, 20)]
B
Arg
Z
(M
)
=
⌘
CdC
7
signatu
Weyl,
Z
0
0
Z
Z
Z
8
scribed classically by the Maxwell equation,2⇡ and that it isormula1now
thequadratWeyl, G
0Z
Z
Zcontributed by Zonly= 28 ⇥ 2⇡A
invariantundertheelectromagnetic
duality
S
:
(E,
B)
!
7
[Witten (89);Monnier-Moore
(18)]
knows
that the electromagnetic
field
is de- flat
top. nontrivial
A3 (M , (Z2 ) )) ◆ 1 Every physicist
H
1
5⇢
TABLE I.
Partitionfunctionsand related dataon
S electromagnetic
/Z .
(B,
E).
The
properties
oftheduality
indep of ⌧ , ⌧¯in the
ZU(1)CS (Mby⌘signatureequation,+ Arf(q).
(17)
q
3) =
scribedtheMaxwell
and
that
it
is
2⇡ Arg classically
8(5)
twistedinby the Squantum theory might not be as well known to physicists
m invariant under theelectromagnetic
duality
S well: (E,B) 7! in the litera-1 Z t Z
general,andin factofare Arf
not veryunderstood
INTRODUCTION
eta
invariantinvariantofquadratic
Let us first cd
Let(B,
us nowthat a chiralboson
can be fermionized.
(QD /~) =E). recall
The properties
oftheelectromagnetic
dualityinthe96⇡ 2 tture. Thisis particularly
true whenofgoingaround a nontrivial⌃
the signature
op refinement
the torsion
SL(2,Z)
backgr)Thenthebulk theory
can
taken
to wellbe the2+1dinto
fermion1
2 (M ,withquantumtheory might
not
as
known
(⋆d
+spacetimed⋆)pairingonHℤ) inpathbeinthebe
results
aphysicists
dualityIn
3 transformation.
⇣
⌘
Z)
(6)
X wellhas
simplyvanishes
1function
nfinitemass, andwhose
partition
phase
general,in fact
areletter,
not
in
the
this
we sign(
uncoverathefeatureof [31]theMaxwell theory
andE ! cos ✓E += very) understood(1) litera2
reg
call
that
ture. This is particularly
truewhen goingarounda nontrivial
its duality
symmetryin sucha situation,
namely thatit has
Ba!a quadrsin ✓E1
1anomaly.
Arg quantum
Zfermion(M⌘fermion.
(18)
path in the2⇡spacetime
resultsin3 )a=dualitytransformation.
Inevaluated on fla
)
U(1)
oa
TK
We arecall
that
quantumtheory ind(2)+ 1 anddimensions
withcana be wri
this letter, wefeatureofthe Maxwell
theory(7) uncover
mentGaussian units)

Department of Physics, Tohoku University, Sendai 980-8578, Japan

(1)

C
2

S
4

ST
3

o

I

2

II

9

4

9

36

4

9

36

32

9

288

4

2

III
IV

f

9

2

2

2

f
2

16

5

k

2

1

symmetry protected topologicalphase acting
in 4+1
symmetry5
I. Partition
functionsrelated of
dataon S /Zk .
sidering
properties
a (5+1)-dimension
where c = c1 (FR) is the first TABLE
Chern classof thegaugetheandbundleity
symmetryofanomalyoftheMaxwelltheory
plays
an
impo
⇡i
(A/2⇡)(F/2⇡)
U(1)
o
Z
=
O(2)
(12)
2
)

Gaussian units)(1)nd q(c) := e.
duce the backgro
Anomalyof
2dchiral
boson
in
terms
of
3d
U(1)
We note that q(c) is simply
thelevel-1 clasU(1) oZ2exponentiated= O(2) INTRODUCTION(12) C which meansthZ(2)
None charge conj. Z2 S-duality ZS
4 ST-sy
Chern-Simons
theoryical action
evaluated at a flat
A. AsMaxwell
recalled
above,
definingowhen0 going arou
0
0
Z
(M
)
=
3 orientedI manifold and thet) requires something moreU(1)CSthan anMaxwell0Z2- partition
"
#
of
the
f
Z
II
R
R
X
Maxwell0 a quadratic
Z2
ntegration
on it.(3)Mathematically,
q
is
known
as
⇡i
(A/2⇡)(F/2⇡)⇡i (A/2⇡)(F/2⇡)
n 2 Z ZU(1)CSIII[DA]top.triviale.1(M3 ) =
Maxwell0 ⇥ 1 Z2e 2
2"
Arg
#)
efinement
H (MIV3 ,(p
Z).
The
Arf
intrR
(1
A:flat2⇡
Z of the torsionR pairing onMaxwell
2
1Z=
Z
Z
2
4
8⇡
Xf R2
⇡i (A/2⇡)(F/2⇡))
variant
Arf(q)
is definedby the equation
above
is known
Weyl,⇥G ⇥Z2e⇡iand0 (A/2⇡)(F/2⇡)
0
Z4 (13)[DA]e.
top.trivial◆
wherethe eta in
f
Weyl,
G
0
Z
Z
16
32
E
A:flato take values in(4)one eighth
ofphysicist
an integer.
Wethatend
with the field is deEvery
knows
theup
electromagnetic
acting on the dif
The phase of Z isscribedgiven classicallyby [Hsieh-Tachikawa-Yonekura(19, 20)]B(13)by theI.Maxwell
equation, andandrelated
that itdata
is on S 5 /Z
ormula1
1 functions
TABLE
Partition
Arg
Z
(M
)
=
⌘
+
Arf(q)
mod
1 (2
now
the
quadrat3
signature2⇡ underU(1)CS
8 duality
invariantthe electromagnetic
S : (E, B) 7!
knows that the electromagnetic field is de32) ◆ 1 Every physicist (B,
H
(M
,
(Z
)⇢ )
1 properties of the electromagnetic duality
5
E).
The
in
the
Arg
Z
(M
)
=
⌘
+
Arf(q).
(17)q
U(1)CS3signaturescribedclassicallyby the Maxwell
equation, and INTRODUCTION
that it is2⇡
8(5)
twistedinby the Squantumtheory
might
not
be
as
well
known to physicists[Atiyah-Patodi-Singer(75);
Brumfiel-Morgan(73)]
m invariant✓◆Z
under thegeneral,
electromagnetic
duality S : (E, B) 7!
and in fact are not very well understood in the Let
litera-us first cLet(B,
us now
recallthatachiralboson
canbefermionized.
11E). The properties
of is
the
electromagnetic
duality
in
the
L
(X
)
+
1
4
ture. =
This
particularly
true
when
going
aroundanontrivial
8
8 (X4 ) . (3SL(2,Z) backgr
X
@X
=M3to
)
Then
the
bulk theory
can
taken
to well
be4 ;the2+1d
4
1quantumtheory might
not
as
known
in
pathbeinthebespacetime
results
infermionaphysicists
duality withtransformation.
In
2
1
Z)
(6)
p1/3in (p=trR) andvanishes
simply2
1nfinitemass, andwhose
partitionfunction
has
the
phase
[31]
general,in fact
are
not
very
well
understoodthelitera8⇡ theory
this letter, we uncover a feature of the Maxwell
Zsymmetrycall
thatture. This is particularly
truewhen goingarounda nontrivial
its duality
in sucha situation,
namely thatit hasa a quadr
1
1 1 Arg Z= (MÂ.1 = ⌘Dirac =) fla(4anomaly.
Arg quantum
Zfermion⌘fermion(18)
path in the2⇡spacetime
resultsin3 )a=dualitytransformation.
Inevaluatedferm (M3on
2⇡
)
Xthat
1a quantum
4 ; @X
4 =M3
We arecall
theoryin
d=+ 1 1anddimensions
a bemod
Arg
Z
(M
)
+with
Arf(q)
this letter, wefeatureoftheMaxwelltheory(7) uncoverU(1)CS
3canwri
2⇡
8 ⌘signaturement

TABLE I. Partition functions
relatedtopologicaldatasymmetryon S phase/Zk . acting
symmetryand
protected
in 4+1
5
I. Partition
functionsrelated of
dataon S /Zk .
sidering
properties
a (5+1)-dimension
where c = c1 (FR) is the first TABLE
Chern classof thegaugetheandbundleity
symmetryofanomalyoftheMaxwelltheory
plays
an
impo
⇡i
(A/2⇡)(F/2⇡)
U(1)
o
Z
=
O(2)
(12)
2
)nd q(c) := e.
duce the backgro
INTRODUCTION
Anomalyof
2dchiral
boson
in
terms
of
3d
U(1)
We note that q(c) is simply
thelevel-1 clasU(1) oZ2exponentiated= O(2) INTRODUCTION(12) C which meansthZ(2)
None charge conj. Z2 S-duality ZS
4 ST-sy

Gaussian units)(1)

Chern-Simons
theoryical action
evaluated at a flat
A. AsMaxwell
recalled
above,
definingowhen0 going arou
0
0
Z
(M
)
=
3 orientedI manifold and thet) requires something moreU(1)CSthan anMaxwell0Z2- partition
"
#
of
the
f
Z
II
R
R
X
Maxwell0 a quadratic
Z221
ntegration
on it.(3)Mathematically,
q
is
known
as
⇡i
(A/2⇡)(F/2⇡)⇡i(A/2⇡)(F/2⇡)
n 2 Z ZU(1)CS
(p
=
trR
)
III
1 0
[DA]top.triviale⇥ 2 Z2e. 1 (1)(M3 ) =
8⇡
Maxwell2"2#1Arg
efinement
H (MIV3 ,(p
Z).
The
Arf
intrR
)
(1
A:flat2⇡
Z of the torsionR pairing onMaxwell
2
1Z=
Z
Z
2
4
8⇡
Xf R2
⇡i (A/2⇡)(F/2⇡))
variant
Arf(q)
is definedby the equation
above
is known
Weyl,⇥G ⇥Z2e⇡iand0 (A/2⇡)(F/2⇡)
0
Z4 (13)[DA]e.
top.trivial◆
wherethe eta in
f
Weyl,
G
0
Z
Z
16
32
E
A:flato take values in(4)one eighth
an integer.
Wethatend
with the field is deEveryknows
the1upelectromagnetic
1 ofphysicist
actingon 1the(2)
dif
Arg
Z
⌘signatureArf(q)
mod
U(1)CS (M3 ) =The phase of Z isscribed
given
by[Hsieh-Tachikawa-Yonekura(19,+20)]
B
(13)
2⇡
8
by theI.Maxwell
equation, andandrelated
that itdata
is on S 5 /Z
ormula1 classicallyTABLE
1 functions
Partition
Arg
Z
(M
)
=
⌘
+
Arf(q)
mod
1 (2
now
the
quadrat3
signature2⇡ underU(1)CS
8 duality
invariantthe electromagnetic
S : (E, B) 7!
knows that the electromagnetic field is de32) ◆ 1 Every physicist (B,
H
(M
,
(Z
)⇢ )
1 properties of the electromagnetic duality
5
E).
The
in
the
Arg
Z
(M
)
=
⌘
+
Arf(q).
(17)q
U(1)CS3signaturescribedclassicallyby the Maxwell
equation, and INTRODUCTION
that it◆is
2⇡
8✓Z
(5)
twistedquantumtheory
might
not
be
as
well
known to physicists
in by the S
1 ✓Z
1m invariant under the electromagnetic◆
duality L
S 1well
: (E,
B)
= and
(X
(X4 ) . (3)4 ) 7!+ in the
8 in fact are not very
8 Letgeneral,understoodliteraus first cLet(B,
us nowthat a=chiralfermionized.
1 boson1
X4 ; @Xcan
4 =Mbe3E). recall
The properties
oftheelectromagnetic
dualityinthe
L1 going(Xaround4 ) +a nontrivial
ture. This is
8 particularly true when
8 (X4 ) . (3SL(2,Z) backgr
X
@X
=M3to
)
Then
the
bulk theory
can
taken
to well
be4 ;the2+1d
4
1quantumtheory might
not
as
known
in
pathbeinthebespacetime
results
infermionaphysicists
duality withtransformation.
In
2
1
Z)
(6)
Z
p1/3in (p=trR) andvanishes
simply2
1nfinitemass, andwhose
partitionfunction
has
the
phase
[31]
general,in fact
are
not
very
well
understoodthelitera8⇡ theory
this letter, we uncover a feature of the Maxwell
1=trueÂaround⌘Dirac=namely
Zcall
(M
(4)
Zsymmetry
that
1 =
ferm
ture. This is particularly
when goinga nontrivial
its duality
in sucha situation,it hasa3 )a quadr
2⇡ Argthat
X4 ; @X4 =M3
1
1 1 Arg Z= (MÂ.1 = ⌘Dirac =) fla(4anomaly.
Arg quantum
Zfermion⌘fermion(18)
path in the2⇡spacetime
resultsin3 )a=dualitytransformation.
Inevaluatedferm (M3on2⇡)/24 in d + 1 1dimensions with aXthat
1a quantum
4 ; @X
4 =M3 -p1theory
We arecall
Arg
Z(M3theory) = 8and⌘signaturement+ Arf(q)this letter, wefeatureoftheMaxwell(7) uncoverU(1)CScan bemodwri2⇡

functions
on S /Zk . acting
Weyl,TABLE
G ⇥ Z2I. Partition0
0 and related dataZ4 symmetry
Z9
5
TABLE
I. Partition
functionsrelateddata on S /ZkZ. 9
where c = c1 (FR) is the first
Chern
gauge
bundleWeyl,
Gf class
0 of theZ16 andZ32
ity symmetry of⇡i(A/2⇡)(F/2⇡)
U(1)
o
Z
=
O(2)
(12)
2
)nd q(c) := e.
duce the5 backgro
INTRODUCTION
Anomalyof
2dchiral
boson
in
terms
of
3d
U(1)
TABLE
I.=Partition
functionsandclasrelated
data on S /Zk .
We note that q(c) is simply
theexponentiatedlevel-1
U(1)
o
Z
O(2)
(12)
2
which means th
Z(2)
INTRODUCTION

Gaussian units)(1)

Chern-Simons
theoryical action
evaluated at a flat
A. As recalled above, defining
when going arouZ(M)=
3 oriented manifold and thet) requires something moreU(1)CSthan an"#of the partitionf
INTRODUCTION
Z
R
R
X
2⇡i (A/2⇡)(F/2⇡)
1
ntegration
on it.(3)Mathematically,q is known
as a ⇥
quadratic
n 2 Z ZU(1)CS
(p1 =
trR
)
2
[DA]top.trivial e⇡i (A/2⇡)(F/2⇡)e. 1 (1)(M3 ) =
8⇡2"1
efinement
Z).
Arf
in-2#)(1
A:flat3 ,(p2⇡ ArgZ of the torsionR pairing on H (M
2 trR
1 =The8⇡RX(13)⇡i (A/2⇡)(F/2⇡)⇡iand(A/2⇡)(F/2⇡))
variant
Arf(q)
is definedby the equation
above
is known .
[DA]
e
⇥
e
top.trivial◆
where the eta in
E
1
1
A:flato take values in(4)one eighth
ofphysicist
anZU(1)CS
integer.
We
endwith the
Every
knows
that
the1upelectromagnetic
field
is de(M)=⌘+Arf(q)Arf(q)
mod
1(2)
(
1 Arg
3
signatureacting
on
the
dif
2⇡
8
Arg
Z
(M
)
=
⌘
+
mod
1
U(1)CS
3
The phase of Z isscribedgiven
by [Hsieh-Tachikawa-Yonekura(19, 20)]B(13)
2⇡ classically
8 signatureby the Maxwell
ormula1
1 equation, and that it isZ(M3 ) = 8 ⌘signature + Arf(q)mod
1 (2now thequadrat2⇡ ArginvariantunderU(1)CSthe electromagnetic
duality S : (E, B) 7!
knows that the electromagnetic field is de32) ◆ 1 Every physicist (B,
H
(M
,
(Z
)⇢ )
1 properties of the electromagnetic duality
5
E).
The
in
the
✓
◆
Arg
Z
(M
)
=
⌘
+
Arf(q).
(17)Zq
U(1)CS3signaturescribedclassicallyby the Maxwell
equation, and that it◆is
2⇡
8✓Z
(5)
twistedquantumtheorymight
not be as well known to physicists
in by the S
1 1 (X ) . (m invariant under the electromagnetic== 181 ✓Z duality L
L
(X
)
+
◆
S 1: (E,B)(X(X4 ) 4 . (3)4 )4 7!+ in the
8litera8 in fact are not very1well understood
8 Let
general, andus first cX=Mbe
3 fermionized.
4 =M
Let(B,
us nowthat a=chiral1 boson1
X44;; @X@Xcan
43E). recall
The properties
oftheelectromagnetic
dualityinthe
L1 going(Xaround4 ) +a nontrivial
ture. This is
8 particularly true when
8 (X4 ) . (3SL(2,Z) backgr
X
@X
=M3to
)
Then
the
bulk theory
can
taken
to well
be4 ;the2+1d
4
1quantumtheory might
not
as
known
in
pathbeinthebespacetime
results
infermionaphysicists
duality withtransformation.
In
Z)
(6)
ZZfunction
simplynfinitemass, andwhose
partition
phase
general,in fact
areletter,notvery
wellhasunderstood
in
the
this
we uncoverathefeatureof [31]the literaMaxwell theory
andvanishes
11Â⌘⌘Dirac==namely
Zcall
(M
(4)(
Zsymmetry==trueÂaround=Arg
Zitferm
(M
that
11 =
ferm
Dirac
ture. This is particularly
when goinga nontrivial
its duality
in sucha situation,thathasa3 )a3 )quadr
2⇡2⇡Arg
X44;; @X
@X44=M=M33
X
1
1 1 Arg Z= (MÂ.1 = ⌘Dirac =) fla(4anomaly.
Arg quantum
Zfermion⌘fermion(18)
path in the2⇡spacetime
resultsin3 )a=dualitytransformation.
Inevaluatedferm (M3on
2⇡
)
Xthat
4 ; @X
4 =M3
We arecall
aofquantum
theory ind + 1 anddimensions
withcana be wri
this letter, wefeaturethe Maxwell
theory(7) uncovermentGaussian units)(1)
the spacetime
involves the action of the duality transformation exchanging theU(1) o Z2 =WeO(2)(12) of this sy
field, as well as its SL(2, Z) generalizations.
find that the anomaly
is 56 times
ofchirala Weyl boson
fermion. This
result is ofderivedin two independent
Anomaly
ofthat2dU(1)
in
terms
3d
U(1)
o
Z
=
O(2)
(12)
2
(2) protected topologicalsymmetryphase in 4+1 dimensions characterizing the an
Chern-Simons
theorysidering the propertiesof a (5+1)-dimensional superconformal field theory kno
ZU(1)CS (M3 ) =" role in the consistency# of str
anomaly of theZMaxwell theory plays an important

)

Z

)

)

n 2 Z ZU(1)CS (M(3)3 ) =◆Z[DA]top.trivial e

R

R⇡i (A/2⇡)(F/2⇡)

INTRODUCTION[DA]top.triviale⇡i (A/2⇡)(F/2⇡) ⇥

"

X

e

⇥

X

A:flateR⇡i (A/2⇡)(F/2⇡)

R⇡i (A/2⇡)(F/2⇡)

#

.

.

(13)Noneo
Maxwell0E
A:flatEveryphysicist
knows
that
the
electromagnetic
fieldisdeMaxwellI0
The phase (4)of Z isscribedgiven classicallyby [Hsieh-Tachikawa-Yonekura(19, 20)]B(13)by the Maxwell equation, and that it isII
Maxwell0
invariant under the electromagnetic duality S : (E, B) 7!
knows that the electromagnetic field is de- MaxwellIII) ◆ 1 Every physicist (B,01The properties of the electromagnetic duality in theE).
ZU(1)CS (Mby
Arg Zfermion(M3 )and that
(1)it isq3) =scribedthe2⇡
Maxwellequation,2⇡ Arg classically
Z2
MaxwellinIV(5)quantumtheory
might
not
be
as
well
known
to
physicists
m invariant under the electromagnetic duality S : (E, B) 7!
G ⇥ Zf2 0
general, and in fact are not very well understood Weyl,in the literaØ
means
chiral boson(0-form
gauge field)
and 2dchiralf(B,ThisE).
The 2dproperties
of istheparticularlyelectromagnetic
duality
the afermion
ture. This
true whengoinginaroundnontrivial
Weyl,
Ghave
0
the same
anomaly(perturb
grav
anomaly);ittois physicists
as expectedinsince the two)
1
quantumtheorymight
notmbespacetimeas 5wellknown
path
results
in a duality transformation.
In
5 in thenZ)(6)
theories
actually
identical
in2d4understood(traditional
sense(2)of “bosonization”).
Z(X)=inZ(S/Zletter,) veryZ(S/Z) a ·feature··3
general,andarefactthis
are
not
well
in
we uncoverof thethe literaMaxwell theory
and I. Partit
TABLE
ture. This is particularly
truesymmetrywhen goingarounda nontrivial
its duality
in sucha situation,
namely that it has aquantum
anomaly.
path in the spacetime
results
in a duality transformation.1 In)⌘ aofquantum
We arecall
that
theory ind + 1 anddimensions with athis letter, wefeaturethe Maxwell
theory(7) uncover
1 ⇣XGaussian units)(1)
the spacetime
involves the action of the duality transformation exchanging theU(1) o Z2 =WeO(2)(12) of this sy
field, as well as its SL(2, Z) generalizations.
find that the anomaly
is 56 times
ofchirala Weyl boson
fermion. This
result is ofderivedin two independent
Anomaly
ofthat2dU(1)
in
terms
3d
U(1)
o
Z
=
O(2)
(12)
2
(2) protected topologicalsymmetryphase in 4+1 dimensions characterizing the an
Chern-Simons
theorysidering the propertiesof a (5+1)-dimensional superconformal field theory kno
ZU(1)CS (M3 ) =" role in the consistency# of str
anomaly of theZMaxwell theory plays an important

)

Z

)

)

n 2 Z ZU(1)CS (M(3)3 ) =◆Z[DA]top.trivial e

R

R⇡i (A/2⇡)(F/2⇡)

INTRODUCTION[DA]top.triviale⇡i (A/2⇡)(F/2⇡) ⇥

"

X

e

⇥

X

A:flateR⇡i (A/2⇡)(F/2⇡)

R⇡i (A/2⇡)(F/2⇡)

#

.

.

(13)Noneo
Maxwell0E
A:flatEveryphysicist
knows
that
the
electromagnetic
fieldisdeMaxwellI0
The phase (4)of Z isscribedgiven classicallyby [Hsieh-Tachikawa-Yonekura(19, 20)]B(13)by the Maxwell equation, and that it isII
Maxwell0
invariant under the electromagnetic duality S : (E, B) 7!
knows that the electromagnetic field is de- MaxwellIII) ◆ 1 Every physicist (B,01The properties of the electromagnetic duality in theE).
ZU(1)CS (Mby
Arg Zfermion(M3 )and that
(1)it isq3) =scribedthe2⇡
Maxwellequation,2⇡ Arg classically
Z2
MaxwellinIV(5)quantumtheory
might
not
be
as
well
known
to
physicists
m invariant under the electromagnetic duality S : (E, B) 7!
G ⇥ Zf2 0
general, and in fact are not very well understood Weyl,in the literaØ
the analysis
hereelectromagnetic
can be generalized
to higher(B,Nevertheless,E). The properties
of is
the
duality
the dimensions,ture. Thisparticularly true whengoinginarounda Weyl,nontrivial
Gf0
where onecanmight
still relatetheasanomaliesof p-form
gauge fields)
1 ofquantumtheorynotmbewell known
in to thosepath
spacetime
results
intoaphysicists
duality transformation.
In
5 in the
5nZ)(6)fermions,
thougharewelldifferent
Z(X)/Zletter,)they
Z(S/Z4understood) a ·feature·theories!
·(2)
3
general,and=even
inZ(S
fact
are
not
very
in
this
we uncoverof thethe literaMaxwell theory
and I. Partit
TABLE
ture. This is particularly
truesymmetrywhen goingarounda nontrivial
its duality
in sucha situation,
namely that it has aquantum
anomaly.
path in the spacetime
results
in a duality transformation.1 In)⌘ aofquantum
We arecall
that
theory ind + 1 anddimensions with athis letter, wefeaturethe Maxwell
theory(7) uncover
1 ⇣XGaussian units)(1)
the spacetime
involves the action of the duality transformation exchanging theU(1) o Z2 =WeO(2)(12) of this sy
field, as well as its SL(2, Z) generalizations.
find that the anomaly
is 56 times
ofchirala Weyl boson
fermion. This
result is ofderivedin two independent
Anomaly
ofthat2dU(1)
in
terms
3d
U(1)
o
Z
=
O(2)
(12)
2
(2) protected topologicalsymmetryphase in 4+1 dimensions characterizing the an
Chern-Simons
theorysidering the propertiesof a (5+1)-dimensional superconformal field theory kno
ZU(1)CS (M3 ) =" role in the consistency# of str
anomaly of theZMaxwell theory plays an important

)

Z

)

)

n 2 Z ZU(1)CS (M(3)3 ) =◆Z[DA]top.trivial e

R

R⇡i (A/2⇡)(F/2⇡)

INTRODUCTION[DA]top.triviale⇡i (A/2⇡)(F/2⇡) ⇥

"

X

e

⇥

X

A:flateR⇡i (A/2⇡)(F/2⇡)

R⇡i (A/2⇡)(F/2⇡)

#

.

.

(13)Noneo
Maxwell0E
A:flatEveryphysicist
knows
that
the
electromagnetic
fieldisdeMaxwellI0
The phase (4)of Z isscribedgiven classicallyby [Hsieh-Tachikawa-Yonekura(19, 20)]B(13)by the Maxwell equation, and that it isII
Maxwell0
invariant under the electromagnetic duality S : (E, B) 7!
knows that the electromagnetic field is de- MaxwellIII) ◆ 1 Every physicist (B,01The properties of the electromagnetic duality in theE).
ZU(1)CS (Mby
Arg Zfermion(M3 )and that
(1)it isq3) =scribedthe2⇡
Maxwellequation,2⇡ Arg classically
Z2
MaxwellinIV(5)quantumtheory
might
not
be
as
well
known
to
physicists
m invariant under the electromagnetic duality S : (E, B) 7!
G ⇥ Zf2 0
general, and in fact are not very well understood Weyl,in the literaØ
the analysis
hereelectromagnetic
can be generalized
to higher(B,Nevertheless,E). The properties
of is
the
duality
the dimensions,ture. Thisparticularly true whengoinginarounda Weyl,nontrivial
Gf0
where onecanmight
still relatetheasanomaliesof p-form
gauge fields)
1 ofquantumtheorynotmbewell known
in to thosepath
spacetime
results
intoaphysicists
duality transformation.
In
5 in the
5nZ)(6)fermions,
thougharewelldifferent
Z(X)/Zletter,)they
Z(S/Z4understood) a ·feature·theories!
·(2)
3
general,and=even
inZ(S
fact
are
not
very
in
this
we uncoverof thethe literaMaxwell theory
and I. Partit
TABLE(A formalof genericp-formgauge
theories
is by using differential
ture.
This is treatment
particularly
truesymmetry
when
going
arounda nontrivial
its duality
in sucha situation,
namely that it has acohomology [Cheeger-Simons (85); Hopkins-Singer (02); Córdova
et al. (19);quantum
anomaly.
path in the spacetime
results
in a duality transformation.1 In
Hsieh-Tachikawa-Yonekura (20); etc.]))⌘ aofquantum
We arecall
that
theory ind + 1 anddimensions with athis letter, wefeaturethe Maxwell
theory(7) uncover
1 ⇣XOutline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions

2d
2. 4d
3. 6d
1.

• Conclusion

(3)[DA]top.trivial e⇡i(A/2⇡)(F/2⇡)⇥e⇡i(A/2⇡)(F/2⇡).

A:flat(13)
Anomaly of Maxwell theory in terms of 5d top BdC(4) theory= to⌘fermionmod1 is a TQFT w/ two(14)
The 5d bulk theory corresp
Maxwelltheory2form gauge fields: [Verlinde (95); Kravec-McGreevy-Swingle (13]R
(5)⇡i [(B/2⇡)d(C/2⇡) (C/2⇡)d(B/2⇡)]
BdC theory Maxwell
Every physicist knows that the electromagnetic field is detheoryØAtscribed
the levelofdifferential
form,
this
5d
theoryhasanclassically by the Maxwell equation, and that it isSL(2,ℤ) symmonthe(B, electromagneticC), corresp to duality
symm
invariantunder
duality
S of: (E, B) 7!
(6)(B,E). Theproperties of the electromagnetic duality in thetheMaxwelltheory.
quantum theory might not be as well known to physicists ingeneral, toandin fact
not sensible
very wellunderstoodin the B,
literaØHowever,maketheareaction
when
top nontrivial
C areture. This we
is particularly
true whenarounda nontrivial
require the action
to begoinga quadratic
refinementof(7) considered,1
path in the spacetime
in i.e.
a dualitytransformation.
(diff-cohomology)
paringresultsof B, C,2"#q(B,C), which mightIn
break
the SL(2, ℤ) symm in general.
(8)Anomaly of Maxwell theory in terms of 5d top BdC
theory• The choice of quadratic refinement q(B, C) is not unique, and ingeneral we have

• For X=Y=0, q(0, 0) (B, C) is SL(2, ℤ) invariant only on spin-manifolds.

Its boundary theory is the ordinary Maxwell theory. [Witten (98); Gomi (04)]• q(X, Y) (B, C) can be made SL(2, ℤ) invariant on any 5d manifold if wetake X/2" = Y/2" = w2 (2nd Stiefel-Whitney class of M5). In this casethe corresp boundary theory is all-fermion ED.

symmetry acting on (B,C), which corresponds to the dualfermion
R ature; see, e.g., [? ].
X
Wecan
considerthe(3+1)-dimensional
Maxwe⇡i (A/2⇡)(F/2⇡)⇡i (A/2⇡)(F/2⇡)
ity symmetry of
the Maxwell
theory.
We
andwill
introLetus
firstconsider
the
case
where
we
do
not
have
the
e
⇥
e
.
op.trivial
the spacetime involves the action of the duali
bundleSL(2,In thiscase,
the signature eta invariant
gauge field⇢Z)forbackground.
this
SL(2,
Z)
symmetry,
Rduce the backgroundfield,
as
well
as
its
SL(2,
Z)BdC
generalizations. W
A:flatAnomalyofMaxwelltheoryterms
oftop
simplyinvanishes,and only
the5d
Arf invariant
contributes. Re⇡i which[(B/2⇡)d(C/2⇡)(C/2⇡)d(B/2⇡)]
is 56 timestransformationthat of a Weyl fermion. This resu
means that there is a nontrivial
dualitycall that a quadratic (13)refinement is simply the classical action
symmetry protected topological phase in 4+1
theory-1 clasevaluated
B and C. Thena general
quadratic refinewhen going around a nontrivial
looponinflat
spacetime.
The
phase
sidering the properties of a (5+1)-dimensiona
ment can be written as
definingofthepartitionfunctionisthenMaxwell
Z anomaly ofZ the(19,Z theory plays an impo
• The phase of the part funcis[Hsieh-Tachikawa-Yonekura
20)]
(15)
B dC
dB C
B dC
= ⌘fermion mod 1 ⌘signature = 0
(14)
+
+
,
(48)
and the
2⇡ 2⇡
2⇡ 2⇡
2⇡ 2⇡
1
1
Arg
Z
(M
)
=
⌘ C 2 H 2 (M+ Arf(q)(21)
BdC
5
uadratic
INTRODUCTION
2⇡
where4 B,signature
5 , R/2⇡Z) are the background fields
for the electric and magnetic 1-form U(1) symmetry of the
Arf inZ/2⇡)d(C/2⇡)(C/2⇡)d(B/2⇡)]
Maxwelltheorysignature[? ], which
we chose to⇤dbelet’s
flat. Its Arf
1 isto now• Beforecoupling
thissystemany(duality)
symm
background,
where
the
eta
invariantfortheoperatorRknown
Arf(q) =(X/2⇡)d(Y/2⇡).
(16)where is theZ 3 (C/2⇡)
invariant is computed
to be 2 (B/2⇡)
see what
can tell2 us.forms
In Bockstein
thissituation,actingtheonphasethe differential
tensoredwith (Z 1)⇢ ,2(X/2⇡)
andq is!/2⇡).
homomorphism(M
H 3 (M5 , Z);
5 , R/Z)(Y
Arf(q) = : H
with the2 pairinghomomorphism
can roughly
now the quadratic refinement theof Bockstein
the natural
torsiononbe regarded as⌘= 0 and 2 the exterior derivative(15) d when it acts on torsion elements of3 signature 2
H (M5 , (Z )⇢ ). Here (Z )⇢cohomology
signifiesgroups.
the coefficient
The end result issystem
that
2
2 Z
X, Y (2 H (M5 , U (1)))
X,
Y Z)
2H
(M5field(17)are flat
twistedtheelectromagnetic
SL(2,bundle⇢., U(1))
hysicistthatbythe
is de(18) knows1(B/2⇡) (C/2⇡).
(49)
2⇡ Arg ZBdC (M5 ) =Let Maxwell
us first consider
the andcase that
whereassically by theequation,it weis do not have the• Taking X/2! = Y/2! = w2 (i.e. all-fermionED),
we result.
get Indeed, the mixed anomalyThis reproduces
a knownSL(2,Z)
background.
In
this
case,
the
signatureeta
R invariant
onized.the electromagnetic duality S :is known
under(E, B)
!
7
1
1
to
be
of
the
form
2⇡i
(B/2⇡)d(C/2⇡),
for which)=R ZU(1)CS (M3M
2⇡ Arg2⇡ Arg Zfermion (M3 )simplyvanishes,theinvariantcontributes.
⇡i thew2precisew
on withtheArfformulation Re[? ] reduces to3
The
propertiesof theelectromagnetic
dualityin
ZBdC(M5and) =only|ZBdC(Mmathematically
)|e
(18)
5
Eq. (49)
when we only
considerflat fields.
Furthermore, we
call
that
a
quadraticrefinementissimplytheclassical
action31]
heory might not be as well known to canphysicists
take B/2⇡ in= C/2⇡ = w2 , where w2 is the Stiefel5 anomaly)m5=±1(a Zquadratic=1
2 gravevaluatedon flat
B and
C.the
Then
aclassgeneral
Z(X)
Z(S
/Zand
Z(S
/Z
)n element
···
Every
physicist
knows
that
electromagnetic
fieldisdeWhitneyof the=spacetime
regarded
as4 an3 ) refinend in fact are not very well understood ofinHthe
litera2
5 , R/Z) by using Z2 ! R/Z. The Maxwell theory
(19)
ment
can
be
writtenasscribed classically by the Maxwell(Mequation,and that it isR

5

Anomaly of Maxwell theory in terms of 5d top BdC
theory• Now we consider the case when a nontrivial SL(2, ℤ) background ispresent. Note that there are multiple choices of the symm structure,depending on the value of charge-conjugation square C2 (=S4)

(6)onal theory on a torus. The above result for the anomaly, n = c/2, was derived

Z

Anomaly of Maxwellin terms/2⇡).
of 5d top (16)BdC
Arf(q)theory=(X/2⇡)d(Y
2 the choice of spin structure. A simple exa
uch a compactification does depend ontheory(7)

he assumption that there is no spin structure
dependence on the torus. Howeve1

he free 6d Weyl fermion compactified on a torus.
• Nowconsideron
thethecase
whenSL(2, ℤ)
backgroundThe
spin westructures
torus
falla nontrivial
into two classes,
threeeven spinis structu

X,multiple
Y 2 H 2 (M(17)
present. Note that there arechoicesof the symm structure,
5 , U(1))
single odd spin structure. In the even spin structures the2 spinors
have anti-p4)depending
on
the
value
of
charge-conjugationsquareC(=S(8)

oundary condition around some of the directions; the direction does not matter b(9)R
hey can be exchanged by SL(2, Z). In the odd spin structure
the spinors have p

ZBdC (M5 ) = |ZBdC (M5 )|e⇡i w2 w3(18)
oundary conditions in all directions.
• Let’s focus on the case C2 = (−1)F, the ferm number parity. The
Thecorresp
odd spinstructure
is moresymm
structure
is interesting, because after the compactification
orus it(10)leads to massless fermions. Also, it is unique and therefore we expect SL(
spin- Mp(2, Z) := (spin ⇥ Mp(2, Z))/Z2(19)

ct simply. However, there are some changes from the above discussions.

First,
SL(2,should be groupreplacedby its
spin
where
theZ)
metaplectic
Mp(2,
ℤ) is
thedouble
doublecover,
cover definedof SL(2,asℤ)
M p(2, Z) := !S, T | S 2 = (T −1 S)3 , S 8 = 1" .

he one dimensional representations of this group are as in (2.8), except that now

Anomaly of Maxwell theory in terms of 5d top BdC
theory• Canonical examples are 5d lens spaces S5/ℤk, k = 2, 3, 4, 6, where

going around the generator of π1(S5/ℤk) = ℤk comes with the duality
action by an element of order k in SL(2, ℤ)

ory. The natural generalization in this case is to considerR
the bulk theory with the action ⇡i [(B/2⇡)d(C/2⇡)(C/2⇡)d(B/2⇡)],
B andtheory
C are two
2-form ofgauge
Anomaly of where
Maxwellin terms
5d top BdC
fields to be path-integrated over. This action has the SL(2, Z)
theorysymmetry acting on (B, C), which corresponds to the duality• symmetry
the Maxwell
theory.
We canS5and
introCanonicalofexamples
are 5dlens spaces/ℤk, willk = 2,
3, 4, 6, where
ducegoing
the backgroundfield ⇢offorπ1thisSL(2,Z)k comes
symmetry,around thegauge
generator(S5/ℤ
with the dualityk) = ℤ
actionby thatan element
order k induality
SL(2, ℤ)
which
means
there is ofa nontrivial
transformation
when going around a nontrivial loopin spacetime.
spin-Mp(2,Z) (:=The
spinphase⇥Z2 Mp(2, Z)of •thepartition functionis thenObservation:on (some)
S5/ℤk’s, we have the following fact
1
2⇡ Arg ZBdC (M5 ) =

1
= 56⌘
(21)
fermion4 ⌘signature + Arf(q)mod 1even k5 the signature
555⇤dwhere the eta invariant is now for
operator
S /Z2 S /Z3 S /Z4 S /Z6
beddinfi
31
1(Z2 ) ,knows
14 the electromagnetic
physicist
that
acting on the differential
forms tensored
with
and
q
is
⌘signature
0Every
⇢
9
2
9
tions
3
2
2
scribedclassicallybytheMaxwellequation,andfo
H refinement(M5 , (Z )⇢of) the
(Z2natural)
ZtorsionZ
Z1
now the quadratic
pairingon
3
2
1
invariantundertheduality choiceS : (Arf(q)+ 12 the coefficient+ 18electromagnetic0
H 3 (M5 , (Z2 )⇢ ). Here(Z2 )⇢ signifies
system4
1
1
2
1
7
refinem(B,
E).
The
properties
oftheelectromagnetic
dua
Arg
Z
+
+
+
2⇡
2
9
4
18
twisted by the SL(2, Z) bundle ⇢. 1
1
5
35
⌘fermionquantumtheorymight
notbeto elephy
16
144 as well knownfor
Let us first consider the case where
we 9do not 32
have thegeneral, and in fact are not very well understoodWhein
SL(2, Z) background. In this case, the signature eta invariant

X (w/ a spin⇥Zn str) is bordant to Y (w/ a spin⇥Zn str) if @W = X t⌘R mod Z

!
X
1
2X
2
3
Spin ⇥ Znn: str)
↵R :=is bordantn + 3nto+ 2X1 tsX
si2 t
mod
,
Any 5d X (w/ a spin⇥Zt · Z,· · Xn2 t X· · ·Z (wi 1mod6n

Anomaly of Maxwell theory in terms of 5d top BdC
theoryAnomaly of the Electroii

SpinZ2m :

"
#
X
X
X
1
1
2
3
↵
˜ R :=(2mgenerators+ m + 1)s̃i the(mbordism+ 3)
s̃i group
mod Z,m w/
s̃3i
Such X1 and X
calledof(associated
2 are48m
2m
1, 2• Well, it might just be a coincidenceiiChang-Tse Hsieh,iY

1
But we would really like to know whether such an identity holds
for for the Ph
Kavli Institute✓◆1
2 University of Tok
any 5-manifold w/ a spin-Mp(2, ℤ) structure↵ :=n + 3n + 2
s
mod Z
s
mod Z ,
6n
n
2
,

R

2

3

1

for Solid
Physics,exp(
2⇡i⌘))Statepart• Actually, exp( 2⇡i⌘R (X)) =isInstituteaninvertible
TQFT
func (R (Y3of Physics,
5✓
cobordism
invariant)on a (class of) 5dDepartment
closed manifoldwithØThe answer is YES, once we knowthese
1 ⇥ S /ℤk’s are, under⇤ (co)bordism,1↵
˜ associated:=(2m+ m + 1)[Wittens̃(m +3) 16]s̃
mod Z,(m s̃ + s̃ )structure
15,
48m
2m
One
of
our
main
messagesis thWe
considerthe(3+1)-dimensionalMaxw
generators of any 5d spin-Mp(2, ℤ) manifold!
X
S /ℤ
R

2

3

1

3

1

5 1

spin- Mp(2, Z) := (spin ⇥ Mp(2, Z))/Z2
spin-Mp(2, Z) (:= spin
W ⇥Z2 Mp(2,
Y Z)

X

3
sues concerning
the self-dualfi
the spacetime
involves the actionof the dua(20)
field, as wellZ) generalizations.
manifest
themselvesin the cas–X3 –3as its SL(2,
X
1
5/ℤ
q̃
=
2km,
S
3
i
is 56 times
thatof a Weylbackgroundfermion.
Thisforresthenon-trivial
isymmetryX
protected
topological. phase in 4+q̃i = 2k 0 m.
..
on.
i properties of a (5+1)-dimension
sidering the
Before proceeding,weannote
2X
anomaly of the Maxwell theory
plays
imp
S5X/ℤ
4

ity togroupin=thequantumtheor
X is bordant
Y if @W
X tY.
(21)
the lattice Z2 of the electric an
S5X/ℤ2 4
INTRODUCTION
on
the Maxwell theory
on a cX (w/ a spin⇥Zn str) is bordant to Y (w/ a spin⇥Zn str) if @W = X t Y .
analyzed in [20] and. it was int
bordism: [X] = [Y ] if @W = X t Y.. in [21].
gravitational anomaly
Every physicist knows that the electromagnetic field is deAny 5d X (w/ a spin⇥Zn str) is bordantto X1 t X1 t
X2 tdeterminatio
X2 t · · · (w/ theconsideredas· · ·thePart.
func. Z of an IFT, e.g. ZBdC andscribed classically by the Maxwell
equation, and that Z(X)
it is = Z(S 5 /Z3 )m Z(S 5 /Z4 )n · · ·the anomaly.
e(−2$%&), isthea (co)bordism
invariantinvariant ferm
underelectromagnetic
duality S : (E, B) 7!
Our computation shows that= 56⌘fermion

mod 1

be
Since this explanation of (30) requires a lot of informatio
55
5
ate Anomaly
from stringandM-theory,
it
would
be
of
independentinteresofMaxwelltheory
in
terms
of
5d
top
BdC
1didimensions.
It alsoex-ex- structure.
alreadyin [50].
1+1
dimensions.
It alsostructure.This
Thispoint
pointwaswasdirect
alreadyessentiallyessentially madeto checktheex-equality(30)
analysis3+1
and 4+
in 1+1theory
dimensions.
It alsostructure.
This bypointawas
alreadyessentially inmadein [50].
ethe
spin
structure
could
Since
spin
structure
couldbebe
Sincethisthisexplanation
explanationofof(30)(30)requires
requires aa lot
lot of information
information
hy
the
spin
structure
could
be
ons
Since
this
explanationof(30)
requiresa
lotof
information t
dimensions.
Totranslatetheanalysisin
5+1
dimensions
nt necessary
to toformulatefromstringandM-theory,
it
would
be
of
independentinterest
ment
necessary
formulatestringandM-theory,
it it
would
bebeofofindependentinterestWell, it mightjust befrom
afrom
coincidence
nement• necessary
to formulate
stringandM-theory,
would
independentinterestow
mons
theory.
Instudy
3+1
di-ditheoftheMaxwelltheory,weneedtorequirethat
Simonstheory.
In
3+1
to
check
the
equality(30)
byadirectanalysisin
3+1
and
4+1 thto tocheckthetheequality(30)
byby
a directanalysis4+1
rn-Simonstheory.
In
3+1dicheck
equality(30)
adirectanalysis
in
3+1
and
4+1
But
we
would
reallylike
toknow
whethersuchanidentity
holds
for
ory
and
56
Weyl
fermions2
heory
andand56
Weylfermions
dimensions.
Totranslatetheanalysisindimensions
toucdimensions.
ToTotranslatetheanalysis
inin5+1
5+1
dimensions
toTbundleoverMspecified
by
the
SL(2,
Z)
backgroundll theory56
Weylfermions
5
dimensions.
translatetheanalysis
5+1
dimensions
to iany
5-manifoldw/aspin-Mp(2,ℤ)
structurehatis therelation?
How
Whatis therelation?
How
theofoftheMaxwelltheory,
we
need
to
that
the
thestudy
study
thetheMaxwelltheory,wewe
needtotorequire
the
es.
What
is
the
relation?
How
the
study
of
Maxwelltheory,needrequirethatthelaequipped
with
a
spin
structure.
This
means
that
the
symmetr
2
2
sely
the
spin-Z) struc2k 2k
thespin-Z)2kstrucbundle
MM5these
the
SL(2,
Z)Z)
backgroundisis
2 weoverbundleover
specifiedthetheSL(2,backgroundisØThe
answeris) strucYES,T Tonce
know
S5/ℤby
’sby
are,
under
(co)bordism,
eeciselyprecisely
thespin-Z
5specifiedTbundleoverMspecifiedSL(2,background
kby
5
ric
structure
we
in
3+1
and
4+1
dimensions
is a spin
refinement?
One
explaticrefinement?
One
explaequippedwitha aspinstructure.
This
means
that
symmetryequippedwithspinstructure.
This
means
that
draticrefinement?
One
explageneratorsofany
5d consider
spin-Mp(2,ℤ)
manifold!
equippedwitha spinstructure.
This
means
thatthethesymmetry
symmetry
ngly, bybysupersymmetricrisingly,
supersymmetricerstructure
weconsiderininto3+1
and
4+1
dimensions
isisa aspinspinstructure
wewe
considerand
4+1dimensions
urprisingly,
bysupersymmetric
Mp(2,Z) structure.
Accordingthecobordism
classificatio
structure
consider
in3+1
3+1
and
4+1
dimensions
spinosnks
thebetterunderto therecentbetterunderZ)Z)structure.
Accordingthecobordism
torecent
the
recentbetterunder- Mp(2,Mp(2,structure.
Accordingthethecobordism
classificationMp(2,Z)
structure.
Accordingtocobordism
classification
theorem[51–54],theanomalyofanytotosystemwithclassification
this
symme
Math
fact:
all
spin-Mp(2,ℤ)
5-mflds
areclassifiedbyanabeliangroup
theorem[51–54],theanomalyofanysystemwithsymme.
theorem[51–54],theanomalyofZ)
anysystemtheorem[51–54],theanomalyofanysystemwiththisthissymmesymmespin-Mp(2,spin-Mp(2,Z)
Its
spin-Mp(2,Z)
spin-Mp(2,tryis classified
Hom(⌦, U(1))
==ZZ
Z
in
5+1
dimensions.
ItsItsIts trytryby
istry
by
,Z)
5Hom(⌦
ldin
5+1
dimensions.
field
in
5+1
dimensions.
isclassifiedclassified
byHom(⌦,U(1))
U(1))
isclassifiedby
Hom(⌦, U(1))=
Z
9 99 ZZ
3232 32
55 5
32
spin-Mp(2, Z)
spin-Mp(2,Z)
spin-Mp(2,spin-Mp(2,Z) Z)
riserise
torisethethethehegivestotheMaxwell
theZ
.
Here
⌦
==Z
Z
the
group
es
to
MaxwelltheZMaxwell
.
Here
⌦
=
Z
bordism
grou
Z
.
Here
⌦
=
ZZ322ZZis
isthebordism
bordismgroup
group
Z
.
Here
⌦
Z32
Z32
is
the
2
99Z9 Z
22Zis
2
2the
5
2
32
2
9
5
5
5
structiongeometrizes
thethethe forforclosed
s construction
geometrizes
5-manifolds
withspin-Mp(2,Z)
structuresandisis ionstruction
geometrizesforclosed5-manifolds
withspin-Mp(2,structuresand
andandclosed5-manifolds
withspin-Mp(2,
Z)Z)
is
the
for
closed5-manifolds
withspin-Mp(2,Z)
structures
5 5S 5 /Z 5, 5S 5 /Z , and [(S55 5 /Z 00 )0 + 9(S
5 5 /Z )], reMaxwell
theory.
Corre- generated
Maxwelltheory.
Corregenerated
by
by
S
/Z
,
S
/Z
,
and
[(S
/Z
)
+
9(S
/Z
reetheMaxwelltheory.
Corregenerated
by
S
/Z
,
S
/Z
,
and
[(S
/Z
)
3
4
4
4 re3
4
4
4 )],
generators:
3
4
40
5
5
5
5
regenerated
by Sspectively,/Z/Z(S4(S4,(Sand/Z
9(S8 8structure/Z
4 0 0 [(S
0
4/Z
3, S
4) +
4 )], reonal
BdC
theoryBdC
theoryonon
Monspectively,
where
)has
has
naturalspin-Z
structure
5 cou- spectively,where)4 )4hasthenaturalspin-Z
structurel BdC
theoryM
5 couwhere/Z4/Z
thethenatural
5Mcou4from the0 embedding55S 5 /Z ⇢ 3C3
e dimensional
reduction
the coming
oucoming/Zwithdifensional
reduction
ofof
theof
spectively,(S/Z)embedding
has thenaturalspin-Z
from
the
SS /Z
with
difimensional
reduction
thewhere4
4 butcomingfromthe/Z44 ⇢⇢CC3/Z/Z
a adif44 but
4embedding
8 structur
2 2T 2 bunyn 7on
M
,
which
is
the
ferentactiondiag(i,i).We
Wehavehavenot
notdidi73i, i).
M
,
which
is
the
T
4 givenactionofofZofgiven
diag(i,
i,C
i,i,
diM
,
which
is
the
Tbunbun- ferent
ferentaction
Z4 4Z
given
byby
diag(i,
i,
i,
i).
the 7
comingfromtheembedding
S 5by
/Z
⇢
/Z
but
with
a dif
4
4

sues
mani
= 56⌘fermion mod 1(21)
Anomaly of Maxwell theory in terms of 5d top BdC the non.
theoryBe
Therefore,
ity g
1
1
Arg
Z
(M
)
=
56
⇥
the laBdC
5
2⇡
2⇡ Arg Zfermion (M5 ) mod 1
(22)
on th
is true on any 5-manifold w/ a spin-Mp(2, ℤ) structure
analy
[X] =
] ifduality@W =ofXthet Maxwell
Y
Namely,bordism:
the anomaly
of [YEM
theory is 56 times gravi
physicist
knows that the electromagnetic field is dethat of aEvery
4d chiral
fermionconsi
scribed classically by the Maxwell equation, and that it isthe ainvariantunderelectromagnetic
duality
S : Where(E, B)does
7! this Ou
It is still
abstract(andthesomehowmysterious),
however.
(B,56E).
The
properties of the electromagnetic duality in thenumber
come
from?
theor
quantummight
not thebe property
as well knowninØ Weprovidetheory
an answer
using
of someto6dphysicists
SCFT, known
cise
and infact arenot very wellunderstoodin theliteraasgeneral,
the E-string
theory[Ganor-Hanany(96);
Seiberg-Witten
(96)]
come
ture. This is particularly true when going around a nontrivial
5+1
1
path in the spacetime results in a duality transformation. In
knowOutline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions

2d
2. 4d
3. 6d
1.

• Conclusion

is
ell
ote
ed
reto

Chang-Tse Hsieh,1, 2 Yuji Tachikawa,1 and1Kavli Institute for the Physics and Mathematics
University of Tokyo, Kashiwa, Chiba 27
2
Institute for Solid State Physics, University of Tokyo, Ka
3
Department of Physics, Tohoku University, Se

Anomaly of Maxwell theory in terms of 6d SCFTThe E-string theory has two branches of vacua, the tensor branch andWe consider the (3+1)-dimensional Maxwell theory in the situati
the Higgs branch
the spacetime involves the action of the duality transformation exch
ØOn the Higgs branch theEasis SL(2,Higgsedto E7, whichon the28anomalfield,
well as itsZ) generalizations.
Weactsfind that
8 symm56 times that ofrepa Weyl fermion. This result is derived in two infermions via its 56 dimisfundamental
symmetry protected topological phase in 4+1 dimensions characteriØWhen one moves to the tensor branch, the E8 symm is restored and asidering the properties of a (5+1)-dimensional superconformal fieldself-dual tensor field appears
anomaly of the Maxwell theory plays an important
4 role in the consis
Tensor branch
5+1d
T2
3+1d

∪
Self-dual tensor

E-string
INTRODUCTION

Higgs branch∪
28 fermions

1
1
2⇡ Arg ZCdC (M7 ) = continuous
8 ⌘signature + Arf(q) = 28⌘Dirac
Maxwell
56 fermions
1=deformation
28 ⇥ 2⇡Arg Zferm (M7 )indep of ⌧ , ⌧¯

[Hsieh-Tachikawa-Yonekura (20)]

(1)

Maxw
MaxwellI
M
M
M
We
W

is
ell
ote
ed
reto

Anomaly of Maxwell theory in terms of 6d SCFT
By compactifying this system on T2, one finds that one Maxwell field is
continuously connected to 56 chiral fermions, showing that they shouldhave the same anomaly. The EM duality is formulated as the SL(2, ℤ)
acting on this T2

4
Tensor branch
5+1d∪
Self-dual tensor

T2
3+1d

E-string

Maxwell

Higgs branch∪
28 fermionscontinuous
deformation

56 fermions• So far I have discussed the anomaly of SL(2, ℤ) duality of one

version of Maxwell theory (i.e. all-fermion electrodynamics).
• It is interesting to find out anomalies of any subgroups of SL(2,ℤ) in other versions of Maxwell theory.

We consider the (3+1)-dimensional Maxwell theory in the situation where going a
the spacetime involves the action of the duality transformation exchanging the electrifield, as well as its SL(2, Z) generalizations. We find that the anomaly of this system in
is 56 times that of a Weyl fermion. This result is derived in two independent ways:
symmetry protected topological phase in 4+1 dimensions characterizing the anomaly
sidering the properties of a (5+1)-dimensional superconformal field theory known asanomaly of the Maxwell theory plays an important role in the consistency of string theHere I listed the result w/o details:

theory,symm str

S
ST
None charge conj. ZC
full EM duality SL(2, Z)
2 S-duality Z4 ST-symm. Z3o
3 + 1dMaxwell000Z9Z9spin×G(20)
MaxwellI
0 spin- Mp(2,Z2 Z) := (spinZ4 ⇥ Mp(2,- Z))/Z2
sodualityII
0
Z2(G=none) Maxwell
techniq
III&
Maxwell0Z2
spin-Mp(2,Z) (:= spin ⇥Z2 Mp(2, Z)IV
One ofspin-Gf
Z2
Z2
Z4
Z9
Z36
Maxwellspin×G Weyl, G ⇥ Zf2 0
0
Z4
Z9
Z36
sues confWeyl, G
0
Z16
Z32
Z9
Z288
spin-Gf
manifes= 56⌘fermion

mod 1

(21)

anomaly“-”: noI. symm
TABLE
Partition functions“0”:and norelateddata on S 5 /Z“ℤ. # ”: mod-k anomaly

the non(p1
on.
Anomalies of 4d Weyl ferm under the same symm was determined in [Hsieh (18)],
and we have the following
result in general (on either spin×G or spin-Gf mflds) Befo
INTRODUCTION
ity grou
1
Arg
Z(M
U(1)CS
3) =
1
1
2⇡
Arg
Z
(M
)
=
56
⇥
Arg
Z
(M
)
mod
1
the latti
TK : No2⇡onsite rep.
BdC
5
fermion
5
2⇡
TK : ! i y(22)
on the✓Z
analyzekOutline• Maxwell theory × EM duality × Anomaly• Anomalies: self-dual fields vs. chiral fermions(1+1)d2. (3+1)d3. (5+1)d1.

• SummarySummary
We consider various versions of 4d Maxwell theory and their duality
symmetries, and compute the corresp ’t Hooft anomaliesØIn particular, we foundAnomaly of duality symm of Maxwell = 56 times that of a chiral fermion
Ø The interpretation is twofold: one is by the 5d bulk SPT(top. BdC theory) phase characterizing the anomaly, andthe other is by the properties of a 6d SCFT (E-string theory)

BdC Maxwell
theory theory

Ø Our result reproduces, as a special case, the known anomaly of the all-fermion electrodynamics discovered in the last few years

Thank you for your attention!