Discrete Gauge Anomalies Revisited
Chang-Tse Hsieh
Kavli IPMU & Institute for Solid States Physics

Gong Show presentation, Strings 2018, OIST
June 25, 2018

Anomalies in chiral gauge theories• Cancellation of gauge anomalies – in a chiral theory such as thestandard model – is a fundamental constraint on a consistent
quantum field theory.

A U(1) chiral gauge theory is anomalous if the anomaly
cancellation condition

is not satisfied. Here {"# } and {"% } are U(1) charges of Weylfermions.

ny closed five-manifold X ⇤ endowed with spin and Zn structuresAnomalies
in chiral gauge theories) = ,= 12 (15iiX• Cancellation of gauge
anomalies – in a chiraltheory such as thei↵qi//LWeyl =@ + qi A) i ,i !ei
standard model– iis ai (ifundamental
constraint
on a consistent
quantum field theory.
LWeyl =X/ i,i i@i !e2⇡isi /ni• A U(1) chiral gaugei theory is anomalous if the anomaly

(26)

(27)

cancellation condition

Purely gauge :

XqL3

left

Mixed gauge and grav :

Xleft

X3qR
=0rightqL

XqR = 0(28)

right

****
is not satisfied. Here {"# } and {"% } are U(1) charges of Weyl
a compact connected semi-simple Lie group G, the structure of perturbative gauge andfermions.
auge-gravitational anomalies in the theory has been well understood for a long time (there
re gravitational anomalies in four dimensions). [? ]Q: While anomalies of cont. symm are well understood, how about
the case of gauge anomalies associated with discrete symm?

Q: While anomalies of cont. symm are well understood, how about
the case of gauge anomalies associated with discrete symm?
ØIn this case, there are only global (non-perturbative) anomalies, andone can not use a “usual method” to calculate them

Q: While anomalies of cont. symm are well understood, how about
the case of gauge anomalies associated with discrete symm?
ØIn this case, there are only global (non-perturbative) anomalies, andone can not use a “usual method” to calculate themØIn a paper by Krauss and Wilczek (1989), they also mentioned
VOLUME 62, NUMBER 11

PHYSICAL REVIEW LETTERS
Discrete Gauge Symmetry in Continuum
Lawrence M. Krauss

13 MARCH 1989

Theories' and Frank Wilczek

of California at Santa Barbara, Santa BarbaraC, alifornia 93I06(Received 16 December 1988)PHYSICAL REVIEW LETTERS
11VOLUME 62, NUMBER

Institute for Theoretical Physics, University

We point out that local symmetries can masquerade as discrete global symmetries to an observer
mention
there local
are gauge
discrete
how-Lazarides andinvariance can, 'G.
The existence First,of the underlying
low-energy twoprobes. caveats.
equipped with only
2A review containeA'ects. withtherefore carry discrete gauge
Aharonov-Bohm-type
Black holes
can anomalies—
those associatedever, result in observable
symmetries
global"Neither black-hole
wormholes, norof nonclassicala form thatevaporation,
charges
Phys.elseRep. 150, 1 (1cannot "hair.
be consistently
Identificationof anythinggauged.
can a review of t
can violate discrete gauge symmetries. In supersymmetric unified theories such discretesuch anomalies is a di%cult but well developed art, '" intosymmetries For
forbid proton-decay amplitudes that might otherwise be catastrophic.
ject, with references,which we shall not enter here. Second, it is not quite
nia at Santa Barbara
PACS numbers: 11.30.— 13.30.Ce, 97.60. Lf——true j,that the identifications we envisage in field space are
Advanced Study ins
the discrete transformations, in most inlocally triviallished).
teresting cases, have fixed points, leading to conical
4See especially Gconto
Although it is a common and fruitful practiceiscal
astatement
thatvariables
theused in asymmetry
It is conceivable that these singularities
singularities.
1524(tobepublishe
sider local gauge invariance under discrete groups in latIn la.nguage that may
bephysical theory are redundantlead to subtle problems that have not yet been discerned.

—

Q: While anomalies of cont. symm are well understood, how about
the case of gauge anomalies associated with discrete symm?
ØIn this case, there are only global (non-perturbative) anomalies, andone can not use a “usual method” to calculate themØIn a paper by Krauss and Wilczek (1989), they also mentioned
VOLUME 62, NUMBER 11

PHYSICAL REVIEW LETTERS
Discrete Gauge Symmetry in ContinuumLawrence M. Krauss

13 MARCH 1989

Theories' and Frank Wilczek

of California at Santa Barbara, Santa BarbaraC, alifornia 93I06(Received 16 December 1988)PHYSICAL REVIEW LETTERS
11VOLUME 62, NUMBER

Institute for Theoretical Physics, University

We point out that local symmetries can masquerade as discrete global symmetries to an observer
mention
there local
are gauge
discrete
how-Lazarides andinvariance can, 'G.
The existence First,of the underlying
low-energy twoprobes. caveats.
equipped with only
2A review containeA'ects. withtherefore carry discrete gaugeAharonov-Bohm-type
Black holes
can anomalies—
those associatedever, result in observable
symmetries
global"Neither black-hole
wormholes, norof nonclassicala form that
evaporation,charges
Phys.elseRep. 150, 1 (1cannot "hair.
be consistently
Identificationof anythinggauged.
can a review of t
can violate discrete gauge symmetries. In supersymmetric unified theories such discretesuch anomalies is a di%cult but well developed art, '" intosymmetries For
forbid proton-decay amplitudes that might otherwise be catastrophic.
ject, with references,which we shall not enter here. Second, it is not quite
nia at Santa Barbara
PACS numbers: 11.30.— 13.30.Ce, 97.60. Lf——true j,that the identifications we envisage in field space are
Advanced Study ins
the discrete transformations, in most inlocally trivial
lished).
a difficult
but wellart!to conicalfixed developed
terestingcases, havepoints, leading
4See especially G
conto
Although it is a common and fruitful practiceiscal
astatementthatvariables
theused in asymmetry
It is conceivable that these singularities
singularities.
1524(tobepublishe
sider local gauge invariance under discrete groups in latIn la.nguage that maybe
physical theory are redundantlead to subtle problems that have not yet been discerned.

—

Q: While anomalies of cont. symm are well understood, how about
the case of gauge anomalies associated with discrete symm?

• For example, how do we couple Weyl fermions consistently to a(topological) ℤ" gauge theory in 4d?

Q: While anomalies of cont. symm are well understood, how about
the case ofX
gauge anomalies associated with discrete symm?
/ + qch,i A)/ ch,i ,ch,i (i@Lchiral =i,ch=L,Rch,i ! ei↵qch,ich,i• For example,
X how do we couple Weyl fermions consistently to a2⇡isi /nLLeft =@/ L,i , theorysi 2 Zn(topological)ℤL,i" igaugein e4d?
L,i !
L,i ,iØIn some cases, we might be able to write down such a theory asZ X
i/ + qi A)/ i+i (i@in2⇡Z

he introductory paragraph ********ipn
B ^ dA +4⇡ZB^B(1(1(1[Kapustin-Seiberg 14]

it is argued that the anomaly constraints on Zn charges of a set of massless ch
ns in 3+1 dimensions can be deduced by embedding the Zn gauge symmetry inoup; these constraints are derived basing on the anomaly cancellation condition
metry, which involve the perturbative gauge and mixed gauge-gravitational anoma

Q: While anomalies of cont. symm are well understood, how about
X discrete symm?
the case ofX
gauge anomalies associated with
L/chiral =/(i@+qch,i A) ch,i ,ch,iLchiral =i,ch=L,R/ ch,i ,(i@/ch,i+ qch,i A)
i↵qch,i!
ech,ich,ii,ch=L,Rch,i(1X2⇡isi /ni↵qch,i/• For example,couple
Weylconsistentlytoa=A)i@,!
e/fermions
Lchiral(i@/L+Left
qch,i,!
eL,iL,iL,
X =how do wech,iL,ich,ich,ich,i2⇡is/nLLeft =@/ L,i , theory(1(topological)ℤL,iin e4d? i i L,i , si 2 ZnL,i !
i,ch=L,R" igaugeXiZ toZ asØIn some cases, wemight be ablewrite down such a theoryXXinip2⇡is/niA)///(i@+q+B^
dA+si 2 Zn
iL,i ,L,i ,L,i i! e Zi
L,i i@ Z
Z X LLeft =2⇡
4⇡in
ipnii/ + qi A)/ i+B ^ dA +B^B
(1i (i@2⇡4⇡
Z XZ
Ziin
ipn
//Spin(4)⇥^Z14](i@+qA)+B^
dA+B[Kapustin-SeibergnBiii2⇡4⇡
he introductory paragraph ********iØThere are two kinds of ℤ" chiral gauge theories, depending on theit is argued
the anomalyon⇥ZZof a set
massless chn ncharges⇥ Zofsymm thatof ferm,
which canconstraints
be Spin(4)or (Spin(4)2m )/Z2
ns in 3+1 dimensions can be deduced by embedding the Zn gauge symmetry in******* The introductory
paragraph ********“untwisted”“twisted” condition
oup; these constraints are derived basing
on the anomaly cancellation(Spin(4)⇥ Z2m)/Z2gauge-gravitational anoma
metry, which involve the perturbative
gauge andmixed

Previous works• There have been several attempts to tackle this problem, such asthe works by Ibanez-Ross (91), Banks-Dine (91), CsakiMurayama (97), Araki et al. (08), etc.
• Let’s review some of these works

Ibanez-Ross
Their argument (only for untwisted ℤ" symm):
U(1) anomaly cancel. cond.
Zn anomaly cancel. cond. =+charge constraints on massive states through SSB of U(1)

References

Ibanez-Ross (BZ ) andDetails of the derivation ofSpin
5n5Spin (BZn )1. Derivation of Eq. (2.14)

Their argument(only andfor untwistedℤ" symm):
2. Derivationof Eqs. (2.21)(2.23)
U(1) anomaly cancel. cond.
Zn anomaly cancel. cond. =1.

+INTRODUCTION

charge constraints on massive states through SSB of U(1)************************The result (a necessary cond.):X

nqi3 = pn + r3,8X
nq i = p0 n + r 0 ,2iip, r 2 Z; p 2 3Z if n 2 3Z,p0 , r0 2 Z.

Contribution from Dirac and Majorana masses, respectivelyX

s3 = 0 mod a ,Banks-Dine
Comments on Ibanez-Ross:

Banks-Dine
Comments on Ibanez-Ross:• Only the linear constraint should be satisfied
Ø It can be argued by considering the violation of the low energy Zn
symm in the presence of a grav instanton which is a spin manifold

Banks-Dine
Comments on Ibanez-Ross:• Only the linear constraint should be satisfied
Ø It can be argued by considering the violation of the low energy Zn
symm in the presence of a grav instanton which is a spin manifold• The nonlinear (cubic) constraint might be too restrictive andmight not be required for consistency of the low energy theory
Ø It is not solely from the low energy considerations and would
depend on assumptions about UV embedding theories

Csaki-Murayama
Argument by ’t Hooft anomaly matching. Two types of discrete
anomalies are involved:

Csaki-Murayama
Argument by ’t Hooft anomaly matching. Two types of discrete
anomalies are involved:• For Type I anomalies, the matching conditions have to bealways satisfied regardless of the details of the massive boundstate spectrum.

Csaki-Murayama
Argument by ’t Hooft anomaly matching. Two types of discrete
anomalies are involved:• For Type I anomalies, the matching conditions have to bealways satisfied regardless of the details of the massive boundstate spectrum.
• The Type II anomalies have to be also matched except if there

are fractionally charged massive bound states in the theory.

Our approach• Here we revisit this problem from a more modern perspective

based on the concept of symmetry protected topological (SPT)phases (from condensed matter physics) and also from a refined
definition of global anomalies by Witten (2016)

Our approach• Here we revisit this problem from a more modern perspective

based on the concept of symmetry protected topological (SPT)phases (from condensed matter physics) and also from a refined
definition of global anomalies by Witten (2016)• Our approach is based on geometrical and topological considerations

Our approach• Here we revisit this problem from a more modern perspective

based on the concept of symmetry protected topological (SPT)phases (from condensed matter physics) and also from a refined
definition of global anomalies by Witten (2016)• Our approach is based on geometrical and topological considerations

ØWe compute the ’t Hooft anomaly of ℤ" (global) symm, deduced by
the consistency of formulating the theory on a generic manifold w/ auntwisted/twisted spin structure and a background ℤ" field

Main result• The anomaly cancel. cond. (necessary & sufficient) ofuntwisted/twisted ℤ" symm:

i (i@ + qi A) i +i

/ L,i ,L,i i@2⇡

L,i ! eB ^ dA +2⇡isi /nXL,i ,s3i = 0B^B

4⇡

(1.14)

si 2 Zn(1.13)

mod anSpin(4)Main
ZresultZX ⇥Zinipni(1.15)

ni =+0 modBb^
n B/ i++ qi A)
B ^ sdA(1.14)
2⇡
4⇡i• The anomaly
cancel. cond. (necessary & sufficient) of

untwisted/twisted(Spin(4) ⇥ Z2mℤ)/Z" 2symm:
Spin(4) ⇥ Zn2(n + 3n + 2)

(1.16)

si 2 ZX

s3i = 0(1.15)(1.19)

mod 6niXX
X2(Spin(4) ⇥ Z2m )/Z3 2= 0 +mod2m + m + 1) 2 s̃ si (m3) n s̃ = 0ii(1.18)iX
X32+ 3n + 2)
sis3i==00 modmod 6nani(1.17)
mod 48m (1.16)s̃i = 0

mod 2miX(1.20)

ii i

X
Xmod nbni =00 mod2
sis=s̃i 2 2Z + 1iiuctory paragraph ********(1.18)(1.17)(1.21)

i (i@ + qi A) i +i/ L,i ,L,i i@2⇡

L,i ! eB ^ dA +2⇡isi /n

X

L,i ,

s3i = 0mod anSpin(4)Main
Zresult
ZX ⇥Zinipn
ini =+0B ^ sdA

B(1.14)
i ^B
X
00nq=
pn+r, (1.13)
p0 , r0 2 Z.
si 2 Zn i24⇡i(1.15)

XnmodBb^n B q 3 = pn + r, (1.14)
p, r 2 Z; p 2 3Z if (1.18)
n 2 3Z,i
84⇡
i3/ i++ qi A)
2⇡i• The anomaly
cancel. cond. (necessary& sufficient) ofX

untwisted/twisted(Spin(4) ⇥ Z2mℤ)/Z" 2symm:Spin(4) ⇥ Zn2(n + 3n + 2)

Xinq i = p0 n + r 0 ,2si 2 Zs3i = 0(1.15)

X

mod 6niXXX⇥ Z2m2 )/Z= 0 +mod2m2(Spin(4)+ m + 1)s̃3i s2i (m3) niii

XX32+ 3n + 2)
sis3i==00 modmod 6nan0p0 , r(1.16)
2 Z.

Xn
s3i = pn + r(1.19)
3s̃i = 08Xn0= p(1.17)
n + r0
mod 48msi (1.16)
2i
consistent w/ Ibanez-Ross cond!

s̃i = 0

mod 2mi(1.20)ii i

XX
mod nbni =00 mod2
sis=s̃i 2 2Z + 1i
i

uctory paragraph ********(1.18)(1.17)(1.21)

B ^ dA(Spin(4)+(1.14)i Z^⇥B)/Z22mB4⇡
s=0modbXi i↵qch,in0i
2⇡isi ,/n0n0 0//(i@+qA)!
e(1.12)/q=pn+r,p, r 2 Z.
ch,ich,ich,ich,ii@,!
e,s2
Z(1.13)ch,ii
inL,iL,iL,iL,ii2
X=L,R

X

i (i@ + qi A) i +(1.12⇡s3i = 0mod an

MainresultinipniZn2 + 3nZ + 2)
Z Spin(4)X ⇥ (niX3(1.15)si = 0 mod 6nX2⇡isi /niXn3/s=0modb3
i@,!
e,s2Zs2ZininL,iL,iL,iX/ i++ qiL,iA)
B ^ dA +B i^ B qi = pn + r , (1.14)
p,(1.13)r 2 Z; p 2 3Z if (1.18)
n 2 3Z,82s=
0modn(1.12⇡4⇡
iii• The anomaly cancel. cond. (necessary& sufficient)
ofiXn0qi = p0 n + r0 , p0 , r(1.16)
2 Z.
untwisted/twistedsymm:(Spin(4))/ZZ ⇥ Z2mℤ
Z"22inipni/ i+(i@/ + qi A)
B ^ dAX+ XB^B X(1.14)
23Spin(4)⇥Z(1.15)s2Z(1.19)3ni
4⇡=0mod as̃ni = 0 mod 48m(2m2⇡ + m + 1)s̃i si(m+ 3)iii
XXn3
X
323X
si = pn + r(n + 3n + 2)
si = 0 mod s6n(1.1i = 0 mod bn8s̃i = 0i mod 2mSpin(4)Zn(1.15) nX⇥XXi
02(Spin(4) ⇥ Z2m )/Z3 2(1.16)
n + r0= 0 +mod
2m + m + 1) 2 s̃ si (m3) n s̃i = 0 mod 48msi = p(1.17)

2iiXiii

X si 2 Zs̃i = 0X(Spin(4)s̃i i2 2Z + 1X3⇥ Z2m )/Z23+ 3n + 2)
sisi==00 modmod 6nanXi i2consistent w/ Ibanez-Ross cond!
ii

mod 2mX(1.16)

X (2mparagraphX
23The introductory********+m+1)s̃(m + 3)s̃i = 0 mod 48mimod nbn(1.18)
i =00 mod2 X
sis=(1.17)
ii(n2 + 3n + 2)i ii

2Z +s3i =s̃0i 2mod6n1

Xuctory paragraph2s ********= 0 mod nXis̃i = 0mod 2m(1.17)(1.1(1.20)(1.21)(1.2Conclusion• We propose a new formula for evaluating the anomalies (andthe corresponding cancel. cond.) of an underlying chiral gauge
theory.
• While agreeing with previous works by Ibanez and Ross and by

Csaki and Murayama using anomaly matching argument, ourresult provides, from a purely low-energy perspective, a more
complete aspect of discrete symmetry anomalies, respecting theviewpoint in the work of Banks and Dine.

Thank You!