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Anomalies and topological phases in relativistic QFT Overview Kazuya Yonekura Kyushu University 1

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Page 1: Anomalies and topological phases in relativistic QFT Overviewevents.kias.re.kr/ckfinder/userfiles/201901/files/Lecture... · 2019-01-14 · 2/ 77 These lectures are about symmetries

Anomalies and topological phases in relativistic QFT

Overview

Kazuya Yonekura

Kyushu University

1

Page 2: Anomalies and topological phases in relativistic QFT Overviewevents.kias.re.kr/ckfinder/userfiles/201901/files/Lecture... · 2019-01-14 · 2/ 77 These lectures are about symmetries

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These lectures are about symmetries and anomaliesin quantum field theory (QFT).

Recent developments are stimulated bytopological considerations in condensed matter physics:Classification of topological phases of matter.

However, I will just focus on relativistic QFT relevant for high energy physics (particles physics & strings).

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This lecture: Brief overview (but biased by my own preference)

Later lectures: More specific topics (Some topological properties of gauge theories)

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Contents

4

1. Introduction

2. Symmetry

3. Anomaly

4.Applications: String theory

5. Applications: Strong dynamics

6. Summary

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Symmetry and anomaly

5

Usual symmetries and perturbative anomalies arewell-treated in standard textbooks in QFT and stringtheory.

famous triangle diagram in 4-dim.

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Anomalies: examples

6

Anomaly of Global symmetry: [’t Hooft, 1979]Very convenient for the study of dynamics

Anomaly of Dynamical Gauge symmetry: Need to be cancelled for a consistent theory

• Selection of possible gauge non-abelian groups , in 10d supergravity

• Chiral symmetry in QCD:

E8 ⇥ E8 SO(32)

• Anomaly cancellation in SU(3)⇥ SU(2)⇥ U(1)

SU(Nf )L ⇥ SU(Nf )R

Remark: These two types of anomalies will not be distinguished.

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Symmetry and anomaly

7

What is symmetry?

A textbook answer

�(x) : fields S[�] : action

Symmetry means that the action is invariant undertranformation

�(x) ! g · �(x)

S[g · �] = S[�]

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Symmetry and anomaly

8

What is anomaly?

A textbook answer

S[�]The classical action is invariant under transformation of fields

But quantum mechanically it is violated (e.g. by path integral measure).

�(x) ! g · �(x)

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Symmetry and anomaly

9

The concepts of symmetry and anomaly are organized and generalized more and more in recent years.

I will review some of those developments.

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Contents

10

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: String theory

5. Applications: Strong dynamics

6. Summary

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Symmetry

11

What is symmetry in modern understanding?

Actually I don’t know how to treat them most generally.

The terminology “symmetry” is not appropriate in some of generalizations.

I feel more abstract language is necessary for aunified treatment of several generalizations.

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Symmetry

12

• Usual symmetry (continuous, discrete, spacetime)

• Higher form symmetry [Kapustin-Seiberg 2014, Gaiotto-Kapustin-Seiberg-Willett 2014]

• 2-group [Kapustin-Thorngren 2013, Tachikawa 2017, Cordova-Dumitrescu-Intriligator-2018, Benini-Cordova-Hsin2018]

• Duality group [Seiberg-Tachikawa-KY 2018]

• Topological defect operator [Bhardwaj-Tachikawa 2017, Chan-Lin-Shao-Wang-Yin 2018]

• …

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Some properties

13

Let me describe some properties of some of them.

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Some properties: Topology

14

Let us recall the case of usual continuous symmetry.

rµJµ = 0

: conserved current: conservation equation

In the language of diffential forms, it is more beautiful.J := Jµdx

µ

⇤J =1

(d� 1)!Jµ✏µµ1···µd�1dx

µ1 ^ · · · dxµd�1

d(⇤J) = 0 : conservation equation

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Some properties: Topology

15

: charge operator

is invariant under continuous deformation of by

⌃ : codimension-1 (dimension ) surface

d(⇤J) = 0

Q(⌃) =

Z

⌃⇤J

Q(⌃)

• Stokes theorem• is closed: ⇤J

In this sense, is topological.Q(⌃)

d� 1

This is total charge on the surface ⌃

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Some properties: Topology

16

⌃0

Q(⌃) = Q(⌃0)d(⇤J) = 0Stokes &

topological(charge conservation)

Q(⌃) =

Z

⌃⇤J

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Some properties: Topology

17

• It is topological in the sense that it is invariant under continuous deformation of the surface ⌃

• The operator exists for each group element

Symmetry operator:

ei↵ = g 2 G

U(⌃,↵)

U(⌃,↵) = exp(i↵Q(⌃)) = exp(i↵

Z

⌃⇤J)

g : element of group: element of Lie algebra

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Some properties: Topology

18

So the usual symmetry is implemented by operators

U(⌃, g) : topological operator

⌃ : surface

g : “label” of the operator (group element)

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Some properties: Topology

19

U(⌃, g) : topological operator

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Some properties: Topology

20

Q: Does it need to be an exponential of ? A: No. Discrete symmetry has without .

QU Q

U(⌃, g) : topological operator

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Some properties: Topology

21

Q: Does it need to be an exponential of ? A: No. Discrete symmetry has without .

Q: Does the surface need to be codimension-1?A: No. Higher form symmetry uses higher codimension .

QU Q

⌃⌃

U(⌃, g) : topological operator

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Some properties: Topology

22

Q: Does it need to be an exponential of ? A: No. Discrete symmetry has without .

Q: Does the surface need to be codimension-1?A: No. Higher form symmetry uses higher codimension .

Q: Do we need group elements ?A: No. Topological defect operator is just topological without any group.

QU Q

⌃⌃

U(⌃, g) : topological operator

g 2 G

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Some properties: Topology

23

Remarks:

• Sometimes these operators cannot be written explicitly in “elementary ways” by using fields.

• These operators are sometimes described by abstract mathematical concepts such as fiber bundles, algebraic topology, and so on.

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Some properties: Background

24

Quite generally, operators can be coupled tobackground fields.

The most basic case of an operator coupledto a background field

O(x)A(x)

Z[A] = hexp(iZ

A(x)O(x))i

: called generating functional or partition function.I will use the terminology partition function.

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Some properties: Background

25

For the current operator , we have a backgroundgauge field . The coupling between them isAµ(x)

Jµ(x)

ZddxAµJ

µ =

ZA ^ ⇤J

In a similar (but more abstract) way, the symmetrycorresponding to can be coupled to background fields.

U(⌃,↵)

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Some properties: Background

26

Z[A]

I will write

A : abstract background fields for the symmetry U

: partition function in the presence of the background fields A

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Some properties: Background

27

Examples of background field :

• For discrete symmetry ,A

G

: principal bundle G

• For higher form symmetry such as p-form symmetry with abelian group G = ZN

(cohomology group)

• Parity, Time-reversal symmetryA : non-orientable manifold (e.g. Klein-bottle)

Hp+1(M,ZN )A 2

A

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Some properties: Background

28

Remarks:

• As the previous examples show, abstract description of the background fields requires mathematical concepts from topology and geometry.

• But if you don’t like mathematics, some simple cases can be treated in elementary ways.

• For example, p-form field = BF theory. E.g. [Banks-Seiberg, 2010]

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Contents

29

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: String theory

5. Applications: Strong dynamics

6. Summary

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Anomaly

30

What is anomaly in modern understanding?

There is now a way which is believed to describealmost all anomaly.

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Anomaly

31

Z[A]

A : abstract background fields for the symmetry

: partition function in the presence of the background field

: -dimensional spacetime manifold (part of background fields for spacetime symmetry)

M d

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Anomaly

32

• First of all, anomaly means that the partition function is ambiguous.

• However, if we take a (d+1)-dimensional manifold whose boundary is the spacetime and on which the background fields are extended, then the partition function is fixed without any ambiguity.

NM

A

MN @N = M

Z[A]

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The dependence on : anomaly

Anomaly

33

MN

@N = M

Z[N,A]A

: no ambiguity, if we are given and extension of into N

N

N

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Anomaly

34

This description of anomaly may look quite abstract,but there is a very natural motivation from condensed matter physics / domain wall fermion.

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Topological phase

35

Some material:Topological phase

(Manifold )N

Anomalous theory on the surface/domain wall(Manifold )M

• Cond-mat/Lattice systems are not anomalous as a whole.

• It is anomalous if we only look at surface/domain-wall.

• The bulk topological phase is gapped. No degrees of freedom. Almost trivial theory.

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Example: quantum Hall system

36

The simplest example is given by quantum Hall system.

d = 2, d+ 1 = 3

symmetry: U(1) (QED)

Anomalous theory: chiral fermion in d = 2

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Example: quantum Hall system

37

Quantum Hall state(2+1-dimensions)

Anomalous chiral fermion(2-dimensions)

On the surface of matter exhibiting quantum Hall effect,there appears chiral fermions which have anomaly under U(1) symmetry.

: U(1) background gauge fieldA = Aµdxµ

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Example: quantum Hall system

38

S = ik

4⇡

Z

bulkAdA

The action of the quantum Hall system in 3-dimensions

�↵S = ik

4⇡

Z

bulkd↵dA = i

k

4⇡

Z

boundary↵dA

Gauge transformation

Cancels against boundary fermionanomaly

�A = d↵

( : background. No dynamical degrees of freedom.)A

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Example: quantum Hall system

39

det(PL�µDµ) · exp(

ik

4⇡

Z

bulkAdA)

boundary fermionpartition function

bulk topological phase partition function

Product of bulk and boundary partition functions: gauge invariant

Z[N,A] =

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Characterization of anomaly

40

More generally, anomalies are completely characterized by (d+1)-dimensional topological phases.

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Characterization of anomaly

41

MN

@N = M

MN 0

@N 0 = M

A manifold: Another manifold:

Gluing the two manifold:

N

Closed manifold

X = N [N 0N 0

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Characterization of anomaly

42

Z[N ]

Z[N 0]= Z[X] ( )X = N [N 0

Anomaly is characterized by (d+1)-dimensionalpartition function on the closed manifoldZ[X]

• If , there is no anomaly because means that the partition function is independent of

Z[X] = 1 Z[N ] = Z[N 0]

• is really the partition function of (d+1)-dim. theory. This (d+1)-dim theory is called symmetry protected topological phases (SPT phases) or invertible field theory

Z[X]Z[X]

N

X

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Perturbative anomaly

43

Example : Perturbative anomaly Usual perturbative anomaly is described by the so-calleddescent equation for the gauge field

: anomaly polynomial in (d+2)-dimensions

Id+2 = dId+1 Id+1 : Chern-Simons

Id+2 ⇠ tr F ( d+22 )

Z[X] = exp(i

Z

XId+1)

F = dA+A ^A

: Chern-Simons

Quantum Hall system: I2+1 ⇠ AdA

I2+2 ⇠ F 2 = dAdA

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General fermion anomaly

44

More general fermion anomalies

Id+1 can be used only for perturbative anomalies ofcontinuous symmetries.

More generally: global anomaly

Examples of global anomaly:

• SU(2) with a doublet Weyl fermion in 4-dimensions• All anomalies of discrete symmetries ( , time-reversal,..)ZN

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General fermion anomaly

45

General fermion anomaly formula [Witten, 2015]

Z[X] = exp(�2⇡i⌘)

: Atiyah-Patodi-Singer invariant ⌘ ⌘

[Atiyah-Patodi-Singer 1975]

Perturbatively , but the invariant containsmore information. (Details omitted)

⌘ ⇠ Id+1 ⌘

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All global anomalies

46

All global anomalies (not restricted to fermions)

Classified by the cobordism groups.

Conjectured in[Kapustin 2014, Kapustin-Thorngren-Turzillo-Wang 2014]

Essentially proved in[Freed-Hopkins 2016, Yonekura 2018]

By looking at the cobordism groups, we can see what anomaly is possible in a given dimension with a given symmetry group.

Details omitted.

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Applications

47

There are many applications.I sketch only a few of them (related to my own works).

In the rest of this lecture, I will give a sketch of applications of the refined concepts of symmetries and anomalies.

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Contents

48

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: String theory

5. Applications: Strong dynamics

6. Summary

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Anomalies in string theory

49

It is well-known how perturbative anomalies are cancelledin string theory.

This does not mean that we have the complete picture of anomaly cancellation in string theory.

More subtle global anomalies and more subtletopological structures in string theory, as I now discuss.

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Their fluxes, , are often said to have integralperiods due to Dirac quantization condition:

Flux quantization

50

String theory contains several higher form gauge fields,such as RR p-form field :C

C = Cµ1···µpdxµ1 ^ · · · ^ dxµp

F = dC

Z

(p+1)-cycleF 2 Z

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O-plane charges

51

Another well-known fact:

O -plane RR charges are given by (sign neglected)p

p < 5

2p�5

This means that the integral of around the O-plane is given by

F = dC

Z

around OpF = 2p�5

This is not integer for

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Dirac quantization

52

Let us recall the argument of Dirac quantization.

It seems that Dirac quantization condition is violated.

Is string theory inconsistent?

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Dirac quantization

53

O-plane

D-brane worldvolume

exp(i

Z

MC)

M

Coupling to RR field

C : RR-field

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Dirac quantization

54

O-plane

exp(i

Z

MC)

M

Coupling to RR field

N

@N = M

= exp(i

Z

NF )

C : RR-field F = dC

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Dirac quantization

55

O-plane

exp(i

Z

MC)

M

Coupling to RR field

N 0

@N 0 = M

= exp(i

Z

N 0F )

C : RR-field F = dC

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Dirac quantization

56

O-plane

MN 0

@N 0 = M

N

@N = M

exp(iRN F )

exp(iRN 0 F )

= exp(i

Z

XF ) 6= 1 If the charge is

not integer.

X = N [N 0

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Dirac quantization

57

Didn’t we see similar figures before?

Maybe you have forgotten, so let me repeat it.

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Characterization of anomaly

58

MN

@N = M

MN 0

@N 0 = M

A manifold: Another manifold:

NX = N [N 0N 0

Gluing the two manifold:Closed manifold

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Characterization of anomaly

59

Z[N ]

Z[N 0]= Z[X] ( )X = N [N 0

Anomaly is characterized by (d+1)-dimensionalpartition function Z[X]

Anomaly means Z[X] 6= 1

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Characterization of anomaly

60

The total partition function of D-brane worldvolume is

Z[N ] exp(i

Z

NF )

The partition functionof worldvolume fields(fermions, gauge fields)

Coupling to RR-field

Z[N ] exp(iRN F )

Z[N 0] exp(iRN 0 F )

= Z[X] exp(i

Z

XF )

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Characterization of anomaly

61

The (in)consistency of the worldvolume of D-branes iscontrolled not by or , but by the combinationZ[X]

ZF

Z[X] exp(i

Z

XF )

Anomaly free condition:

Z[X] exp(i

Z

XF ) = 1

[Moore-Witten,1999][Freed-Hopkins,2000]

[Witten,2016][Tachikawa-Yonekura,2018 & work in progress]

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Shifted flux quantization

62

Conclusion:Fluxes are not quantized to be integers, but

ZF 2 q + Z

q : quantity controlled by the anomaly of worldvolume theory of D-brane.

A complete story is not yet understood.More work is necessary.

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Contents

63

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: String theory

5. Applications: Strong dynamics

6. Summary

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’t Hooft anomaly matching

64

Some UV theory with global symmetry F

UV:

IR:

Anomaly of F in UV = Anomaly of F in IR

’t Hooft anomaly matching

Anomalies are very useful because of ’t Hooft matching:

???

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’t Hooft anomaly matching

65

In other words: Anomalies are conserved under RG flows.

Why?

Let’s recall that anomalous theories are realized ona boundary of topological phase.

Topological phase(Manifold )N

Anomalous theory on the boundary(Manifold )M

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’t Hooft anomaly matching

66

Some UV theory UV:

IR: ???

RG flow

Topological phase in one-higher dimensions

Large mass gap:invariant under RG flow

boundary Bulk

UV and IR anomalies are controlled by the same bulktopological phase. Therefore, anomalies are conserved.

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’t Hooft anomaly in QCD

67

’t Hooft anomaly matching is very useful in studyingstrong coupling dynamics such as QCD.

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’t Hooft anomaly in QCD

68

In QCD, there exist perturbative triangle anomalies ofchiral symmetry .

’t Hooft anomaly matching in QCD

qL

qR

: left-handed quarks, rotated by SU(Nf )LSU(Nf )R: right-handed quarks, rotated by

SU(Nf )L,R

SU(Nf )L,R

SU(Nf )L,R

qL,R

qL,R

qL,R

SU(Nf )L ⇥ SU(Nf )R

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’t Hooft anomaly in QCD

69

UV:

If there is no chiral fermion to match the anomaly,the chiral symmetry must be spontaneously broken.

Implications of ’t Hooft anomaly matching in QCD

confinement

The quarks have the ’t Hooft anomaly

IR:

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’t Hooft anomaly in QCD

70

Simple case: Nc = even

All gauge invariant composites are bosons, so there is no fermion to match the anomaly.

Therefore, chiral symmetry must be brokenif the theory is in confinement phase.

Other values of requires more complicateddiscussions.

Nc

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’t Hooft anomaly in QCD

71

Confinement Chiral symmetrybreaking

’t Hooft anomaly matching gives an important relationbetween the two most important concepts in QCD:

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Other theories

72

How about other theories?

• Pure Yang-Mills theories• Pure Super-Yang-Mills theories• …(any gauge theory)

N = 1

They don’t have continuous symmetries.Pure Yang-Mills don’t even have fermions.

Is there no useful anomaly at all?

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Subtler symmetry & anomaly

73

It turns out:

• They have 1-form symmetry (called center symmetry)

• The theories have a mixed anomaly between the 1-form symmetry and a discrete symmetry

• Time-reversal for pure Yang-Mills• Axial symmetry for Super-Yang-Mills

Discrete symmetry:

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Subtler symmetry & anomaly

74

The discrete symmetries (time-reversal or discrete axial symmetry) must be broken after confinement. (More exotic possibilities neglected.)

[Gaiotto-Kapstin-Komargodski-Seiberg,2017][Komargodski-Sulejmanpašić-Unsal, 2017]

[Shimizu-Yonekura, 2017]

The existence of the subtle symmetry and the anomalyimplies that:

More surprisingly, the subtle anomaly can even constrainfinite temperature phase transition.

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I will review more details of this topic in later lectures.

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Contents

76

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: String theory

5. Applications: Strong dynamics

6. Summary

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Summary of the overview

77

The concepts of symmetry and anomaly are organized and generalized in recent years.

They are very useful for the studies of strong dynamics. There are many more applications.

String theory has extremely subtle and sophisticated topological structures related to anomaly which need to be investigated further.