anomalies and topological phases in qft

Anomalies and Topological phases in QFT

Kazuya Yonekura, Kyushu University

1

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Introduction

2

However, what is symmetry, and what is anomaly?

Symmetry and Anomaly:

I hope I don’t need to explain theimportance of these concepts in QFT.

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Introduction

3

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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Introduction

4

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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Introduction

5

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

Do not stick to textbook understandings.

I will review some of those developments, which are particularly related to my own works.

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Contents

6

1. Introduction

2. Symmetry

3. Anomaly

4. Applications: Strong dynamics

5. Applications: String theory

6. Summary

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Symmetry

7

What is symmetry in modern understanding?

Actually I don’t know how to say it in simple words.

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

8

• 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]

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

• …

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Some properties

9

Let me describe some properties of some of them.

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Some properties: Topology

10

Let us recall the case of usual continuous symmetry.

Jµ

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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11

d(⇤J) = 0 : conservation equation

: charge operator

is invariant under continuous deformation of by

⌃ : codimension-1 (dimension d-1) surface

d(⇤J) = 0

⌃

Q(⌃) =

Z

⌃⇤J

Q(⌃)

• Stokes theorem• is closed: ⇤J

In this sense, is topological.Q(⌃)

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12

⌃

⌃0

Q(⌃) = Q(⌃0)d(⇤J) = 0Stokes &

topological(charge conservation)

Q(⌃) =

Z

⌃⇤J

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13

• It is topological in the sense that it is invariant under continuous change 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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14

So the usual symmetry is implemented by operators

U(⌃, g) : topological operator

⌃ : surface

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

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15


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16

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

QU Q


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17


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

QU Q

⌃⌃


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18


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

⌃⌃


g 2 G

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19

Remarks:

• Sometimes these operators cannot be simply 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

20

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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21

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

Jµ(x)

ZddxAµJ

µ =

ZA ^ ⇤J

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Operators can be coupled to a background.

In fact, this operator itself can be seen as a backgroundwhen inserted in the path integral.


22

U(⌃,↵)

exp(i

ZA(x)O(x))

U(⌃,↵) with A(x) ⇠ ↵�(⌃)

�(⌃) : delta function localized on the surface ⌃

Schematically:

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23

Z[A]

I will write

A : abstract background field for the “symmetry” U

: parition function in the presence of the background field A

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24

Example:

• 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

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25

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 abelian symmetry groups can be treated in Lagrangian way.

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

• Altenatively, some of them can also be described by network of operators . U(⌃,↵)

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Contents

26

1. Introduction

2. Symmetry

3. Anomaly



6. Summary

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Anomaly

27

What is anomaly in modern understanding?

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

Remarks:• I don’t know a proof. Or I don’t even know in what

axioms it should be proved.

• “Almost” above means that I personally don’t understand how to treat conformal anomaly in the framework, but probably it is also possible.

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Anomaly

28

Z[A]

A : abstract background fields for the “symmetry” U

: parition function in the presence of the background field

: d-dimensional spacetime manifoldM

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Anomaly

29

• 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 is extended, the partition function is fixed without any ambiguity.

NM

A

MN @N = M

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Anomaly

30

MN

@N = M

Z[N,A]

: it depends on the manifold and extention of into .

N

A N

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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Anomaly

31

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.

• However, it is anomalous if we only look at surface/domain-wall.

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Characterization of anomaly

32

Anomalies are completely characterized by (d+1)-dimensional topological phases as I now explain.

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33

MN

@N = M

MN 0

@N 0 = M

A manifold: Anothor manifold:

Gluing the two manifold:

N

Closed manifold

X = N [N 0N 0

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34

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 parition function is independent of

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

• is really the parition 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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35

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+1 = dId : variation of under gauge transformation

Id

Id+2 ⇠ tr F ( d+22 )

Z

Z[X] = exp(i

Z

XId+1)

F = dA+A ^A

: Chern-Simons

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36

More general fermion anomalies

Id+1

[Witten, 2015]

can be used only for perturbative anomalies forcontinuous symmetries. More general global anomalies:

Z[X] = exp(�2⇡i⌘)

: Atiyah-Patodi-Singer invariant ⌘ ⌘

[Atiyah-Patodi-Singer 1975]

This quantity is closely related to domain wall fermions.I omit details. [Fukaya-Onogi-Yamaguchi 2017]

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37

All global anomalies of usual symmetries(not restricted to fermions)

Classified by the so-called cobordism groups.

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

Proved in[Freed-Hopkins 2016, KY 2018]

By looking at the cobordism groups which mathematicians have computed, we can see what anomaly can occur in a given dimension with a given symmetry group.

I omit details.

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Contents

38

1. Introduction

2. Symmetry

3. Anomaly



6. Summary

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Applications

39

There are too many applications.I only discuss two of my own works.

[Simizu-KY 2017]

[Tachikawa-KY 2018]

Strong dynamics:

String theory:

Unfortunately I will completely omit details. Please see the papers for details.

(Related works: [Tanizaki-(Kikuchi)-Misumi-Sakai])

(Related earlier work: [Witten 2016])

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QCD phase transition

40


41

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• Axion abundance, gravitational waves, etc. • Dark matter from here!? [Witten,1984]

• Around here, there was a phase transition of the Universe due to QCD.

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42

In any case, QCD phase transition is a very importantproblem in particle physics & cosmology.

What happens at the phase transition?

Chiral symmetry SU(Nf )L ⇥ SU(Nf )R

q =

✓qLqR

◆Quark: SU(Nf )L

SU(Nf )R

Gauge invariant composite: meson field� ⇠ qLqR : bifundamental of

Hint:

SU(Nf )L ⇥ SU(Nf )R

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43

T ⌧ ⇤QCD

T � ⇤QCD

Chiral symmetry SU(Nf )L ⇥ SU(Nf )R

Broken at low temperature

Preserved at high temperature

The argument of universality may suggest that the phasetransition is described by linear sigma model of the form

L = tr@µ�†@µ�+ (T � Tc)tr|�|2 + · · ·

� : bifundamental scalar of SU(Nf )L ⇥ SU(Nf )R

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44

L = tr@µ�†@µ�+ (T � Tc)tr|�|2 + · · ·

At , the field is just a scalar without anomaly.�

However, in [Simizu-KY, 2017] we found an ’t Hooft anomaly at finite temperature by using a kind of generalization of higher form symmetry.

Conclusion:The above picture based on universality is wrong!

T ⇠ Tc

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45

Remarks:• I cheated a bit. There are several details and

conditions. If you are interested, please see our paper or ask me in this workshop.

• The field does not have anomaly not because it is boson, but because it is a linear sigma model.

Bosons can have global anomalies.

See [Gaiotto-Kapustin-Komargodski-Seiberg 2017] for an anomaly of bosons and its applications. Our work is based on this paper.

�

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Contents

46

1. Introduction

2. Symmetry

3. Anomaly



6. Summary

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O-plane charges

47

Two standard textbook facts:

• Objects such as branes must satisfy Dirac quantization conditions and hence their RR-charges must be integers.

• O -plane RR charges are given by This is not integer for p �2p�5

The immediate corollary of these two facts:

String theory is inconsistent.

p < 5

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Announcement

48

This workshop:

Strings and fields 2018

Fields 2018

Goodbye string theory!We never forget you!

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Announcement

49

This workshop:

Strings and fields 2018

Fields 2018

Goodbye string theory!We never forget you!

Of course I’m kidding. But how can we resolve theinconsistency?

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Dirac quantization

50

Let us recall the argument of Dirac quantization.

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Dirac quantization

51

O-plane

D-brane worldvolume

exp(i

Z

MC)

M

Coupling to RR field

C : RR-field

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Dirac quantization

52

O-plane

exp(i

Z

MC)

M


N

@N = M

= exp(i

Z

NF )

C : RR-field F = dC

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Dirac quantization

53

O-plane

exp(i

Z

MC)

M


N 0

@N 0 = M

= exp(i

Z

N 0F )

C : RR-field F = dC

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Dirac quantization

54

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

55

Didn’t we see similar figures today?

Maybe you have forgotten, so let me repeat it.

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56

MN

@N = M

MN 0

@N 0 = M

A manifold: Anothor manifold:

N

Closed manifold

X = N [N 0N 0

Gluing the two manifold:

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57

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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58

The total action of D-brane worldvolume is

Z[N ] exp(i

Z

NF )

The partition functionof worldvolume fields RR-coupling

Z[N ] exp(iRN F )

Z[N 0] exp(iRN 0 F )

= Z[X] exp(i

Z

XF ) = 1

Anomaly and ambiguity of RR-coupling cancel each other.

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Contents

59

1. Introduction

2. Symmetry

3. Anomaly



6. Summary

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Summary and outlook

60

The concepts of symmetry and anomaly are refined 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. There are much more to be investigated.

anomalies and topological phases in qft

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