Anomalies and Topological phases in QFT Kazuya Yonekura, Kyushu - - PowerPoint PPT Presentation

Anomalies and Topological phases in QFT

Kazuya Yonekura, Kyushu University

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Introduction

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However, what is symmetry, and what is anomaly? Symmetry and Anomaly: I hope I don’t need to explain the importance of these concepts in QFT.

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Introduction

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What is symmetry? A textbook answer φ(x) : fields S[φ] : action Symmetry means that the action is invariant under tranformation φ(x) → g · φ(x) S[g · φ] = S[φ]

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Introduction

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What is anomaly? A textbook answer S[φ] The classical action is invariant under transformation of fields But quantum mechanically it is violated (e.g. by path integral measure). φ(x) → g · φ(x)

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Introduction

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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

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• 1. Introduction
• 2. Symmetry
• 3. Anomaly
• 4. Applications: Strong dynamics
• 5. Applications: String theory
• 6. Summary

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Symmetry

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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 a unified treatment of several generalizations.

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Symmetry

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• 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

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Let me describe some properties of some of them.

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Some properties: Topology

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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 := Jµdxµ ∗J = 1 (d − 1)!Jµ✏µµ1···µd−1dxµ1 ∧ · · · dxµd−1 d(∗J) = 0 : conservation equation

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Some properties: Topology

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d(∗J) = 0 : conservation equation : charge operator is invariant under continuous deformation of by Σ : codimension-1 (dimension d-1) surface d(∗J) = 0 Σ Q(Σ) = Z

Σ

∗J Q(Σ)

• Stokes theorem
• is closed:

∗J In this sense, is topological. Q(Σ)

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Some properties: Topology

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Σ Σ0 Q(Σ) = Q(Σ0) d(∗J) = 0 Stokes & topological (charge conservation) Q(Σ) = Z

Σ

∗J

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Some properties: Topology

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• It is topological in the sense that it is invariant under

continuous change of the surface Σ

• The operator exists for each group element

Symmetry operator: eiα = g ∈ G α U(Σ, α) U(Σ, α) = exp(iαQ(Σ)) = exp(iα Z

Σ

∗J) g : element of group : element of Lie algebra

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Some properties: Topology

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So the usual symmetry is implemented by operators U(Σ, g) : topological operator Σ : surface g : “label” of the operator (group element in the current case)

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Some properties: Topology

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U(Σ, g) : topological operator

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Some properties: Topology

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Q: Does it need to be an exponential of ? A: No. Discrete symmetry has without . Q U Q U(Σ, g) : topological operator

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Some properties: Topology

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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 U Q Σ Σ U(Σ, g) : topological operator

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Some properties: Topology

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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. Q U Q Σ Σ U(Σ, g) : topological operator g ∈ G

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Some properties: Topology

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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

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Quite generally, operators can be coupled to background fields. The most basic case of an operator coupled to a background field O(x) A(x) Z[A] = hexp(i Z 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

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For the current operator , we have a background gauge field . The coupling between them is Aµ(x) Jµ(x) Z ddxAµJµ = Z A ∧ ∗J

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Operators can be coupled to a background. In fact, this operator itself can be seen as a background when inserted in the path integral.

Some properties: Background

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U(Σ, α) exp(i Z A(x)O(x)) U(Σ, α) with A(x) ∼ αδ(Σ) δ(Σ) : delta function localized

• n the surface Σ

Schematically:

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Some properties: Background

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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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Some properties: Background

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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 symmetry

A : non-orientable manifold (e.g. Klein-bottle) Hp+1(M, ZN) A ∈

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Some properties: Background

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Remarks:

• As the previous examples show, abstract description
• f 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(Σ, α)

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Contents

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• 1. Introduction
• 2. Symmetry
• 3. Anomaly
• 4. Applications: Strong dynamics
• 5. Applications: String theory
• 6. Summary

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Anomaly

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What is anomaly in modern understanding? There is now a way which is believed to describe almost 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

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Z[A] A : abstract background fields for the “symmetry” U : parition function in the presence of the background field : d-dimensional spacetime manifold M

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Anomaly

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• 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. N M A M N ∂N = M

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Anomaly

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M N ∂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

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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

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

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Characterization of anomaly

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M N ∂N = M M N 0 ∂N 0 = M A manifold: Anothor manifold: Gluing the two manifold: N Closed manifold X = N ∪ N 0 N 0

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Characterization of anomaly

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Z[N] Z[N 0] = Z[X] ( ) X = N ∪ N 0 Anomaly is characterized by (d+1)-dimensional partition function on the closed manifold Z[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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Characterization of anomaly

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Example : Perturbative anomaly Usual perturbative anomaly is described by the so-called descent equation for the gauge field : anomaly polynomial in (d+2)-dimensions Id+2 = dId+1 Id+1 : Chern-Simons δId+1 = dId : variation of under gauge transformation Id Id+2 ∼ tr F ( d+2

2

)

Z Z[X] = exp(i Z

X

Id+1) F = dA + A ∧ A : Chern-Simons

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Characterization of anomaly

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More general fermion anomalies Id+1

[Witten, 2015]

can be used only for perturbative anomalies for continuous symmetries. More general global anomalies: Z[X] = exp(−2πiη) : Atiyah-Patodi-Singer invariant η η

[Atiyah-Patodi-Singer 1975]

This quantity is closely related to domain wall fermions. I omit details.

[Fukaya-Onogi-Yamaguchi 2017]

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Characterization of anomaly

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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

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• 1. Introduction
• 2. Symmetry
• 3. Anomaly
• 4. Applications: Strong dynamics
• 5. Applications: String theory
• 6. Summary

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Applications

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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

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QCD phase transition

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{

Age of the Universe Radius of the Visible Universe

Free Electrons Scatter Light Earliest Time Visible with Light Inflation Protons Formed Nuclear Fusion Begins Nuclear Fusion Ends Cosmic Microwave Background Neutral Hydrogen Forms Modern Universe Big Bang

10−32 s 1 µs 0.01 s 3 min 380,000 yrs 13.8 Billion yrs

Quantum Fluctuations

Expansion of the Universe (From Wikipedia)

• 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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QCD phase transition

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In any case, QCD phase transition is a very important problem in particle physics & cosmology. What happens at the phase transition? Chiral symmetry SU(Nf)L × SU(Nf)R q = ✓ qL qR ◆ Quark: SU(Nf)L SU(Nf)R Gauge invariant composite: meson field Φ ∼ qLqR : bifundamental of Hint:

SU(Nf)L × SU(Nf)R

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QCD phase transition

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T ⌧ ΛQCD T ΛQCD Chiral symmetry SU(Nf)L × SU(Nf)R Broken at low temperature Preserved at high temperature The argument of universality may suggest that the phase transition is described by linear sigma model of the form L = tr∂µΦ†∂µΦ + (T − Tc)tr|Φ|2 + · · · Φ : bifundamental scalar of SU(Nf)L × SU(Nf)R

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QCD phase transition

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L = tr∂µΦ†∂µΦ + (T − Tc)tr|Φ|2 + · · · At , the field is just a scalar without anomaly. Φ 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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QCD phase transition

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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. Φ

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Contents

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• 1. Introduction
• 2. Symmetry
• 3. Anomaly
• 4. Applications: Strong dynamics
• 5. Applications: String theory
• 6. Summary

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O-plane charges

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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

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This workshop: Strings and fields 2018 Fields 2018 Goodbye string theory! We never forget you!

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Announcement

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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 the inconsistency?

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Dirac quantization

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Let us recall the argument of Dirac quantization.

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Dirac quantization

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O-plane D-brane worldvolume exp(i Z

M

C) M Coupling to RR field C : RR-field

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Dirac quantization

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O-plane exp(i Z

M

C) M Coupling to RR field N ∂N = M = exp(i Z

N

F) C : RR-field F = dC

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Dirac quantization

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O-plane exp(i Z

M

C) M Coupling to RR field N 0 ∂N 0 = M = exp(i Z

N 0 F)

C : RR-field F = dC

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Dirac quantization

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O-plane M N 0 ∂N 0 = M N ∂N = M exp(i R

N F)

exp(i R

N 0 F) = exp(i

Z

X

F) 6= 1 If the charge is not integer. X = N ∪ N 0

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Dirac quantization

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Didn’t we see similar figures today? Maybe you have forgotten, so let me repeat it.

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Characterization of anomaly

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M N ∂N = M M N 0 ∂N 0 = M A manifold: Anothor manifold: N Closed manifold X = N ∪ N 0 N 0 Gluing the two manifold:

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Characterization of anomaly

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Z[N] Z[N 0] = Z[X] ( ) X = N ∪ N 0 Anomaly is characterized by (d+1)-dimensional partition function Z[X] Anomaly means Z[X] 6= 1

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Characterization of anomaly

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The total action of D-brane worldvolume is Z[N] exp(i Z

N

F) The partition function

• f worldvolume fields

RR-coupling Z[N] exp(i R

N F)

Z[N 0] exp(i R

N 0 F) = Z[X] exp(i

Z

X

F) = 1 Anomaly and ambiguity of RR-coupling cancel each other.

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Contents

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• 1. Introduction
• 2. Symmetry
• 3. Anomaly
• 4. Applications: Strong dynamics
• 5. Applications: String theory
• 6. Summary

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Summary and outlook

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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.