Anomalies and Topological phases in QFT
Transcript of Anomalies and Topological phases in QFT
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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 theimportance of these concepts in QFT.
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Introduction
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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
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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
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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 aunified 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 := 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
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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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Some properties: Topology
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⌃
⌃0
Q(⌃) = Q(⌃0)d(⇤J) = 0Stokes &
topological(charge conservation)
Q(⌃) =
Z
⌃⇤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 ⌃
• 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
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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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Some properties: Topology
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U(⌃, 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 .
QU Q
U(⌃, 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 .
QU Q
⌃⌃
U(⌃, 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.
QU Q
⌃⌃
U(⌃, g) : topological operator
g 2 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 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
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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.
Some properties: Background
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U(⌃,↵)
exp(i
ZA(x)O(x))
U(⌃,↵) with A(x) ⇠ ↵�(⌃)
�(⌃) : delta function localized on the surface ⌃
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 symmetryA : non-orientable manifold (e.g. Klein-bottle)
Hp+1(M,ZN )A 2
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Some properties: Background
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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
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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 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
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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 manifoldM
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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.
NM
A
MN @N = M
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Anomaly
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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
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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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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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Characterization of anomaly
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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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Characterization of anomaly
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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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Characterization of anomaly
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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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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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2017/09/22 午後9(10
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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
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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QCD phase transition
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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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QCD phase transition
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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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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.
�
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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
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
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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 theinconsistency?
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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
MC)
M
Coupling to RR field
C : RR-field
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Dirac quantization
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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
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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
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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Characterization of anomaly
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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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Characterization of anomaly
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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
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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
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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.