1 Anomalies and Topological phases in QFT Kazuya Yonekura, Kyushu University
2 Introduction Symmetry and Anomaly: I hope I don’t need to explain the importance of these concepts in QFT. However, what is symmetry, and what is anomaly? / 60
3 Introduction 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 ] / 60
4 Introduction What is anomaly? A textbook answer S [ φ ] The classical action is invariant under φ ( x ) → g · φ ( x ) transformation of fields But quantum mechanically it is violated (e.g. by path integral measure). / 60
5 Introduction 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. / 60
6 Contents 1. Introduction 2. Symmetry 3. Anomaly 4. Applications: Strong dynamics 5. Applications: String theory 6. Summary / 60
7 Symmetry 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. / 60
8 Symmetry • 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] • … / 60
9 Some properties Let me describe some properties of some of them. / 60
10 Some properties: Topology Let us recall the case of usual continuous symmetry. : conserved current J µ r µ J µ = 0 : conservation equation In the language of diffential forms, it is more beautiful. J := J µ dx µ 1 ( d − 1)! J µ ✏ µµ 1 ··· µ d − 1 dx µ 1 ∧ · · · dx µ d − 1 ∗ J = d ( ∗ J ) = 0 : conservation equation / 60
11 Some properties: Topology d ( ∗ J ) = 0 : conservation equation Z : charge operator Q ( Σ ) = ∗ J Σ Σ : codimension-1 (dimension d-1) surface Q ( Σ ) is invariant under continuous deformation of by Σ • Stokes theorem • is closed: d ( ∗ J ) = 0 ∗ J Q ( Σ ) In this sense, is topological. / 60
12 Some properties: Topology Σ 0 Z Q ( Σ ) = ∗ J Σ Stokes & Q ( Σ ) = Q ( Σ 0 ) d ( ∗ J ) = 0 topological (charge conservation) Σ / 60
13 Some properties: Topology Symmetry operator: Z U ( Σ , α ) = exp( i α Q ( Σ )) = exp( i α ∗ J ) Σ • It is topological in the sense that it is invariant under continuous change of the surface Σ • The operator exists for each group element U ( Σ , α ) e i α = g ∈ G g : element of group : element of Lie algebra α / 60
14 Some properties: Topology So the usual symmetry is implemented by operators U ( Σ , g ) : topological operator : surface Σ : “label” of the operator g (group element in the current case) / 60
15 Some properties: Topology U ( Σ , g ) : topological operator / 60
16 Some properties: Topology U ( Σ , g ) : topological operator Q Q: Does it need to be an exponential of ? Q A: No. Discrete symmetry has without . U / 60
17 Some properties: Topology U ( Σ , g ) : topological operator Q Q: Does it need to be an exponential of ? Q A: No. Discrete symmetry has without . U Q: Does the surface need to be codimension-1? Σ A: No. Higher form symmetry uses higher codimension . Σ / 60
18 Some properties: Topology U ( Σ , g ) : topological operator Q Q: Does it need to be an exponential of ? Q A: No. Discrete symmetry has without . U Q: Does the surface need to be codimension-1? Σ A: No. Higher form symmetry uses higher codimension . Σ g ∈ G Q: Do we need group elements ? A: No. Topological defect operator is just topological without any group. / 60
19 Some properties: Topology 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. / 60
20 Some properties: Background Quite generally, operators can be coupled to background fields. The most basic case of an operator coupled O ( x ) A ( x ) to a background field Z Z [ A ] = h exp( i A ( x ) O ( x )) i : called generating functional or partition function. I will use the terminology partition function. / 60
21 Some properties: Background For the current operator , we have a background J µ ( x ) gauge field . The coupling between them is A µ ( x ) Z Z d d xA µ J µ = A ∧ ∗ J / 60
22 Some properties: Background Operators can be coupled to a background. U ( Σ , α ) In fact, this operator itself can be seen as a background when inserted in the path integral. Schematically: Z exp( i A ( x ) O ( x )) with A ( x ) ∼ αδ ( Σ ) U ( Σ , α ) δ ( Σ ) : delta function localized on the surface Σ / 60
23 Some properties: Background I will write A : abstract background field for the “symmetry” U Z [ A ] : parition function in the presence of the background field A / 60
24 Some properties: Background Example: • For discrete symmetry , G : principal bundle A G • For higher form symmetry such as p-form symmetry with abelian group G = Z N H p +1 ( M, Z N ) A ∈ (cohomology group) • Parity, Time-reversal symmetry A : non-orientable manifold (e.g. Klein-bottle) / 60
25 Some properties: Background 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 ( Σ , α ) / 60
26 Contents 1. Introduction 2. Symmetry 3. Anomaly 4. Applications: Strong dynamics 5. Applications: String theory 6. Summary / 60
27 Anomaly 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 / 60 framework, but probably it is also possible.
28 Anomaly A : abstract background fields for the “symmetry” U Z [ A ] : parition function in the presence of the background field : d-dimensional spacetime manifold M / 60
29 Anomaly • First of all, anomaly means that the partition function is ambiguous. • However, if we take a (d+1)-dimensional manifold N whose boundary is the spacetime and on which M the background fields is extended, the partition A function is fixed without any ambiguity. N M ∂ N = M / 60
30 Anomaly Z [ N, A ] : it depends on the N M manifold N and extention of ∂ N = M A into . N This description of anomaly may look quite abstract, but there is a very natural motivation from condensed matter physics / domain wall fermion. / 60
31 Anomaly Some material: Anomalous theory on Topological phase the surface/domain wall (Manifold ) (Manifold ) N M • Cond-mat/Lattice systems are not anomalous as a whole. • However, it is anomalous if we only look at surface/domain-wall. / 60
32 Characterization of anomaly Anomalies are completely characterized by (d+1)-dimensional topological phases as I now explain. / 60
33 Characterization of anomaly A manifold: Anothor manifold: M N N 0 M ∂ N 0 = M ∂ N = M Gluing the two manifold: Closed manifold N N 0 X = N ∪ N 0 / 60
34 Characterization of anomaly 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 ] X • If , there is no anomaly because Z [ N ] = Z [ N 0 ] Z [ X ] = 1 means that the parition function is independent of N • is really the parition function of (d+1)-dim. theory. Z [ X ] Z [ X ] This (d+1)-dim theory is called symmetry protected topological phases (SPT phases) or invertible field theory / 60
35 Characterization of anomaly Example : Perturbative anomaly Usual perturbative anomaly is described by the so-called descent equation for the gauge field F = dA + A ∧ A : anomaly polynomial in I d +2 ∼ tr F ( d +2 ) 2 (d+2)-dimensions I d +2 = dI d +1 I d +1 : Chern-Simons I d : variation of under Z δ I d +1 = dI d gauge transformation Z Z [ X ] = exp( i I d +1 ) : Chern-Simons X / 60
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