セミナー @5号館

2021. 10. 13.

Anomaly and Superconnection
菅野 颯人
京都大学 基礎物理学研究所 (YITP)

Based on arXiv:2106.01591 [hep-th]
Now, v2 is available.

Work with 杉本茂樹 (基研)What is “anomaly”? (1)Anomaly (Quantum Anomaly)
An classical action has some symmetries, but sometimes
these symmetries disappear in quantum theory.
e.g.) 𝜋 0 → 2𝛾
• In massless QCD, there is a chiral symmetry 𝑈 𝑁𝑓

𝐿× 𝑈 𝑁𝑓𝑅.

𝑁𝑓 : # of flavors• If there is NO anomaly, 𝜋 0 never decays.
• However, 𝜋 0 decays into 2𝛾, because of an anomaly!
𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 ⊃ 𝑈 1 𝐴 has an anomaly.
Introduction
Introduction(1/4)(4)

Derivation (9)Application (7)Index theorem (5)String theory (5)Theories what we want to think (1)Let us consider 4dim action contain fermions.

• This action is massless, so it has a chiral symmetry 𝑈 𝑁𝑓 × 𝑈 𝑁𝑓 .
𝐿𝑅
• There also be a 𝑈 1 𝐴 anomaly.

• Add mass term
• Mass term breaks the chiral symmetry.

• Let the mass depend on the spacetime.
• This mass is almost same as the Higgs field.
• How change the symmetry and the anomaly?
Introduction
Introduction(2/4)(4)

Derivation (9)Application (7)Index theorem (5)String theory (5)The spacetime dependent mass
What is “the spacetime dependent mass”?
• e.g.) Domain wall fermions• One way to realize chiral fermions on the lattice.
• Consider 5dim spacetime, and realize 4dim
fermions on 𝑚 𝑥 = 0 subspace.

𝑥: the fifth
direction• Chiral anomalies with Higgs fields• If Higgs fields change as bifundamental under the𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 chiral symmetry, the action isinvariant for the symmetry.
• It is known that chiral anomalies are not changedby adding Higgs fields.
• See Fujikawa-san’s text book.
Introduction
Introduction(3/4)(4)

Derivation (9)Application (7)Index theorem (5)String theory (5)The spacetime dependent mass
What is “the spacetime dependent mass”?
• e.g.) Domain wall fermions• One way to realize chiral fermions on the lattice.
• Consider 5dim spacetime, and realize 4dim
fermions on 𝑚 𝑥 = 0 subspace.

𝑥: the fifth
direction• Chiral anomalies with Higgs fields• If Higgs fields change as bifundamental under the𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 chiral symmetry, the action isinvariant for the symmetry.
• It is known that chiral anomalies are not changedby adding Higgs fields.
• See Fujikawa-san’s text book.
Introduction
Introduction(3/4)(4)

Derivation (9)Application (7)Index theorem (5)String theory (5)Theories what we want to think (2)How about the spacetime dependent mass?
• The chiral anomaly is changed by the mass!!
• Deference between Higgs and mass• Higgs field : bounded• Spacetime dependent mass : unbounded• If the mass diverges at some points, it contributes tothe anomaly.
• This contribution might be unknown.
• We can find the anomaly in any dimension.

• The anomaly can be written by “superconnection.”
Introduction
Introduction(4/4)(4)

Derivation (9)Application (7)Index theorem (5)String theory (5)Plan
1. Introduction (4)3. Application (7)• What is anomaly?
• Theories what we want to think

2. Derivation (9)• How to calculate anomalies• The anomaly for massless case
• The anomaly for massive case• Superconnection• The result• Kink, vortex• With boundary

4. Index theorem (5)• Index for massive case
• APS index theorem

5. String theory (5)• Tachyon condensation

6. Conclusion (1)Introduction (4)Derivation (9)Application (7)Index theorem (5)String theory (5)How to calculate anomalies[’79 Fujikawa]Fujikawa method• There are several ways to calculate
anomalies.
• Today, we focus on the Fujikawa
method.

e.g.) 𝑈 1 𝑉transformation• Consider path integral for fermions.
• Anomaly = Jacobian comes from path
integral measure• We only consider perturbative
anomalies.

anomaly• We calculate log 𝒥 for anomalies inthe last part of this talk.
Introduction (4)Derivation
Derivation(1/9)(9)

Application (7)Index theorem (5)String theory (5)Chiral symmetry
Anomalous symmetries wecalculate• 𝑈 𝑁𝑓𝐿× 𝑈 𝑁𝑓𝑅• 𝑈(𝑁𝑓 ) flavor symmetry

chiral symmetry• For even dimension• Because chirality operators exist
only even dimensions.
• Weyl fermions couple to 𝑈 𝑁𝑓 𝐿
background gauge field 𝐴𝜇𝐿 and𝑈 𝑁𝑓 background gauge field 𝐴𝜇𝑅 .
𝑅• For odd dimension• No perturbative anomaly as usual.
• Dirac fermions couple to 𝑈(𝑁𝑓 )background gauge field.

For even dimension

For odd dimension
Introduction (4)Derivation
Derivation(2/9)(9)

Application (7)Index theorem (5)String theory (5)The anomaly for massless casese.g.) fermions in 4dim
• Mass less case
• With 𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 chiral sym.
• 𝑈 1 𝑉 anomaly is written by thefield strengths.

• With a Higgs field• With 𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 chiral sym.
• The 𝑈 1 𝑉 anomaly is same formassless case.

• How about the massive case?
Introduction (4)Derivation
Derivation(3/9)(9)

Application (7)Index theorem (5)String theory (5)The anomaly for massive case (1)Let us consider spacetime dependent mass!
• The action for general even dim with 𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 symmetry is,• For odd dim case, there is only 𝑈(𝑁𝑓 ) sym, we put 𝐴𝜇 = 𝐴𝜇𝑅 = 𝐴𝜇𝐿 and 𝑚 = 𝑚† .
• We take 𝒎(𝒙) divergent.
• 𝐼 denotes some directions which 𝑚(𝑥) changes its values.

• We calculated 𝑈 1 𝑉 anomaly for this action by Fujikawa method.

• It is easy to get the anomaly for any dimension.
• It is also easy to get the anomaly for 𝑈 𝑁𝑓 𝐿 × 𝑈 𝑁𝑓 𝑅 , not only for 𝑈 1 𝑉 .

Introduction (4)Derivation
Derivation(4/9)(9)

Application (7)Index theorem (5)String theory (5)The anomaly for massive case (2)e.g.) In 4dim case, the 𝑈 1 𝑉 anomaly is,

Λ is UV cut-off
comes fromheat kernel
regularization.

• This result seems very complicated...
• Can we rewrite it more simple way? → Superconnection!
Introduction (4)Derivation
Derivation(5/9)(9)

Application (7)Index theorem (5)String theory (5)Superconnection (1)[’85 Quillen]• We define the superconnections for even and odd dimensions.
• This is made by Quillen, who is a mathematician, in 1985.

Even dimension• Superconnection• Field strength

Introduction (4)𝐴𝑅 : 𝑈 𝑁𝑓

𝑅

𝐴𝐿 : 𝑈 𝑁𝑓

𝐿

𝑇 : 𝑈 𝑁𝑓

𝐿gauge field (1-form)
gauge field (1-form)× 𝑈 𝑁𝑓𝑅bifundamental scalar field (0-form)• Supertrace

Derivation
Derivation(6/9)(9)

Application (7)Index theorem (5)String theory (5)Superconnection (2)[’85 Quillen]

Odd dimension• Superconnection

𝐴 : 𝑈(𝑁𝑓 ) gauge field (1-form)

𝑇 : 𝑈(𝑁𝑓 ) adjoint scalar field (0-form)• Field strength• Supertrace

We apply superconnection to write the anomaly.
Introduction (4)Derivation
Derivation(7/9)(9)

Application (7)Index theorem (5)String theory (5)The result (1)• We can rewrite the 𝑈 1 𝑉 anomaly by superconnection.

• For odd dimension case, put 𝐴𝜇 = 𝐴𝜇𝑅 = 𝐴𝜇𝐿 and 𝑚 = 𝑚† . Then, we get 𝑈(1)anomaly.
• In odd dimension, the definition of Str is different from even dim case.

Introduction (4)Derivation
Derivation(8/9)(9)

Application (7)Index theorem (5)String theory (5)The result (2)• In this formula, 𝑚(𝑥) is included as 𝑚(𝑥).
෥• Λ is UV cut-off comes from heat kernel regularization.
• If 𝑚(𝑥) is finite, the mass dependence disappears because we need to takeΛ→∞
• The Λ dependence of the anomaly disappears after we integrate Str[eℱ ] overthe spacetime.

• It is easy to check this anomaly is consistent with 4dim massless case.

Introduction (4)Derivation
Derivation(9/9)(9)

Application (7)Index theorem (5)String theory (5)3. Application

Introduction (4)Derivation (9)Application (7)Index theorem (5)String theory (5)How can we apply this anomaly?
Mass means a wall for some cases!
• e.g.) Domain wall
• Can we make domain walls by this 𝑚(𝑥)?
→Yes!

• We can make some systems with boundaries.
• Kink, vortex and general codimension case
• With boundary• We also discuss about some index theorems.
• APS index theorem• Callias type index theorem
Introduction (4)Derivation (9)Application
Application(1/7)(7)

Index theorem (5)String theory (5)Kink (1)Mass kink for our set up• For example, let’s consider 5dim case.
• In our set up, “kink” means this mass configuration.
• This “mass” diverges at 𝑦 → ±∞.
• 5dim fermions with 𝑈(𝑁𝑓 ) sym, and the mass depends on only 𝑦 direction.

• The 𝑈(1) anomaly is,• Recall 4dim 𝑈 1 𝑉 anomaly,Introduction (4)Derivation (9)Corresponds to the sign of 𝑢.

Application
Application(2/7)(7)

Index theorem (5)String theory (5)Kink (2)What is the meaning of the anomaly?
• 4dim Weyl fermions are localized at 𝑦 = 0.
• 𝑢 > 0 corresponds to chirality + (right-handed)fermion, and 𝑢 < 0 corresponds to chirality –(left-handed) fermion.

Domain wall fermion• This Weyl fermions correspond to domain wallfermions.
• But the regularization is different, so that I don’tknow the correspondence in detail.

Introduction (4)Derivation (9)Application
Application(3/7)(7)

Index theorem (5)String theory (5)Vortex
Next, we check codim-2 case.
• Vortex is 2dim topological object.
• Let us consider 2𝑟 + 2 dim.
• 𝑚(𝑧) depends on 2 directions, and it is complex valued “mass”.
• This mass diverges at 𝑧 → ∞.

• For simplicity, we put 𝐴𝐿 = 𝐴𝑅 in 2𝑟 + 2dim.
• The 𝑈 1 𝑉 anomaly is,• This is 2𝑟dim 𝑈(1) anomaly with 𝑈 𝑁𝑓

𝑅gauge field.

• If you want to get chirality – (left-handed) result, use 𝑚 𝑧ҧ = 𝑢𝑧,ҧ instead.
Introduction (4)Derivation (9)Application
Application(4/7)(7)

Index theorem (5)String theory (5)General defects
We can apply this formula to general codimension cases.
• When we think 𝑑 dim system with 𝑛 dim topological defects, we get 𝑑 − 𝑛dim 𝑈(1) anomalies.
• If 𝑑 − 𝑛 is odd, we get nothing because odd dim mass less fermions areanomaly-free.
• The mass configurations for general codimension is,• This results correspond to “tachyon condensation” in string theory.
• We will discuss about it in section 5.
Introduction (4)Derivation (9)Application
Application(5/7)(7)

Index theorem (5)String theory (5)With boundary (1)Let us make some boundaries.
• Fermions are massive = boundary

Odd dimension (2r dim)
• We realize localized fermions at [0,𝐿].
• The bulk is anomaly-free.
• The anomaly is,

Introduction (4)Derivation (9)Application
Application(6/7)(7)

Index theorem (5)String theory (5)With boundary (2)Even dimension• The anomaly is,• 𝜔 is Chern-Simons form.
• Anomaly from bulk + CS
Introduction (4)Derivation (9)Application
Application(7/7)(7)

Index theorem (5)String theory (5)4. Index theorem

Introduction (4)Derivation (9)Application (7)Index theorem (5)String theory (5)Index for massive Dirac op. (1)• We will discuss index theorems for the massive Dirac operator 𝒟.
• We just consider flat spacetime.

Chern character• We need to define Chern character for the superconnection.
• The Chern character for massive case is,

Introduction (4)Derivation (9)Application (7)Indextheorem
theorem(1/5)
(5)
Index

String theory (5)Index for massive Dirac op. (2)• We can write the 𝑈(1) anomaly by the Chern character.

• The index for the massive Dirac operator is,• For closed manifolds, this is Atiyah-Singer index theorem.

• Let us consider 2𝑟 dimensional system with boundary.
• We can make the boundary with the mass.
• The index will be Atiyah-Patodi-Singer(APS) index.
• Let’s check the index!
Introduction (4)Derivation (9)Application (7)Indextheorem
theorem(2/5)
(5)
Index

String theory (5)APS index theorem (1)[’75 Atiyah-Patodi-Singer]

APS index theorem• APS index is the index for open manifold with boundaries.
• APS index = bulk index + 𝜂-invariant on the boundaries• In APS paper, they introduce APS boundary condition, which is non-local
boundary condition for fermions.

• It is known that APS index is realized with local boundary condition.
• [’17 Fukaya-Onogi-Yamaguchi] [hep-th/1710.03379]• If you use domain walls for boundaries, you can use local boundary conditionfor fermions.

• Let us consider both boundary conditions in our set up.
• We can realize boundaries by mass; this is very similar to DW set up.
Introduction (4)Derivation (9)Application (7)Indextheoremtheorem(3/5)
(5)
Index

String theory (5)APS index theorem (2)APS index in our set up• The index is,• This is the APS index theorem for the massive Dirac operators in 2r dim withboundaries at 𝑦 = 𝑦± .
• We just consider 𝑚(𝑥) depends on only one direction 𝑦.

• Let us consider APS index for 2r dim system inprevious section.
• If you take 𝑦+ = 𝐿 and 𝑦− = 0, then you will get APS
index with APS boundary condition.
• But in this case, mass does not work because theDirac op. is massless in 0 < 𝑦 < 𝐿.

Introduction (4)Derivation (9)Application (7)Indextheoremtheorem(4/5)
(5)
Index

String theory (5)APS index theorem (3)APS index with mass
• Let us put 𝑦+ = ∞ and 𝑦− = −∞.
• Boundaries come from the mass.
• APS index is,• This is APS index with “physicist-friendly” boundary condition.
• cf. [’17 Fukaya-Onogi-Yamaguchi]• In our set up, massive ≃ domain wall.

• To apply this form, we get a relation between eta invariant and ChernSimons form 𝜔.
Introduction (4)Derivation (9)Application (7)Indextheorem
theorem(5/5)
(5)
Index

String theory (5)5. String theory

Introduction (4)Derivation (9)Application (7)Index theorem (5)String(5)String theory
theory (5)String theory
Let us check the relation between this anomaly and string
theory.
• Consider type IIA or IIB string theory with D-branes.
• Open strings have their ends on D-branes.
• Excitation modes of these open strings → Fields on D-branes• Open strings on D𝑝 -branes → QFT in 𝑝 + 1 dim• In some cases, excitation modes of the strings have tachyon modes.
• Lowest excitation modes are 𝑚2 < 0. (Tachyon mode)• Non-BPS states have tachyons.
• This tachyonic modes are unstable. → Tachyon condensation• See Sen’s review [hep-th/9904207].
Introduction (4)Derivation (9)Application (7)Index theorem (5)Stringtheory
theory(1/5)(5)
StringTachyon condensation (1)• Tachyonic modes are unstable, so the tachyons have non-zero VEV.
• Non-trivial configuration of tachyon is also realizable.

ഥ -brane) systeme.g.) 𝐷-brane and anti 𝐷-brane (𝐷• Non-BPS stateഥ string.
• Tachyonic modes appear in 𝐷 − 𝐷• The shape of tachyon potential is known.
• If tachyon configuration is trivial, the 𝐷-branes disappear.
Introduction (4)Derivation (9)Application (7)Index theorem (5)Stringtheory
theory(2/5)(5)
StringTachyon condensation (2)ഥ𝑝 system
Kink on tachyon in 𝐷𝑝 − 𝐷• Tachyonic kinks for this system is,• We get 𝐷𝑝−𝑛 -branes from this tachyon.
• If 𝐷𝑝−𝑛 -branes are non-BPS, tachyons still exist on the 𝐷-branes.
• In this case, tachyon condensation occur again.

𝑈 𝑁 × 𝑈(𝑁)symmetry
Introduction (4)Derivation (9)𝑈 𝑁 symmetry
Application (7)Index theorem (5)Stringtheorytheory(3/5)(5)
StringTachyon condensation (3)• The superconnection is used in the context of tachyon condensation.
• This structure comes from RR-coupling of D-branes.
cf.)[’98 Witten] [hep-th/9810188][’99 Kennedy-Wilkins] [hep-th/9905195][’01 Kuraus-Larsen] [hep-th/0012198][’01 Takayanagi-Terashima-Uesugi] [hep-th/0012210]• The tachyon configuration is given by [’98 Witten].
• In this paper, relation between tachyon condensation and K-theory isdiscussed.
• This tachyon configuration comes from [’64 Atiyah-Bott-Shapiro]

Introduction (4)Derivation (9)Application (7)Index theorem (5)Stringtheorytheory(4/5)(5)
StringRelation between the anomaly and string• This tachyon configuration is same for the mass defect in section 3!

• This anomaly can be understood from string theory.
• Fermions are found where 𝐷-branes intersect.
• This is similar to flavor symmetry on holographic QCD model.
(Sakai-Sugimoto model)

fermion

𝑈 𝑁 × 𝑈(𝑁)symmetry
Introduction (4)Derivation (9)𝑈 𝑁 symmetry
Application (7)Index theorem (5)Stringtheory
theory(5/5)(5)
StringConclusion• We discussed about perturbative anomaly with spacetime dependent mass.
• If value of the mass diverge, non-trivial contribution of the mass appears.

• The anomaly can be written by superconnection.

• There are some applications.
• Kink, vortex, ...
• With boundary• Index theorem
Introduction (4)Derivation (9)Application (7)Index theorem (5)String theory (5)