1. Six-dimensional regularization of chiral gauge theories on a lattice I
Hidenori Fukaya (Osaka Univ.)
in collaboration with
Tetsuya Onogi, Shota Yamamoto,
and Ryo Yamamura (Osaka Univ.)
arXiv: (
Thank you very much. I’d like to talk about our recent paper appeared on arXiv on Friday.
which is in collaboration with these members at Osaka univerisity.

2. 1. Introduction
O.K. let me start with introduction.

3. History of lattice chiral symmetry
1981 Nielsen-Ninomiya’s no go theorem 1992 Domain-wall fermion (Kaplan) 1997 Fixed point action (Hasenfratz) 1998 Overlap fermion (Neuberger) Luescher’s proof for existence of U(1) chiral gauge invariant regularization Kikukawa-Nakayama: SU(2)xU(1) also O.K Grabowska & Kaplan : manifestly gauge invariant construction of chiral gauge theory
Since Nielsen Ninomiya theorem appeared, it has been a big challenge to constract lattice chiral symmetry.
Although the problem was partly solved by Kaplan Hasenfratz and Neuberger on the vectorlike chiral symmetries,
It has been still difficult to realize chiral gauge invariance.
But last year, Grabowska and Kaplan proposed a promising approach, keeping a manifest gauge invariance.

4. What’s new in Grabowska-Kaplan, PRL 116 (2016) no.21 211602 ?
[Grabowska Theory Wed, Kaplan, plenary Sat]
Before GK (Neuberger, Luescher, Kikukawa, Suzuki…):
Break gauge sym. explicitly,
Find a counter-term if anomaly free,
(Mainly) studied w/ 4D overlap fermions.
Grabowska & Kaplan PRL 116 (2016) no :
Keep gauge sym. explicitly,
If not anomaly free, no 4D local action,
5D construction is essential.
We can directly listen to their own talks on Wednesday and Saturday.
So let me briefly review what is new in their idea.
Before their work, people tried this. First, break the gauge symmetry explicitly, and tried to find a counter-term if the fermions are anomaly free,
As a lucky exception, mainly studied with 4-dimensional overlap fermions.
But their new approach keeps the gauge symmetry explicitly, but show there is no 4D local action if the theory is not anomaly free as unlucky exception,
where 5 dimensional construction is essential.

5. The key is gradient flow (again).
They put Gauge d.o.f. do not flow ! Links at different t have the same 4D gauge invariance.
The key to achieve this was the Yang-Mills gradient flow.
They showed that the gauge dof do not flow and the links at different flow time have the same 4D gauge invariance.
So that the gauge invariance of the total theory is guaranteed.
Figure by Kaplan

6. Extra-dimension is essential.
They successfully reproduced a picture: gauge anomaly = gauge current missing in extra dim, [Callan & Harvey 1985] keeping total gauge invariance in 5D.
Absorbed by 5D Chern-Simons term.
They successfully reproduced a picture of gauge anomaly as the current missing into the extra dimensions, first proposed by Callan Harvery,
Keeping the total gauge invariance in 5 dimensions and the gauge non-invariance is absorbed into CS term.
4D surface

7. How about global anomaly ?
Global anomaly [Witten 1982]:
Gauge anomaly SU(2) = mod 2 index of 5D
Dirac operator w/ 5-th direction
= gauge non-invariant path
So far so good.
Then how about global anomaly? This is my topic.
As Witten showed a long ago, the SU(2) global anomaly is viewed as mod 2 index of 5dimensional Dirac operator,
taking a gauge non-invariant interpolation to the 5th direction.
This interpolation can create the so-called mod 2 instanton in the 5dimensional bulk,
which is the origin of the global anomaly.
Mod 2 instanton flips the sign of partition function.
4D surface

8. Extra-dim. is essential for global anomaly, too.
Witten’s claim at Strings 2015:
We need “extension” of global anomaly.
Not only for “mapping torus”:
but also for ANY D+1 manifold with D-dim. boundary Weyl fermions, if the determinant has a phase
then the theory has a global anomaly.
Anomaly cannot be understood within 4-dim !
Let me here introduce an important claim by Witten at last string conference.
He claimed that we need an extention of the definition of the global anomalies.
Not only for mapping torus, he first derived the global anomaly, but also any D+1 dimensional manifold with D dim bondary with Weyl fermions,
If the fermion determinant has a non-trivial phase, then, the we should consider thies theory has a global anomaly.
This new notion is very important in that the anomaly cannot be determined withn 4 dimensions alone.

9. We need extra dimension(s) !
We want combine them. 1. Grabowska-Kaplan’s 5D 2. Witten’s 5D But how ?
Gauge inv. gradient flow → cannot detect global anomaly. 4D
boundary
So, we need extra-dimension both for perturbative anomaly and global anomaly.
Now we want combine them. But how ?
In fact, it is difficult to identify the extra dimension of Grabowska and Kaplan and Wittens.
For the former set up, the gauge invariant gradient flow cannot detect the global anomaly.
However, if you want to have gauge non-invariant flow in the 5-th direction, we cannot keep the gauge symmetry.
Gauge non-invariant flow → cannot keep gauge symmetry. 4D
boundary

10. with 2 different domain-walls
Our proposal = 6D
with 2 different domain-walls
GK domain-wall
YM gradient flow →
　W domain-wall
← Linear flow
Then our proposal is to construct a 6-dimensional theory with 2 different domain-walls,
On one domainwall let us call GK domain-wall, we put the Yang-Mills gradient flow,
While on another W domain-wall we put a linear interpolation to a trivial background.
And our 4-dimensional world is put here at the junction of two domain-walls.
We (4dim) are here.

11. GK domain-wall contains Stora-Zumino anomaly descent equations
6DAxial U(1) anomaly
5D Chern-Simons 2
5D Chern-Simons 1
4D perturbative anomaly
GK domain-wall
YM gradient flow →
　W domain-wall
← Linear flow
In this setup, the perturbative anomaly is abosorbed by the Chern-Simons at the 5-dimensional domain-wall, in the same way as Grabowska and Kaplan.
But this is not the end of the story.
The Chern simons term is balanced with another Chern-Simons which originates from the axial U(1) anomaly in 6-dimensional bulk.
This structure is precisely what is known as the Stora-Zumino anomaly descent equations.
[Stora 1983, Zumino 1983,
Alvarez-Gaume & Ginsparg 1984,Sumitani 1984]

12. W domain-wall mediates global anomaly inflow.
6D mod 2 index
SU(2) example
5D mod 2 index
4D global anomaly
GK domain-wall
YM gradient flow →
　W domain-wall
← Linear flow
Moreover, we find that the W domain-wall mediates a similar anomaly ladder of the global anomalies.
The global anomaly is the mod 2 index of the 5 dimensional surface,
Which is balanced with another mod 2 index in the 6 dimensional bulk.
We (4dim) are here.

13. Our 6-dim formulation has
Stora-Zumino anomaly ladder: D U(1)A index → gauge anomaly.
Global anomaly ladder (new finding): 6D exotic index → global anomaly.
3. Anomaly free condition = sign-
problem free condition in 6D:
→ If anomaly free, 6D determinant is
real positive. → Monte Carlo is O.K. !
Let me summarize here the features of our 6-dimensional formulation of chiral gauge theory.
It naturally contains the anomaly ladder by Stora and Zumino,
Global anomaly ladder, which may be new finding at least, in physics,
The anomaly free condition is translated to the cancellation of sign problems so that
If anomaly free, we may apply Monte Carlo method.

14. Contents Introduction “Parity” and axial U(1) anomalies in 6D✔️
Introduction
“Parity” and axial U(1) anomalies in 6D
Two domain-walls
Anomaly ladder through GK domain-wall
Anomaly ladder through W domain-wall
Anomaly free condition
Lattice regularization
Summary and discussion
My talk
This is the plan of my talk and the next talk by Ryo Yamamura.
I will explain the key symmetries in 6-dimensions and how a single Weyl fermion is localized at the junction of two different domain-walls.
Then I will show how the Stora-Zumino’s anomaly ladder through GK domain-wall appears.
Then the next speaker will talk about another anomaly ladder of global anomalies,
What is required to be anomaly free, and finaly regularize our theory on a lattice.
8:30
Next talk by
Ryo Yamamura

15. 2. “Parity” and axial U(1) anomalies in 6D
O.K. let me explain two key anomalous symmetries in 6dimensions.

16. Two anomalous symmetries
1. Axial U(1) symmetry 2. Parity’ symmetry (reflection in 5th direction) Mass-term is not invariant
One is conventional axial U(1) symmetry, with the chiral operator denoted by gamma_7.
Another important symmetry is parity’ symmetry or reflection in 5th direction operated by this.
With both symmetry transformation, the massterm is not invariant.

17. Parity anomaly
Cf. Usual parity (only in even-dim.) : mass is allowed since (in any dim.) has anomaly: massless fermion action is invariant, but (zero-mode part of) fermion measure is NOT :
This parity’ symmetry is not the same as usual parity, existing only in even dimensions for which the mass term is allowed.
Our P’ symmetry, which can be defined in any dimensions, has an anomaly.
Because of the sign flip of the zero mode measure.

18. Two mass terms
:U(1)A and P’ asymmetric. :odd in P’ but U(1)A invariant. ⇒ Dirac fermion w/ periodic boundary
Now we introduce two different kinds of mass terms.
One is the conventional mass term, which breaks both of axial U(1) and P’ symmetries.
Another is odd in P’ but U(1)A invariant. Here R_5 and R_6 are the reflection operators.
Then, the Dirac fermion determinant with Pauli Villars field with opposite signs of the mass terms, can count the zero-modes
Of two different kinds, P is the conventional U(1)A index while, I is exotic index, related to global anomalies.

19. 3. Two domain-walls
Then we introduce two domainwalls with these two mass terms.

20. Two domain-walls Let’s consider a 6D Dirac fermion where we assume
is symmetric under
(* later, gauge field is given by gradient & linear flows)
GK DW W DW
Let us consider this 6dimensional Dirac fermion determinant, where we have kink structures in these two mass terms.
The former represents the GK domain-wall, while the latter is Witten domain-wall.
Here we assume both of M mu are positive and A5 and A6 are zero and 4-dimensional gauge fields are symmetric under 5d and 6d reflections.

21. Fermion determinant is still real !
determinant has Hermiticity.
Indices become non-trivial
[Atiyah-Patodi-Singer 1975]
It is important to note this doubly domain-wall fermion determinant is still real thanks to gamma=5 R=5 Hermiticity,
so that we can assign -1 to the integers.
However, these integers now become non-trivial indices known as the Atiyah-Patodi-Singer indices,
Representing the anomaly ladders.
→　Perturbative anomaly in 4D
→　global anomaly in 4D

22. Massless Weyl fermion appears !
Dirac equation has a localized solution at as
Before going to the anomaly ladder, let us confirm if the massless Weyl fermion really appears at the domain-wall junction.
The Dirac equation has a exponentially localized solution where two conditions guarantee
that the 4-dimensional mode has a plus chirality and satisfy massless Dirac equation.
Note that these conditons commute with each other.
So a single Weyl fermion really appears at the domain-wall and we will see this is not the coincidence.
* Opposite chiral mode appears if

23. 4. Anomaly ladder through GK domain-wall
O.K. now let us trace the anomaly ladder through GK domain-wall.

24. Bulk/edge decomposition
Simple example without W domain-wall where we assume Imaginary part →
Bulk
edge
Let us first consider a simple example without Witten domain-wall, like this, of which determinant is real so we have an integer here.
Inserting the same determinant in the numerator and denominator, we can decompose it into 6D bulk contribution and 5D edge determinant.
Then we compute the complex phase of each determinant and see how the original integer is decomposed.

25. Atiyah-Patodi-Singer index
6D bulk → Axial U(1) anomaly 2nd determinant → 5D Dirac fermion
Fujikawa’s method ↑
In fact, this 6D phase is axial U(1) anomaly, with boundary so it can be written as some integer multiplied by pi and Chern-Simons term at the boundary.
The 2nd determinant converges to a single 5D Dirac fermon determinant with a conventional mass M2 whose phase is known as
The eta-invariant in 5D, Thus, the integer is decomposed into two non-integers, whish is known as the Atiyah-Patodi-Singer index theorem.
[Atiyah-Patodi-Singer 1975]
Integer = non-integer + non-integer

26. Full 6D/5D/4D decomposition
With W domain-wall and
No change in 6D bulk.
(U(1)A cannot feel W-DW.)
Now lets turn on the Witten domain-wall, and decompose the determinant into 6D/5D/4D contributions.
let me skip the technical details, though.
Note that the total determinant is real so we have integer J here.
The first two determinant is the exactly the same as the previous one so that they reproduce the U(1) anomaly and Chern-Simons at the boundary.
The 2nd determinant produces another CS but in the region of negative x5 of which non-gauge invariance is
Absorbed by the phase of the 4-D Weyl fermion determinant that the 3rd determinant converges to.
This part is almost the same as Grabowska and Kaplan.
Weyl fermion !
* 5D/4D decomposition is (almost) the same as Grabowska & Kaplan.

27. Stora-Zumino anomaly ladder
To summarize what we have computed,
(integer)
6D U(1)A anomaly → 5D parity anomaly
Let me summarize what we have computed.
We decompose the phase of the fermion determinant, like this.
In this series, the axial anomaly, parity anomaly , and gauge anomaly all appear
are beautifully balanced to give one integer index.
This is the structure found by Stora, Zumino, including the overall constant determined by Alvares-GaumeGInsparg and Sumitani.
→ 4D gauge anomaly
[Stora 1983, Zumino 1983,
Alvarez-Gaume & Ginsparg 1984,Sumitani 1984]
is hidden.
(Next talk)

28. Stora-Zumino anomaly ladder
6DAxial U(1) anomaly
5D Chern-Simons 2
5D Chern-Simons 1
4Dperturbative anomaly
GK domain-wall
(YM gradient flow →)
　W domain-wall
← Linear flow
So we have confirmed that GK domai-wall mediates the Stora Zumino anomaly ladder.
[Stora 1983, Zumino 1983,
Alvarez-Gaume & Ginsparg 1984,Sumitani 1984]

29. Summary of part I Our 6D determinant w/ 2-different DWs is real,
has a Weyl fermion at 4D junction,
gauge anomaly originates from 6D
U(1)A index [Stora-Zumino anomaly ladder].
Now let me summarize the part one of our talks.
We propose a 6-dimensional massive Dirac fermion determinant having two domain-walls,
Which is real,
Having a Weyl fermion at the junction,
And the Stora-Zumino anomaly ladder is beautifully embeded in the 6-dimensional set-up.
6D U(1)A anomaly→ 5D parity anomaly→ 4D gauge anomaly

30. Next talk Introduction “Parity” and axial U(1) anomalies in 6D✔️
Introduction
“Parity” and axial U(1) anomalies in 6D
Two domain-walls
Anomaly ladder through GK domain-wall
Anomaly ladder through W domain-wall
Anomaly free condition
Lattice regularization
Summary and discussion
My talk ✔️ ✔️ ✔️
I stop here and Ryo Yamamura will give the second part next. Thank you very much.
Next talk by
Ryo Yamamura

31. From Witten’s slide “anomaly revisited” at Strings 2015
Claiming anomaly free ⇔ sign problem free.
Let me finally share with you the slide by Witten showing the equivalence of the anomaly free condition to the sign problem free condition.
Our work is along the line with his idea.

32. Back-up slides
O.K. let me start with introduction.

33. Possible applications
4D→2D : Doubly gapped (M and μ)　topological insulator can exist ?
Higgs : → definition of standard model ?
Higher dim theory : Our world is really 6D ?

34. What are really essential ?
6D : Yes. Stora-Zumino’s solution for consistent anomaly is unique.
Two domain-walls : Yes. At least, need to distinguish U(1)A and P’
Gradient flow : we don’t know. no imaginary part even without it.
Non-locality (R5,R6): probably no. but analysis is easier with them.

35. Phase of 5D determinant Anomaly-free → must be zero !
global anomaly (old definition)
Perturbative anomaly
global anomaly new def. by Witten 2015:
no local 4D action to express the phase.

36. Why 6D ? can be determined only relatively
(direct computation is ill-defined due to UV div.).
is our 6th coordinate ! → We need
5th direction to separate L/R chiral modes,
6th direction to determine
[Alvarez-Gaume et al. 1986]

37. CP restoration
Complex phase of 5D determinant = CP violating lattice artifact (w/o CKM ) Our 6D construction may be automatically giving a counter-term to keep the CP symmetry at finite lattice spacing. [Fujikawa-Ishibashi-Suzuki 2002, Hasenfratz 2005]

38. Global anomaly classification
SU(2) global anomaly : O.K.
Other groups on 4-dim torus: Maybe. Index can be detected by P’.
But higher dim: we don’t know. For example, may require quite non-trivial treatment.

39. Parity anomaly on a lattice
P’ has an anomaly.
On the lattice, we may need Ginsparg-Wilson-type relation for P’ symmetry.
The U(1)A invariant mass term in the kernel of overlap Dirac operator ?

40. Anomaly free condition
1. Axial U(1) cancelation in 6D: cancels perturbative anomaly. 2. “Parity” anomaly cancelation : # fundamental rep. = even cancels global anomaly. ⇒ Our determinant is real positive !
In our formulation, the gauge anomaly free conditions are simplified to the cancellation of two anomalies,
The axial U(1) cancelation cancels perturbative anomaly, while the parity invariance cancels the global anomaly.
As a bonus, our total fermion determinant is assured to be real positive, if the theory is anomaly-free.

41. 5D -> 4D Together with anti-domain-wall, it becomes
Another CS on 5D Weyl fermion !
We can further decompose this 5-dimensional Dirac determinant into 2 pieces and we find that
One produces another Chern-Simons but this time, restricted to negative region of x-5.
Another one converges to our target Weyl fermion determinant, except for some phase factor.
Now we can see that the gauge anomaly of the Weyl fermion is absorbed by this Chern-Simons term,
Which is exactly the same structure as Grabowska Kaplan.

42. Massless Weyl fermion appears !
Dirac equation has a localized solution at as
Before going to the Atiyah-Patodi-Singer index, let us confirm if the massless Weyl fermion really appears at the domain-wall junction.
The Dirac equation has a localized solution where two conditions guarantee that the 4-dimensional mode has a plus chirality.
Note that these conditons commute with each other.
So a single Weyl fermion really appears at the domain-wall and we will see this is not the coincidence.
* Opposite chiral mode appears if

43. Summary of part 1 and 2 : Our 6D formulation has
Stora-Zumino anomaly ladder: 　　6D U(1)A index → gauge anomaly
Global anomaly ladder : 　　6D exotic index → global anomaly.
Gradient flow in x5 + linear interpolation in x6 → mirror fermions are decoupled.
Anomaly free condition = sign-
problem free condition in 6D:　　　　　
Monte Carlo is O.K. !
Let me summarize here the features of our 6-dimensional formulation of chiral gauge theory.
It naturally contains the anomaly ladder by Stora and Zumino,
Global anomaly ladder, which may be new finding at least, in physics,
The cancellation of gauge anomalies is translated to the cancellation of two global symmetries.
And our formulation is sign-problem free, so that the Monte Carlo method is possible.