格子ゲージ理論における指数定理

深谷英則(大阪大学)
スライドURL:
http://www-het.phys.sci.osaka-u.ac.jp/~hfukaya/slides/Fukaya-LatIndex.pdf
1

自己紹介
深谷英則 (ふかやひでのり) hfukaya[at]het.phys.sci.osaka-u.ac.jp
出身: 静岡県浜松市
家族: 妻&こども3人* 深谷賢治さんとは(私の知る限り)縁戚関係にありません。

2006年 京都大学にて博士(理学)取得
2006-2010 理研、Niels Bohr 研究所(デンマーク)、名古屋大などでポスドク&特任助教
2010- 大阪大学理学研究科助教
専門: 素粒子論 (格子ゲージ理論)
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素粒子論の3つの分野何を扱う？

エネルギー(1/波長)
あの世

超弦理論

重力を含む全て(top down)
素粒子標準模型

素粒子現象論を超える物理格子ゲージ理論(主に)素粒子標準理論の強い力(bottom up)

1019 GeV

1000-10000 GeV(現在の加速器の限界)
1 GeV [陽子の質量]
この世
3素粒子論の3つの分野
数学との交流
超弦理論

あの世

多い
(Witten,Seiberg,Gukov…)

素粒子現象論

格子ゲージ理論

少ない

でも実は数学的に一番 well-definedかも。(有限次元ベクトル空間で書ける) この世
4

格子ゲージ理論でやっていること強い相互作用は解析的に理論計算できない(結合が強すぎる)。
→ 格子上に場の量子論を粗視化、スーパーコ
ンピュータにのせてシミュレーション=経路積分
して理論計算する。
世界最速の富岳もガンガン使っています。

ハドロン質量PACS-CS
collaboration

カイラル対称性の自発的破れの検証
JLQCD 20105

格子ゲージ理論でやっていること＝強い相互作用をきちんと計算するでも意外とほかの分野との接点も…
1. Domain-wall fermion [Kaplan 1992] = クォーク場を5次元トポ
ロジカル絶縁体の4次元エッジ状態として扱う手法
2. 数学的にしっかりしている。経路積分が有限でwell-defineな多重積分。また、離散化された空間なのにAtiyah-Singerの指数定理が定式化できたりする ← 古田幹雄氏驚く(2018)6数学者のみなさんとの共同研究
2017 物理屋でもわかるAtiyah-Patodi-Singer (APS)指数(連続理論)
深谷、大野木、山口（阪大）が発見法的に提案(計算も摂動論) [ arXiv:1710.03379 ]2018 古田（東大)：「数学者への挑戦と受け止めました」
2019 物理屋でもわかるAPS指数の数学的証明
深谷、古田、松尾(名大)、大野木、山口、山下（京大） [arXiv:1910.01987]
2019 物理屋でもわかるAPS指数の格子理論への応用
阪大のPD+学生の川井、森、松木、中山らと摂動計算 [arXiv:1910.09675]
2020 物理屋でもわかる格子上のAS, APS指数の数学的定式化
in progress.

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本講演の目標
1. 数学のみなさんに 離散化された格子上の場の理論（有限次元ベクトル空間）でもindex が定義できることに驚いていただく。
2. 物性のみなさんに 格子ゲージ理論で質量ゼロのフェルミオンを扱うにはトポ
ロジカル絶縁体とそっくりな構造を使うほかないことを理解いただく。「この世は5次元トポロジカル絶縁体の4次元エッジ状態かもしれない」
8割は非専門家へのレビュー、残り2割(時間があれば)で最近の研究：
格子理論上のAPS指数定理の話をします。
スライドURL:
http://www-het.phys.sci.osaka-u.ac.jp/~hfukaya/slides/Fukaya-LatIndex.pdf

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2. 格子ゲージ理論とは

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2.1 まずは連続時空の場の理論「場の量子論とはファイバー束の統計力学である。」
時空＝底空間場＝ファイバー束の切断

場の量子論の経路積分＝ファイバー束のありとあらゆる切断のとり方に対して統計平均をとること。
ただし、その確率分布はHamiltonianではなく「作用」で与えられる。
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ゲージ原理: ファイバー束の選び方
1. まずゲージ群を決める＝主束を決める
素粒子標準模型は SU(3)xSU(2)xU(1)束SU(3)の接続 = グルーオン場(私の研究対象)
SU(2)の接続 = W, Z boson場
U(1) の接続 = 電磁場 (* 正確にはSU(2)xU(1)->U(1)の束の簡約化(reduction)後のU(1)束。ゲ ージ対称性の自発的破れ。)
2. 物質場を決める= 主束の同伴束。どんな表現の場であってもいいはずだが自然は基本表現ばかり選んでいる。反粒子=複素共役表現の束。
クォーク場: SU(3), SU(2), U(1)全てにチャージを持つ。
レプトン(電子、ニュートリノなど)場: SU(2),U(1)のチャージ。
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Higgs場 : SU(2)のチャージ

作用原理
作用 S = 接続および 同伴ベクトル束<latexit sha1_base64="f//Q6ARFTocjNexk/6toRC4eoNo=">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</latexit>

(x)

の切断の

テンソル積のうち、trivial 表現になるものの線形結合の空間積分。
ただし係数は定数にとる。
さらにローレンツ変換(空間回転)に対してtrivial であることも要求。

Z<latexit sha1_base64="pPVtH9UwVXm6QaZJ3oTKRwrg96o=">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</latexit>例)dx ¯(x) (x)
X

曲率
反クォーク
12

分配関数(経路積分)
Z
Z<latexit sha1_base64="ifylfKksTsM0OgnVzOlVTrxJZQU=">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</latexit>

Z=

Y
x

d (x)d ¯(x)YxdA(x) exp

: Planck定数 (6.6x10-34 Js)⇥

S(A, , ¯, { i })/~⇤

Boltzmann weight
すべての切断、接続のとりかたについて足し上げる。
非加算無限個の多重積分（数学的にはill-defined. 私見：格子理論の極限として定義すべき）。
作用を通じてψ, Aが互いに影響 = 相互作用。
例) 電子がエネルギーを失うと光子が励起。
作用の最小値(の近似)= 古典場の理論(電磁場ならMaxwell方程式に従う)
13

くりこみ可能性
作用のとりかたは無数にあるはずだが、くりこみ可能性= 波長の短い寄与を正則化したとき係数がコントロール可能* 長波長ではくりこみ不可能な寄与は効かないという見方もできる。
を要求すると、係数の質量次元が非負のものに限られる。
くりこみ可能な作用の選択肢はごくわずか。
<latexit sha1_base64="+1Ez7Dqv1vdNlqcQJ2rN0MAQ+0U=">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</latexit>

SQCD =Z 
X

1
TrF ⇤ F
4¯(D + m) + ✓F F

Dirac operator

なぜか

14
なので無視

量子電磁気学(QED)
数学的にwell-definedでないものがどうして計算できるのか？
(意味あるの？)

実は分配関数はeの項をのぞいてGauss積分。
→
で(漸近)展開して無理やり積分。
の計算は10桁の精度で実験と一致。髪の毛vs.地球一周
数十万個のFeynmann diagramの計算。木下東一郎(コーネル大)のグループが主導。
15

量子色力学(QCD)
強い相互作用(クォークとグルーオンのSU(3)ゲージ理論)では<latexit sha1_base64="7YaRMDw/xDAJobkX2gIGTF0ksZc=">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</latexit>

ZQCD (g) =Z

DAD exp⇢Z 
X

1
TrF ⇤ F4¯ ( µ (@µ + igAµ ) + m)なので摂動展開できない。無理やりやってもクォーク3個の束縛状態として陽子が存在するなどの記述は不可能。
ー＞ 格子理論で粗視化してコンピュータに計算させる。格子QCD
4次元の場の理論は自由度が膨大。スパコンが必要。
* 歴史的にはQCDのためにスパコンが発展。
CP-PACS (1996年世界1を記録)は(ほぼ)QCD専用機。

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2.2 格子ゲージ理論
格子ゲージ理論 = 底空間を正方格子点（with 周期境界条件）上に離散化、格子間隔をフェルミオン場とする。
<latexit sha1_base64="f//Q6ARFTocjNexk/6toRC4eoNo=">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</latexit>(x) を各点(site)に代表させる(ファイバーは連続)。

ゲージ場(gluon場)(接続)をリンクにおく。
となることを期待して、ゲージ変換をで与える。

はsite上におく。

17

Gauge不変な作用の構成
Q. リンク変数を用いて作用(Gのtrivial 表現)を作るにはどうしたらよいか？

A. ループを作ってトレースをとればよい。

Wilson のplaquette 作用(gluonの作用)18フェルミオン(quark)の作用
Fundamental表現の同伴束の切断。同時にSpin(4)の表現であることも要求

2種類の共変差分<latexit sha1_base64="GiGYZYIL32+no26rYr5qGZI9ajE=">AAAB/HicZVDLSgMxFM3UV62vsS7dBEuhLiwzUtRl0Y3LCvYBnaFk0kwbmmSGJCMtQ/0UV4KCuPVHXPk3pu0gtj1w4XDOvdx7TxAzqrTj/Fi5jc2t7Z38bmFv/+DwyD4utlSUSEyaOGKR7ARIEUYFaWqqGenEkiAeMNIORnczv/1EpKKReNSTmPgcDQQNKUbaSD276MWKVsYXyBsinXo8mZ737JJTdeaA68TNSAlkaPTsb68f4YQToTFDSnVdJ9Z+iqSmmJFpwUsUiREeoQHpGioQJ8pP57dPYdkofRhG0pTQcK7+n0gRV2rCA9PJkR6qVW8m/nnlpVU6vPFTKuJEE4EXm8KEQR3BWRKwTyXBmk0MQVhScyzEQyQR1iavgknBXf15nbQuq+5VtfZQK9Vvszzy4BScgQpwwTWog3vQAE2AwRi8gDfwbj1br9aH9blozVnZzAlYgvX1Cy8IlGE=</latexit><latexit sha1_base64="NU+gmSA3YN6/kInD4b5C2GGVwBk=">AAAB/HicZVDLSgMxFM3UV62vsS7dBEuhIpQZKeqy6MZlBfuAzlAyaaYNTTJDkpGWoX6KK0FB3Pojrvwb03YQ2x64cDjnXu69J4gZVdpxfqzcxubW9k5+t7C3f3B4ZB8XWypKJCZNHLFIdgKkCKOCNDXVjHRiSRAPGGkHo7uZ334iUtFIPOpJTHyOBoKGFCNtpJ5d9GJFK+ML5A2RTj2eTM97dsmpOnPAdeJmpAQyNHr2t9ePcMKJ0JghpbquE2s/RVJTzMi04CWKxAiP0IB0DRWIE+Wn89unsGyUPgwjaUpoOFf/T6SIKzXhgenkSA/VqjcT/7zy0iod3vgpFXGiicCLTWHCoI7gLAnYp5JgzSaGICypORbiIZIIa5NXwaTgrv68TlqXVfeqWnuoleq3WR55cArOQAW44BrUwT1ogCbAYAxewBt4t56tV+vD+ly05qxs5gQswfr6BSvolF8=</latexit>

(x

aµ̂)

<latexit sha1_base64="maKhYbKI36mxLJFtiybSCr4VW5g=">AAAB73icZVBdSwJBFL1jX2ZfVo+9DIlgL7IbUj1KvfRokBroIrPjrA7Ozm4zs5Es/oiegoLotd/TU/+mUZdIPXDhcM693HuPHwuujeP8oNza+sbmVn67sLO7t39QPDxq6ShRlDVpJCL14BPNBJesabgR7CFWjIS+YG1/dDP1209MaR7JezOOmReSgeQBp8RYqd2NNa88n/WKJafqzIBXiZuREmRo9Irf3X5Ek5BJQwXRuuM6sfFSogyngk0K3USzmNARGbCOpZKETHvp7NwJLlulj4NI2ZIGz9T/EykJtR6Hvu0MiRnqZW8q/nnlhVUmuPJSLuPEMEnnm4JEYBPh6fO4zxWjRowtIVRxeyymQ6IINTaigk3BXf55lbTOq+5FtXZXK9WvszzycAKnUAEXLqEOt9CAJlAYwQu8wTt6RK/oA33OW3MomzmGBaCvX00Fj2U=</latexit>(x)(x + aµ̂)<latexit sha1_base64="kgU+rBFe1TJe1mkdUb804rPGLCM=">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</latexit><latexit sha1_base64="s7C9GVJ8TDsM1FDZsMnME9zSwNc=">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</latexit>

rfµ (x) = [Uµ (x) (x + aµ̂)rbµ(x) = [ (x)Uµ† (x(x)]/aゲージ変換性<latexit sha1_base64="XX904JAiKzyoUBJhoxXrJDndXjE=">AAACIHicbVDLSsNAFJ3UV62vqEs3g6VQQUoipbosunFZwT6giWUynbRDJ5MwMxFL6J+48lNcCQoirvRrnLRBbOuFYQ7n3Mu953gRo1JZ1peRW1ldW9/Ibxa2tnd298z9g5YMY4FJE4csFB0PScIoJ01FFSOdSBAUeIy0vdFVqrfviZA05LdqHBE3QANOfYqR0lTPrDkceQz1nCC+S/xTb+JEkpYfThwVwkH6/y/3zKJVsaYFl4GdgSLIqtEzP51+iOOAcIUZkrJrW5FyEyQUxYxMCk4sSYTwCA1IV0OOAiLdZOpvAkua6UM/FPpxBafs34kEBVKOA093BkgN5aKWkr9aaW6V8i/chPIoVoTj2SY/ZlC7T9OCfSoIVmysAcKC6mMhHiKBsNKZFnQK9qLnZdA6q9i1SvWmWqxfZnnkwRE4BmVgg3NQB9egAZoAg0fwDF7Bm/FkvBjvxsesNWdkM4dgrozvH0bbou4=</latexit>aµ̂) (xaµ̂)]/a

f,b
rf,b
(x)
!
g(x)r
(x)
µ
µ

Naïve Dirac 演算子(実は不十分)
Quark 作用<latexit sha1_base64="9GuRQ00XGceo2qx7Sm7u1kFx7Po=">AAACFHicZVDLSgMxFM34rPU16tJNsBRahTIjRd0IRUVcVrQP6NQhk6ZtaDIzJBlpGfoPrvwUV4KCuHXjyr8x0xax7YHAuefcy809XsioVJb1YywsLi2vrKbW0usbm1vb5s5uVQaRwKSCAxaIuockYdQnFUUVI/VQEMQ9Rmpe7zLxa49ESBr492oQkiZHHZ+2KUZKS655eOdew3OIHoqOjLjbh46HROyEkg5z/Xzu6ojnk0Jz6JoZq2CNAOeJPSEZMEHZNb+dVoAjTnyFGZKyYVuhasZIKIoZGaadSJIQ4R7qkIamPuJENuPRTUOY1UoLtgOhn6/gSP0/ESMu5YB7upMj1ZWzXiL+edmpVap91oypH0aK+Hi8qR0xqAKYJARbVBCs2EAThAXVn4W4iwTCSueY1inYszfPk+pxwT4pFG+LmdLFJI8U2AcHIAdscApK4AaUQQVg8ARewBt4N56NV+PD+By3LhiTmT0wBePrFyuVnEs=</latexit>

SF = a4Xx¯(x)(D + m) (x):qの複素共役場
:質量

19

経路積分
空間をT4 にとり、有限格子で分配関数を与える。

Z Y<latexit sha1_base64="YuY02XfsjGbla2RLYm3a58reEmU=">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</latexit>d ¯(x)d (x)dUµ (x) exp [ SG

SF ]

x,µ

群積分

* フェルミオンはGauss積分してDirac演算子のdeterminantで書ける。
この多重積分をスパコン上でMonte Carlo法で評価。
より細い格子で実行し、外挿 = 連続極限
より大きい格子で実行し、外挿=熱力学極限
目標はO(1)%の精度を達成、実験と比較。

20

3. Nielsen-Ninomiyaの定理

21Fermion ダブリング
自由フェルミオン粒子のDirac方程式

Fourier変換

1 X
!
2a µ

µ

˜ ipx eipµ a
(p)e

e

ipµ a

<latexit sha1_base64="IKBq1fgFiI1FW3QvfiLrXdJq1r0=">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</latexit>

iX
=
a µ

µ

˜ ipx sin(pµ a) = 0.
(p)eにゼロ点(4方向あるので16点もある)。
でほしいのは<latexit sha1_base64="eLTdYXe0hVpm9agXgBg+Uzu6ozQ=">AAAB73icZVBNSwMxEJ2tX7V+VT16CZaCp5IVQS9C0YvHCvYD2qVk02wbmmTXJCuUpT/Ck6AgXv09nvw3pu0i1j4YeLw3w8y8MBHcWIy/vcLa+sbmVnG7tLO7t39QPjxqmTjVlDVpLGLdCYlhgivWtNwK1kk0IzIUrB2Ob2d++4lpw2P1YCcJCyQZKh5xSqyT2km/J9Nr3C9XcA3PgVaJn5MK5Gj0y1+9QUxTyZSlghjT9XFig4xoy6lg01IvNSwhdEyGrOuoIpKZIJufO0VVpwxQFGtXyqK5+nciI9KYiQxdpyR2ZP57M/HXqy6tstFVkHGVpJYputgUpQLZGM2eRwOuGbVi4gihmrtjER0RTah1EZVcCv7/n1dJ67zm45p/f1Gp3+R5FOEETuEMfLiEOtxBA5pAYQzP8Apv3qP34r17H4vWgpfPHMMSvM8fV/aPZQ==</latexit>

pµ = 0 だけ。

あとの15個は非物理的自由度=ダブラー

22

Nielsen-Ninomiyaの定理 [1981]
格子理論のDirac演算子がいくつかの(場の理論に不可欠な)仮定の下で、chirality 演
算子 (Z2 grading 演算子)と反可換なとき、フェルミオンダブリングは避けられない。
＊その本質的理由は格子化により運動量空間が周期的になること。

23

Wilson fermion [1974]
ダブラーを重くして理論から排除 (励起できない=存在しない)

Fourier変換

For*,
Hermiticity :

, for

,

24

固有値分布anti-Hermitian

Hermitian

連続理論でmassless Diracの固有値は虚軸に分布。質量項は固有値を実軸方向へずらす。
Naïve Diracは虚軸に16本分布

Wilson fermion
1

4

6

4

1

25

Figure by M. Creutz結論：格子ゲージ理論において、カイラル対称性は破らなけ
ればならないNielsen-Ninomiya の定理(1981):ダブラーを取り除くた
めには質量項が必要= カイラル対称性を破るしかない。

そもそも格子上でトポロジーなんて議論できそうもない。
「指数定理なんて絶対無理！」と考えられてきた。
n+

1
n =
32⇡ 2

Dirac方程式のカイラルな解の数<latexit sha1_base64="CA6Aq6Ax87oz8ZvSHvDTZgm4FpE=">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</latexit>Z

d4 x✏µ⌫⇢ trc F µ⌫ F ⇢ (x)X

ゲージ場の巻きつき数

26注) 物性理論の運動量空間は連続

Ginsparg-Wilson relation [1982]「もしカイラル対称性を保つ連続理論」で格子点 “以外”の部分を積分しきれたとしたら、その格子理論のDirac
演算子はどのようなものになるかの考察がなされた。

この論文は20年近く忘れ去られる。。。
しかし・・・
27

Ginsparg-Wilson relation [1982]
全1184 citationの年分布

Figure from Inspire

1998年に何かが起きた。
28

ここまでのまとめ
1. 格子理論は有限格子点上のファイバー束の統計力学として定式化される。ゲージ場(接続)-> link 変数。フェルミオン場（切断）-> site変数。
2. Nielsen-Ninomiyaの定理は格子理論のDirac演算子がchirality 演算子(Z2 grading)と反可換になれないことを要求する。＝left-rightが混ざってカイラル対称性は必ず壊れる。) 指数定理なんて無理っぽい。
[参考文献]
青木慎也「格子上の場の理論」シュプリンガーフェアラーク

29

4. Chiral symmetry on lattice30カイラル対称性とは連続理論のカイラル対称性 ：質量ゼロのフェルミオンの作用Z<latexit sha1_base64="AOdzTlSAku7dEKtmGuhb3UuAwGM=">AAACHHicZVDLSgMxFM3UV62vUZdugqXgqsyUom6EoiIuK9oHdMaSSTNtaOZBckcsQ//DlZ/iSlAQV4Lg35g+ENseSDiccy/33uPFgiuwrB8js7S8srqWXc9tbG5t75i7e3UVJZKyGo1EJJseUUzwkNWAg2DNWDISeII1vP7FyG88MKl4FN7BIGZuQLoh9zkloKW2Wbptp44M8NUQn2GHh4A79+VH7AjmQ8vxiEydWPHh5eh3JO/2wG2beatojYEXiT0leTRFtW1+OZ2IJgELgQqiVMu2YnBTIoFTwYY5J1EsJrRPuqylaUgCptx0fNsQF7TSwX4k9dPbjdX/HSkJlBoEnq4MCPTUvDcS/7zCzCjwT92Uh3ECLKSTSX4iMER4lBTucMkoiIEmhEqul8W0RyShoPPM6RTs+ZsXSb1UtI+L5ZtyvnI+zSOLDtAhOkI2OkEVdI2qqIYoekIv6A29G8/Gq/FhfE5KM8a0Zx/NwPj+Bf66oUY=</latexit>

SF =は Dirac 演算子がカイラル変換 ¯(x) ! ¯(x)ei↵ 5⇥

d4 x ¯Dと反可換なので、<latexit sha1_base64="Kmzy4yw/M9rIyX4axBa36S4GBFA=">AAACEXicZVDLSgMxFM34rPU16tJNsBTrpsxIfSyLblxWsA/o1HInTdvQZGZIMmIZ+gmu/BRXgoK4denKvzFtB7HtgcDhnHu5OcePOFPacX6speWV1bX1zEZ2c2t7Z9fe26+pMJaEVknIQ9nwQVHOAlrVTHPaiCQF4XNa9wfXY7/+QKViYXCnhxFtCegFrMsIaCO17WMvUqzweII9HWJ6nzAPeNQHrwdCQPsMj1K/beecojMBXiRuSnIoRaVtf3udkMSCBppwUKrpOpFuJSA1I5yOsl6saARkAD3aNDQAQVUrmQQa4bxROrgbSvMCjSfq/40EhFJD4ZtJAbqv5r2x+OflZ07p7mUrYUEUaxqQ6aVuzLFJP64Hd5ikRPOhIUAkM5/FpA8SiDYlZk0L7nzmRVI7LbrnxdJtKVe+SvvIoEN0hArIRReojG5QBVURQU/oBb2hd+vZerU+rM/p6JKV7hygGVhfv/SInF8=</latexit><latexit sha1_base64="PVhMxrJ3xv67nkiRC+LLNTi4GGc=">AAACGXicZVDLSgMxFM34rPVVdekmWAp1U2fE17LoxmUF+4DOONxJ0zY0mRmSjFiGfoYrP8WVoCBuu/JvTB9IHwcC555zLzf3BDFnStv2r7Wyura+sZnZym7v7O7t5w4OaypKJKFVEvFINgJQlLOQVjXTnDZiSUEEnNaD3t3Irz9TqVgUPup+TD0BnZC1GQFtJD935gYg3Vix4sspdnWEZ2r6lDIXeNwF7HZACPAvB34ub5fsMfAycaYkj6ao+Lmh24pIImioCQelmo4day8FqRnhdJB1E0VjID3o0KahIQiqvHR82AAXjNLC7UiaF2o8VmcnUhBK9UVgOgXorlr0RuK/V5hbpds3XsrCONE0JJNN7YRjk8AoJtxikhLN+4YAkcx8FpMuSCDahJk1KTiLNy+T2nnJuSo5Dxf58u00jww6RieoiBx0jcroHlVQFRH0it7RJ/qy3qwP69v6mbSuWNOZIzQHa/gHXOefzg==</latexit>の下で不変。カイラル表示では<latexit sha1_base64="2yHWIl0Z7fS7DuDpLM7HrvVD7Pw=">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</latexit>

5 =

✓

12⇥2
0

0
12⇥2

0

◆
<latexit sha1_base64="kILF5h5z0TMptmkCEf9jwGiKqrU=">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</latexit>

B
=B
@

1

1

0

(x) ! eos
1

1

C B | "i C
C=B
C
A
@
| #i A
3
2
4

⇤

os
4

i↵ 5

(x)

* os : off-shell mode (classically zero)<latexit sha1_base64="5i0yXEjjTMLavJ1xpEYT7MBLGhA=">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</latexit>

5 = 1 ! right-handed<latexit sha1_base64="xCG464/ptAVhALaJlWdsNruw2OE=">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</latexit>

5 =1 ! left-handed31カイラル対称性とは質量（行列）は<latexit sha1_base64="1D5UcqXskTFY1lTz2KgJ5owQA+Q=">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</latexit><latexit sha1_base64="tml9bxGLjU5c1XgLHAaqzfohnrU=">AAAB8HicZVDLSgNBEOyNrxhfUY9eBkPAU9iV+DgGvXiMYB6QLKF3MpsMmdldZmaFEPITngQF8ervePJvnCSLmKSgoajqprsrSATXxnV/nNzG5tb2Tn63sLd/cHhUPD5p6jhVlDVoLGLVDlAzwSPWMNwI1k4UQxkI1gpG9zO/9cyU5nH0ZMYJ8yUOIh5yisZK7e4ApcTeVa9YcivuHGSdeBkpQYZ6r/jd7cc0lSwyVKDWHc9NjD9BZTgVbFroppolSEc4YB1LI5RM+5P5vVNStkqfhLGyFRkyV/9PTFBqPZaB7ZRohnrVm4l/XnlplQlv/QmPktSwiC42hakgJiaz70mfK0aNGFuCVHF7LKFDVEiNzahgU/BWf14nzcuKd12pPlZLtbssjzycwTlcgAc3UIMHqEMDKAh4gTd4d5Tz6nw4n4vWnJPNnMISnK9fOleP8Q==</latexit>

5 と可換なので、カイラル対称性を破る。

¯M (x) ! ¯e2i↵ 5 M (x) = ¯L e2i↵ M

2i↵
¯
(x)
+
e
M
R
R

L (x)

left-rightが混ざる。
Wilson 項も全く同様。
<latexit sha1_base64="EV1N1f3wMOIRQrf2nGOkNLXR+xE=">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</latexit>¯W=aX¯ f b
rµ rµ
2 µ

!

a X ¯ 2i↵ 5 f b
e
rµ rµ
2 µ

ダブラーを排除するため質量を与える＝カイラル対称性を破る

32

Nielsen-Ninomiyaの定理再考
フェルミオンダブリングを避けるにはカイラル対称性を破る= left, right が混ざる効果が必要 (Cf. Wilson fermion)。
->物理的に遠ざけたらどうか？
5次元方向に遠ざけてみよう＝Left-handed, right-handed fermion がエッジモードとして現れる
5次元トポロジカル絶縁体を考えよう= domain-wall fermion [Kaplan, 1992]

33

Domain-wall fermion (連続理論)
DDW = D5D
<latexit sha1_base64="BKi6lMa2YM/XCfK9naYieOmnMEE=">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</latexit>

M ✏(x5 ) =5 (@5Jackiw and Rebbi 1976,
Callan and Harvey 1985

4D
M
✏(x
))
+
D
5
5
✏(x5 ) = signx5<latexit sha1_base64="ImluhtRDXcPUI5B+ncCSeUMb/fE=">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</latexit>

は edge-localized 解を持つ :<latexit sha1_base64="ZGNVnRDxlw5rZmmeRyzNFO4zwdk=">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</latexit>(x) = c exp( M |x5 |)D4D 4D = 0.
<latexit sha1_base64="dUFSg/EqVdVuC7pAEsGnb+790Io=">AAACoHichVE9LwRRFD3G11pfi0ai2dgQ1eaOCCKRbFDoWCzCspkZDy/mKzOzm7BRaP0BhYpERDRUfoDGH1D4CaIk0SjcnZ0QBHfy5p173j33vZOru6b0A6KHGqW2rr6hMdYUb25pbWtPdHQu+k7RM0TOcEzHW9Y1X5jSFrlABqZYdj2hWboplvSdycr5Ukl4vnTshWDXFWuWtmXLTWloAVOFRM/UejnvWcmhqf1k3t2WH9k4pQuJFKUpjORPoEYghShmncQN8tiAAwNFWBCwETA2ocHnbxUqCC5zaygz5zGS4bnAPuKsLXKV4AqN2R3+b3G2GrE255Wefqg2+BaTl8fKJProni7ome7okh7p7dde5bBH5S27vOtVrXAL7Yfd86//qizeA2x/qv5Q6Fz9t6cAmxgNvUj25oZMxaVR7V/aO3qeH5vrK/fTKT2xvxN6oFt2aJdejLOsmDtGnAekfh/HT7A4mFaH02p2KJWZiEYVQw96McDzGEEG05hFju89wDmucK30KtPKjJKtlio1kaYLX0JZeQeYvpmg</latexit><latexit sha1_base64="M/ivo/Uu2Oz39CqwncgSUf+bYUE=">AAACvHichVHLSiNBFD22r5jxEXUjuAkGxY2hWnwhKKIuZukrKtgaujuVWKRfdFeC2uQH/AEX4mIGXMh8gB+QjQtdzsJPGFw64MaFN50GUVFvU12nzr3nVh2u4VkikIzdtyitbe0dnYmu5I/unt6+VP/AduBWfJPnTNdy/V1DD7glHJ6TQlp81/O5bhsW3zHKK438TpX7gXCdLXns8X1bLzmiKExdEpVPzWol3bb1/HRa8w7FQaj5dnpqtbagSX4ko/ahzwu1cKL2Jp/NpzIsy6JIfwRqDDKIY81NXUNDAS5MVGCDw4EkbEFHQN8eVDB4xO0jJM4nJKI8Rw1J0laoilOFTmyZ/iU67cWsQ+dGzyBSm3SLRcsnZRqj7C+7Yo/shv1h/9jzp73CqEfjLce0G00t9/J9p0ObT9+qbNolDl9VXygMqv7ak0QRc5EXQd68iGm4NJv9qydnj5vzG6PhGPvNHsjfL3bP6uTQqf43L9f5xjmSNCD1/Tg+gu3JrDqTVdenMkvL8agSGMYIxmkes1jCT6whR/deoI5b3CmLSkEpK3azVGmJNYN4E0r1BVLipmw=</latexit>

5

4D

=

4D

4D
normal insulator.

topological insulator

x5<latexit sha1_base64="ncI+yDKll7VKs0OU7KisUh9ATR4=">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</latexit>

この状態はleft-handedで質量ゼロ!
質量の符号を反転させればright-handed。

34

Domain-wall fermion (連続理論)
L5 が有限だと L/R mixing で質量(residual mass)が生じるが、それはL5 とともに指数関数的に減衰。
normal insulator

normal insulator
topological insulator<latexit sha1_base64="o6W0wMUZJAOFtI58042iSjVXBAU=">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</latexit><latexit sha1_base64="MquWK21IKUX+wh7maJU2Gwz+pnw=">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</latexit>

L

R

L5 x5
<latexit sha1_base64="l0vWs0yrGfu+MICqeMZ2TgEMCLg=">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</latexit><latexit sha1_base64="ncI+yDKll7VKs0OU7KisUh9ATR4=">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</latexit>

35Lattice Domain-wall fermion [Kaplan 1992]
5D Wilson Dirac 演算子でdomain-wall fermion を構成、カイラル対称性が L5 ! 1 で成り立つことを確認。
<latexit sha1_base64="Xm9jiKmxkkME/Dkh5fDzi1opgGY=">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</latexit>

lat.
5D
DDW
= DW
<latexit sha1_base64="RW1C6o449FyaHEtwo22lIvmJD+8=">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</latexit>⇤✏(x5

a/2)✏(L5x5a/2)

格子サイズは L4 ⇥ 2L5 ですべての方向に<latexit sha1_base64="W6Iv6FY660qIB1yYCcJwxpicrVM=">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</latexit>

周期境界条件を課す

5次元方向にゲージ場は入れない。 U5 (x) = 1.
<latexit sha1_base64="5ym0GEhaX16AiSvYMJap8QK+2Qc=">AAACjHichVG7SgNBFD2u7/hI1EawEUMkNmFWfKEEgoJY+spDYgi76yQu7ovdTVCDP2BrYaEWChbiB/gBNv6ART5BLCPYWHh3syAa1DvMzJkz99yZw5UtTXVcxuptQntHZ1d3T2+or39gMBwZGs44ZsVWeFoxNdPOyZLDNdXgaVd1NZ6zbC7pssaz8sGKd5+tcttRTWPbPbJ4QZfKhlpSFcklaiddnI0fTiXFRDESZQnmx3grEAMQRRDrZuQBu9iDCQUV6OAw4BLWIMGhkYcIBou4AmrE2YRU/57jBCHSViiLU4ZE7AGtZTrlA9ags1fT8dUKvaLRtEk5jhh7ZneswZ7YPXthH7/Wqvk1vL8c0S43tdwqhk9Ht97/Vem0u9j/Uv2hkCn7b08uSljwvajkzfIZz6XSrF89Pm9sLW7GapPshr2Sv2tWZ4/k0Ki+KbcbfPMCIWqQ+LMdrSAznRDnEuLGTDS1HLSqB2OYQJz6MY8U1rCONL2r4wyXuBIGhRlhSUg2U4W2QDOCbyGsfgLjQpIb</latexit>

4次元方向のゲージ場はx5依存性を持たせない(5-軸は内部空間と解釈もできる)。

figure by Kaplan
36

Shamir domain-wall fermion [Shamir 1993]
質量を非対称にとり、
lat.
5D
DDW = DW
<latexit sha1_base64="7EPBpIY3dg7S9l3IAMHKlmjHGi8=">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</latexit>⇤ + ⇤2 ! +1,<latexit sha1_base64="kXpIPM8noYRE4AMckQU5Wpbul6I=">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</latexit>⇤✏ + ⇤2⇤⇤2 = M fixed.

の極限をとる。normal insulator region のフェルミオンをdecouple.
x5 < 0,<latexit sha1_base64="oGErDvVTCnAD1NsDSuDQHgsPGHQ=">AAACm3ichVHLSsNAFD3Gd320KoggQrEoLqTciC/URdGNiAtfVcFKSOJYg2kSkrRYqz/gD7hwpeBCdO8HuPEHXPgJ4lLBjQtv0oCoqHeYmTNn7rkzh6s5puH5RI81Um1dfUNjU3OspbWtPZ7o6Fz37KKri6xum7a7qameMA1LZH3DN8Wm4wq1oJliQ9ufC+43SsL1DNta88uO2C6oecvYNXTVZ0pJdB8oYzM0nJsOBuNcXiQXlTElkaI0hZH8CeQIpBDFkp24RQ47sKGjiAIELPiMTajweGxBBsFhbhsV5lxGRngvcIwYa4ucJThDZXaf1zyftiLW4nNQ0wvVOr9i8nRZmcQAPdAVvdA9XdMTvf9aqxLWCP5S5l2raoWjxE96Vt/+VRV497H3qfpDoXH235587GIy9GKwNydkApd6tX7p8PRldWploDJIF/TM/s7pke7YoVV61S+XxcoZYtwg+Xs7foL1kbQ8npaXR1OZ2ahVTehFP4a4HxPIYB5LyPK7R7jANW6kPmlOWpAWq6lSTaTpwpeQsh+7S5cd</latexit>

x5

L5 の格子ははじめから無視できる。

注) Dirichlet 境界条件とは違う(後述)。
edge modeの存在も非自明。

figure by Kaplan
37

Shamir domain-wall fermion 詳細<latexit sha1_base64="3HGLBBTm/Kdyw6wOaisYhwCWa9Y=">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</latexit>

1 5 f
M = DW M +(@5 + @5b )2
shift 演算子 S5± (x) = (x ± aµ̂5 ) で書き直すと、

5D
DSDW = DW

a f b
@5 @5
2

<latexit sha1_base64="kQW/BXmwTnKQmkT1DhQvGC5Qy58=">AAACqnichVFNSxtRFD2OVmNaNepG6CY0pChIeCNGRRDEbrrUxKglo8PM5GkezhczL0EN/gH9AS66UuiidNWVoMtu/AMu/AnSpYIbF96ZDIiKeoc397zz7rnvHa7p2yKUjF11KJ1dH7p7Ur3pj5/6+gcyg0MrodcILF6xPNsL1kwj5LZweUUKafM1P+CGY9p81dz+Fp2vNnkQCs9dlrs+X3eMLVdsCsuQROmZfFkvbmi+k9X8uhjdGZtr54gxtLohNaehF8f0TI4VWBzZl0BNQA5JLHqZU2iowYOFBhxwuJCEbRgI6atCBYNP3DpaxAWERHzOsY80aRtUxanCIHab/lu0qyasS/uoZxirLbrFphWQMos8u2S/2Q27YH/YNbt/tVcr7hG9ZZey2dZyXx84GCnfvatyKEvUH1VvKEyqftuTxCZmYi+CvPkxE7m02v2be0c35dlSvvWVnbD/5O+YXbF/5NBt3lq/lnjpJ9I0IPX5OF6ClYmCOlVQlyZz8wvJqFL4jC8YpXlMYx7fsYgK3XuIvzjDuTKulJQfSrVdqnQkmmE8CaX2ALdTneI=</latexit><latexit sha1_base64="+kOB4IX151D+DlAc8KeOLs8Vmyo=">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</latexit>@5f + @5b =

S5+

1+1
a

S5

=

S5+

S5
a

<latexit sha1_base64="3lTyE6mGlBij4BagSeaGXc+u/do=">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</latexit>

aDSDW = 1 + DW a
|
{z

=Dk

(S5+
f b
a@5 @5 =<latexit sha1_base64="8xyXnycgaj/CUZaSvOW0XjpaV5Y=">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</latexit>

Ma
}

1

5

2
| {z }
=P

1)(1
a

+
S5

S5 )

S5+ + S5
=
a

1+ 5
S5
2
| {z }=P+

ただし、 (x5 < 0) = (x5 > L5 ) = 0 の境界条件(x5=0ではP+にだけDirichlet.P-はfree)<latexit sha1_base64="QyGN2UHoIZMkeUNv+O9h+ykhf7k=">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</latexit>

238

Shamir domain-wall fermion 詳細
5D方向だけ行列表示にすると、
0
Dk
P
0
0
···
B P+ D kP0···
B
P+ D kP···
aDSDW = B
B 0
@ ···
···
···
···
···
0
···
P+
Dirac 方程式の解 を ± chirality ごとに求める。
<latexit sha1_base64="1eM18cXcSK0bB2S36ik14kRtHaY=">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</latexit><latexit sha1_base64="bUM+3cRMZKKwNH+6zbAqmHBQ16E=">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</latexit>

aDSDW<latexit sha1_base64="MC9zYzRuPAdTmGPoyy0bLV/CyPM=">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</latexit>

=0

1

0
0 C
C
0 C
C
··· A
Dk

x5 = 0<latexit sha1_base64="KKDdmR/2hf8AE+xcoQCaF2Xh8jA=">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</latexit>

x5 = L5<latexit sha1_base64="lpl/0r64WLaHrwUo3P02wywnJRs=">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</latexit>

+
S5 ]

[Dk
= 0.
[ S5 + Dk ] + = 0.
<latexit sha1_base64="/GzSVQ/bdOJ+9zmqdgjOPY13//c=">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</latexit>

39

Shamir domain-wall のedge 解
1X
Dnaive =2 µ<latexit sha1_base64="q+5a+0YpXUFR2PEflJUv9H4f+S0=">AAACznichVHPT9RAFP6ooriAu+jFxEvjBoIx2UyJAWJiQtSD3hZwgYRCM62z64RO2/RXgk3j1fgPePAECQfiwaN/gBf/AQ5c4Gw4QsKFA6/dJqgEfM10vvfN+97Ml2cHroxixvYHtBs3B2/dHrpTGx4ZvVtvjN1bivwkdETH8V0/XLF5JFzpiU4sY1esBKHgynbFsr3xsjhfTkUYSd97G28GYk3xnie70uExUVbjzSsrM0Ole1ymItef62Y35E5m5NlUbkaJskyV6GaPK8XXCU6aHrddXrDr3Sd/JPZjq9FkLVaGfhkYFWiiirbf+AET7+DDQQIFAQ8xYRccEX2rMMAQELeGjLiQkCzPBXLUSJtQlaAKTuwG/XuUrVasR3nRMyrVDt3i0gpJqWOc7bFddsx+sW/sNzu7sldW9ijeskm73deKwKp/frB4+l+Voj3G+wvVNQqbqq/3FKOL2dKLJG9ByRQunX7/9MOX48VnC+PZBNtmR+Rvi+2zn+TQS0+cnXmx8BU1GpDx7zgug6WpljHdMuafNudeVKMawkM8wiTNYwZzeI02OnTvd+zhAIdaW0u1XPvYL9UGKs19/BXap3N/M62m</latexit>+
S5 ]<latexit sha1_base64="bUM+3cRMZKKwNH+6zbAqmHBQ16E=">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</latexit>

[Dk

= 0.

µ

(rfµ + rbµ )+S5 ]<latexit sha1_base64="NbFM8yQlTIhVvy2FAqBn2oyQ/rY=">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</latexit>[Dnaive + W a + 1 M a= 0.
x5= (1 M a + W a) 4D
は x5 = 0 に局在する解が存在
s.t. Dnaive 4D = 0 if |1 M a + W a| < 1.
M a < 0 なら解なし(normal insulator)
0 < M a < 2 なら1つだけ解あり[doublerは排除]
M a > 2 ではdoublerの寄与もedge modeに(Chern数>1)。
<latexit sha1_base64="9UrQD/lcX0uDiIufM1xpy6/neSs=">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</latexit><latexit sha1_base64="KKDdmR/2hf8AE+xcoQCaF2Xh8jA=">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</latexit><latexit sha1_base64="VIXO5BT55rjovG816EZX789Jxow=">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</latexit><latexit sha1_base64="1n8eN8xsZPhUr/HD1xvCuFBxYgM=">AAACj3ichVG7SgNBFD2u7/hIoo1gEwyKIIZZESMiErTRQjDRGEGD7K4THdwXu5uAJv6AH6CFhQ+wED/AD7DxByz8BLFUsLHw7mZBVIx3mJkzZ+65M4er2rpwPcaemqTmlta29o7OSFd3T280Fu9bd62yo/G8ZumWs6EqLteFyfOe8HS+YTtcMVSdF9T9Bf++UOGOKyxzzTuwedFQdk1REpriEbVVk8eXlbGCUpuVU9uxJEuxIBK/gRyCJMJYsWJ32MIOLGgowwCHCY+wDgUujU3IYLCJK6JKnENIBPccR4iQtkxZnDIUYvdp3aXTZsiadPZruoFao1d0mg4pExhmj+yGvbIHdsue2ceftapBDf8vB7SrdS23t6PHA6vv/6oM2j3sfakaKFTKbuzJQwnTgRdB3uyA8V1q9fqVw9PX1ZnccHWEXbEX8nfJntg9OTQrb9p1lufOEKEGyT/b8RusT6TkqZScnUxm5sNWdWAQQxilfqSRwSJWkKd3bZzgHBdSXEpLc1Kmnio1hZp+fAtp6RPskJNt</latexit><latexit sha1_base64="vOsV9D6Usyc1LnyTnaA1UQ7WG6U=">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</latexit><latexit sha1_base64="D56IQSK4UbvUYPN001TOPCJBcZI=">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</latexit><latexit sha1_base64="VeJ7kD/9pEU6Y6OkenJi9NzK4Dw=">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</latexit>

ただし、 (x5 = L5 ) = 0 を満たすためには L5 = 1 が必要。
<latexit sha1_base64="Y9BV8kIYTq0I13VMv68tjEsu+Ow=">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</latexit><latexit sha1_base64="x0PvaVpbYaG7CqZJx52d2EtXAIE=">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</latexit>

40

Shamir domain-wall のedge 解<latexit sha1_base64="/GzSVQ/bdOJ+9zmqdgjOPY13//c=">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</latexit>[ S5 + Dk ] + = 0.

同様に

は 0 < M a < 2 のとき<latexit sha1_base64="D56IQSK4UbvUYPN001TOPCJBcZI=">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</latexit>

x5 = L5 に局在する解を1つ持つ。
<latexit sha1_base64="lpl/0r64WLaHrwUo3P02wywnJRs=">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</latexit>

topological insulator<latexit sha1_base64="MquWK21IKUX+wh7maJU2Gwz+pnw=">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</latexit><latexit sha1_base64="o6W0wMUZJAOFtI58042iSjVXBAU=">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</latexit>

R

L

x5
<latexit sha1_base64="ncI+yDKll7VKs0OU7KisUh9ATR4=">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</latexit>

L5<latexit sha1_base64="l0vWs0yrGfu+MICqeMZ2TgEMCLg=">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</latexit>

L5が有限のとき解にはならないが、そのギャップ=residual質量は指数関数的に小さい。
41

Shamir domain-wall fermion 行列式0<latexit sha1_base64="xvAePOjwA0qbUCJUBhzfwHWcvdU=">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</latexit>BB
det[aDSDW ] = det B
B
@

Dk
P+
0
···
0

P
Dk
P+
···
···

0
P
Dk
···

0
0
P
···

···
···
···
···
P+

1

0
0 C
C
0 C
C
··· A
Dk

欲しいのは edge mode だけ。 bulk massive modeを打ち消したい
のでPauli-Villars 場(on anti-periodic lattice)を用意して割り算。
<latexit sha1_base64="eGGfJ7PUisBYK21FywkKOg3LGNY=">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</latexit>

0BB
det[aDPV ] = det B
B
@

Dk
P+
0
···
P

P
Dk
P+
···
···

0
P
Dk
···

0
0
P
···

···
···
···
···
P+

1

P+
0 C
C
0 C
C
··· A
Dk

<latexit sha1_base64="7bD+SP5nJ7jXWMTVXOtu2H6zEeY=">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</latexit>

det[aDSDW ]det[aDPV ]42

5次元 SDWは実は4次元の理論と等価<latexit sha1_base64="mAfcsE/zN8Btv6iYTg51TLO3JNk=">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</latexit>

det[aDSDW ]= det[(DPV ) 1 DSDW ]det[aDPV ]いろいろ線形代数を駆使(UDL分解etc.)して変形していくと…
0
1
4D
aDe↵
B 0
B
= det BB 0
@ 00<latexit sha1_base64="2ziyK9Dpeyn/8xK6INb5pRJ0lvs=">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</latexit><latexit sha1_base64="8tC4fW35WgPrdM9aWzbwWDvfaVw=">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</latexit>



1
4D
De↵ =
1+
a

0
1
0
···
···

0
T
1
···

0
0
T
···

···
···
···
···
0

(1 + HT )L5 /a (15(1 + HT )L5 /a + (1

0
0 C
C
4D
0 C
=
det[aD
e↵ ]
C
··· A
1

HT )L5 /a
HT )L5 /aHT =<latexit sha1_base64="RNh0u0omMyINi/yakvK8ABxCN+Y=">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</latexit>

up to an numerical const.

5 [a(DW

T = (1 + HT )(1<latexit sha1_base64="9KjFf21eHaMxfThi4fAxWeUlFBc=">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</latexit>M )][2 + a(DW

M )] 1HT ) 1

詳細は青木慎也「格子上の場の理論」付録J および
私の手書きのノート http://www-het.phys.sci.osaka-u.ac.jp/~hfukaya/slides/Fukaya-LatIndex-note.pdf

435次元方向の無限大＆連続極限<latexit sha1_base64="8tC4fW35WgPrdM9aWzbwWDvfaVw=">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</latexit>

1
4D
De↵ =
1+
a

(1 + HT )L5 /a (15(1 + HT )L5 /a + (1HT )L5 /a
HT )L5 /aHT =<latexit sha1_base64="RNh0u0omMyINi/yakvK8ABxCN+Y=">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</latexit>

5 [a(DWM )][2 + a(DW

M )] 1

L5→無限大極限で<latexit sha1_base64="OXvvPOxHc3qgTQYEkZDnLs+i6yw=">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</latexit>(1 + HT )L5 /a (1(1 + HT )L5 /a + (1HT )L5 /aHT! sgnHT = p 2
L/aHT ) 5HT

5次元方向の格子間隔を a5 ! 0 とすると、<latexit sha1_base64="urtKVbC2ug+0P3K97kvZ5Csz3tg=">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</latexit><latexit sha1_base64="VXHeNejJUTiGLCdCxAB4l3L/0HM=">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</latexit>

HT =5 [a(DWM )][2 + a5 (DWM )] 1 !

5 [a(DWM )]/2

通常 M a = 1 にとる。
<latexit sha1_base64="vNn5RbA1+mApjymoVw0/dLMCGpM=">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</latexit>

44

Overlap Dirac operator [Neuberger 1998]<latexit sha1_base64="5rVzKgL79mtmtGQ5FI3ofMg6460=">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</latexit>

DSDW= det Dov
L5 -> ∞ （& a5 ! 0 [とらなくてもよい]）極限で lim det
L5 !1
DPV
<latexit sha1_base64="urtKVbC2ug+0P3K97kvZ5Csz3tg=">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</latexit>を得る。これをoverlap Dirac operatorとよぶ。HW は負の質量を持つ<latexit sha1_base64="Y4eANPBlkQq3pKAL7huJcz7QByk=">AAAB63icZVBNSwMxEJ2tX7V+VT16CZaCp7IrpXoseumxov2AdinZNNuGJtklyQpl6U/wJCiIV3+RJ/+NabuIbR8MPN6bYWZeEHOmjev+OLmt7Z3dvfx+4eDw6PikeHrW1lGiCG2RiEeqG2BNOZO0ZZjhtBsrikXAaSeY3M/9zjNVmkXyyUxj6gs8kixkBBsrPTYGnUGx5FbcBdAm8TJSggzNQfG7P4xIIqg0hGOte54bGz/FyjDC6azQTzSNMZngEe1ZKrGg2k8Xp85Q2SpDFEbKljRoof6fSLHQeioC2ymwGet1by7+eeWVVSa89VMm48RQSZabwoQjE6H542jIFCWGTy3BRDF7LCJjrDAxNp6CTcFb/3mTtK8rXq1SfaiW6ndZHnm4gEu4Ag9uoA4NaEILCIzgBd7g3RHOq/PhfC5bc042cw4rcL5+AX4xjco=</latexit>

重い Wilson Dirac 演算子。一見まともなDirac演算子には見えないが、、、<latexit sha1_base64="E88Se6xPSnHRa1t3fP+fs2WihAc=">AAACDHicZVDLSgNBEJyNrxhfqx69DAmBeEjYlfi4CKIePCqYByRh6Z3MJkNmdpeZWSEsuXvyUzwJCuLVH/Dk3ziJQUxS0FBUddPd5cecKe0431ZmaXlldS27ntvY3NresXf36ipKJKE1EvFINn1QlLOQ1jTTnDZjSUH4nDb8wdXYbzxQqVgU3uthTDsCeiELGAFtJM/O33gNfI7L7UACSd1RCqN2D4QA77jkluHaaxx6dsGpOBPgReJOSQFNcevZX+1uRBJBQ004KNVynVh3UpCaEU5HuXaiaAxkAD3aMjQEQVUnnfwywkWjdHEQSVOhxhP1/0QKQqmh8E2nAN1X895Y/POKM6t0cNZJWRgnmobkd1OQcKwjPE4Gd5mkRPOhIUAkM8di0gcTizb55UwK7vzPi6R+VHFPKtW7auHicppHFh2gPCohF52iC3SDblENEfSIntErerOerBfr3fr4bc1Y05l9NAPr8weRkJlS</latexit>q<latexit sha1_base64="j++6gNpkbb1VyL0myxeVQ96rPyQ=">AAACDHicZVDLSsNAFJ3UV62vqEs3Q0uhRShJKepGKLrpzgr2AU0sk+mkHTp5ODMRSsjelZ/iSlAQt/6AK//GaRvEtgcuHM65l3vvcUJGhTSMHy2ztr6xuZXdzu3s7u0f6IdHbRFEHJMWDljAuw4ShFGftCSVjHRDTpDnMNJxxtdTv/NIuKCBfycnIbE9NPSpSzGSSurreUs8cBk3+p37agIvoeVyhGMziVFSMk9vSqhc7usFo2LMAFeJmZICSNHs69/WIMCRR3yJGRKiZxqhtGPEJcWMJDkrEiREeIyGpKeojzwi7Hj2SwKLShlAN+CqfAln6v+JGHlCTDxHdXpIjsSyNxX/vOLCKule2DH1w0gSH883uRGDMoDTZOCAcoIlmyiCMKfqWIhHSMUhVX45lYK5/PMqaVcr5lmldlsr1K/SPLLgBORBCZjgHNRBAzRBC2DwBF7AG3jXnrVX7UP7nLdmtHTmGCxA+/oF7zaZhA==</latexit>

HW =

1
5 (1
a

aDW )

を代入してみると、、、

2 =
HW

1
(1 + O(a))
a<latexit sha1_base64="G4sPtsxKddLiEwzdsE0c/Eu1lJM=">AAACAHicZVDLSsNAFJ3UV62vqODGzWApVISSSFE3QtEu3FnBPqANYTKdtENnkjAzKZTYjZ/iSlAQt/6GK//GaRvEtgcuHM65l3vv8SJGpbKsHyOzsrq2vpHdzG1t7+zumfsHDRnGApM6DlkoWh6ShNGA1BVVjLQiQRD3GGl6g9uJ3xwSIWkYPKpRRByOegH1KUZKS655VHWTjuAwHI7hNay6TXh2X0Snrpm3StYUcJnYKcmDFDXX/O50QxxzEijMkJRt24qUkyChKGZknOvEkkQID1CPtDUNECfSSab3j2FBK13oh0JXoOBU/T+RIC7liHu6kyPVl4veRPzzCnOrlH/lJDSIYkUCPNvkxwyqEE7SgF0qCFZspAnCgupjIe4jgbDSmeV0Cvbiz8ukcV6yL0rlh3K+cpPmkQXH4AQUgQ0uQQXcgRqoAwyewAt4A+/Gs/FqfBifs9aMkc4cgjkYX7+OupRZ</latexit>

Dov = DW + O(a)45

なんとこれがGinsparg-Wilson relationを満たす[HTのままでもO.K.]<latexit sha1_base64="ZPCdkI9fmRfDV3laKiRPp0nBD6I=">AAACMnicbVDLSgMxFM34rPVVdekmWAqCUGakPjZCURddVrAPaMtwJ03b0GRmSDKFMvSXXPknulFQELd+hOmD1rYeCBzOOZebe7yQM6Vt+81aWV1b39hMbCW3d3b39lMHh2UVRJLQEgl4IKseKMqZT0uaaU6roaQgPE4rXvdu6Fd6VCoW+I+6H9KGgLbPWoyANpKbKty7cV0KHPQG9TYIAe4FPpuymYlvMPwTnUluKm1n7RHwMnEmJI0mKLqpl3ozIJGgviYclKo5dqgbMUjNCKeDZD1SNATShTatGeqDoKoRjy4e4IxRmrgVSPN8jUfq34kYhFJ94ZmkAN1Ri95QnHqZuVW6dd2ImR9GmvpkvKkVcawDPOwPN5mkRPO+IUAkM5/FpAMSiDYtJ00LzuLNy6R8nnUus7mHXDp/O+kjgY7RCTpFDrpCeVRARVRCBD2hV/SBPq1n6936sr7H0RVrMnOE5mD9/AITjKld</latexit>

Dov 5 +

5 Dov = aDov 5 Dov
46格子上のカイラル対称性 [Luscher 1998]
オーバーラップフェルミオン作用は以下の “カイラル”変換の下で不変。
と反可換ではないのでNielsen-Ninoomiyaは回避。

さらにGinsparg-Wilson 関係式の左右からをかけて座標表示にすると、カイラル対称性の破れは同じsiteに集中し O(a)であることがわかる。
47

オーバーラップ演算子の固有値分布
複素固有値は中心が 1/a にある半径1/aの円上に並ぶ。
15個のダブラーはすべて2/aに存在。
Cf. Wilson Dirac operator

Figure by M. Creutz48

5. Atiyah-Singer index on the lattice49

実は破れてほしいカイラル対称性＝ カイラル量子異常。
<latexit sha1_base64="ZPCdkI9fmRfDV3laKiRPp0nBD6I=">AAACMnicbVDLSgMxFM34rPVVdekmWAqCUGakPjZCURddVrAPaMtwJ03b0GRmSDKFMvSXXPknulFQELd+hOmD1rYeCBzOOZebe7yQM6Vt+81aWV1b39hMbCW3d3b39lMHh2UVRJLQEgl4IKseKMqZT0uaaU6roaQgPE4rXvdu6Fd6VCoW+I+6H9KGgLbPWoyANpKbKty7cV0KHPQG9TYIAe4FPpuymYlvMPwTnUluKm1n7RHwMnEmJI0mKLqpl3ozIJGgviYclKo5dqgbMUjNCKeDZD1SNATShTatGeqDoKoRjy4e4IxRmrgVSPN8jUfq34kYhFJ94ZmkAN1Ri95QnHqZuVW6dd2ImR9GmvpkvKkVcawDPOwPN5mkRPO+IUAkM5/FpAMSiDYtJ00LzuLNy6R8nnUus7mHXDp/O+kjgY7RCTpFDrpCeVRARVRCBD2hV/SBPq1n6936sr7H0RVrMnOE5mD9/AITjKld</latexit>

Dov 5 +
右辺が5 Dov = aDov 5 Dov
でも物理に効いてほしい(Cf. pion の2光子decay)。

＊連続理論ではどうなっているか？[藤川の方法]

作用は不変。でも 積分測度がJacobian を出す。
50

カイラル変換（連続）の下で作用は不変。でも 積分測度がJacobian を出す。
無限積なので正則化が必要。

ログをとると、
Heat-kernelで正則化すると、

要するに、Atiyah-Singer 指数 が L/Rの対称性の破れ＝量子異常の帰結として現れた。

51

Overlap フェルミオンの量子異常
格子理論のJacobian

連続理論との整合性が完璧ならば

のはず。
52

Overlap Dirac演算子の指数 [Hasenfratz et al. 1998]
Traceをの固有関数の完全系で評価する。

Ginsparg-Wilson 関係式よりが成り立つ。 よって 任意の固有状態に対し、となるペアが存在し、
ただし、例外は53

Overlap Dirac演算子の指数
Traceをの固有関数の完全系で評価する。

Doublerの寄与がうまく消されている。

= n+<latexit sha1_base64="BRY5qrhBsE0xGBf4bSRDKD44SDQ=">AAAB8HicZVBNS8NAEJ3Ur1q/oh69LJaCIC2JiHoRil48VrAf0Iaw2W7apZtN2N0IJfRPeBIUxKt/x5P/xm0bxLYPBh7vzTAzL0g4U9pxfqzC2vrG5lZxu7Szu7d/YB8etVScSkKbJOax7ARYUc4EbWqmOe0kkuIo4LQdjO6nfvuZSsVi8aTHCfUiPBAsZARrI3VuhX9eFX7Vt8tOzZkBrRI3J2XI0fDt714/JmlEhSYcK9V1nUR7GZaaEU4npV6qaILJCA9o11CBI6q8bHbvBFWM0kdhLE0JjWbq/4kMR0qNo8B0RlgP1bI3Ff+8ysIqHd54GRNJqqkg801hypGO0fR71GeSEs3HhmAimTkWkSGWmGiTUcmk4C7/vEpaFzX3quY+Xpbrd3keRTiBUzgDF66hDg/QgCYQ4PACb/BuSevV+rA+560FK585hgVYX79ReI9X</latexit>n54

Geometric index との等価性
オーバーラップ演算子のindex

連続理論のindex

heat kernel の役割を担っていそう。
実際、リンク変数をとおいて格子間隔について展開、density を計算してみると、[Kikukawa-Yamada 1998, Fujikawa 1998, Adams 1998 Suzuki 1998]

55

Topological 不変量？
リンク変数は隣と完全に独立にとれるので、このindexはジャンプしうる＝そのままではトポロジカル不変量ではない。
でも plaquetteに滑らかさを要求＝admissibility condition:

すると、十分小さい有限のとでトポロジカルになる。

に対し、indexはジャンプできなくなる＝この制限のも[Hernandez et al. 1998]

56

Topological (geometric) index
plaquetteに滑らかさを要求＝admissibility condition:

すると、Atiyah-Singer 指数の右辺もトポロジカル不変量として定義でき
る[Luescher 1982]。
Z

n+

1
n =
32⇡ 2

<latexit sha1_base64="CA6Aq6Ax87oz8ZvSHvDTZgm4FpE=">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</latexit>

Overlap 演算子のゼロ固有値の数

d4 x✏µ⌫⇢ trc F µ⌫ F ⇢ (x)X

ゲージ場の巻き付き数を与えるように単位格子の境界 hypercube上に変換関数を一意的に内挿できる。
* でも等式自体は数学的に未証明。

57

ここまでのまとめ
1. 格子理論は有限格子点上のファイバー束の統計力学として定式化される。ゲージ場(接続)> link 変数。フェルミオン場（切断）-> site変数。
2. Nielsen-Ninomiyaの定理は格子理論のDirac演算子がchirality 演算子(Z2 grading)と反可換になれないことを要求する。＝left-rightが混ざってカイラル対称性は必ず壊れる。) 指数定理なんて無理っぽい。[1日目]3. Domain-wall fermion[Kaplan, 1992]は5次元のトポロジカル絶縁体のメカニズムでleft,rightを5次元方向に引き離し、カイラル対称性を取り戻す。
4. 5次元方向を無限大にとったものの行列式は 4次元のoverlapDirac演算子の行列式と等価。Overlap 演算子はGinsparg-Wilson 関係式を満たす。
5. Overlap 演算子はカイラル対称性およびその量子異常を連続理論と遜色ない形で実現。
6. しかもoverlap Dirac演算子（with なめらかなplaquettes)は指数定理を定式化可能。

スライドURL:
http://www-het.phys.sci.osaka-u.ac.jp/~hfukaya/slides/Fukaya-LatIndex.pdf

58補足: 素粒子論のトポロジーは基本的に実空間で議論
素粒子論の並進対称性はフェルミオン&ボゾンの複合系で満たされる。
どちらかだけを見てもまともな運動量空間は定義不可能。
例) 運動量 p の電子と運動量 q の陽電子が対消滅して光子が励起など。
<latexit sha1_base64="30tItoivFnh1OnH1yhYrH6bT+JA=">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</latexit>

格子ゲージ理論は各リンクにばらばらの電磁場がかかっているようなもの。

¯ ( µ (@µ + igAµ (x)))

domain-wall fermion= bulk-edge 対応は“運動量”が定義できない系でも有効。
59

6. Atiyah-Patodi-Singer(APS) index[境界付き多様体上の指数定理] on the lattice
並進対称でないフェルミオン系のbulk-edge対応[F, Kawai,Matsuki,Mori,Nayakama,Onogi,Yamaguchi, 2019]59

Atiyah-Patodi-Singer指数[1975]とはboundarybulk

curvatureη不変量* Here we consider 4-d flat Euclidean spacewith 3-d boundary at x4=0.
ただし、非局所的なAPS境界条件を課す。

⌘(iD) =

X

sgn i

i

<latexit sha1_base64="DUE8TwAU6F3bYK2KLVRIpiFib1E=">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</latexit><latexit sha1_base64="46vRFZjMDeygAEhACDWGDM/UQkE=">AAACiXichVG7ThtBFD1sHhhDYgMNUpoVthGVdRdFCnKFIAUljxgjYWTtLmNn5NmHdtcWxvIP+AdSpAIpihANLWnT5AdS8AkRpZFoUuR6vRICK3BHM3PmzD13zsxYvpJhRHQ9ob14+er1ZGoqPT3z5m0mOzu3F3qtwBZl21NesG+ZoVDSFeVIRkrs+4EwHUuJitXcGO5X2iIIped+ijq+OHTMhivr0jYjpmrZfFVx8pFZkyW96ljecVfIhnDbpmoJ3avrefkx36tlc1SkOPRxYCQghyS2vOx3VHEEDzZacCDgImKsYCLkdgADBJ+5Q3SZCxjJeF+ghzRrW5wlOMNktsljg1cHCevyelgzjNU2n6K4B6zUUaDfdE4D+kUX9If+/rdWN64x9NLh2RpphV/L9Bd2755VOTxH+HyvetJzhDpWY6+SvfsxM7yFPdK3T74Mdks7he4SndEN+z+la/rJN3Dbt/a3bbHzFWn+AOPxc4+DvZWiQUVj+31ubT35ihTeYRHL/N4fsIZNbKHM5/ZxiSv80KY1Q1vVSqNUbSLRzONBaBv/AKkvlyg=</latexit>i : eigenvalue of iD

APS 指数が4次元bulkと3次元edgeの対応を記述する[Witten 2015]。 61

Path integral of massless edge fermion
Z

Zedge =
<latexit sha1_base64="ykm+8r0FDG++vd8Gwm8sVOFjlUA=">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</latexit>d d ¯ exp

= lim detµ!1

<latexit sha1_base64="Y5IAaIRmQ5YnpRFdlb+XjTtkiOw=">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</latexit>

= lim

µ!1

η不変量

<latexit sha1_base64="X5Jxwf34jJjY/HzIAi7ns4WGLX0=">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</latexit>⌘(iD) =
<latexit sha1_base64="DUE8TwAU6F3bYK2KLVRIpiFib1E=">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</latexit>

X
i

Y
i

sgn i

Z

✓

D=

¯D
edge

D
D+µ

<latexit sha1_base64="mvhGR/H3jhRxUv/hII1AscOUYqI=">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</latexit>i (@i + Ai )i=1,2,3

ランダムなゲージ場◆ we consider Euclidean space.
Y

i i
⇠ lim
i i + µ µ!1 i<latexit sha1_base64="46vRFZjMDeygAEhACDWGDM/UQkE=">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</latexit>

X

Pauli-Villars regulator(breaking T symmetry)

i i
/ exp
µ

i : eigenvalue of iDh⇡i ⌘(iD)2

T (⌘(iD)) =

i

⌘(iD)

<latexit sha1_base64="9+JxwsH+fSRC8xyOLTJHv91owdE=">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</latexit>

Edge mode has T anomaly.

Path integral of massive bulk fermion<latexit sha1_base64="AeWzys69vodRzzCPkGikpEtEcDA=">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</latexit>

Zbulk =

Zd d ¯ exp

<latexit sha1_base64="AbDYFrTuRMHxep/lz4FZmtZzfo8=">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</latexit>= lim det
M !1

Z

✓

¯ D4D
bulk

D4D M
D4D + MM◆

<latexit sha1_base64="c8Zpg7jz0vVNhYIHnhF89rhbE8s=">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</latexit>

D

4D

=

X

µ

(@µ + Aµ )µ=1,2,3,4

Pauli-Villars regulator

We can flip the sign of the mass by chiral rotation:<latexit sha1_base64="RbF66wWjG9SmJmZXhWaI5gSiYrk=">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</latexit>

M¯ !

M ¯e

i⇡
2

5

e

i⇡
2

5

=M¯

Zbulk = 1?
<latexit sha1_base64="pqSQtpg3+Y+Pt3vV7XZ49vu5kpk=">AAACdnichVHLSsNAFD2N7/poqxtBkGCpuio3IiqCKLpx6au2+KAkcdTQvEjSihZ/wB9wIS4UVMTPcOMPuPATxKWCLlx4mwZERb3DzJw5c8+dMzOaaxp+QPQQkxoam5pbWtvi7R2dXYlkqnvVd8qeLnK6YzpeQVN9YRq2yAVGYIqC6wnV0kyR10pztf18RXi+4dgrwb4rNi11xza2DV0NmComU2vF6oZnyVrZLB3KU7IyXUymKUthyD+BEoE0olhwklfYwBYc6CjDgoCNgLEJFT63dSgguMxtosqcx8gI9wUOEWdtmbMEZ6jMlnjc4dV6xNq8rtX0Q7XOp5jcPVbKyNA9XdMz3dENPdL7r7WqYY2al32etbpWuMXEUe/y678qi+cAu5+qPz0H2MZE6NVg727I1G6h1/WVg+Pn5cmlTHWQzumJ/Z/RA93yDezKi36xKJZOEOcPUL4/90+wOpJVxrLK4mh6Zjb6ilb0YQDD/N7jmME8FpDjc/dwiktcxd6kfikjDdVTpVik6cGXkOgDZ4SP/g==</latexit>

Path integral of massive bulk fermion
No, since the chiral U(1) symmetry has anomaly.
<latexit sha1_base64="5p2bgUKwe3LltGuumfOjKlWJAfM=">AAAD7XicrVFbaxNBFD7Z9VLjpan6IIgyGBoiYpiNQUUoFC3FF6EX0xYzSdjdTNIhe2N2ElKXfRZ8Fx8UQUGK+DN88Q/40J8ggi8VfPHBs5vFamoVxFl25pvznW/Od2aswBGhonQnp+mHDh85OnUsf/zEyVPThZnTa6E/kDav277jyw3LDLkjPF5XQjl8I5DcdC2Hr1v9Owm/PuQyFL53X20FvOmaPU90hW0qDLVnco8etCMmXWINnH5MSnOECU+RDgtCgbNlyiiBMWF8FDCHd1UjSfhJs5eT0uWFVsrVFuKr95gUvU11OWHHsMlYvjT3+xI/KpQFU3yk0uYiyTsx8kh3pWlHRhxdq+K2VY0nfJBOqzYii4tZTfIvjifrXokPaoH8Z7f5dqFIKzQdZD8wMlCEbCz5hW1g0AEfbBiACxw8UIgdMCHErwEGUAgw1oQIYxKRSHkOMeRRO8AsjhkmRvs493DXyKIe7pMzw1RtYxUHf4lKArP0A31Dd+l7+pZ+pN8OPCtKz0i8bOFqjbU8aE8/Prf69a8qF1cFm3uqP3pW0IWbqVeB3oM0knRhj/XDh093V2+tzEYl+op+Qv8v6Q59hx14wy/262W+8gySBzAmr3s/WKtWjOsVY7lWnL+dPcUUnIdLUMb7vgHzcBeWoA527rN2VrugXdR9/Yn+XH8xTtVymeYM/DL07e9D7x7q</latexit>

Zbulk =

Zd d ¯ exp

Z

¯ D4D

M

bulk
✓
◆
Z
Z
Z
1
4
¯
¯ D4D +M
= d d exp i⇡
d
xF
F
exp
32⇡ 2 bulk
bulk
✓
◆
Z
1
4
= exp i⇡d
xF F
232⇡ bulk<latexit sha1_base64="dOaeagTf5Mkh/Fr9LhI3ErL+e3M=">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</latexit>

F F = ✏µ⌫⇢ tr[F

µ⌫

F

⇢

<latexit sha1_base64="Wxv+3XDYiI/g26d3dioBiqnNPyE=">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</latexit>]=E·B<latexit sha1_base64="zivabxGITNdT0fkJewbxO4KDjHY=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="u6PuMmZOZafK1YIT7Mpuu9NDooQ=">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</latexit>
sha1_base64="E2zn7e/creIH/WtVoo9Agr7YWng=">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</latexit>= d(CS)near boundary

This is also T-odd.

APS index protects T symmetry.
<latexit sha1_base64="BQhwXlk9jxxiuWPugHqSG7ng1cc=">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</latexit>

Zbulk Zedge / exp i⇡<latexit sha1_base64="RV/KDU+WNOFhyyVRMbkB/aZ0cPI=">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</latexit>= ( 1)

✓

1
32⇡ 2

IAPS

Z

4

d xF Fbulk

1
3D⌘(iD |edge )2◆Under T transformation, each phase is odd:
<latexit sha1_base64="NXUlu/68RavN8E9uNzQQnUxtVgM=">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</latexit>

but<latexit sha1_base64="SQa9QmYoPodYaQ7dY4DPr0QhuC4=">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</latexit>

FF !

( 1)

F F,

IAPS

⌘(iD3D |edge ) !

= ( 1)⌘(iD3D |edge )IAPS T anomaly is canceled
between bulk and edge.

“運動量”が定義できなくてもedgeモードの存在がわかる。

[See Dai-Freed 1994, and related works: Metlitski 15, Seiberg-Witten 16,
Tachikawa-Yonekura 16&18, Freed-Hopkins 16, Witten 16, Yonekura 16&19, Witten-Yonekura 19…]

APS指数をそのまま格子理論で扱うのは難しい。
非局所的なAPS境界条件がボトルネック。

格子理論では

1. Overlap Dirac演算子でBをかんたんに分離できない。
2. 仮に何かできたとしてもその境界条件はGinsparg-Wilson関係式をやぶる。
[周期、反周期以外あらゆる境界条件がGWを破る。Luescher 2006]
66

もう一度格子のAtiyah-Singer 指数をよく見てみる…

<latexit sha1_base64="LKR/uOoryJvZpuQLKKxqZimUxNA=">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</latexit>⌘(H) =

X

sgn i

i

格子上のindex定理は、

Cf. Itoh-Iwasaki-Yoshie 1982, Adams 2001

1. 指数がmassiveなDirac演算子のη不変量で書けること
2. その定義にカイラル対称性は必要ないこと(Wilson Dirac で十分)を“知っていた”！！
67

連続理論の massive Dirac 演算子再訪の固有値を考える。
For

For
with

any

make a pairsinceMoreover, fromwe obtain68

AS 指数= massive Diracのspectral flow

古田さんに教わったK理論では<latexit sha1_base64="ZaxQWkhz+NmYBW5loiSbqKHO9j8=">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</latexit>

K 0 (point) = K 1 (line, ୺఺)<latexit sha1_base64="ZK7INebWpYKE798f1nWYG3msLck=">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</latexit><latexit sha1_base64="ZK7INebWpYKE798f1nWYG3msLck=">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</latexit>

5 あり

5 なし
69

= number of crossing from negative to positive= number of crossing from positive to negative= spectral flow of
On the other hand,increases/decreases

by 2 when one eigenvalue cross zero from negative/
positive to positive/negative.
→
Topological insulator

Normal insulator (trivially
zero)

70

Massive (bulk) evaluation is stable against
chiral symmetry breaking (on a lattice).
古田さんに教わったK理論では<latexit sha1_base64="ZaxQWkhz+NmYBW5loiSbqKHO9j8=">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</latexit>

K 0 (point) = K 1 (line, ୺఺)<latexit sha1_base64="ZK7INebWpYKE798f1nWYG3msLck=">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</latexit><latexit sha1_base64="ZK7INebWpYKE798f1nWYG3msLck=">AAACjXichVG7SgNBFD2u7/hI1EawCQbFKkwkPhCRoIWWJhoNqITddRKH7O4su5tgDPkBSxuL2ChYiD9gb+MPWOQTxFLBxsKbzYKoqHdg5s6Zew5zOJptCNdjrNmhdHZ19/T29YcGBoeGw5GR0R1Xlh2dZ3VpSCenqS43hMWznvAMnrMdrpqawXe10lrrfbfCHVdIa9ur2vzAVIuWKAhd9QjK7RdV01Tzc/lIjMWZX9GfTSJoYghqU0busI9DSOgowwSHBY96AypcWntIgMEm7AA1whzqhP/OUUeIuGWa4jShElqivUi3vQC16N7SdH32MU1IYkpSqSOKKfbIbtgLe2C37Im9/6pW81Vav6nSqbW53M6HT8e33v5lmXR6OPpk/cHQaPpvVx4KWPTdCHJn+0jLp97Wr5ycv2wtZaZq0+yKPZO/S9Zk9+TQqrzq12meaSBEESW+B/Kz2ZmNJ+bjyXQylloNwurDBCYxQ4ksIIUNbCLrJ3GGBi6UsDKnLCsr7VGlI+CM4Usp6x9cUpM8</latexit>

5 あり

5 なし

We cannot define

線で数えるほうが安定<latexit sha1_base64="9S/1E76lnWNieVEcm10JQ1uqzZc=">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</latexit>

IndDov =1⌘( 5 (DW
2

M ))
71

η不変量＝index (連続理論)
物理屋フレンドリーなAPS指数の再定式化[F-Onogi-Yamaguchi 2017, F-Furuta-Matsuo-Onogi-Yamaguchi-Yamashita 2019]<latexit sha1_base64="DTbBEymbhLARq12UMPWJ1K1l/To=">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</latexit>

IndAPS D =1
reg.
⌘( 5 (D + M "))2
4次元のdomain-wall fermion

APS指数はmassive Dirac演算子のeta不変量で書き直せる。
ただし、数学的なsetupはもともとのAPSと大きく異なる。
1. HDW =

5 (D + M ") は境界のない多様体で定義(APS境界条件不要!)。

<latexit sha1_base64="RQmIm7/uELiprsvzLIz3f95MyEc=">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</latexit>

2. 右辺は符号関数が負： " =<latexit sha1_base64="jC+VVTC/F1Mw8ly6ct91Tf0SsZ0=">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</latexit>

1 の領域のみのAPS指数を与える。

3. 符号が変わる境目にedge-localized modeが生じる。
<latexit sha1_base64="jC+VVTC/F1Mw8ly6ct91Tf0SsZ0=">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</latexit>

4. 全領域で " =1 ならば AS indexに帰着。

72

Unification of index theorems
Index with massless Diraccontinuum

latticeAS
APS

w/ APS bc

Not known

Index with massive Dirac on a closed manifold
continuum

lattice

AS
APS
YES !
[F, Kawai, Matsuki, Mori, Nakayama, Onogi, Yamaguchi, 2019]73

格子理論におけるAPS指数
F, Kawai, Matsuki, Mori, Nakayama, Onogi, Yamaguchi, arXiv:1910.09675

On 4-dimensional Euclidean lattice with periodic
boundaries (T4), we have shown with

* Bulk part is similar to that of AS index [H.Suzuki 1998].

Note that LHS is always an integer.
See our paper for the details.

74おまけ: Curved space の指数定理に向けて[S. Aoki & HF, arXiv:220X.XXXXX]
AS指数定理、APS指数定理、格子理論でも定式化できたがここまではすべて正方格子、domain-wall もまっすぐ 。
つまり、flat な時空に限られていた。

75

先行研究: curved Lattices[Brower et al. 2016, 2017,Ambjorn et al. 2016, Caterall et al. 2018…]

Curved space は三角格子で扱われてきた[多様体の三角形分割のイメージ]。
* systematicにやるのは難しい [三角形の最小単位(面積、辺、角度、コ
ントロールするべき自由度がたくさん)]* 連続極限も非自明(正方格子では格子間隔a だけでcontrolできた)<latexit sha1_base64="j2tKQklDzMcJUMK2uynowapH8YY=">AAAB6XicZVBNS8NAEJ3Ur1q/qh69LJaCp5KIqMeiF48t2A9oQ9lsJ+3SzSbsboQS+gs8CQri1Z/kyX/jtg1i7YOBx3szzMwLEsG1cd1vp7CxubW9U9wt7e0fHB6Vj0/aOk4VwxaLRay6AdUouMSW4UZgN1FIo0BgJ5jcz/3OEyrNY/lopgn6ER1JHnJGjZWadFCuuDV3AbJOvJxUIEdjUP7qD2OWRigNE1Trnucmxs+oMpwJnJX6qcaEsgkdYc9SSSPUfrY4dEaqVhmSMFa2pCEL9e9ERiOtp1FgOyNqxvq/Nxd/verKKhPe+hmXSWpQsuWmMBXExGT+NhlyhcyIqSWUKW6PJWxMFWXGhlOyKXj/f14n7cuad13zmleV+l2eRxHO4BwuwIMbqMMDNKAFDBCe4RXenInz4rw7H8vWgpPPnMIKnM8fQ+CNFg==</latexit>* 2D sphere を扱った研究: 回転対称性の破れは自明には回復しない。

figures from Brower et al. 201676

Curved domain-wall fermion
Nash のembedding theorem [1956]: 任意のRiemann多様体は平坦な時空に等長に埋め込める。
正方格子に曲がったdomain-wallを埋め込んだらどうなる？
edge-mode がdomain-wall上に拘束され、その運動は加速運動となる。
Einsteinの等価原理よりこの加速は重力として働くはず＝ spin connectionが誘導されるはず。
* 物性理論[ただし連続理論]ではtopological
絶縁体のcurved表面に“重力”が確認されている。
[Lee 2009, Parente et al. 2011,
Imura et al. 2012, Takae et al. 2013…]
意外と格子ではやられていない。

77

2次元正方格子中の円形 domain-wall fermion
domain-wall の置き方の ambiguityは円とsiteが重なったときの処方くら
いしかない[Our choice=重なったら+の質量]。
<latexit sha1_base64="OsoF24C9D4Z3ftM3HQXg7xyGbds=">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</latexit>

質量の符号の分布

HDW =

2D
3 (DW

M ")

の最低固有値の波動関数の絶対値分布

edgeに局在。
でもガタガタしている。

78

3次元正方格子中の2次元球面domain-wall<latexit sha1_base64="sn4UG5wr78tDWRGfIohL4mDTDEU=">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</latexit>

HDW =

3⌦1

3D
⌦
D
1
W

の最低固有値の|波動関数|の分布

M"＊ 3D massive Diracは一般にcomplexなので
2-flavorに拡張して Hermitianにしている。

輪切りにしたもの

79

S1上の重力場をdetectする方法
1次元の円に拘束された粒子のDirac 演算子 [連続理論]<latexit sha1_base64="wIdP7S8J9HAQe29coDpxhmK3zdE=">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</latexit>

1

S
iDe↵
=
<latexit sha1_base64="K0/oMnUlqipcjerLs8qEte6VPRY=">AAACiHichVE7S8NQGD3Gd3y06iK4iEVxKl+k+JqkLo59WBVUShKvNTQvkrSopX9AcLWDk4KD+AfcXfwDDv4EcVRwcfBLGhAV9Qvce+6553y5h09zTcMPiB47pM6u7p7evn55YHBoOJEcGd3wnZqni5LumI63pam+MA1blAIjMMWW6wnV0kyxqVVXw/vNuvB8w7HXgyNX7FpqxTb2DV0NmCp6ZSonU5SmqCZ/AiUGKcSVc5K32MEeHOiowYKAjYCxCRU+f9tQQHCZ20WDOY+REd0LNCGzt8YqwQqV2SqvFT5tx6zN57CnH7kPWeGw0+EuTUximh7oml7onm7oid5/7daIuoSvOeJda3uFW06cjBff/nVZvAc4+HT94dBY/XeqAPtYjNIYnM6NmDCn3u5fP269FJcL040ZuqRnzndBj3THCe36q36VF4VzyDwi5ftAfoKNubQyn87kM6mVbDysPkxgCrM8kQWsYA05lPi/FZziDC1JlkhakJbaUqkj9ozhS0nZD3CFkRg=</latexit>1ir0

r0 : domain-wallの半径

✓

@
1
+i
@✓
2

◆

固有値

2n+1
E=±
(n = 0, 1, 2, · · · )2r0<latexit sha1_base64="MKOuVzC/4cFp62kt6reqPtiFvTE=">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</latexit>

重力によるベクトルポテンシャル[Spin-c connection]
重力の効果は固有値のギャップとして現れる。
Cf. 消滅定理[非負の曲率を持つ多様体上でDirac方程式は解を持たない]80

S2上の重力場をdetectする方法
2次元の球に拘束された粒子のDirac 演算子 [連続理論]<latexit sha1_base64="hKdrpBvixiY+yj+4USK+74kljO0=">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</latexit>

S2
iDe↵ =<latexit sha1_base64="K0/oMnUlqipcjerLs8qEte6VPRY=">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</latexit>

1
i
r0



@
+
1
@✓

2

✓

1 @
1
+i
sin ✓ @
2 sin ✓cos ✓i
2 sin ✓

3◆

r0 : domain-wallの半径

重力によるベクトルポテンシャル[Spin(-c) connection]
固有値<latexit sha1_base64="kBMh0RxUYIWjhmaMOHSCR5ekdeo=">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</latexit>n+1
E=±(n = 0, 1, 2, · · · )r0

重力の効果は固有値のギャップとして現れる。
Cf. 消滅定理[非負の曲率を持つ多様体上でDirac方程式は解を持たない]81

格子domain-wall Dirac 演算子の固有値分布
S1上の固有値分布

S2上の固有値分布
bulk modes

bulk modes

edge modes

edge modes
gapが存在gapが存在bulk modes

bulk modes
固有値のラベル

固有値のラベル固有値分布は連続理論とよく一致(重力によるギャップも開いている)。

82

連続極限＆無限体積極限
最低固有値の連続理論からのずれ

格子間隔

最低固有値の体積依存性

格子全体のサイズ

回転対称性の回復

mesh を細かくしていくとガタガタがなめらかに。
(*最低固有状態は軌道角運動量ゼロ)

ガタガタの格子間隔依存性付録のまとめ
1. 正方格子に曲がったdomain-wallを埋め込むとエッジモードは重力を感じる。
2. 誘導された重力効果＝固有値のギャップを確認。
3. 単純な連続極限をとるだけでdomain-wallの回転対称性が回復してそう。
4. 指数定理をやるにはゲージ場を入れるなどしてゼロ固有値が存在する系を考える必要がある。
5. 詳細は近日発表の論文を見てください＆ぜひ青木匠門くん（阪大）をセミナ
ーに！
85

まとめ
1. 格子理論は有限格子点上のファイバー束として定式化される。ゲージ場(接続)-> リンク変数。
2. 格子上のDirac演算子はカイラル対称性(Z2 grading operator)を失うことを要求される(NielsenNinomiya)。[1日目]
3. ドメインウォールフェルミオンは５次元のトポロジカル絶縁体のエッジ状態としてカイラル対称性を回復。
4. それと等価なオーバーラップフェルミオンのDirac演算子はGinsparg-Wilson関係式によりカイラル対称性とAS指数定理を実現。[2日目]5. でも実はeta不変量を使えばカイラル対称性を破るWilson Dirac演算子で十分。
6. Wilson Dirac演算子とdomain-wall 質量を使えばAPS指数を定式化可能。
7. domain-wall を曲げるのは簡単なので curved spaceも扱える。
86

今後の展望Index with massive Dirac on a closed manifold
continuum

latticeAS<latexit sha1_base64="N6D1HRpTWuvRJZ+anPlmsPrQGXw=">AAACF3icZVDLSgNBEJz1bXxFPXoZDEI8JOyKr6OoBy+CgjGCG5beSW8cMjO7zMwKYclXePJTPAkK4lU8+TdOYhAfBQ1FVTfdXXEmuLG+/+GNjU9MTk3PzJbm5hcWl8rLK5cmzTXDBktFqq9iMCi4woblVuBVphFkLLAZd48GfvMWteGpurC9DFsSOoonnIF1UlSu1cJEAyuCfrHVD9FCNeyAlBDtVI+jZi28BY2Z4SJV9HRzMypX/Lo/BP1PghGpkBHOovJ72E5ZLlFZJsCY68DPbKsAbTkT2C+FucEMWBc6eO2oAommVQzf6tMNp7RpkmpXytKh+nOiAGlMT8auU4K9MX+9gfjtbfxaZZP9VsFVlltU7GtTkgtqUzoIiba5RmZFzxFgmrtjKbsBl5N1UZZcCsHfn/+Ty616sFsPzrcrB4ejPGbIGlknVRKQPXJATsgZaRBG7sgDeSLP3r336L14r1+tY95oZpX8gvf2CQLDnnQ=</latexit>

APS

1⌘( 5 (DW2"M ))格子理論への応用は正方格子(T4)のみ。数学的な検証も不十分。
数学的な定式化: F-Furuta-Matuo-Onogi-Yamaguchi-Yamashita, in progress.
Cf. Yamashita 2020, Kubota 2020
Curved domain-wall Dirac 演算子 : Aoki-F, in progress87本講演の目標
1. 数学のみなさんに 離散化された格子上の場の理論（有限次元ベクトル空間）でもindex が定義できることに驚いていただく。
2. 物性のみなさんに 格子ゲージ理論で質量ゼロのフェルミオンを扱うにはトポロジカル絶縁体とそっくりな構造を使うほかないことを理解いただく。「この世は5次元トポロジカル絶縁体の4次元エッジ状態かもしれない」
3. 素粒子、原子核理論のみなさんに「domain-wall fermion すごい！」 の再確認。
88

But the index itself is physicist-unfriendly.

IAPS = n+ n
n± : number of solutions w/<latexit sha1_base64="oAnChUu560x7ZcECezYim3dWJpE=">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</latexit><latexit sha1_base64="tw0lC0OIxpQsuAjgandlNCLtb7A=">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</latexit>

chirality
4Dof massless Dirac equation DX=0
with a non-local boundary condition on theedge.
<latexit sha1_base64="YxR8LE+LauSYMikzIZorLIYXPYw=">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</latexit>

But topological insulator is massive(gapped) anddescribed by a local field theory…

Difficulty with boundary
If we impose local and Lorentz (rotation) invariant boundary
condition, + and – chirality sectors do not decouple any more.

+

For reflecting particle,
momentum flips, butangular momentum does not→ chirality flips.

and the index do not make sense.

Atiyah-Patodi-Singer boundary condition[Atiyah, Patodi, Singer 75]

gives up the locality and rotational symmetry to keep the chirality.

Eg. 4 dimgauge

They impose a non-local b.c.
boundaryindex = n+<latexit sha1_base64="esrqkDvu22FjxU7ro8/y0gLZqzs=">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</latexit>nMathematically beautiful!
But physicist-unfriendly.

More physical set-up?
Lab
topologicalinsulatorLab
topologicalinsulator

Topological insulator is nontrivial because its outside(our lab) is made of normal insulator.