ℤ𝑵 1-from ゲージ場に付随する't Hooftアノマリーの格子定式化
阿部 元一（九大）
熱場の量子論とその応用 @ KEK
2023/08/17〜/21
Topology of SU(N) lattice gauge theories coupled with ℤ𝑵 2-form gauge fields
M. Abe, O. Morikawa, S. Onoda, H. Suzuki and Y. Tanizaki
arXiv:2303.10977[hep-th]
Fractional topological charge in lattice Abelian gauge theory
M. Abe, O. Morikawa and H. Suzuki
PTEP 2023 (2023) 2, 023B03 [arXiv:2210.12967[hep-th]]Symmetry and Anomaly Ⅰ• Classical Theory：Conservation low

Symmetry（Noether Theorem）• Quantum Theory：The conservation low may be broken (Anomaly).
Ø Focus on the Partition function

Z[A] =!

[D(field)] eS[field,A]

Ø How to distinguish the anomaly : Whether the 𝑍 is invariant or not under a transformation?

Z ! =Z!
→ Z ! [A + ∂θ] = [D(field)] eS[field,A+∂θ]!
S[field,A][D(field)]
e=e"A[θ,A]#$ %#$%’t Hooft anomaly "=Z

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Symmetry and Anomaly Ⅱ
Ø We can predict low energy dynamics of the gauge theory.
※Gauge theory : Theory which describes the Standard Model of particlesü e. g., we decided the theory for the strong interaction is the SU(3) gauge theory
because the theory and the experiment are well matched.
Particle ExperimentParticle TheoryPredict
https://www.icepp.s.utokyo.ac.jp/information/20220426.html

Low

HighEnergy
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Anomaly and Quantum Field Theory• The anomaly is a peculiar phenomenon in quantum field theory (QFT).
Ø QFT has the infinite degree of freedom.
Ø To define QFT correctly, we let the infinite degree be finite (Regularization).
Ø This regularization breaks the symmetry (Anomaly).
Anomaly！
Regularization

The infinite DOF
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The Finite DOF
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Recent Developments in Anomalies• Recently, Gaiotto et al. has expanded the concept of symmetry. : Higher Form Symmetry(Gaiotto, Kapustin, Seiberg, Willett, arXiv:1412.5148[hep-th])Ø By anomalies with higher form (and discrete) symmetries, the low energy dynamics of gauge
theories has been discussed. (Gaiotto, Kapustin, Komargodski, Seiberg, arXiv:1703.00501)
Ø Many new anomalies have been discovered and related studies has been done.
ü Yamaguchi, arXiv:1811.09390[hep-th]
ü Hidaka, Hirono, Nitta, Tanizaki, Yokokura, arXiv:1903.06389[hep-th]

Lattice Gauge Theorya<latexit sha1_base64="MIM7ZFbLfAacbpeDO2Kb44ZaT04=">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</latexit>ü Honda, Tanizaki, arXiv:2009.10183[hep-th]
ü etc.

☆Motivation : Understand these anomalies
in the lattice field theory where we treat
the regularization well.
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L (nnµ̂)
µ̂

L (n)

n5/16

Anomaly of the 𝑆𝑈 𝑁 gauge theory with 𝜃 term
• The 𝑆𝑈 𝑁 gauge theory with the 𝜃 term has the time reversal (𝒯) symmetry at 𝜃 = 𝜋.

!

!

Da eSSU (N ) [a] eiθQ[a] , Q ∈ Z!
!
−−−−−−−−−→ Z ! = Da eSSU (N ) [a] eiπ(−Q[a]) = Da eSSU (N ) [a] eiπ(+Q[a]) e"−i2πQ[a]#$ % = Z
Z=Da eS[a] =θ=π, T trans.

=1• Then, we construct the 𝑆𝑈 𝑁 gauge theory with the higher form symmetry (ℤ! 1-form gauge
symmetry). This means we couple ℤ! 2-form gauge field to the theory.
Ø The topological charge (TC) becomes fractional, so it is not invariant under the 𝒯 transformation.

Important!!

e−i2πQ != 1

Ø This theory at 𝜃 = 𝜋 has the mixed anomaly between the ℤ! 1-form gauge and 𝒯 symmetry.

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Topological Charge on the Lattice• How to calculate the topological charge 𝑄,

Ø 𝑣",$ (𝑥) is the gauge translation function (transition function).

• On the lattice, topological values are ill-defined.
Ø Restricting the size of plaquette (Admissibility condition), Lüscher constructed
integer TC on the lattice (Lüscher, Commun. Math. Phys. 85 (1982)).
Ø We aim to construct the fractional TC on the 𝑆𝑈(𝑁) lattice by extended the Lüscherʼs
topological charge.
ü Itou, arXiv:1811.05708[hep-th]ü Anosova, Gattringer, Göschl, Sulejmanpasic, Törek, arXiv:1912.11685 [hep-lat]

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Fiber Bundle• The fiber bundle describes the gauge theory.
Ø Covering a manifold 𝑀 by patches 𝑈" , each patch has the 𝑆𝑈(𝑁) gauge field 𝑎" andthe matter field 𝜙" with the irreducible representation 𝜌.

• Gauge fields at 𝑈!" = 𝑈! ∩ 𝑈" areconnected by the gauge transformation
function 𝑔!" .
aj = gij −1 ai gij − igij −1 dgij

• At 𝑈!"# = 𝑈! ∩ 𝑈" ∩ 𝑈# , the cocycle conditionis satisfied.

φj = ρ(gij −1 )φi𝑈%

𝑈"

𝑈&
𝑈$transition function 𝑔!"
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𝑈#𝑈0128

Transition Function on the Lattice• The manifold is divided by hyper cubes 𝑐 𝑛 .
• Fiber bundle describes the gauge theory by transition function for gaugetransformation.
• e. g., in the 3dInterpolate from cite 𝑛

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Lüscher, Commun. Math. Phys. 85 (1982)

Interpolate Function in 𝑆𝑈(𝑁) Gauge Theory• In 𝑥 ∈ 𝑓(𝑛, µ),𝑠'𝑠)𝑠(𝑠,𝑥𝑥.

𝑠*𝑥/28/08/23

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𝑠%𝑠&𝑠+
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Transition Function• By the interpolate function made from the new link variable, we define thetransition function as continuum function on the lattice .

𝑛 − 𝜇̂

𝑛

Image of 𝑣90,2𝑥
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Transition Function for Fractional TC• Coupling ℤ$ 2-form field to the theory, the structure of fiber bundle becomes rich.
vn−ν̂,µ (n)vn,ν (n)vn,µ (n)−1 vn−µ̂,ν (n)−1 = 1l.
2πiṽn−ν̂,µ (n)ṽn,ν (n)ṽn,µ (n)−1 ṽn−µ̂,ν (n)−1 = e N Bµν (n−µ̂−ν̂) 1l.
• We find that the ℤ$ 1-form gauge invariance plays the center role.

Ø Admissibility condition

Ø Components of transition function!1l − Ũµν (n)! < ε2πiŨµν (n) ≡e− N Bµν (n)× U (n, µ)U (n + µ̂, ν)U (n + ν̂, µ)−1 U (n, ν)−128/08/23

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Fractional TC• By the ℤ$ 1-form invariant transition function, we calculate TC.
!
zµν =Bp
mod N.

Qtop = −1N!

p∈(T 2 )µν

1 εµνρσ zµν zρσ
1P2 (Bp ) mod 1 ∈ −+ZN
8T4 2P2 (Bp ) = Bp ∪ Bp + Bp ∪1 dBp• In the 𝑈(1) lattice gauge theory, we make sure that (cf. Abe, Morikawa, Suzuki, arXiv:2210.12967[hep-th])
1 ! !
1Z+ZQtop =εF̃(n)F̃(n+µ̂+ν̂)∈µνρσ µνρσ2232π
Nµ,ν,ρ,σn∈Λ28/08/23

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Anomaly Ⅰ• The action on the lattice isS[Ul , Bp ] ≡ −SW [Ul , Bp ] + iθQtop [Ul , Bp ]• The topological charge is

Qtop = −1N!

1P2 (Bp ) + Z ≡ frac[Bp ] + int[Ul , Bp ]T4 2

✧ Invariant under the ℤ! one-form gauge transformation✧ Odd under the 𝒯 transformation on the lattice,T

Qtop → −QtopØ We discuss the anomaly at 𝜃 = 𝜋 between the ℤ! 1-form gauge and 𝒯 symmetry.

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Anomaly Ⅱ• At 𝜃 = 𝜋, the partition function is, under 𝒯 transformation,!
!
Z[Bp ] = DUl eS[Ul ,Bp ] = DUl e−SW [Ul ,Bp ] eiθQtop [Ul ,Bp ]!
!
top [Ul ,Bp ]−−−−−−−−−→ Z " [Bp ] = DUl e−SW [Ul ,Bp ] eiπ(−Qtop [Ul ,Bp ]) = DUl e−SW [Ul ,Bp ] eiπQtop [Ul ,Bp ] "e−i2πQ#$%
θ=π, T trans.

=e−i2πfrac[Bp ]=e−i2πint[Ul ,Bp ] e−i2πfrac[Bp!

DUl e−SW [Ul ,Bp ] eiπQtop [Ul ,Bp ] #=Z[Bp ]
"
#$%
=Z

• This means that there is the anomaly at 𝜃 = 𝜋 between the ℤ$ 1-form gauge and 𝒯symmetry.

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Conclusion and Future Work✩Conclusion• We construct the fractional topological charge on the S𝑈(𝑁) lattice gauge theory.
• By this topological charge, we construct the anomaly at 𝜃 = 𝜋 between the ℤ$ 1form gauge and 𝒯 symmetry on the lattice.
✩Future work
• Construct the magnetic operator under the admissibility condition on the latticeü cf. Abe, Morikawa, Onoda, Suzuki, Tanizaki, arXiv:2304.14815 [hep-lat]Ø Parallel session2 by Onoda tomorrow• Construct non-invertible symmetries on the lattice
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