Symmetries and Strings of Adjoint QCD2 Kantaro Ohmori (Simons - - PowerPoint PPT Presentation

Symmetries and Strings of Adjoint QCD2

Kantaro Ohmori (Simons Center for Geometry and Physics)

based on arXiv:2008.07567 with Zohar Komargodski, Konstantinos Roumpedakis, Sahand Seifnashri @ Strings and Fields 2020 (Kyoto)

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Introduction and summary

1+1d Adj. QCD was studied extensively in '90s:

[Klebanov, Dalley '93], [Gross, Klebanov, Matytsin, Smilga ’95], [Gross, Klebanov, Hashimoto ’98]... [Kutasov '93][Boorstein, Kutasov '94],[Kutasov, Schwimmer '95],

When massless, claimed to be in deconfined phase, although fermion cannot screen a probe in fundamental representation.

[Cherman, Jacobson, Tanizaki, Unsal ’19] analyzed symmetry (incl. one-form) and its anomaly.

Concluded it is in the confined (or partially deconfined) phase when . Ordinary symmetry is not enough. Non-invertible topological line accounts for deconfinement. First (non-topological) gauge theory example of non-invertible top. op. Precise (putative) IR TQFT description string tension with small mass.

N ≥ 3 ⇒

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1+1d massless Adjoint QCD

1+1d gauge theory with with single massless Majorana fermions ( ) Gapped (from central charge counting). 0-form (ordinary) symmetry: 1-form (center) symmetry: (center of ) charged object: Wilson line "Symmetry operator" : for each , Topological and local op. .

G = SU(N ) (ψ

L i j ˉ R i j ˉ

∑i

L,R ii ˉ

L = Tr −

+ iψ

(

4g2 1 2 L L R∂

ˉ

R L z R z ˉ)

j

L,R i j ˉ

∑k

L,R ik ˉ L,R k, j ˉ

Z

2 C

Z

2 χ

Z

2 F

Z

N (1)

SU (N ) W

λ

U

k

ω

k

Z

N

U

k λ

Tr

λ k λ k

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One-form symmetry in 1+1d and "Universe"

An energy-eigenstate satisfying cluster decomposition (on ) diagonalizes : ( topological local op.) : two states on approximating states and . Even on , and does not mix if : No domain wall between and with finite tension. SSB of ordinary discrete symmetry allows domain walls. Separated sectors even on compact space: "universes" labelled by eigenvalue of (

• f

them).

∣p⟩ R U

k

U

k

e ∣p⟩

N 2πikp

U

k

∣p

1 S

1

2 S

1

S1 ∣p

1

∣p

2

S1 ∣p

1 S 1

∣p

2 S 1

p

1  p 2 mod N

∣p

1

∣p

2

p U

1 N

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"Universe" and (de)confinement

Wilson line (worldline of infinitely heave partible) separates "universes": Wilson loop contains another "universe" in it: area law, confinement perimeter law, deconfinement Non-invertible topological lines forcing universes to completely degenerate!

U

k p

e W

N 2πikp

p k

E

p  E

p+1

E

p

E

p+1

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Symmetry and top. op.s

Symmetry Topological codim.-1 op for For , is invertible: Not all topological operators have its inverse: non-invertible top. op.s.

G ⟹ U (g)[Σ] g ∈ G e ∈

iα

U(1) U (e )[Σ] =

iα

eiα

∫Σ

μ μ

U (g)[Σ] U (g)[Σ]U (g )[Σ] =

−1

1

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Non-invertible topological lines

• Top. lines have fusion rule:

, Data of lines = Fusion category Should be regarded as generalization of symmetry, as they shares key features with symmetry (+anomaly): gauging, RG flow invariance. : "Category symmetry"

[Brunner, Carqueville, Plencner ’14],[Bhardwaj, Tachikawa, ’17],[Chang, Lin, Shao, Wang, Yin, ’18]

Category symmetry constrains the IR physics of strongly coupled system like adj. QCD.

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Topological lines in adj QCD

Topological lines in adj QCD (ignoring charge conjugation) = preserving (commutes with ) top. lines in free fermions. (No possible anomalies since is simply-connected.) No classification of top. lines in general 1+1d free theory.

[Fuchs, Gabrdiel, Runkel, Schweigert '07] for

theory Majorana fermions non-diagonal (spin-)RCFT General theory on top. lines in RCFT

[Fuchs,Runkel,Schweigert '02]...

SU(N)

su(N ) j SU (N ) S1 N −

2

1 ⊃

− spin

2

1)

1

(N ) su

N

su(N )

N

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Topological lines in N=3 adj QCD

Generated by 2 invertible lines , and 8 non-invertible lines (up to charge conjugation) , etc.. ( ) has one-form charge, e.g.: "Top. line - mixed anomaly" If is a vacuum: is a degenerate vacuum in a different universe! Deconfinement ( follows from unitarity) Furthermore, minimally, there has to be 4 vacua:

χ

L (−1)F

L

i

L

1

L

2

2(trivial line) + 2χ

L

L

i i = 1, ⋯ 6

U

1 1

e L

2πi/N 1 1

Z

2 (1)

∣0⟩ L

1

⟹ L

1

 0

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General

• lines. The precise determination of the category is hard for large

. Always completely deconfined (i.e. with unit one-form charge). Probably we need vacua to accommodate those lines: Hagedorn behavior with . [Kutasov '93] A "natural" candidate : TQFT (with further gauging) Assuming this, for , -string tension behaves at the first order of the mass

• f the adjoint quark

(new computation! conjecture for )

N

O(2 )

2N

N L O(2 )

N

T

H

Spin(N −

2

1)/SU (N ) Z

2

N = 3, 4, 5 k T

k

∣m∣ sin(πk/N ) m N ≥ 6

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Summary and prospect

1+1d massless adj. QCD has many ( ) topological line operators, most of which are non-invertible. Topological line is an interface between different "universes" due to "top. line - mixed anomaly" deconfinement vacua With small mass, the -string tension would be Higher dimensions? Concrete examples of non-invertible topological operators in higher dimensional non- topological QFT?

O(2 )

2N

Z

N (1)

⟹ O(2 )

N

k T

k

∣m∣ sin(πk/N )

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Thank You For Your Attention!

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