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) 1

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. N ≥ 3 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. 2

1+1d massless Adjoint QCD ˉ ˉ i j i j G = SU( N ) ( ψ , ψ ) 1+1d gauge theory with with single massless Majorana fermions ( L R ˉ = 0 ∑ i ii ) ψ L , R ( ˉ ) ˉ 1 2 L = Tr − + i ψ ∂ ψ + i ψ R ∂ + j + j F ψ A A L L R L z R z 4 g 2 ˉ ˉ ˉ k , i j j = ∑ k ik j ψ ψ L , R L , R L , R Gapped (from central charge counting). Z Z Z χ C × × F 0-form (ordinary) symmetry: 2 2 2 (1) Z SU ( N ) 1-form (center) symmetry: (center of ) N charged object: Wilson line W λ Z ( p ) ∈ "Symmetry operator" : for each , Topological and local op. U ω k k N = Tr ( ω ) W . U W U k λ λ k λ k 3

One-form symmetry in 1+1d and "Universe" R ∣ p ⟩ An energy-eigenstate satisfying cluster decomposition (on ) diagonalizes : U k 2 π i kp ∣ p ⟩ = ∣ p ⟩ ( topological local op.) U e U N k k S 1 ∣ p ⟩ , ∣ p ⟩ ∣ p ⟩ ∣ p ⟩ : two states on approximating states and . 1 S 2 S 1 2 1 1 S 1 ∣ p ⟩ ∣ p ⟩ 1  p = 2 mod N Even on , and does not mix if : p 1 S 1 2 S 1 ∣ p ⟩ ∣ p ⟩ No domain wall between and with finite tension. 1 2 SSB of ordinary discrete symmetry allows domain walls. Separated sectors even on compact space: "universes" labelled by eigenvalue of ( of p U 1 N them). 4

"Universe" and (de)confinement Wilson line (worldline of infinitely heave partible) separates "universes": 2 π i kp = U W e W U N k p p k Wilson loop contains another "universe" in it: p  E = ⟹ area law, confinement E p +1 = ⟹ perimeter law, deconfinement E E p +1 p Non-invertible topological lines forcing universes to completely degenerate! 5

Symmetry and top. op.s G ⟹ U ( g )[Σ] g ∈ G Symmetry Topological codim.-1 op for μ i α i α e i α ∫ Σ d S ∈ U(1) U ( e )[Σ] = J For , e μ −1 1 U ( g )[Σ] U ( g )[Σ] U ( g )[Σ] = is invertible : Not all topological operators have its inverse: non-invertible top. op.s. 6

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. 7

SU ( N ) Topological lines in adj QCD su ( N ) Topological lines in adj QCD (ignoring charge conjugation) = preserving (commutes with ) top. lines in free fermions. j SU ( N ) (No possible anomalies since is simply-connected.) No classification of top. lines in general 1+1d free theory. S 1 [Fuchs, Gabrdiel, Runkel, Schweigert '07] for theory 2 2 − 1 ⊃ ( N − 1) ⊃ ( N ) Majorana fermions N spin su 1 N su ( N ) non-diagonal (spin-)RCFT N General theory on top. lines in RCFT [Fuchs,Runkel,Schweigert '02] ... 8

Topological lines in N=3 adj QCD L (−1) F Generated by 2 invertible lines , and 8 non-invertible lines χ L i (up to charge conjugation) ⊗ = 2(trivial line) + 2 χ , etc.. L L 1 2 L 2 πi / N i i = 1, ⋯ 6 = ( ) has one-form charge , e.g.: L U L e L U 1 1 1 1 (1) Z " Top. line - mixed anomaly " 2 ∣0⟩ ∣0⟩ If is a vacuum: is a degenerate vacuum in a different universe! L 1 ⟹ ∣0⟩ =  0 Deconfinement ( follows from unitarity) L 1 Furthermore, minimally, there has to be 4 vacua: 9

General N 2 N O (2 ) lines. The precise determination of the category is hard for large . N Always completely deconfined (i.e. with unit one-form charge). L O (2 ) Probably we need N vacua to accommodate those lines: = 0 Hagedorn behavior with . [Kutasov '93] T H 2 Spin( N − 1)/ SU ( N ) A "natural" candidate : TQFT (with further gauging) Z 2 N = 3, 4, 5 k Assuming this, for , -string tension behaves ∼ ∣ m ∣ sin( πk / N ) at the first order of the mass of the adjoint quark T m k N ≥ 6 ( new computation! conjecture for ) 10

Summary and prospect 2 N O (2 ) 1+1d massless adj. QCD has many ( ) topological line operators , most of which are non-invertible. (1) Z Topological line is an interface between different "universes" due to "top. line - mixed N ⟹ anomaly" deconfinement O (2 ) N vacua ∼ ∣ m ∣ sin( πk / N ) With small mass, the -string tension would be k T k Higher dimensions? Concrete examples of non-invertible topological operators in higher dimensional non- topological QFT? 11

Thank You For Your Attention! 12