1+1d Adjoint QCD and non-invertible topological lines Kantaro - PowerPoint PPT Presentation

1+1d Adjoint QCD and non-invertible topological lines Kantaro Ohmori (Simons Center for Geometry and Physics) based on WIP with Zohar Komargodski, Konstantinos Roumpedakis, Sahand Seifnashri @ East Asian String Webinar 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] revisited the problem. They analyzed symmetry (incl. one-form) and its anomaly. Concluded it is in N ≥ 3 confined (or partially deconfined) phase when . Symmetry is not enough. Non-invertible topological line accounts for deconfinement. First (non-topological) gauge theory example of non-invertible top. op. 2

1+1d massless Adjoint QCD G = SU( N ) 1+1d gauge theory with with massless Majorana fermions ˉ ˉ ˉ ij ij ( ψ , ψ ) = 0 ii ∑ i ( ) ψ L , R L R ( ˉ ) ˉ 1 2 L = Tr − + i ψ ∂ ψ + i ψ R ∂ + j A + j A F ψ L L R L z R z 4 g 2 ˉ ˉ ˉ k , j ij = ik ∑ k j ψ ψ L , R L , R L , R F × Z N (1) Z Z × Z 2 χ × C Symmetry: ( : one-form (a.k.a. center) symmetry) 2 2 3

Quartic couplings Two independent classically marginal couplings preserving all the symmetry L = c O + c O 1 1 2 2 q O = Tr( ψ ψ ψ ψ ) = Tr j j , 1 + + − − L R 2 2 O = (Tr( ψ ψ )) − Tr( ψ ψ ψ ψ ) ( − ) 2 + − + + − N 2 − 1 SU ( N ) N In the free fermion theory, O 2 is a sum of primaries N ⟨ O (0) j ( z ) j ( z ) ⋯ j ( w ) j ( w ) ⋯ ⟩ free ψ = 0 Fusion rule : 2 1 2 1 2 L L R R μ ⟩ ⟨ − S [ A ] μ ∫ ⟨ 2 ⟩ adj QCD = = 0 j A ∫ O D A e O e Y M +cntr 2 free ψ j A + No Feynman diagram that can generate O 2 with and O 1 coupling j A ˉ L z R z in adj QCD! 4

Protection by non-invertible line What protects from radiative generation? O 2 There is no symmetry that O 2 violates. We claim that non-invertible top. lines protects it. The same set of lines also explains deconfinement . Parameter space: We expect that def. breaks all the non-invertible lines and thus leads us to the O 2 picture of [Cherman, Jacobson, Tanizaki, Unsal ’19] but have not succeeded to proof. 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 J d S ∫ Σ ∈ U(1) U ( e )[Σ] = For , e μ −1 1 U ( g )[Σ] U ( g )[Σ] U ( g )[Σ] = is invertible : ⟺ "Higher-form" symmetry invertible top. op. with higher codimension . [Gaiotto,Kapustin,Seiberg,Willett '14] 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 and topological junctions = Fusion category Should be regarded as generalization of symmetry , as they shares key features with symmetry (+anomaly): gauging, RG flow invariance . [Brunner, Carqueville, Plencner ’14] , [Bhardwaj, Tachikawa, ’17] , [Chang, Lin, Shao, Wang, Yin, ’18] 7 ′ c = 10 E.g. Tricritical ( ) Ising + relevant perturbation preserves line with σ W 7 7 , 16 16 2 = 1 + W ⟹ fusion asymmetric 2 vacua (First noticed by integrability) W [Chang, Lin, Shao, Wang, Yin, ’18] Massless Adj. QCD is another example, without (known) integrability. 7

Outline Charge massless Schwinger model q Non-abelian bosonization Non-invertible lines and confinement in 1+1d massless adj. QCD 8

Charge massless Schwinger model q U(1) Ψ q q > 1 1+1d gauge theory with charge massless Dirac fermion ( ), q i qα Ψ → Ψ e q q Z Z q × C χ (Ordinary) Symmetry: 2 (1) Z ⊂ Ψ q Z q U(1) acts trivially on : one-form (a.k.a. center) symmetry q e 2 π i p ∮ W = A Wilson line : worldline of heavy probe with charge p p Ψ q is screened by and deconfined. W q p =  0 mod q How about when ? W p 9

One-form symmetry in 1+1d and "Universe" 1 Z Ψ q The electric field (classically in ) fluctuates because of , but jumps only F 01 e 2 by . q 2 π i k F Z ⊂ = 01 U(1) qe 2 is valued topological local (codim-2) operator U e k q (1) Z q Interpreted as the symmetry operator for 2 π i kp R U k U ∣ p ⟩ = ∣ p ⟩ Clustering energy eigenstates (on ) diagonalizes : e q k S 1 ∣ p ⟩ ∣ p ⟩ p = 1  p 2 mod q Even on , and does not mix if 1 2 ∣ p ⟩ ∣ p ⟩ No domain wall between and with finite tension 1 2 Separated sectors even on compact space: "universes" labelled by eigenvalue of p . U 1 10

"Universe" and (de)confinement Wilson line (worldline of infinitely heave partible) separates "universes": 2 π i kp U W = e W U q k p p k Wilson loop contains another "universe" in it: E = p  E ⟹ area law, confinement p +1 E = ⟹ perimeter law, deconfinement E p +1 p 11

Abelian bosonization A way to study the charge Schwinger model is the bosonization: q Ψ ⟺ ϕ 2 π Dirac fermion , where : periodic scalar (set to be ) ϕ ~ e inϕ + iwϕ = O n , w 2 S = nw Q = wq Ψ = 2 1 Δ = n + , , , O w 1 ,1 q 4 2 1 1 2 2 q (∂ ϕ ) − + The dual description is: F ϕF 8 π 4 e 2 2 π 2 πk Z k ∈ Z q : ϕ → ϕ + χ χ for . q q e → ∞ G / G Naively, IR limit seems equivalent to . If true, theory is theory ( BF G = U (1) q ⟹ TQFT with ) describing vacua deconfined . q UV reason? 12

(1) Z × Z q χ anomaly and deconfinement q ~ 1 ϕ Z q e i q = χ Q = 1 : defect operator at the edge of line in free boson, . O L 1 1 0, q 2 π i n 2 π i ( z ) O (0) = ( e z ) O (0) O O e q n , w n , w 1 1 0, q 0, q U(1) O 0, q After gauging , connects and : W 1 L 1 1 2 π i kp 2 π i kp (1) k Z × Z q U W = ⟹ U L = χ : anomaly e W U e L U q q k p p k k p p q is topological : L 1 L ∣ ψ ⟩ ∣ ψ ⟩ and have degenerate energy: 1 SU( N ) adj QCD has a smiliar story but requires to consider non-invertible top. N ≥ 3 lines when . 13

Nonabelian bosonization SU( N ) We would like to repeat a similar analysis for massless adjoint QCD with gauge group. 2 − 1 SU( N ) Dualize Maj. fermions while keeping the symmetry manifest. N ⟹ Nonabelian bosonization [Witten '84] /(−1) F ⟺ Spin( n ) 1 (Maj.) WZW model [Ji, Shao, Wen '19] n ψ 2 2 PSU( N ) ⊂ Spin( N − 1) ( N ) ⊂ ( N − 1) 1 , : su spin N 2 c ( ( N ) ) = c ( ( N − 1) ) conformal embedding ( su spin 1 N ∞ /(−1) F ⟺ Spin( N 2 → − 1) /SU( N ) (Adj. QCD with ) coset TQFT g YM 1 N spinor Z 2 (−1) F Arf "Gauge back" by gauging with twist. [Alvarez-Gaume, Bost, Moore, Nelson, Vafa '87] ,... [Thorngren '18] , [Karch, Tong, Turner '19] spinor 2 Z → ∞ ⟺ Spin( N − 1) /SU( N ) / Adj. QCD with g YM 1 Arf N 2 Precise version of bosonization prediction by [Kutasov '93] , [Boorstein, Kutasov '94] , [Kutasov, Schwimmer '95] 14

spinor 2 Z Spin( N − 1) /SU( N ) / 1 Arf N 2 ⟹ 2 N −1 Coset counting vacua . : [Kutasov '93] Most of them are not because of SSB (1) Z N All the universes (due to ) are degenerate = deconfined . N g → ∞ Naively IR limit = , as is super-renormalizable. However it is not very g clear whether the flow generate other terms in the strongly coupled regime. UV reason of deconfinement and exponentially many vacua? : Topological lines 15

Topological lines in adj QCD su ( N ) Topological lines in adj QCD = preserving (commutes with ) top. lines in j free fermions: μ ⟩ ⟨ μ − S [ A ] ∫ ⟨ L , O , ⋯ ⟩ adj QCD = L , O , ⋯ e j A ∫ D A e Y M +cntr free ψ 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 spin su N 1 N su ( N ) N non-diagonal (spin-)RCFT General theory on top. lines in RCFT [Fuchs,Runkel,Schweigert '02] ... SU ( N ) ⟹ Much easier in diagonal RCFT ( WZW models) Verlinde lines 16

Verlinde lines in diagonal RCFT Diagonal RCFT = CS theory on a interval. O i , i : Line bridging boundaries L i Chiral alg. pres. topological line in RCFT = topological line Σ 2 O (2 ) N in CS along : Verlinde line ( of them) ( is the topological Wilson line of the auxiliary gauge field in 3d bulk. Not to be L i confused with the Wilson line of the physical gauge field in adj QCD.) W i L ⊗ L = ⨁ k k N L i j i , j k l V ⊗ ⨁ k , l Defect operator at the edge of : : L i N V l i , k k 17

su ( N ) N Fermions as RCFT su ( N ) Topological lines in adj QCD = preserving top. lines in fermions su ( N ) N non-diagonal (spin-)RCFT Non-diagonal RCFT = CS theory on a interval with surface op. insertion: [Kapustin Saulina '10] , [Fuchs, Schweigert, Valentino '12] , [Carqueville, Runkel, Schaumann '17] 18

ˉ SU ( N ) N ψ ij preserving topological lines in ± Subset of topological lines : defined by L i + V ⊗ ⨁ k , l , m k Defect operator at the edge of : L i N Z V m k , m l , i l O ∈ V ⊗ In particular, always exists. V 0 19 i i