Vacuum structure of 2 d adjoint QCD anomaly, mod 2 index, and - PowerPoint PPT Presentation

Vacuum structure of 2 d adjoint QCD – anomaly, mod 2 index, and semiclassics – Yuya Tanizaki North Carolina State University Sep, 2019 @ ICTP, Trieste Collaborators: Aleksey Cherman, Theodore Jacobson (UMN), Mithat ¨ Unsal (NCSU) References: 1908.09858[hep-th] Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 1 / 23

2 d adjoint QCD 2 d adjoint QCD We consider 2 d SU ( N ) YM + one adjoint Majorana fermion: � � 1 S = tr[ G ∧ ⋆G ] + tr[ ψ + D + ψ + + ψ − D − ψ − + mψ + ψ − ] 2 g 2 M 2 M 2 In this talk, we shall elucidate its ground-state properties based on careful analysis of symmetries and ’t Hooft anomalies, and explicit exploration of dynamics with semiclassical analysis on small R × S 1 . Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 2 / 23

2 d adjoint QCD Motivations There are some similarities with 4 d confining gauge theories: Theory is not solvable. Theory has a Z N center symmetry. Confined or deconfined? ◮ It’s an interesting question if 1 -form symmetry in 2 d can be spontaneously broken. ◮ Indeed, most likely, 1 -form symmetry in 2 d is unbroken, unless anomaly requires. (cf. Gaiotto, Kapustin, Seiberg, Willet, ’14) 2 d pure YM is also good, but it does not have any propagating modes by gauge invariance. 2 d adjoint QCD has O ( N 2 ) microscopic DOF thanks to adjoint Majorana fermions ψ . In large- N , there are infinitely many Regge-like trajectories. (’t Hooft model (i.e. 2 d YM + fundamental) only has one.) Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 3 / 23

2 d adjoint QCD Main result We find various Z 2 anomaly for the symmetry at m = 0 : Z [1] G = ⋊ ( Z 2 ) C × ( Z 2 ) F × ( Z 2 ) χ . N � �� � � �� � ���� � �� � center sym. charge conj. ( − 1) F discrete chiral Minimal requirement of anomaly matching shows Spontaneous chiral symmetry breaking, ( Z 2 ) χ → 1 , for N = 4 n, 4 n + 2 , 4 n + 3 but not for N = 4 n + 1 . For odd N , center symmetry is unbroken. For even N , partial deconfinement is required, Z [1] N → Z [1] N/ 2 Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 4 / 23

2 d adjoint QCD Some remarks 2 d adjoint QCD has the marginally relevant four-fermion interactions, (tr[ ψ + ψ − ]) 2 , tr[ ψ + ψ + ψ − ψ − ] . Adding these does not break any symmetry of the Lagrangian. Since our analysis is based only on symmetry, our result generalizes to any local deformations like this! (assuming mass gap) Moreover, the semiclassical analysis on small R × S 1 prefers minimal scenario of anomaly matching. Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 5 / 23

Symmetry and Anomaly Symmetry and Anomaly Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 6 / 23

Symmetry and Anomaly Symmetry of adjoint QCD (Internal) Symmetry of 2 d adjoint QCD with m = 0 : Z [1] G = ⋊ ( Z 2 ) C × ( Z 2 ) F × ( Z 2 ) χ . N ���� � �� � � �� � � �� � center sym. charge conj. ( − 1) F discrete chiral The first three factors are the vector-like symmetry: Z [1] N : W ( C ) �→ e 2 π i /N W ( C ) . ( Z 2 ) C : a ij,µ �→ − a ji,µ , ψ ij �→ ψ ji . ( Z 2 ) F : ψ �→ − ψ . The last one is the chiral symmetry: ( Z 2 ) χ : ψ + �→ ψ + and ψ − �→ − ψ − . Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 7 / 23

Symmetry and Anomaly Mixed Anomaly We will see that the partition function Z transforms as ( Z 2 ) χ : Z → ( − 1) ζ Z , under the background gauge field (or twisted b.c.) of vector-like symmetry. List of mixed ’t Hooft anomalies: � if ( − 1) ζ = − 1 . Anomalous symmetry N = 4 n 4 n + 1 4 n + 2 4 n + 3 Z [1] N × ( Z 2 ) χ � � ( Z 2 ) F × ( Z 2 ) χ � � ( Z 2 ) F × ( Z 2 ) C × ( Z 2 ) χ � � (Note: Z [1] N × ( Z 2 ) χ -anomaly was partly discovered in Lenz, Shifman, Thies (hep-th/9412113)) Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 8 / 23

Symmetry and Anomaly Absence of familiar Dirac index In 4 d gauge theory (also in 2 d U(1) gauge theory), we are familiar with the index theorem, stating that Imbalance of chirality = Topological charge BUT SU ( N ) gauge field is traceless, and this index theorem is not useful: #( Zero modes with + chirality ) − #( Zero modes with - chirality ) � = 1 tr( G ) = 0 2 π ⇒ No imbalance between ψ + and ψ − Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 9 / 23

Symmetry and Anomaly Mod 2 index & Z 2 mixed anomaly Theorem (Mod 2 index theorem) Let us set ζ = #( Zero modes with + chirality ) = #( Zero modes with - chirality ) . Then, ζ is a topological invariant mod 2 . The Z 2 topological invariant ( − 1) ζ determines the mixed anomaly! D ψ ∼ (d ψ +(0) ) ζ (d ψ − (0) ) ζ � d ψ + i d ψ − i , i : λ i � =0 ( Z 2 ) χ : D ψ �→ ( − 1) ζ D ψ. Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 10 / 23

Symmetry and Anomaly Proof of mod 2 index theorem The Dirac operator / D in 2 d adjoint fermions is real anti-symmetric. / Du = i λu. For λ � = 0 , we get (cf. Witten 1508.04715) : (i γ 1 γ 2 ) u ∗ +i λ u u ∗ − i λ (i γ 1 γ 2 ) u Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 11 / 23

Symmetry and Anomaly Z 2 anomalies We can find mixed anomalies with chiral symmetry. ( Z 2 ) χ With anti-periodic (AP) B.C. on T 2 : Z AP/AP − − − → Z AP/AP . With periodic (P) B.C. on T 2 = Background flux on ( Z 2 ) F : ( Z 2 ) χ → ( − 1) N − 1 Z P/P . Z P/P − − − � B = 2 π With AP B.C. with the minimal ’t Hooft flux N : ( Z 2 ) χ → ( − 1) N − 1 Z AP/AP [ B ] . Z AP/AP [ B ] − − − With P/ C -twisted B.C.: ( Z 2 ) χ → ( − 1) N ( N − 1) / 2 Z P/ C . Z P/ C − − − Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 12 / 23

Anomaly matching Anomaly matching and low-energy behaviors Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 13 / 23

Anomaly matching Anomaly matching We cannot gauge G with anomaly. We regard our theory as a boundary of ( d + 1) -dim. SPT phase protected by G (Wen, ’13, Kapustin, Thorngren, ’14, Cho, Teo, Ryu, ’14, ...) : ⇒ Low-energy DOF must also cancel the anomaly inflow from bulk. Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 14 / 23

Anomaly matching Anomaly matching: Chiral symmetry breaking Anomaly matching Low-energy DOF reproduce the same anomaly. Possible options of low-energy physics: massless excitations, spontaneous symmetry breaking, or topological order. The list of our ’t Hooft anomaly is Anomalous symmetry N = 4 n 4 n + 1 4 n + 2 4 n + 3 Z [1] N × ( Z 2 ) χ � � ( Z 2 ) F × ( Z 2 ) χ � � ( Z 2 ) F × ( Z 2 ) C × ( Z 2 ) χ � � ⇒ Chiral symmetry breaking seems to be a natural option. (For even N , this is indeed the unique option. For N = 4 n + 3 , C -breaking is also a possibility) Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 15 / 23

Anomaly matching What about deconfinement? In the following, let’s assume chiral symmetry breaking. Can we say anything useful about confinement/deconfinement? Yes, domain wall physics tells us that Z [1] N → Z [1] N/ 2 is required for even N . Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 16 / 23

Anomaly matching Massive adjoint QCD as nontrivial SPT To see the nontrivial property of the wall, we first consider the massive deformation m � = 0 . D − mγ ) = m ζ � ′ ( λ 2 i + m 2 ) . Pf(i / i This means that m < 0 is a nontrivial SPT compared with m > 0 if ( − 1) ζ = − 1 : Z m = − M = ( − 1) ζ . Z m = M Since Domain wall ≃ Boundary of nontrivial SPT, there must be gappless excitations on the domain wall with appropriate charge. Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 17 / 23

Anomaly matching Partial deconfinement for even N Recall that, for even N , � + N πζ = πζ free B . 2 � �� � � �� � ( − 1) F Z [1] N Thus, boundary excitation is fermionic, and has N -ality N/ 2 . ⇒ In order for two vacua having the same energy density, N/ 2 -string tension must vanish: σ N/ 2 = 0 . We have no symmetry reasonings for deconfinement of other strings, so that we propose � � 4 πk �� σ k ∼ Ng 2 1 − cos . N Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 18 / 23

Semiclassical analysis Semiclassical analysis Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 19 / 23

Semiclassical analysis Analysis on small R × S 1 With gL ≪ 1 , the semiclassical treatment becomes reliable (Smilga hep-th/9402066, Lenz, Shifman, Thies, hep-th/9412113) With AP B.C., the Polyakov-loop potential has N minima, P = e 2 π i k/N ( k = 0 , 1 , . . . , N − 1) . Tunneling between them is associated with fermionic zero modes with ζ = N − 1 . Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 20 / 23

Semiclassical analysis Tunneling and Mod 2 index Note that the fermionic zero modes ζ is protected only mod 2 . Tunneling is possible if ζ = 0 mod 2 . Odd N ⇒ Unique ground state. No SSB. Even N ⇒ Two vacua: Chiral SSB. Also, � � σ k = ∆ E 1 − cos 4 πk . L N Yuya Tanizaki (NCSU) 2d adjoint QCD Sep 3-6, 2019 @ Trieste, Italy 21 / 23