't Hooft anomalies, 2-charge Schwinger model, and domain walls in hot super Yang-Mills theory 1 MOHAMED ANBER BASED ON ARXIV:1807.00093, 1811.10642 WITH ERICH POPPITZ (U OF TORONTO)
Outline Overview on domain walls (DW) in hot Super Yang-Mills 2 t Hooft anomaly matching conditions, the new anomalies DW in hot SYM, the worldvolume theory is a Schwinger model t Hooft anomalies in Schwinger model and bulk/dw Physics
Overview Why we study DW in hot SYM: 1. Prototype of DW in other gauge theories: QCD(adj) for model building 3 Sannino et al Anthenodorou, Bennett, Bergner, Lucini 2014 +many more 2. Provide a lucid understanding of the new t Hooft anomalies in a weakly coupled setup.
Overview 3. DW in hot SYM (and other YM theories) are the high-t counterparts of center vortices (may play a role in confinement). 4 see, e.g., Greensite book 4. The physics of hot YM parallels that of YM on a small spatial circle (see Erich Poppitz talk). MA, Poppitz, Unsal+many others 2008-now
What is an t Hooft anomaly t Hooft anomalies: nonperturbative phenomena in QFT, specially the strongly coupled. They put sever constraints on the IR spectrum of asymptotically free theories. t Hooft anomalies of 0 form symmetries were known since the 80s. t Hooft anomalies of 1-form symmetries non-trivial constraints on the spectrum of a theory. 5 Gaiotto, Kapustin, Komargodski, Seiberg, 2017
What is an t Hooft anomaly? Given a global symmetry of a QFT, we may try to gauge G. (turn on a background gauge of G ) If obstructed (anomalous), the theory has an t Hooft anomaly. t Hooft anomaly is RG invariant, useful for asymptotically free theories. 6 G t Hooft, 1980 UV N F 32π 2 µν F! µν energy scale IR N F 32π 2 µν F! µν
What is an t Hooft anomaly? The UV/IR matching of the anomaly: composite fermions or Goldstone bosons. 7 E.g. 4-D QCD with 2 or 3 fundamental flavors. G = SU (3) L SU (3) R are 0 form global symmetries (act on local operators) q L U L q L
Gauging : What is an t Hooft anomaly? 8 G Frishman, Schwimmer, Banks, Yankielowicz 1981 UV IR SU (3) L SU (3) R SU V (3) GB F quarks fermions
What is an t Hooft anomaly? 9 Given two symmetries G 1 G 2, we can examine one in the background gauge of the other. The obstruction is a mixed t Hooft anomaly. This provides more constraints on the IR spectrum. The lore is that we should check all the t Hooft anomalies of a theory between the UV and IR.
What is an t Hooft anomaly? There are also non-local operators, e.g., line operators (Wilson s loop) (1) It transforms under 1 form global symmetry: Z N 10 C generalized global symmetries Gaiotto, Kapustin, Seiber, Willett, 2014 tr F e i! C A e i2π N tr F e i! C A U x,µ U x,µ Z µ U x,µ Z µ = e i2π N k = 0 confined phase
What is an t Hooft anomaly? It was realized that gauging 1 form symmetries provides us with extra t Hooft anomalies. 11 Gaiotto, Kapustin, Komargodski, Seiberg, 2017 We use the new anomalies to study the physics on the DW in hot super Yang-Mills (SYM).
SYM 12 4D QCD(adj), including SYM, and symmetries (UV): S = 1 2 tr F g 2 µν!" # $# + i tr λ I σ µ D µ λ I!## "## $ SU C ( N ) ( 0) dχ SU (n f ), U R (1) Z 2 n f N (0)dχ The breaking U R (1) Z 2n f N can be understood in the BPST instanton background. (0)dχ is a 0 form discrete chiral symmetry. Z 2n f N Z (0)dχ 2n : λ e f N 2π i 2n f N λ
SYM The global symmetries of QCD(adj): (1)C Z N We may gauge : 13 (1)C (0)dχ Z N Z 2 Nn f (1) couple QCD(adj) to a TQFT (2) turn on twisted t Hooft fluxes on T 4 Kapustin, Seiberg 2014 t Hooft 1979, Van Baal 1982
The end result: under SYM 14 (0)dχ Z 2n f N we find: Z SYM + Z N (1)C 2π gauge N ei Z SYM + Z N (1)C gauge Fate of symmetries for SYM at T = 0 1. (DW with Chern-Simons theory) 2. Z 4 (0)dχ Z 2 (1)C mixed t Hooft anomaly SU (2)
DW in hot SYM Hot SU C (2) SYM: T Λ QCD 2,β 15 Z (1) I L Battacharia, Gocksch, Korthals-Altes, Pisarski 1992 ( 1 S ) 3 1 L S β β = 1 T (0)dχ Z 4 A 0 3 = 0 DW x 3 A 3 0 = 2πT i! A 0 i A 0 (1) β tr F e = 2 tr e! (1) β Z F = 2 (1) 2,β Z 2,β Z 2,β (1) (1) Z 2,L 2,L Z (0)dχ Z 4
DW in SYM The fermions are adjoint+anti-pbc. 16 MA, Poppitz, 2018 Recall A k = a k τ 3 +W k+ τ + +W k τ λ = λ 3 τ 3 + λ + τ + + λ τ heavy W-bosons λ p ± = ± λ p,1 ± λ p,2 Matsubara modes
DW in SYM Two fermion zero modes on the DW. 17 DW theory (2-D) is axial Schwinger model: l = 1 4e 2 + +iλ p= 1,1 ( F ) 2 + kl + iλ p=0,2 SU (2) U (1) 1 + i 2 + i2 a 1 + ia 2 ( ) 1 i 2 i2 a 1 ia 2 ( ) λ + p= 1,1 λ + p=0,2 axial with charge 2
q-charge Schwinger model We can relabel the fermions (vector-axial duality in 2-D: ε kl γ l = γ k γ 5 ). 18 Thus, q = 2-charge vector Schwinger model as EFT on the DW: l = 1 4e 2 ( F ) 2 kl + iψ + 1 i 2 + iq a 1 ia 2 ( ) +iψ 1 + i 2 + iq a 1 + ia 2 ψ ( ) ψ +
q-charge Schwinger model This theory has a non-trivial topology+exactly solvable via bosonization. It enjoys two global symmetries: Under U (1) A the measure transforms as D[ψ ] e i2qχt D[ψ ] 19 U (1) A :ψ ± e ±iχ ψ ± Z (1)C q : e i! A q=1 by Manton 1985, Iso, Murayama 1990 e i 2π q e i! A sum over in path integral T = 1 2π F 12 dx2 =! quantized flux on the 2-torus
q-charge Schwinger model 20 Thus, only (0)dχ Z 2q survives. The global symmetries are Z (0)dχ (1)C 2q Z q For q = 2 these are exactly the global symmetries of the bulk.
q-charge Schwinger model Now, we can gauge by allowing fractional fluxes ( t Hooft twists): (0)dχ :ψ e i2π Z 2q Z 2q (0)dχ Then, under : Z q (1)C Consistent with the bulk/wall anomaly inflow (see Erich Poppitz s talk) 21 2q ψ D[ψ ] e i2πt D[ψ ], T = 1 2π D[ψ ] e i 2π k q D[ψ ] F 12 dx2 = k q mixed t Hooft anomaly
q-charge Schwinger model The q-charge Schwinger model is exactly solvable: 22 1. Construct Dirac sea states (gauge fixing, solution of Dirac s equation in a background gauge) Z q (1)C Y q n = n +1 (0)dχ 2. Under :, : 3. The physical states are (cluster decomposition): n Z 2q X 2q n = ω q n,ω q = e i 2π q P,θ 1 q q 1 k=0 ω q kp θ,k, P = 0,1,...,q 1 physical states θ,k = e i(k+qn)θ k + qn, k = 0,1,...,q 1 n! P',θ P,θ = δ P,P' MA, Poppitz, 2018
q-charge Schwinger model Thus, there are q degenerate ground states (they don t communicate with each other) They satisfy the t Hooft algebra 23 Fate of symmetries: (0)dχ Z 2q (1)C Z q X 2q Y q = ω q Y q X 2q 1. is broken:. P,θ ψ + ψ P,θ 0 central extension, sign of anomaly 2. is broken: string tension vanishes.
q-charge Schwinger model The theory is empty in the IR (no massless excitations) 24 The mixed t Hooft anomaly is matched by a TQFT, as is evident from the anomaly inflow. This picture doesn t change for QCD(adj) (adding more adjoint fermions)
Conclusion DW in hot SYM provide the first QFT exactly solvable model to understand the new t Hooft anomalies and the bulk/anomaly inflow. 25 DW worldvolume theory is rich. Center vortices are similar? DW physics calls for lattice confirmation (specially near ) T C