1. with Erich Poppitz (u of Toronto)
't Hooft anomalies, 2-charge Schwinger model, and domain walls in hot super Yang-Mills theory
mohamed Anber
Miami 2018
Based on arxiv: ,
with Erich Poppitz (u of Toronto)
M. Anber, Lewis & Clark College
Miami 2018

2. Outline Overview on domain walls (DW) in hot Super Yang-Mills
‘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
M. Anber, Lewis & Clark College
Miami 2018

3. Overview Why we study DW in hot SYM:
Prototype of DW in other gauge theories: QCD(adj) for model building
Provide a lucid understanding of the new ‘t Hooft anomalies in a weakly coupled setup.
Sannino et al
Anthenodorou, Bennett, Bergner, Lucini 2014
+many more
M. Anber, Lewis & Clark College
Miami 2018

4. Overview
DW in hot SYM (and other YM theories) are the high-T counterparts of center vortices (may play a role in confinement).
The physics of hot YM parallels that of YM on a small spatial circle (see Erich Poppitz’ talk).
see, e.g., Greensite book
MA, Poppitz, Unsal+many others
2008-now
M. Anber, Lewis & Clark College
Miami 2018

5. 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 symmetries were known since the 80s.
‘t Hooft anomalies of 1-form symmetries
non-trivial constraints on the spectrum of a theory.
Gaiotto, Kapustin, Komargodski, Seiberg, 2017
M. Anber, Lewis & Clark College
Miami 2018

6. What is an ‘t Hooft anomaly?
Given a global symmetry of a QFT, we may try to gauge (turn on a background gauge of )
If obstructed (anomalous), the theory has an
‘t Hooft anomaly.
‘t Hooft anomaly is RG invariant, useful for asymptotically free theories.
‘t Hooft, 1980 UV IR
energy scale
M. Anber, Lewis & Clark College
Miami 2018

7. What is an ‘t Hooft anomaly?
The UV/IR matching of the anomaly: composite fermions or Goldstone bosons.
E.g. 4-D QCD with 2 or 3 fundamental flavors.
are global symmetries (act on local operators)
M. Anber, Lewis & Clark College
Miami 2018

8. What is an ‘t Hooft anomaly?
Gauging :
Frishman, Schwimmer, Banks, Yankielowicz 1981 UV IR GB
F quarks
fermions
M. Anber, Lewis & Clark College
Miami 2018

9. What is an ‘t Hooft anomaly?
Given two symmetries , 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.
M. Anber, Lewis & Clark College
Miami 2018

10. What is an ‘t Hooft anomaly?
There are also non-local operators, e.g., line operators (Wilson’s loop)
It transforms under global symmetry:
generalized global symmetries
Gaiotto, Kapustin, Seiber, Willett, 2014 C
confined phase
M. Anber, Lewis & Clark College
Miami 2018

11. What is an ‘t Hooft anomaly?
It was realized that gauging symmetries provides us with extra ‘t Hooft anomalies.
We use the new anomalies to study the physics on the DW in hot super Yang-Mills (SYM).
Gaiotto, Kapustin, Komargodski, Seiberg, 2017
M. Anber, Lewis & Clark College
Miami 2018

12. SYM 4D QCD(adj), including SYM, and symmetries (UV):
The breaking can be understood in the BPST instanton background.
is a discrete chiral symmetry.
M. Anber, Lewis & Clark College
Miami 2018

13. SYM The global symmetries of QCD(adj): We may gauge :
(1) couple QCD(adj) to a TQFT
(2) turn on twisted ‘t Hooft fluxes on
Kapustin, Seiberg 2014
‘t Hooft 1979, Van Baal 1982
M. Anber, Lewis & Clark College
Miami 2018

14. SYM The end result: under we find: Fate of symmetries for SYM at
(DW with Chern-Simons theory)
mixed ‘t Hooft anomaly
M. Anber, Lewis & Clark College
Miami 2018

15. DW in hot SYM Hot SYM: Battacharia, Gocksch, Korthals-Altes, Pisarski
1992 DW
M. Anber, Lewis & Clark College
Miami 2018

16. DW in SYM The fermions are adjoint+anti-PBC. Recall MA, Poppitz, 2018
heavy W-bosons
Matsubara modes
M. Anber, Lewis & Clark College
Miami 2018

17. DW in SYM Two fermion zero modes on the DW.
DW theory (2-D) is axial Schwinger model:
axial with charge 2
M. Anber, Lewis & Clark College
Miami 2018

18. q-charge Schwinger model
We can relabel the fermions (vector-axial duality in 2-D: ).
Thus, charge vector Schwinger model as EFT on the DW:
M. Anber, Lewis & Clark College
Miami 2018

19. q-charge Schwinger model
This theory has a non-trivial topology+exactly solvable via bosonization.
It enjoys two global symmetries:
Under the measure transforms as
q=1 by Manton 1985, Iso, Murayama 1990
quantized
flux on the
2-torus
sum over in path integral
M. Anber, Lewis & Clark College
Miami 2018

20. q-charge Schwinger model
Thus, only survives.
The global symmetries are
For these are exactly the global symmetries of the bulk.
M. Anber, Lewis & Clark College
Miami 2018

21. q-charge Schwinger model
Now, we can gauge by allowing fractional fluxes (‘t Hooft twists):
Then, under :
Consistent with the bulk/wall anomaly inflow (see Erich Poppitz’s talk)
mixed ‘t Hooft anomaly
M. Anber, Lewis & Clark College
Miami 2018

22. q-charge Schwinger model
The q-charge Schwinger model is exactly solvable:
Construct Dirac sea states (gauge fixing, solution of Dirac’s equation in a background gauge)
Under : , :
The physical states are (cluster decomposition):
physical
states
MA, Poppitz, 2018
M. Anber, Lewis & Clark College
Miami 2018

23. q-charge Schwinger model
Thus, there are degenerate ground states (they don’t communicate with each other)
They satisfy the ‘t Hooft algebra
Fate of symmetries:
is broken:
is broken: string tension vanishes.
central extension,
sign of anomaly
M. Anber, Lewis & Clark College
Miami 2018

24. q-charge Schwinger model
The theory is empty in the IR (no massless excitations)
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)
M. Anber, Lewis & Clark College
Miami 2018

25. Conclusion
DW in hot SYM provide the first QFT exactly solvable model to understand the new ‘t Hooft anomalies and the bulk/anomaly inflow.
DW worldvolume theory is rich. Center vortices are similar?
DW physics calls for lattice confirmation (specially near )
M. Anber, Lewis & Clark College
Miami 2018