confinement, de- confinement, andย ยท [kapustin, ns; gaiotto, kapustin, ns, willett] โข make...
TRANSCRIPT
Confinement, De-confinement, and 3๐๐ Topological Quantum Field
Theory
Nathan SeibergIAS
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐Lore
14๐๐2
โซ ๐๐๐๐ ๐น๐น โงโ ๐น๐น +๐๐ ๐๐
8๐๐2โซ Tr(๐น๐น โง ๐น๐น)
โข ๐๐ is 2๐๐-periodic (on a closed manifold)
โข Generic ๐๐
โ Unique vacuum (no TQFT at low energies), gapped spectrum
โ Confinement: ๐๐ = Tr ๐๐๐๐โฎ ๐๐ โผ ๐๐โ๐ด๐ด๐ด๐ด๐ด๐ด๐๐ โ 0 at long distances
โข ๐๐ multiple of ๐๐: ๐๐๐๐ symmetry (equivalently, time-reversal)
โข ๐๐ odd multiple of ๐๐
โ ๐๐๐๐ is spontaneously broken โ 2 vacua
โ Domain walls
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐Additional observables
[Kapustin, NS; Gaiotto, Kapustin, NS, Willett]
โข Study ๐๐๐๐๐๐ ๐๐ bundles that are not ๐๐๐๐ ๐๐ bundles โnontrivial ๐ค๐ค2 of the bundle.
โข Can describe using a โค๐๐ classical background (two-form) gauge field ๐ต๐ต that sets ๐ค๐ค2 = ๐ต๐ต.
โข ๐ต๐ต can be interpreted as a classical background gauge field of a โค๐๐ one-form global symmetry. More below.
โข This is not a ๐๐๐๐๐๐(๐๐) gauge theory, where ๐ต๐ต is summed over. More below.
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐The operators [โฆ; Kapustin, NS]
โข Wilson lines ๐๐ ๐ถ๐ถ = Tr(๐๐๐๐ โฎ๐ถ๐ถ ๐๐) ๐๐๐๐ โซฮฃ ๐ต๐ต with ๐ถ๐ถ = ๐๐ฮฃ
โข Charges: topological surface operators ๐๐๐ธ๐ธ ๐๐ = ๐๐๐๐ โฎ๐๐ ๐ข๐ข ๐๐
โข The โt Hooft operator ๐๐ is not a genuine line operator.โข Since a Wilson line can detect the Dirac string emanating
from the โt Hooft operator, the Dirac string is visible and sweeps a surface ฮฃ
๐๐ ๐ถ๐ถ ๐๐๐๐ โซฮฃ ๐ข๐ข(๐๐)
It is an open version of ๐๐๐ธ๐ธ. โข ๐๐๐ธ๐ธ can be interpreted as the worldsheet of a Dirac string.
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐๐ต๐ต-dependent counterterm
[Kapustin, NS; Gaiotto, Kapustin, NS, Willett; Gaiotto, Kapustin, Komargodski, NS]
โข Can add to the action a counterterm in the classical fields: 2๐๐ ๐๐ ๐๐2๐๐
โซ ๐๐ ๐ต๐ต
๐๐ = 1,2, โฆ ,2๐๐, ๐๐๐๐ โ 2โค (๐๐ is the Pontryagin square)โข For nonzero ๐ต๐ต there are fractional instantons and therefore ๐๐
is not 2๐๐-periodic. Instead, ๐๐,๐๐ โผ ๐๐ + 2๐๐,๐๐ + ๐๐ โ 1โข ๐๐ โ ๐๐ + 2๐๐ leads to different theories
โ Different contact termsโ Different behavior on boundaries
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐๐๐๐๐ at ๐๐ = ๐๐ [Gaiotto, Kapustin, Komargodski, NS]
๐๐ ๐๐8๐๐2
โซ Tr ๐น๐น โง ๐น๐น +2๐๐ ๐๐ ๐๐2๐๐
โซ ๐๐ ๐ต๐ต๐๐,๐๐ โผ ๐๐ + 2๐๐,๐๐ + ๐๐ โ 1
๐๐๐๐: ๐๐,๐๐ โ โ๐๐,โ๐๐ โผ (๐๐,โ๐๐ + ๐๐ โ 1)โข For ๐๐ even, no value of ๐๐ preserves the symmetry โ mixed
anomaly between ๐๐๐๐ and the โค๐๐ one-form symmetryโข ๐ถ๐ถ๐๐ is spontaneously broken with 2 vacua โข Nontrivial domain wall between the two vacua at ๐๐ = ๐๐
โ Anomaly inflow from the bulkโ ๐๐๐๐ ๐๐ 1 Chern-Simons theory on the wall.
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๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface [Gaiotto, Kapustin, Komargodski, NS]
More generally, consider a space-dependent ๐๐ interpolating between ๐๐ = 0 and ๐๐ = 2๐๐๐๐ for some integer ๐๐โข If the interpolation is very slow (compared with the
confinement scale of the theory), we have ๐๐ copies of ๐๐๐๐ ๐๐ 1
localized where ๐๐ crosses an odd multiple of ๐๐.
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๐๐ = 2๐๐๐๐
๐๐๐๐ ๐๐ 1 โ ๐๐๐๐ ๐๐ 1 โโฏ
๐๐ = 0
โข If the interpolation is fast compared with the confinement scale of the theory, we can have another TQFT ๐ฏ๐ฏ๐๐.
โข Need better dynamical control to determine ๐ฏ๐ฏ๐๐. One option is ๐ฏ๐ฏ๐๐ = ๐๐๐๐ ๐๐ ๐๐. This is consistent with the anomaly flow from the bulk.
โข If the interface is sharp (discontinuous), the result is not universal, but it must have the same anomaly.
๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface
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๐๐ = 0
๐๐ = 2๐๐๐๐
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
โข Can think of it also as separating regions with the same ๐๐, but different values of ๐๐.
๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface
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๐๐ = 0๐๐โ
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐(๐๐ โ 1)
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
๐๐ = 0
๐๐ = 2๐๐๐๐
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
โข On one side of the interface monopoles condense and lead to confinement. On the other side dyons condense and lead to โoblique confinement.โ
โข By continuity, none of them condense on the interface and hence no confinement there.โ The ๐๐๐๐ ๐๐ ๐๐ Wilson lines are Wilson lines of the
microscopic gauge theory.โ Surprise: they have nontrivial braiding โ probe quarks are
anyons.
๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface
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๐๐ = 0๐๐โ
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐(๐๐ โ 1)
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
โ Consider a โค๐๐ charge operator ๐๐๐ธ๐ธ ๐๐ = ๐๐๐๐ โฎ๐๐ ๐ข๐ข ๐๐ that pierces the interface.
โ We interpreted it as the worldsheet of a Dirac string.โ It is associated with a monopole on one side and a dyon on
the other side. Therefore, it has electric charge ๐๐ on the wall. It is a Wilson line there.
โ This explains why there is braiding between Wilson lines on the wall.
๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface [Hsin, Lam, NS]
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๐๐ = 0๐๐โ
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐(๐๐ โ 1)
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
๐๐๐๐๐๐(๐๐) pure gauge theory in 4๐๐Gauge the โค๐๐ one-form global symmetry
[Kapustin, NS; Gaiotto, Kapustin, NS, Willett]
โข Make ๐ต๐ต dynamical and denote it by ๐๐. This amounts to summing over ๐ค๐ค2 of the ๐๐๐๐๐๐(๐๐) bundles
โข The Wilson line is not a genuine line operator ๐๐ ๐ถ๐ถ, ฮฃ = Tr(๐๐๐๐ โฎ๐ถ๐ถ ๐๐) ๐๐๐๐ โซฮฃ ๐๐
with ๐ถ๐ถ = ๐๐ฮฃ
โข The correlation functions of ๐๐๐ธ๐ธ ๐๐ = ๐๐๐๐ โฎ๐๐ ๐ข๐ข ๐๐ are trivial.โข For ๐๐ = 0 the โt Hooft operator is a genuine line operator
๐๐ ๐ถ๐ถ ๐๐๐๐ โซฮฃ ๐ข๐ข(๐๐)
because there is no dependence on ฮฃ. (For other values of ๐๐it should be multiplied by a Wilson line.)
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๐๐๐๐๐๐(๐๐) pure gauge theory in 4๐๐Gauge the โค๐๐ one-form global symmetry
โข New surface operator ๐๐๐๐ = ๐๐๐๐ โฎ๐๐ ๐๐ = ei โฎ๐๐ ๐ค๐ค2
โข It generates a magnetic โค๐๐ one-form symmetry
โข ๐๐๐๐ ๐๐ ๐๐ ๐ถ๐ถ = ๐๐2๐๐๐๐๐๐ ๐๐,๐ถ๐ถ ๐๐ ๐ถ๐ถ
๐๐,๐ถ๐ถ the linking number.
โข The Wilson line ๐๐ ๐ถ๐ถ, ฮฃ = Tr(๐๐๐๐ โฎ๐ถ๐ถ ๐๐) ๐๐๐๐ โซฮฃ ๐๐ is an open version of ๐๐๐๐
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๐๐๐๐๐๐(๐๐) pure gauge theory in 4๐๐Dynamics
[Aharony, Tachikawa, NS; Kapustin, NS; Gaiotto, Kapustin, NS, Willett]
๐๐ ๐๐8๐๐2
โซ Tr ๐น๐น โง ๐น๐น +2๐๐ ๐๐ ๐๐2๐๐
โซ ๐๐ ๐๐๐๐,๐๐ โผ ๐๐ + 2๐๐,๐๐ + ๐๐ โ 1
Now the lack of 2๐๐-periodicity in ๐๐ is more important.โข For ๐๐ = 0 monopoles condense, the โt Hooft operator has a
perimeter lawโข Gapped spectrum, but a nontrivial TQFT at low energies
(not merely an SPT phase) โ a โค๐๐ gauge theoryโข More generally, the low energy theory is a โค๐ฟ๐ฟ gauge theory
(could be twisted on nonspin manifolds) with๐ฟ๐ฟ = gcd ๐๐,๐๐
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In the ๐๐๐๐(๐๐) theory
In the ๐๐๐๐๐๐(๐๐) theory
๐ฟ๐ฟยฑ = gcd ๐๐ยฑ,๐๐
Cannot have ๐ฏ๐ฏ๐๐โค๐๐
= ๐๐๐๐๐๐ ๐๐ ๐๐? on the interface โ it is not consistent!
๐๐๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface
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๐๐ = 0๐๐โ
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐ ๐๐ โ 1
๐ฏ๐ฏ๐๐ (= ๐๐๐๐ ๐๐ ๐๐? )
๐๐ = 0๐๐โ
โค๐ฟ๐ฟโ gauge theory
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐ ๐๐ โ 1โค๐ฟ๐ฟ+ gauge theory
? ? ?
๐ฟ๐ฟยฑ = gcd ๐๐ยฑ,๐๐In order to figure out the theory on the interface we need to understand better the one-form global symmetry, its anomaly, and its gauging.
๐๐๐๐๐๐(๐๐) pure gauge theory in 4๐๐Interface
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๐๐ = 0๐๐โ
โค๐ฟ๐ฟโ gauge theory
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐ ๐๐ โ 1โค๐ฟ๐ฟ+ gauge theory
? ? ?
There are line operators ๐๐๐๐ with ๐๐ integer modulo ๐๐ such that
โข ๐๐๐๐๐๐๐๐โฒ = ๐๐๐๐+๐๐โ
โข Any line ๐๐ โ ๐ฏ๐ฏ has a โค๐๐ charge ๐๐ W (โค๐๐ representation)
= ๐๐2๐๐๐๐๐๐ ๐๐ ๐๐
๐๐
U๐๐
๐๐ ๐๐
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry [Gaiotto, Kapustin, NS, Willett]
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=
๐๐๐๐ ๐๐๐๐โฒ U๐๐+๐๐โฒ
Examplesโข ๐๐๐๐ ๐๐ ๐๐ has a โค๐๐ one-form symmetry associated with the
center of the gauge group. It is generated by a line in a representation of ๐๐ symmetric fundamentals.
โข ๐๐ 1 ๐๐ has ๐๐ lines (for ๐๐ even) realizing a โค๐๐ one-form symmetry, generated by the line of charge one.
โข โค๐๐ gauge theory has a โค๐๐ โ โค๐๐ one-form symmetry (electric and magnetic)
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry [Gaiotto, Kapustin, NS, Willett]
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Consistency implies that the spins of the charge lines are
โ ๐๐๐๐ =๐๐ ๐๐2
2๐๐mod 1
Here ๐๐ = 0,1, โฆ ,2๐๐, ๐๐๐๐ โ 2โค. (In a spin TQFT ignore the second condition and ๐๐ โผ ๐๐ + ๐๐. By changing the โค๐๐ generator we can relate different values of ๐๐.)
Using the braiding we interpret ๐๐ mod ๐๐ = ๐๐ ๐๐1
as the โค๐๐ charge of the generator of the โค๐๐ symmetry.If ๐๐ โ 0, we cannot gauge the one-form symmetry. ๐๐ characterizes the โt Hooft anomaly of the symmetry.
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry [Gomis, Komargodski, NS; Hsin, Lam, NS]
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โ ๐๐๐๐ =๐๐ ๐๐2
2๐๐mod 1
๐๐ characterizes the โt Hooft anomaly of the symmetry.Coupling to a background โค๐๐ gauge field ๐ต๐ต the corresponding anomaly is our 4๐๐ bulk term
2๐๐ ๐๐ ๐๐2๐๐
โซ ๐๐ ๐ต๐ต
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry
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โ ๐๐๐๐ =๐๐ ๐๐2
2๐๐mod 1
For ๐๐ = 0โข the lines ๐๐๐๐ are โค๐๐ neutral โข their spins vanish modulo an integerโข their braiding is trivialโข there is no โt Hooft anomalyโข We can gauge the symmetry
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry with ๐๐ = 0
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For ๐๐ = 0 we can gauge the symmetry (known in the condensed matter literature as โanyon condensationโ) โข Remove from ๐ฏ๐ฏ all the โค๐๐ charged lines (๐๐ ๐๐ โ 0 mod ๐๐)โข Identify the lines ๐๐ โผ ๐๐1๐๐โข If a line ๐๐ is the same as ๐๐1๐๐, it appears multiple times
For our problem with the interface in ๐๐๐๐๐๐(๐๐) we need to gauge a TQFT with nonzero ๐๐.
Gauge the โค๐๐ one-form global symmetry when ๐๐ = 0 [Moore, NS]
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โ ๐๐๐๐ =๐๐ ๐๐2
2๐๐mod 1
For ๐๐ = ๐๐ we can use essentially the standard gauging (except the identification) to find a spin TQFT.
For ๐ฟ๐ฟ = gcd ๐๐,๐๐ โ ๐๐ the lines ๐๐ = ๐๐๐ฟ๐ฟ๐๐ lead to a โค๐ฟ๐ฟ โ โค๐๐
subgroup, whose ๐๐ is ๏ฟฝฬ๏ฟฝ๐ = ๐๐๐๐๐ฟ๐ฟ
= 0 mod ๐ฟ๐ฟ. It can be gauged as above. The resulting theory has โค๐๐โฒ one-form symmetry with anomaly ๐๐โฒwith ๐๐โฒ = ๐๐/๐ฟ๐ฟ, ๐๐โฒ = ๐๐/๐ฟ๐ฟ and hence ๐ฟ๐ฟโฒ = gcd ๐๐โฒ,๐๐โฒ = 1.
So how should we deal with ๐ฟ๐ฟ = gcd ๐๐,๐๐ = 1?
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry with ๐๐ โ 0
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โ ๐๐๐๐ =๐๐ ๐๐2
2๐๐mod 1
For every ๐๐ โ ๐ฏ๐ฏ with charge ๐๐ ๐๐ the line ๐๐1๐ด๐ด๐๐ = Wโฒ (need to show that ๐๐๐ is unique) with ๐๐๐๐ + ๐๐(๐๐) = 0 mod ๐๐ is โค๐๐ neutral. โข Hence, ๐ฏ๐ฏ = ๐ฏ๐ฏโฒ โ๐๐๐๐,๐๐
โ ๐ฏ๐ฏโฒ includes all the neutral lines ๐๐โฒ
โ ๐๐๐๐,๐๐ is a minimal TQFT with โค๐๐ symmetry with anomaly ๐๐.โ The factorization is also guaranteed by a theorem of [Muger;
Drinfeld, Gelaki, Nikshych, Ostrik]
โข This is quite surprising. All the information about the symmetry is in a decoupled universal sector ๐๐๐๐,๐๐!
A 3๐๐ TQFT ๐ฏ๐ฏ with a โค๐๐ one-form global symmetry with ๐ฟ๐ฟ = gcd(๐๐,๐๐) = 1 [Hsin, Lam, NS]
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Examplesโข ๐๐ 1 ๐๐ = ๐๐๐๐,1
โข ๐๐๐๐ ๐๐ 1 = ๐๐๐๐,๐๐โ1
โข ๐๐ 1 ๐๐๐๐ = ๐๐๐๐,๐๐ โ๐๐๐๐,๐๐ for gcd ๐๐,๐๐ = 1 (this generalizes โค๐๐๐๐ = โค๐๐ โ โค๐๐)
๐๐๐๐,๐๐ โ๐๐๐๐,โ๐๐ has ๐๐2 lines with an anomaly free diagonal โค๐๐one-form symmetry. Gauging it leads to (๐๐๐๐,๐๐โ๐๐๐๐,โ๐๐)/โค๐๐, which is a trivial theory.
The minimal 3๐๐ TQFT with a โค๐๐ one-form global symmetry with anomaly ๐๐ with gcd ๐๐,๐๐ = 1
๐๐๐๐,๐๐ [Moore, NS; Hsin, Lam, NS]
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Starting with a theory ๐ฏ๐ฏ with a โค๐๐ one-form symmetry with anomaly ๐๐ such that gcd ๐๐,๐๐ = 1, we cannot gauge the symmetry.However, since
๐ฏ๐ฏ = ๐ฏ๐ฏโฒ โ๐๐๐๐,๐๐
we can extract ๐ฏ๐ฏโฒeither by dropping ๐๐๐๐,๐๐, or by tensoringanother factor and then gauging an anomaly free โค๐๐ symmetry
๐ฏ๐ฏ๐ = (๐ฏ๐ฏ โ๐๐๐๐,โ๐๐)/โค๐๐Since ๐๐๐๐,๐๐ is minimal, this procedure is canonical.
Using ๐๐๐๐,๐๐ [Hsin, Lam, NS]
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๐ฟ๐ฟยฑ = gcd ๐๐ยฑ,๐๐
For simplicity, let ๐ฟ๐ฟยฑ = 1.Then the bulk theories are trivial and there must be a 3d TQFT on the interface.
(The analysis for generic ๐ฟ๐ฟยฑ is more subtle and is explained in the paper.)
Back to the interface in 4๐๐ ๐๐๐๐๐๐(๐๐)[Hsin, Lam, NS]
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๐๐ = 0๐๐โ
โค๐ฟ๐ฟโ gauge theory
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐ ๐๐ โ 1โค๐ฟ๐ฟ+ gauge theory
? ? ?
For simplicity, let ๐ฟ๐ฟยฑ = gcd ๐๐ยฑ,๐๐ = 1.
๐ฏ๐ฏ๐๐ โ๐ฏ๐ฏ๐๐ โ๐๐๐๐,โ๐๐โ โ๐๐๐๐,๐๐+
โค๐๐Since ๐๐+ โ ๐๐โ = โ๐๐ ๐๐ โ 1 = โ ๐๐ โค๐๐ โ ๐ฏ๐ฏ๐๐ , the diagonal โค๐๐ in the numerator is anomaly free and can be gauged.
Can interpret ๐๐๐๐,โ๐๐โ โ๐๐๐๐,๐๐+as arising from the bulk on the left and the right such that we can perform the gauging.
Back to the interface in 4๐๐ ๐๐๐๐๐๐(๐๐)[Hsin, Lam, NS]
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๐๐ = 0๐๐โ
๐๐ = 0๐๐+ = ๐๐โ โ ๐๐ ๐๐ โ 1
? ? ?
4d ๐๐๐๐(๐๐) gauge theory โข For generic ๐๐ the spectrum is gapped with a trivial
lowโenergy theory and at ๐๐ = ๐๐ there are two vacua.โข โค๐๐ one form global symmetry
โ It is unbroken (the theory is confining) โ We can couple the theory to a background two-form โค๐๐
gauge field ๐ต๐ต and add a counterterm 2๐๐ ๐๐ ๐๐2๐๐
โซ ๐๐ ๐ต๐ต
โ Keeping track of this term, ๐๐ is 2๐๐๐๐-periodic (4๐๐๐๐-periodic for even ๐๐ on a non-spin manifold).
โข Steep interface from ๐๐ = 0 to ๐๐ = 2๐๐๐๐ has a TQFT (e.g. ๐๐๐๐ ๐๐ ๐๐) on it
Conclusions
29
4d ๐๐๐๐๐๐ ๐๐ gauge theory is obtained by gauging the โค๐๐ one-form global symmetry of the ๐๐๐๐(๐๐) theory.โข The low energy theory is a โค๐ฟ๐ฟ gauge theory with ๐ฟ๐ฟ =
gcd ๐๐,๐๐โข Interfaces have more subtle TQFTs on them
Conclusions
30
A 3d TQFT with a โค๐๐ one-form global symmetry โข It is characterized by an integer ๐๐ mod 2๐๐, which determines
โ The charge of the generating lineโ The spins of the charge linesโ The โt Hooft anomaly
โข For gcd ๐๐,๐๐ = 1 there is a minimal TQFT with โค๐๐ one-form symmetry and anomaly ๐๐, ๐๐๐๐,๐๐
โ Any theory ๐ฏ๐ฏ with a โค๐๐ one-form symmetry with anomaly ๐๐, such that gcd ๐๐,๐๐ = 1, factorizes ๐ฏ๐ฏ = ๐ฏ๐ฏโฒ โ๐๐๐๐,๐๐
Conclusions
31