QED3Nathan Seiberg
IAS

Analyzing a Lagrangian QFT
Semiclassical physics, mostly in the UV (reliable,straightforward, but can be subtle)• Global symmetry and its ‘t Hooft anomalies• Weakly coupled limits: flat directions, small
parameters, …
Quantum physics in the IR (mostly conjectural)• Consistency with the global symmetry (including ‘tHooft anomalies) and the various semiclassical limits• Approximate methods: lattice, bootstrap, 𝜖𝜖, 1/𝑁𝑁, …
• Integrability• String constructions2

QED3 [Many references using various methods]
Simple, characteristic example, demonstrating surprisingphenomena. Many applications.
• 𝑈𝑈(1) gauge field 𝑎𝑎𝜇𝜇
• 𝑁𝑁𝑓𝑓 fermions 𝜓𝜓 𝑖𝑖 with charges 𝑞𝑞𝑖𝑖 and masses 𝑚𝑚𝑖𝑖

• A bare Chern-Simons term.
– Label the theory as 𝑈𝑈 1 𝑘𝑘 with a parameter 𝑘𝑘.
– When all the fermions are massive, at low energies
1a TQFT 𝑈𝑈 1 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙 with 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑘𝑘 + ∑𝑖𝑖 sign 𝑚𝑚𝑖𝑖 𝑞𝑞𝑖𝑖2 .
2Since 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙 ∈ ℤ ,1
𝑘𝑘 + � 𝑞𝑞𝑖𝑖2 ∈ ℤ .
23𝑖𝑖

Global symmetries• Charge-conjugation 𝒞𝒞: 𝑎𝑎𝜇𝜇 → −𝑎𝑎𝜇𝜇(with appropriate action on the fermions)• 𝑈𝑈 1 0 , 𝑚𝑚 = 0 time-reversal 𝒯𝒯: 𝑎𝑎0 (𝑡𝑡, 𝑥𝑥) → 𝑎𝑎0 −𝑡𝑡, 𝑥𝑥
𝑎𝑎𝑖𝑖 (𝑡𝑡, 𝑥𝑥) → −𝑎𝑎𝑖𝑖 −𝑡𝑡, 𝑥𝑥(with appropriate action on the fermions)• Standard algebra on all the fundamental fields
𝒯𝒯𝒯𝒯 = 𝒞𝒞𝒞𝒞
𝒯𝒯 2 = −1 𝐹𝐹
𝒞𝒞𝒞𝒞 2 = −1 𝐹𝐹• For equal charges and masses more symmetries, e.g.
𝑆𝑆𝑆𝑆 𝑁𝑁𝑓𝑓 .

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Global magnetic (topological) symmetry• 𝑈𝑈 1 𝑀𝑀symmetry: 𝑗𝑗𝜇𝜇1 𝜇𝜇𝜇𝜇𝜇𝜇= 𝜖𝜖 𝜕𝜕𝜈𝜈 𝑎𝑎𝜌𝜌 .
2𝜋𝜋• The charged operators are monopole operators (likea disorder operator).
– Remove a point from spacetime and specify
boundary conditions around it.
• Massless fermions have zero modes, which can leadto “funny” quantum numbers.

5Global magnetic (topological) symmetry
In many applications, the magnetic symmetry isapproximate or absent• In lattice constructions• When the gauge 𝑈𝑈(1) is embedded at higher energies
in a non-Abelian gauge group• When the gauge 𝑈𝑈 1 is emergent• In the generalization of the gauge 𝑈𝑈 1 ≅ 𝑆𝑆𝑆𝑆 2 to𝑆𝑆𝑆𝑆(𝑁𝑁) with higher 𝑁𝑁 (only a magnetic ℤ2 symmetry).
It is natural to break 𝑈𝑈 1 𝑀𝑀 explicitly by adding amonopole operator to the Lagrangian.

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Example: 𝑈𝑈 1 𝑘𝑘 , 𝑁𝑁𝑓𝑓 = 1 with 𝑞𝑞 = 1

1𝑘𝑘 ∈ ℤ +2
• Since 𝑘𝑘 cannot vanish, 𝒯𝒯 is violated (parity anomaly).
• All gauge invariant polynomials in the fundamental
fields are bosons.
• All gauge invariant monopole operators are bosons.
– 𝑈𝑈 1 1 simplest monopole operator has spin 0
2– 𝑈𝑈 1 3 simplest monopole operator has spin 127

Example: 𝑈𝑈 1 0 , 𝑁𝑁𝑓𝑓 = 2 with 𝑞𝑞 = 1[Cordova, Hsin, NS]• All gauge invariant polynomials in the fundamental fields
are bosons with integer flavor (isospin) 𝑆𝑆𝑆𝑆(2).
• All gauge invariant monopole operators are bosons with𝑀𝑀flavor 𝑆𝑆𝑆𝑆(2) isospin = mod 1.
22• On a single monopole 𝒯𝒯 = −1. More generally,
𝒯𝒯 2 = 𝒞𝒞𝒞𝒞 2 = −1 𝑀𝑀rather than the standard 𝒯𝒯 2 = 𝒞𝒞𝒞𝒞 2 = −1 𝐹𝐹 = +1.
Related, but distinct statements in [Wang, T. Senthil;
Metlitski, Fidkowski, Chen, Vishwanath; Witten] and see also
Hsin’s poster.

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Examples: 𝑈𝑈 1 0 , 𝑁𝑁𝑓𝑓 = 1 with 𝑞𝑞 = 2[Cordova, Hsin, NS]• All gauge invariant polynomials in the fundamental
fields are bosons.
• All gauge invariant monopole operators have𝑀𝑀spin = mod 1, i.e.
2
−1 𝑀𝑀 = −1 𝐹𝐹– In the previous example “fractional” isospin. Here“fractional” spin.
• Standard 𝒯𝒯-symmetry
𝒯𝒯 2 = −1 𝐹𝐹

9Break the magnetic 𝑈𝑈(1) symmetry[Cordova, Hsin, NS; Gomis, Komargodski, NS]• Add to the Lagrangian a charge 2 monopole operator.
– Cannot add a charge 1 monopole – it is a fermion.
• It breaks the– magnetic symmetry 𝑈𝑈 1 𝑀𝑀 → 𝒁𝒁2 : −1 𝑀𝑀– 𝒯𝒯-symmetry• But it preserves another time-reversal symmetry (asubgroup of the original symmetry)
𝑖𝑖𝑖𝑖
𝑀𝑀′2
𝒯𝒯 = 𝒯𝒯𝑒𝑒

10Break the magnetic 𝑈𝑈(1) symmetry[Cordova, Hsin, NS; Gomis, Komargodski, NS]
𝑖𝑖𝑖𝑖
𝑀𝑀′2
𝒯𝒯 = 𝒯𝒯𝑒𝑒• Its algebra is non-standard

𝒯𝒯 ′ 𝒞𝒞 = 𝒞𝒞𝒞𝒞 ′ (−1)𝑀𝑀
𝒞𝒞𝒞𝒞’ 2 = −1 𝑀𝑀 = −1 𝐹𝐹
𝒯𝒯’ 2 = 1 ≠ −1 𝐹𝐹
𝒞𝒞 and 𝒯𝒯𝒯 do not commute.

𝒯𝒯′ is not a conventional time-reversal symmetry, but

𝒞𝒞𝒞𝒞′ is.

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What is the long distance behavior?
• What are the phases?
– Gapped? Topological?
– Symmetry breaking?
• What happens at the phase transitions? First orsecond order?
– And if second order, free or interacting?
• It is clear for large 𝑚𝑚−𝑈𝑈 1 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙

𝑚𝑚 < 0?

𝑈𝑈 1 𝑘𝑘 +𝑙𝑙𝑙𝑙𝑙𝑙

𝑚𝑚 > 0 12

CONJECTURES
AHEAD

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Vary 𝑚𝑚 at large |𝑘𝑘| or at large 𝑁𝑁𝑓𝑓

• Large 𝑘𝑘. Essentially free fermions with a modified
Gauss law constraint• Large 𝑁𝑁𝑓𝑓 . Second-order transition as a function ofthe fermion mass 𝑚𝑚 [Appelquist, Nash, Wijewardhana]• Conjecture that this is the case for all 𝑘𝑘, 𝑞𝑞, and 𝑁𝑁𝑓𝑓 .
𝑈𝑈(1)1𝑘𝑘−2𝑁𝑁𝑓𝑓 𝑞𝑞 2𝑚𝑚 < 0𝑈𝑈(1)1𝑘𝑘+2𝑁𝑁𝑓𝑓 𝑞𝑞 2𝑚𝑚 > 0

Second order transition at 𝑚𝑚 = 014

𝑈𝑈 1 𝑘𝑘 , 𝑁𝑁𝑓𝑓 = 1 with charge one (𝑞𝑞 = 1)

𝑈𝑈 1 1/2 flows to the 𝑂𝑂(2) Wilson-Fisher fixed point[Chen, Fisher, Wu; Barkeshli, McGreevy; NS, T. Senthil, Wang,Witten; Karch, Tong]• The spin-zero monopole operator of the gauge theoryis the order parameter of the 𝑂𝑂(2) model.
• Emergent 𝒯𝒯-symmetry in the IR
𝑈𝑈 1 3/2 flows to a fixed point with 𝑆𝑆𝑆𝑆(3) symmetry[Aharony, Benini, Hsin, NS; Benini, Hsin, NS]• The spin-one monopole operator of the gauge theory
becomes a conserved current in O 2 → 𝑆𝑆𝑆𝑆(3).
• Several dual bosonic and fermionic descriptions15

𝑈𝑈 1 0 , 𝑁𝑁𝑓𝑓 = 2 with 𝑞𝑞 = 1[Xu, You; Karch, Tong; Hsin, NS; Benini, Hsin, NS; Wang,Nahum, Metlitski, Xu, T. Senthil]• Microscopically: global 𝑈𝑈(2), 𝒞𝒞 (they do notcommute), and 𝒯𝒯-symmetry– 𝒯𝒯 and 𝒞𝒞𝒞𝒞 have a non-standard algebra
𝒯𝒯 2 = 𝒞𝒞𝒞𝒞 2 = −1 𝑀𝑀• Conjectured IR behavior: fixed point with enhanced
global 𝑂𝑂(4) symmetry• Dual fermionic description: 𝑈𝑈 1 0 with 𝑁𝑁𝑓𝑓 = 2– Its 𝑆𝑆𝑆𝑆(2) flavor symmetry includes the original
𝑈𝑈 1 𝑀𝑀 , and vice versa.

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𝑈𝑈 1 0 , 𝑁𝑁𝑓𝑓 = 2 with 𝑞𝑞 = 1[Benini, Hsin, NS; Komargodski, NS]
Break the magnetic symmetry by adding to the Lagrangiana double monopole operator• This explicitly breaks the flavor 𝑆𝑆𝑆𝑆 2 → 𝑈𝑈 1 and themagnetic 𝑈𝑈 1 𝑀𝑀 → ℤ2 .
• For some range of 𝑚𝑚 the remaining flavor 𝑈𝑈 1symmetry is spontaneously broken.
– The UV theory is massive, but the IR theory is gapless!
Goldstone boson𝑚𝑚 < 0𝑂𝑂(2) Wilson-Fisher fixed points

𝑚𝑚 > 017

𝑈𝑈 1 0 , 𝑁𝑁𝑓𝑓 = 1 with 𝑞𝑞 = 2[Cordova, Hsin, NS]• Flows to a free Dirac fermion 𝜒𝜒 and a decoupled
𝑈𝑈 1 2 TQFT

– Similar, but not identical to [Son; Wang, T. Senthil;
Metlitski, Vishwanath; NS, T. Senthil, Wang, Witten].

• The monopole operator in the UV becomes the freefermion 𝜒𝜒 in the IR.
• The Wilson line with charge 1 in the UV is describedas the Wilson line of 𝑈𝑈 1 2 (semion).

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Break the magnetic 𝑈𝑈(1) symmetry[Cordova, Hsin, NS; Gomis, Komargodski, NS]

Add to the Lagrangian a charge 2 monopole operator.
• It preserves 𝒯𝒯 ′ = 𝒯𝒯𝑒𝑒𝑖𝑖𝑖𝑖𝑀𝑀
2with a nonstandard algebra• In the IR it splits the fixed point with a massless Diracfermion to two points with massless Majorana fermions
𝑈𝑈 1 2𝑚𝑚 < 0𝑈𝑈 1 2𝑈𝑈 1 2Massless Majorana fermions𝑚𝑚 > 019Summary
QED3: 𝑈𝑈 1 𝑘𝑘 gauge theory with charged fermions.
• Subtle global symmetry, especially in the action on
monopoles– Fractional (global symmetry) charges and spins– Unusual algebras involving 𝒞𝒞 and 𝒯𝒯• Conjectures about the IR behavior of these systems– Enhanced global symmetry– Dual descriptions
This is a tiny part of a long story about the dynamics of2+1 dimensional QFT.
Most of it is still unknown…

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