Anomaly of the Electromagnetic Duality of Maxwell Theory $ N # - PowerPoint PPT Presentation

Anomaly of the Electromagnetic Duality of Maxwell Theory $ N # − + # S $ Chang-Tse Hsieh Kavli IPMU & Institute for Solid States Physics, Univ of Tokyo East Asian String Webinar May 29, 2020

Collaboration Yuji Tachikawa Kazuya Yonekura (Kavli IPMU) (Tohoku Univ) CTH-Tachikawa-Yonekura, arXiv: 1905.08943 , PRL 123, 161601 (2019) CTH-Tachikawa-Yonekura, arXiv: 2003.11550

Outline • Maxwell theory × EM duality × Anomaly • Anomalies: self-dual fields vs. chiral fermions 2d 1. 4d 2. 6d 3. • Summary

Outline • Maxwell theory × EM duality × Anomaly • Anomalies: self-dual fields vs. chiral fermions 2d 1. 4d 2. 6d 3. • Summary

Maxwell theory and EM duality From Jackson, we know 2020 now

Maxwell theory and EM duality p Z 0 = µ 0 / ✏ 0 (in SI units) or 1 (in Gaussian units) and

Maxwell theory and EM duality • Classical Maxwell theory: SO(2) duality symmetry (It can be extended to a larger symm, e.g. SL(2, ℝ ) [Gaillard-Zumino (81] ) • Quantum mechanically, we know the electric and magnetic charges must obey the Dirac quantization condition q m qm = 2 ⇡ n ~ , n 2 Z or more generally, the Dirac-Zwanziger-Schwinger quantization condition ( q 2 ,m 2 ) ( q 1 ,m 1 ) q 1 m 2 � q 2 m 1 = 2 ⇡ n ~ , n 2 Z dyons

Maxwell theory and EM duality A heuristic derivation ( not Dirac’s original argument in 1931) : [Jackson, Ch. 6.12] L em = 1 Z angular momentum ~ x × ( ~ E × ~ H ) d 3 x R 3 ~ of the EM field: c 2 m q ~ q r ~ m r 0 ~ ~ E = H = r 3 4 ⇡✏ 0 4 ⇡ µ 0 r 0 3

Maxwell theory and EM duality A heuristic derivation ( not Dirac’s original argument in 1931) : [Jackson, Ch. 6.12] L em = 1 Z angular momentum ~ x × ( ~ E × ~ H ) d 3 x R 3 ~ of the EM field: c 2 m q ~ R ~ L em = qm R ~ 4 ⇡ R ~ q r ~ m r 0 ~ ~ E = H = r 3 4 ⇡✏ 0 4 ⇡ µ 0 r 0 3

Maxwell theory and EM duality A heuristic derivation ( not Dirac’s original argument in 1931) : [Jackson, Ch. 6.12] L em = 1 Z angular momentum ~ x × ( ~ E × ~ H ) d 3 x R 3 ~ of the EM field: c 2 ( q 1 ,m 1 ) ( q 2 ,m 2 ) ~ R ⌘ ~ ⇣ q 1 m 2 − m 1 q 2 R ~ L em = 4 ⇡ 4 ⇡ R ~ q r ~ m r 0 ~ ~ E = H = r 3 4 ⇡✏ 0 4 ⇡ µ 0 r 0 3

Maxwell theory and EM duality A heuristic derivation ( not Dirac’s original argument in 1931) : [Jackson, Ch. 6.12] L em = 1 Z angular momentum ~ x × ( ~ E × ~ H ) d 3 x R 3 ~ of the EM field: c 2 ( q 1 ,m 1 ) ( q 2 ,m 2 ) ~ R ⌘ ~ ⇣ q 1 m 2 − m 1 q 2 R ~ L em = 4 ⇡ 4 ⇡ R n ∈ Z ✓ R = 1 But QM tells us ( L em ) ˆ 2 n ~ ,

Maxwell theory and EM duality The quantum EM duality must preserve the charge quan. cond. ✓ q 1 ◆ q 2 q 1 m 2 − q 2 m 1 = det = 2 ⇡ n ~ , n ∈ Z m 1 m 2 and thus is represented by an SL(2, ℤ ) group, namely ✓ q ◆ ✓ q ◆ ✓ a b ◆ ✓ a b ◆ → m c d m ∈ SL(2 , Z ) c d ! ◆ ! i.e. ~ ~ a, b, c, d ∈ Z ✓ a b E E → ad − bc = 1 ~ ~ c d H H

Maxwell theory and EM duality The quantum EM duality must preserve the charge quan. cond. ✓ q 1 ◆ q 2 q 1 m 2 − q 2 m 1 = det = 2 ⇡ n ~ , n ∈ Z m 1 m 2 and thus is represented by an SL(2, ℤ ) group, namely ✓ 0 ◆ 1 standard EM duality (S-duality) S : � 1 0 generators ✓ 1 1 ◆ Witten effect (2 " shift of top. # −term) T : 0 1

More on Maxwell theory In ordinary Maxwell theory, there is no relation btw charges ( q , m ) and spin/statistics. We could have both neutral boson and fermion as the “origin” of the charge lattice Open circles : primitive vectors, representing single-particle states (SL(2, ℤ ) invariant)

More on Maxwell theory In ordinary Maxwell theory, there is no relation btw charges ( q , m ) and spin/statistics. We could have both neutral boson and fermion as the “origin” of the charge lattice Open circles : primitive vectors, representing single-particle states (SL(2, ℤ ) invariant) ( q =1, m =0) and ( q =0, m =1) span the charge lattice

More on Maxwell theory In ordinary Maxwell theory, there is no relation btw charges ( q , m ) and spin/statistics. We could have both neutral boson and fermion as the “origin” of the charge lattice Open circles : primitive vectors, ✓ � 1 ◆ 0 C : representing single-particle 0 � 1 states (SL(2, ℤ ) invariant) C ( q =-1, m =0) and ( q =0, m =-1) span the charge lattice

More on Maxwell theory In ordinary Maxwell theory, there is no relation btw charges ( q , m ) and spin/statistics. We could have both neutral boson and fermion as the “origin” of the charge lattice Open circles : primitive vectors, ✓ � 1 ◆ 0 C : representing single-particle 0 � 1 states (SL(2, ℤ ) invariant) ✓ 0 ◆ 1 S S : � 1 0 ( q =0, m =-1) and ( q =1, m =0) span the charge lattice

More on Maxwell theory In ordinary Maxwell theory, there is no relation btw charges ( q , m ) and spin/statistics. We could have both neutral boson and fermion as the “origin” of the charge lattice Open circles : primitive vectors, ✓ � 1 ◆ 0 C : representing single-particle 0 � 1 states (SL(2, ℤ ) invariant) ✓ 0 ◆ 1 T S : � 1 0 ( q =1, m =0) and ( q =1, m =1) span the charge lattice ✓ 1 1 ◆ T : 0 1

More on Maxwell theory Now let’s image a world where neutral particles are bosons. Then the charge lattice can be associated w/ various “charge-spin relations” , resulting extra four versions of Maxwell theory. Maxwell IV Maxwell I Maxwell II Maxwell III b b b b b b b b b b b b b b f b f b f b f f b f b f b f f b f b f b f b f b f b f b b b b b b b b f f f f f f f b b b b b b b b b b b b b b f b f b f b f f b f b f b f b f b f b f b f b f b f b f b b b b b b b f f f f f f f b b b b b b b b b b b b b b f b f b f b f f b f b f b f (1, 0): b (1, 0): b (1, 0): f (1, 0): f (0, 1): b (0, 1): f (0, 1): b (0, 1): f (1, 1): f (1, 1): b (1, 1): b (1, 1): f

More on Maxwell theory Now let’s image a world where neutral particles are bosons. Then the charge lattice can be associated w/ various “charge-spin relations” , resulting extra four versions of Maxwell theory. Maxwell IV Maxwell I Maxwell II Maxwell III = all-fermion electrodynamics b b b b b b b b b b b b b b f b f b f b f f b f b f b f f b f b f b f b f b f b f b b b b b b b b f f f f f f f b b b b b b b b b b b b b b f b f b f b f f b f b f b f b f b f b f b f b f b f b f b b b b b b b f f f f f f f b b b b b b b b b b b b b b f b f b f b f f b f b f b f S T (1, 0): b (1, 0): b (1, 0): f (1, 0): f T S (0, 1): b (0, 1): f (0, 1): b (0, 1): f (1, 1): f (1, 1): b (1, 1): b (1, 1): f permuted by SL(2, ℤ ) SL(2, ℤ ) invariant

All-fermion electrodynamics • It seems natural to have these modified Maxwell theories • However, Maxwell IV = all-ferm ED is anomalous (while the other three are not), in the sense that it cannot exist in purely 4d if microscopic DOF are only bosons [Wang-Potter-Senthil (13); Kravec- McGreevy-Swingle (14); Thorngren (14); Wang-Wen-Witten (18)] Such a theory (in the absence of neutral fermions) Ø does not have a bosonic regulator (e.g. 4d U(1) lattice gauge theory) Ø does not have a well-defined part. func. on some spacetime (e.g. CP 2 ) CP 2 Ø is the IR theory of some anomalous UV theory (e.g. ferm of isospin 4" + 3/2 w/ a refined SU(2) anomaly) R M 5 w 2 w 3 Ø must live on the boundary of a 5d bulk (w/ part. func. ) ( � 1)

Symmetries of Maxwell theory Symmetries of Maxwell theory might also be anomalous, and we’d like to know which symm is anomaly-free and thus can be gauged. Ø One example is to consider Maxwell theory w/ extra dynamical gauge fields, e.g. Alice electrodynamics [Schwarz (82)] a “Alice string/loop” with the “Cheshire charge” [Wilczek et al (90)] Alice EM U(1) o Z 2 = O(2) “local” charge conjugation along C q, m ! � q, � m � Fig from [Bucher-Lo-Preskill (92)]

Symmetries of Maxwell theory Symmetries of Maxwell theory might also be anomalous, and we’d like to know which symm is anomaly-free and thus can be gauged. Ø Another example is to consider the Janus configuration [Bak et al. (92); Giaotto-Witten (08)] where the spacetime has a duality twist E ( x + L, y, z ) = B ( x, y, z ) B ( x + L, y, z ) = � E ( x, y, z ) [Ganor et al (08 ,10, 12, 14)] Janus from wiki: God of beginnings, gates, transitions, time, duality, doorways, passages, and ending