Insights into Axion Dynamics and Chern-Simons Theory

Axion dynamics and topics in Chern-Simons theory Hajime Fukuda (LBL) References: 2010.02221 HF & K. Yonekura (See also 2010.02763)

Outline Axions are gauge Bosons Chern-Simons term and anomaly inflow Examples: Magnetic monopoles and the Witten effect Axion strings and charges on the string

Axion Definition A scalar Boson with 2? periodicity: ? ? + 2??? e.g. QCD axion =1 ?? 8? ? ?? ? ? ?? ? ?? ? ? 2??2 ??? 4? The theory is indeed invariant under ? ? + 2??? Note: A shift symmetry ? ? + ? is not exact at all. It is approximately required for the QCD axion (quality problem)

Symmetry and redundancy 2? periodicity is not a symmetry, but a redundancy It can be easily seen from the UV completion of QCD axion = ? ?????? + ? ???? where the axion resides like ??? Here, the 2? shift does not change anything. In other words, any physical state must be invariant under the 2? shift (Otherwise, the UV description becomes inconsistent!) ?? 0, ?? ? ?? ?? 0

Redundancy and gauge symmetry Such a redundancy is called a gauge symmetry ; the Boson which nonlinearly transforms under the symmetry is called a gauge Boson In this sense, the 2? shift is a gauge symmetry and the axion is a 0-form gauge Boson. Differential form Let s see a comparison with a familiar example: U(1) 1-form gauge symmetry (QED)

Photon and Axion Photon A 1-form gauge Boson ? = ??? ??? with a gauge transformation ? ? + ? 1 where is an exact 1-form up to 2? Axion A 0-form gauge Boson ? = ?(?) with a gauge transformation ? ? + ?? where is an exact 0-form (=0) up to 2? = + 2? d?, + 2? so that ???/??is invariant so that ??? ? is invariant

Electric & magnetic charge (Photon) The field strength is ? = d? =1 ???? ????d??d?? 2-form 2 Its magnetic dual is ? = d ? ? =1 (D-2) = 2-form 4????????d??d?? ? ? Around electron: 1 ? ? Around monopole: ? 2? ? ? = ?? ? = ??

Electric & magnetic charge (Axion) The field strength is ? = d? = ??? d?? 1-form Its magnetic dual is ? = d ? ? = 1 (D-1) = 3-form 3!???????? d??d??d?? ? (3-dim surface) Around monopole = axion string: 1 2??? Around electron = instanton: ? ? = ?? ? (1-dim surface) ?? ? = ?? ?

Gauge interaction with matter Photon Axion 0-form ? ?? d?? d?d? electron ? ? = ?? instanton ? ? ? 2-form 2? ? ? ? ? 2??? axion string monopole ? ? ? ?,? The interaction is used to estimate the axion emission from strings

Chern-Simons term A non-trivially gauge invariant interaction is the Chern- Simons term ???=2???1 ?? ?! ??1 ??2 ???,????= 2?? normalization: ? , ? 2? ?? is a p-form gauge field (you can think p=0,1, D=4 in practice) ?2 ? ? Chern-Simons term introduces non-trivial deformation on the dual EM fields

Chern-Simons term and duality Without CS Action: With CS (3-terms) Action: ? 2?=?0 Equations: ???= 0 (Bianchi id.) dual e. o. m ? ??= 2 dual Bianchi (no more 0!) 2?+2?? ?0 2?= ?? Equations: ?? = 0 (Bianchi id.) duality ? ? ? 2?? ?? 3! dual e. o. m ???? ? ? = ? ? = 0 (e. o. m.) ?? ??(e. o. m.) dual Bianchi

Modified Bianchi and gauge transformation The dual field strength satisfies modified Bianchi id. d ??=???? ?? ?? 2 The field strength cannot be a total derivative even locally: ??= d ??+???? ?? ?? ? new term is not gauge invariant The dual field receives additional gauge transformation: ????= d ?,?? ??=???? ???? ?

Matter and anomaly Let s add a monopole = dual electron ??????= ?? Now, the problem is clear - ?????? is not gauge invariant. ????????= ???? ??? 2 This is a gauge anomaly (not chiral anomaly) and VERY bad. What s wrong?

Anomaly inflow and matching We want for the anomaly to be cancelled, since we start from a gauge invariant theory. To cancel the anomaly by ??, there must be another source of the anomaly on the world-sheet. ??????= ???? ? = ??????+ ????, ???= ???????? 2 ????= world-sheet action of monopole of ?? Anomaly inflow Callan & Harvey 84

Current on the world-sheet In general, ??gauge tr can be written using a current ??????= d ? ??? where ??? is a gauge current of ?? on ??? Now, due to the anomaly matching, This is covariant anomaly Bardeen & Zumino 84, Naculich 87, HF & Yonekura 20 ????= ?????? The ? monopole world-sheet theory must have a ? current The gauge current must be anomalous up to ??

Ok, lets see examples As a familiar example, let s take the axion-photon system ???=2? 2 ?? ? The rule is The ? monopole world-sheet theory must have a ? current The gauge current must be anomalous up to ?? Take i, j = photon, k = axion

Witten effect revisited The ? monopole world-sheet theory must have a ? current The gauge current must be anomalous up to ?? According to the rule, The monopole world-sheet theory has a photon (=electric) current The electric current is anomalous, d?0= ?? A monopole carries an electric charge which depends on the axion background This is nothing but the Witten effect: A monopole carries an electric charge ? ?? axion string ? = d?0= ? 1 ?0= 1 HF & Yonekura, 20 monopole

Another examples ???=2? 2 ?? ? With the rule: The ? monopole world-sheet theory must have a ? current The gauge current must be anomalous up to ?? Take i = axion, j, k = photon

Charge on the axion string The ? monopole world-sheet theory must have a ? current The gauge current must be anomalous up to ?? According to the rule, The axion string has a photon (=electric) current The electric current is anomalous, d?1= ?? A axion string carries an electric charge & current which depends on the background electric field ??? = ? ? Axion strings are actually superconducting ! (Just like a heterotic string and the Green- Schwarz mechanism) 2?? Witten 85; HF Manohar Murayama Telem20

Summary Axion is a 0-form gauge field The Chern-Simons term introduces anomaly inflow on axion strings and magnetic monopoles Accordingly, Magnetic monopoles obtains electric charge (Witten effect) Axion strings are superconducting