Discrete symmetries, t Hooft anomalies, and adiabatic continuity

Transcription

1 iscrete symmetries, t Hooft nomlies, nd dibtic continuity Erich Poppitz ORONTO An overview nd some newer results. Work with Mohmed Anber Lewis & Clrk College, Portlnd no time for [hep-th] [hep-th]

2 QFTs, nonperturbtively, re hrd to del with. Exct results: SUSY, often extended but rel world: SM, BSM? Lttice: globl chirl symmetries-hrd guged chirl symmetries-confusion reigns! ny new nlytic (trustble!) pproch should be met with excitement nd studied!

3 min messge of this tlk - combine two new (-ish) pproches Around 007, Unsl relized tht lrge clss of 4d ories cn be nlyticlly studied using nonperturbtive semiclssicl mens if compctified on smll circle: NL 1 number of colors mny studies over pst 10 yers incomplete lphbeticl list Aitken Anber Argyres Bergner Chermn Li Knzw Misumi Piemonte EP Simic Schefer Shifmn Shuryk Sulejmnpsic Tnizki Thoms Virhinos Voloshin Unsl Yffe Zhitnitsky

4 min messge of this tlk - combine two new (-ish) pproches Around 007, Unsl relized tht lrge clss of 4d ories cn be nlyticlly studied using nonperturbtive semiclssicl mens if compctified on smll circle: NL 1 number of colors mny studies over pst 10 yers incomplete lphbeticl list Aitken Anber Argyres Bergner Chermn Li Knzw Misumi Piemonte EP Simic Schefer Shifmn Shuryk Sulejmnpsic Tnizki Thoms Virhinos Voloshin Unsl Yffe Zhitnitsky Mechnism of confinement: belin, loclly 4d generliztion of Polykov s 77 but inherits much of 4d! [eg nomlies I ll tlk bout!] Much closer to rel world YM thn Seiberg-Witten confinement, or ory with clculble Abelin confinement! - ls not this tlk - Anber, Sulejmnpsic, EP 015 Shlchin, EP 017

5 min messge of this tlk - combine two new (-ish) pproches Around 007, Unsl relized tht lrge clss of 4d ories cn be nlyticlly studied using nonperturbtive semiclssicl mens if compctified on smll circle: NL 1 number of colors mny studies over pst 10 yers incomplete lphbeticl list Aitken Anber Argyres Bergner Chermn Li Knzw Misumi Piemonte EP Simic Schefer Shifmn Shuryk Sulejmnpsic Tnizki Thoms Virhinos Voloshin Unsl Yffe Zhitnitsky Mechnism of confinement: belin, loclly 4d generliztion of Polykov s 77 but inherits much of 4d! [eg nomlies I ll tlk bout!] Much closer to rel world YM thn Seiberg-Witten confinement, or ory with clculble Abelin confinement! - ls not this tlk - Anber, Sulejmnpsic, EP 015 Shlchin, EP 017 Mny or results interesting on ir own (rise mny new properties nd issues in QFTs), but when do (nd which of) results extend to R^4?

6 min messge of this tlk - combine two new (-ish) pproches Around 007, Unsl relized tht lrge clss of 4d ories cn be nlyticlly studied using nonperturbtive semiclssicl mens if compctified on smll circle: NL 1 number of colors Around , Giotto, Kpustin, Komrgodski & Seiberg relized tht guging discrete symmetries, including ones not visible in perturbtive continuum formultion of guge ories, cn led to new constrints on IR behviour: discrete t Hooft nomly mtching mny studies over pst 10 yers incomplete lphbeticl list Aitken Anber Argyres Bergner Chermn Li Knzw Misumi Piemonte EP Simic Schefer Shifmn Shuryk Sulejmnpsic Tnizki Thoms Virhinos Voloshin Unsl Yffe Zhitnitsky Mny or results interesting on ir own (rise mny new properties nd issues in QFTs), but when do (nd which of) results extend to R^4?

7 min messge of this tlk - combine two new (-ish) pproches Around 007, Unsl relized tht lrge clss of 4d ories cn be nlyticlly studied using nonperturbtive semiclssicl mens if compctified on smll circle: NL 1 number of colors mny studies over pst 10 yers incomplete lphbeticl list Aitken Anber Argyres Bergner Chermn Li Knzw Misumi Piemonte EP Simic Schefer Shifmn Shuryk Sulejmnpsic Tnizki Thoms Virhinos Voloshin Unsl Yffe Zhitnitsky Mny or results interesting on ir own (rise mny new properties nd issues in QFTs), but when do (nd which of) results extend to R^4? Around , Giotto, Kpustin, Komrgodski & Seiberg relized tht guging discrete symmetries, including ones not visible in perturbtive continuum formultion of guge ories, cn led to new constrints on IR behviour: discrete t Hooft nomly mtching Upshot of tlk: discrete nomlies + smll L results suggest new interesting phses on R^4. Cn be studied on lttice. More generl morl: - py ttention to new consistency conditions - my mention some results on hot domin wlls

8 Consider QC(dj): SU(N) with n_f djoint Weyl fermions (ech one like gugino) n_f < 6 for symptotic freedom cn be solved nonperturbtively t smll-l! - confinement nd chirl symmetry breking n_f 1 is SUSY: one cse where continuity for ll L is gurnteed (Witten index) n_f is one I will focus on mostly, for SU() guge group n_f 3,4,5 somewhere trnsition to conforml window? (unknown: 4?) n_f nd 4 relted to N SUSY nd N4 SUSY, respectively for those firmly rooted in rel world, motivtion is, for me, more of oreticl interest thn pplictions - rre exmple of nonperturbtively solvble QFTs! never mind MWT So, wht is known from smll-l? QC(dj) hs SU(n_f) x U(1) clssicl chirl symmetry (U(n_f) rottes guginos) instnton hs N n_f gugino zero modes: hence U(1) Z_{ N n_f} nomly free discrete chirl symmetry (new feture s opposed to ories with fundmentls) from smll-l: SU(n_f) unbroken, Z_{ N n_f} Z_{ n_f}, so N vcu Unsl, 007 for n_f 1 (SUSY) this is exctly wht is known on R^4 (gurnteed) for n_f > 1 well? seems like SU(n_f) to SO(n_f) is more QC like nd expected

9 So, wht is known from smll-l? QC(dj) hs SU(n_f) x U(1) clssicl chirl symmetry (U(n_f) rottes guginos) instnton hs N n_f gugino zero modes: hence U(1) Z_{ N n_f} nomly free discrete chirl symmetry (new feture s opposed to ories with fundmentls) from smll-l: SU(n_f) unbroken, Z_{ N n_f} Z_{ n_f}, so N vcu Unsl, 007 for n_f 1 (SUSY) this is exctly wht is known on R^4 (gurnteed) for n_f > 1 well? seems like SU(n_f) to SO(n_f) is more QC like nd expected thus, we hve believed tht for n_f>1, re is phse trnsition s L->infinity in fct, QC(dj) hs SU(n_f) x Z_{ N n_f} exct chirl symmetry nd Z_N 1-form center symmetry - not visible to nked eye, well known to lttice folks, but thought - prt from some oreticl studies - lrgely irrelevnt: mod N integer, one per spcetime direction 1-form!

10 in fct, QC(dj) hs SU(n_f) x Z_{ N n_f} exct chirl symmetry nd Z_N 1-form center symmetry - not visible to nked eye, well known to lttice folks, but thought - prt from some oreticl studies - lrgely irrelevnt (- but is not!) mod N integer, one per spcetime direction 1-form! Z_N 1-form globl center symmetry: - only cts on fundmentl representtion Wilson line opertors, infinite or wrpping round T^4 Pe i H dx 1 A 1! e i n 1 N Pe i H dx 1 A 1 - only preserved in ories with zero N-lity representtions: pure YM, QC(dj) - explicitly broken in ories with mssless or light fundmentls (emergent if hevy) - in ories with two-index tensors only (AS, S) Z_ 1-form center is exct, etc. morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form

11 morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form whenever we hve globl symmetries, we cn imgine guging m if we fil to mintin guge invrince, we sy re s t Hooft nomly t Hooft nomly is n nomly w.r.t bckground guge trnsforms it does not represent n inconsistency of ory, but gives strong constrints on possible IR dynmics: this is becuse t Hooft nomly is n RG invrint - is sme t ll scles - so we cn compute it in UV of n symptoticlly free ory (using qurk nd gluon d.o.f.) nd demnd tht it be sme in IR (using whtever IR d.o.f. re) clssic exmple: SU(n_f)_L x SU(n_f)_R in chirl limit of QC hs t Hooft nomly; cn be mtched, in unbroken mode, by mssless bryons, both L nd R or in Goldstone mode by pions (nture s choice) reiterte crucil point: for our purposes best formultion: t Hooft nomly is n RG invrint not to vrition of 4d locl CT (wouldn t be n nomly) but to 4d boundry vrition of 5d locl term depending only on bckground fields ( ones tht guge globl symmetry, nondynmicl) this term does not cre bout scle nd guge ory dynmics (represents formlly t Hooft s wekly coupled nomly cncelling sector)

12 morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form whenever we hve globl symmetries, we cn imgine guging m if we fil to mintin guge invrince, we sy re s t Hooft nomly non vnishing t Hoof nomlies in QC(dj) re [SU(n_f)]^3 Z_{ N n_f} [SU(n_f)]^ [Z_{ N n_f}]^3 Z_{ N n_f} [grv.]^ clssic t Hooft (guginos N^-1 fundmentls) Cski-Murym 97 think of Z_{ N n_f} s U(1) chirl subgroup AN new stuff: Z_{ N n_f} [Z_N]^ 0-form 1-form Giotto et l 14-17

13 morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form AN new stuff: Z_{ N n_f} [Z_N]^ Giotto et l form 1-form under chirl U(1):! e i Nn f R F F e i Nn f Q top. Nn f hence, Z_{ N n_f} is symmetry (s Q_topinteger) e i Q top. Z_{ N n_f} but guging Z_N center mens Z_{ N n_f}: (you hve to buy this; cn give lttice/continuum story, du(1) )! e i k N Q_topk/N: explin!

14 troduction!e Contents Contents 1 The W zero modes nd xil Schwinger model Contents R zero 1 W The W modes zero modes nd Schwinger xil Schwinger 1 The nd xil model model i N nf iscrete F F i N n Q i Q top. top. f nomly mtching in chrge-q Schwinger mod 1 The W zero modes nd xil Schwinger model iscrete nomly mtching in chrge-q Schwinger model iscrete nomly mtching in chrge-q Schwinger N model nf e e iscrete nomly mtching in chrge-q Schwinger model Y Y Y i Fplquette F i plquette U U e i Fplquette plquette Uplquette Y U link e link U U e top. plquette link remember i F Q Y 1 Uplquette Ulink e link plquette linkplquette link plquette plquette Y Y link plquette iftlux thruif T lux thru T lux thru YUplquette e 1 Uplquette eifu 1 1 luxethru T plquetteif e 1 U f lux thru T fthru f n, Z n, fn lux T nt n, n d nzu(1) nflux thru Z nq_top! lux thru T n, Z plquette ll plquettes ll plquettes ll plquettes ll plquettes Y Y Y i iftlux if lux ithru ithru thru T i if lux T in N ei N N Y U e e N Ne U e plquette U e e e plquette plquette -form Z_N guge field plquette bsed Z_N field; ex.: in if lux thru T 1 k Uplquette e e ll plquettes ll plquettes Qtop. ll plquettes N The W1 zero nd xil model model Themodes W (you zero modes nd Schwinger hve to buy Schwinger this;xil cn give ll plquettes e i N 1 The mens W zero modes nd xil Schwinger model but guging Z_N center Q_topk/N: lttice/continuum story, du(1) ) 1 The W zero modes nd xil Schwinger model Eucliden () djoint fermion ction erlier erlier deconfinement ppers: Eucliden SU () djoint fermion ction tken deconfinement ppers: ppers: Eucliden SU () SU djoint fermion ction tkentken fromfrom erlier deconfinement kfrom i j Z_{ N n_f}: from (@S0 + fermion i[ tr 0, N (@])0 ction +i i[j0(@, tken ]) i (@j]) erlier + ]). Eucliden SU djoint j, deconfineme S () tr + i[. i[ (1. j S tr (@0 + i[0, ]) j i explined! (@jj, + i[, ]) j (for. d U(1)!e A unitisx isinnot shown Eucliden time direction; 1,,sp 3 l j 1,, 3j lbels (@in0 A unit x mtrix notmtrix shown Eucliden time direction; j S tr + i[, ]) i (@ + i[, ]) 0 time direction; j A unit x mtrix nd is not shown in Eucliden j j1,, 3.l components 0 lbels Eucliden time. The fermions re two-component, 4d SU(N) orthogonl plnes ) components nd 0 lbels (for Eucliden time. Theneed fermions re two-component, nd nd components nd 0 lbels Eucliden time. The fermions re two-component, n re independent vribles in Eucliden spce; where re SU () gene A unit vribles x mtrix is not shown direction; re independent in Eucliden spce; in where retime SU () genertors,j th Eucliden where re independent vribles in Eucliden spce; re SU () gene Puli mtrices. Similrly, 0, ,...3 Puli mtrices. Similrly,. 0,...3 components nd 0 lbels Eucliden time. The fermionsonly) re istwo-com 0,...3 Puli mtrices. 0,...3 0,...3. The Similrly, fundmentl Polykov loop (keeping constnt mode is re S There fundmentl Polykov loop (keeping constnt mode only) independent Eucliden spce; where The fundmentlvribles Polykov in loop (keeping constnt mode only) is ntum field ory (QFT) is universl prdigm for writing down nture. In mny situtions, however, QFTs re strongly coupled onperturbtive behvior becomes dunting tsk. One of p i 3 P ei 0i 30 dig(e 30, Puli mtrices. Similrly,. i 30 i 0, ,e i 3 )

15 morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form AN new stuff: Z_{ N n_f} [Z_N]^ Giotto et l form 1-form under chirl U(1):! e i Nn f Z_{ N n_f} is symmetry if Q_top1 R F F e i Nn f Q top. Nn f e i Q top. Z_{ N n_f} but guging Z_N center mens Z_{ N n_f}: (you hve to buy this; cn give lttice/continuum story, du(1) )! e i k N Q_topk/N: mixed discrete chirl-center t Hooft nomly discrete Z_N phse in chirl trnsform phse independent on volume of T^4 (used to compute Q_top), so sme t ll scles phse boundry vrition of 5d bulk locl term [not 4d one!]; dep. only on bckground discrete chirl nd discrete center guge fields [best on lttice/tringultion!] cn be written; hs to be mtched in IR long with or t Hooft nomlies

16 Z_{ N n_f}:! e i k N mixed discrete chirl-center t Hooft nomly discrete Z_N phse in chirl trnsform phse boundry vrition of 5d bulk locl term [not 4d one!]; only with bckground discrete chirl nd discrete center guge fields [best on lttice/tringultion!] cn be written; hs to be mtched in IR long with or t Hooft nomlies Z_{ N n_f} or Z_N-center cn be broken in IR, or mtched by CFT, or some TFT bck to IR spectrum of QC(dj) - tke N, n_f : for smll-l: SU(_f) unbroken, Z_{8} Z_{4}, so vcu, Z_N - center unbroken we found solution of ll bove t Hooft mtching condition on R^4 Anber EP for infinite-l: SU(_f) unbroken, Z_{8} by nonzero vev of 4-fermi opertor Z_{4}, so vcu, Z_N - center unbroken O (1) i j i 0 j ii0 jj0 IR: single mssless SU(_f) Weyl doublet O () [ij]k [i b j] c k bc. for N, n_f lttice hs been seeing strnge things, inconsistent with SU() SO() y sy will check bove for bryon Anodorou, Bennett, Bergner, Lucini, 015 dibtic continuity from title sme symmetry reliztion t smll nd lrge L???

17 morl: QC(dj) hs SU(n_f) x Z_{ N n_f} x Z_N exct globl symmetry 0-form 1-form whenever we hve globl symmetries, we cn imgine guging m if we fil to mintin guge invrince, we sy re s t Hooft nomly non vnishing t Hoof nomlies in QC(dj) re [SU(n_f)]^3 clssic t Hooft (guginos N fundmentls) [SU(n_f)]^ [Z_{ N n_f}]^3 Z_{ N n_f} [grv.]^ Cski-Murym 97 think of Z_{ N n_f} s U(1) chirl subgroup AN new stuff: Z_{ N n_f} [Z_N]^ 0-form 1-form AN more new stuff: - Z_ X; Z_4 X; 1-form Giotto et l non-spin mnifold X for N n_f only (twisted N SYM) umitrescu-cordob lst week! clim our proposl modified by dding n emergent Z_ guge ory in IR not cler yet (to me!) how to probe for it (sy on lttice t Hooft loop?) sty tuned! dibtic continuity from title sme symmetry reliztion t smll nd lrge L???

18 Upshot of tlk: discrete nomlies + smll L results suggest new interesting phses on R^4. Cn be studied on lttice. no time to More generl morl: - py ttention to new consistency conditions - my mention some results on hot domin wlls (domin wlls in one phse mimic bulk behvior in nor: high-t W low-t bulk nd v.v.)