Non-hiral urrent algeb ras fo r defo rmed sup ergroup WZW - PowerPoint PPT Presentation

Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels Anatoly K one hny Heriot-W att Universit y Septemb er 21, 2011, EMPG semina r, Edinburgh Based on a joint w o rk with Thomas Quella (JHEP03(2011)124)

Outline Motivation Results obtained using p erturbation theo ry Non-p erturbative refo rmulation of the mo del Dis ussion Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

Motivation and p rio r w o rk Confo rmally inva riant 2D sigma-mo dels whose ta rget spa e is a sup erspa e emerge in diso rdered systems of ondensed matter theo ry ova riant quantization of sup erstrings on AdS d × S d ba kgrounds The string ba kground with pure Neveu-S hw a rz �ux an b e fo rmulated in terms of a WZW mo del on the sup ergroup (Berk ovits, V afa, Witten, 1999). In su h a fo rmulation one an des rib e defo rmations o rresp onding to swit hing on a mixture of N-S and R-R �uxes whi h p reserves the full isometry of whi h is isomo rphi to G × G . In the op erato r des ription this is des rib ed b y p erturbing the WZW mo del b y an op erato r where φ ab is the p rima ry �eld o rresp onding to the adjoint PSU(1 , 1 | 2) rep resentation. Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels G = PSU(1 , 1 | 2) J b : ( z, ¯ : J a φ ab ¯ z )

Geometri ally this defo rmation is des rib ed as a p rin iple hiral mo del on G with a W ess-Zumino term. It w as �rst a rgued in Bershadsky , Zhuk ov, V aintrob (1999) that su h p rin ipal hiral mo dels a re onfo rmal (fo r any value of the WZ term) when the sup ergroup has a vanishing Killing fo rm. This happ ens fo r and fo r its generalizations PSL(n | n) . Their a rguments w ere generalized to general sup ergoups with vanishing Killing fo rm and to their osets:Kagan, Y oung (2005); Babi henk o(2006) One an think of sup ermatri es from SL(n | n) as of blo k 2 n × 2 n matri es with PSU(1 , 1 | 2) the ondition Sdet( M ) = det( A ) / det( B ) = 1 . Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels   A | B M = −− −− −−   C | D

Proje ting out the multiples of identit y one obtains the sup ergroup . The sup ergroup PSU(1 , 1 | 2) is a real fo rm o r PSL(2 | 2) . Another interesting defo rmation of the WZW theo ries on these sup ergroups is an isotropi urrent- urrent defo rmation: This defo rmation only p reserves the diagonal sub roup of the isometry sup ergroup G ⊂ G × G . Both defo rmations a re exa tly ma rginal due to the vanishing PSL( n | n ) Killing fo rm. The onserved urrents o rresp onding to the global symmetries in the defo rmed mo dels do not split into holomo rphi and antiholomo rphi omp onents. The onservation equations a re J ) = κ ba : J a ¯ J b : ( z, ¯ Str( J ¯ z ) Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels ∂ ¯ z ) + ¯ J a ( z, ¯ ∂J a ( z, ¯ z ) = 0 .

The ab ove is usually the ase in massive theo ries with onserved urrents. In CFT's this only happ ens when unita rit y is violated. Supp ose φ is a CFT p rima ry of w eights ( n, 0) then An OPE algeb ra fo r onserved urrents in massive theo ries w ere investigated in M.Lues her (1978); D. Berna rd (1991) . The onstraints imp osed b y urrent onservation w ere explo red and an in�nite to w er of non-lo ally onserved ha rges w ere onstra ted. The general onstraints on OPE w ere mo re re ently generalized to pa rit y-nonp reserving theo ries b y S.Ashok, R. Beni hou and J. T ro ost (2009). The urrent algeb ra in the G × G -p reserving L − 1 | φ �|| 2 = � φ | ¯ || ¯ L 1 ¯ L − 1 | φ � = � φ | 2¯ L 0 | φ � = 0 . defo rmation des rib ed ab ove w as studied in S.Ashok, R. Beni hou and J. T ro ost (2009), (2009); R. Beni hou and J. T ro ost (2010); R. Beni hou (2011). An in�nite set of onserved urrents w ere put fo rw a rd fo r this defo rmation. Those onserved quantities stem from the quantum Maurer-Ca rtan equation. Our w o rk w as mu h inspired b y those pap ers. Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

Many of the results obtained fo r the G × G -p reserving defo rmation dep end on ertain assumptions involving the adjoint p rima ry op erato r φ ab ( z, ¯ whose p rop erties at p resent a re not under analyti ontrol. W e study the se ond defo rmation where no su h p roblem a rises. W e fo us on the algeb ra generated b y the defo rmed urrents whi h is of interest at least fo r three reasons one w ould hop e that as in the massive ase studied b y D. Berna rd there is an in�nite to w er of onserved non-lo al ha rges onstru ted out of the urrents. The interpla y b et w een onfo rmal symmetry and the integrable stru ture (Y angian) z ) w ould b e interesting to explo re one ould hop e that the non- hiral urrent algeb ra ma y b e useful fo r o rganizing the sp e trum it w ould b e a new OPE algeb ra generalizing vertex op erato r algeb ras in unita ry CFT's Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

F o r the urrent- urrent defo rmation w e found that despite the absen e of geometri reasons (no Maurer-Ca rtan equation) the quantum equation of motion tak es the same fo rm as in D. Berna rd (1991) whi h mak es p ossible the onstru tion of Y angian ha rges. Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels

The mo del The WZW mo del on a sup ergroup G has lo al urrents J a ( z ) and ob eying the OPE where κ ab is the non-degenerate inva riant 2-fo rm and f ab a re the stru ture onstants. They satisfy ¯ J b (¯ z ) ( z − w ) 2 + if ab kκ ab c J c ( w ) J a ( z ) J b ( w ) = + . . . z − w c ¯ kκ ab w ) 2 + if ab J c ( ¯ - the graded Ja obi identit y . The vanishing of the Killing fo rm w ) ¯ z ) ¯ J a (¯ J b ( ¯ w ) = + . . . implies that (¯ z − ¯ z − ¯ ¯ w c Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels κ ab = ( − 1) a κ ba , f ba c = − ( − 1) ab f ab c f ab d f dc e + ( − 1) c ( a + b ) f ca d f db e + ( − 1) a ( b + c ) f bc d f da e = 0 f acd f b dc = 0

Exa t t w o-p oint fun tions of urrents In the defo rmed theo ry the op erato rs rep resenting the defo rmed urrents a re the ba re WZW urrents inserted into p erturbation theo ry series. The dimension and spin a re onserved and there a re no �elds with whi h the urrents ould fo rm a Joa rdan blo k. F o rmally the defo rmed t w o-p oint fun tions a re T w o rema rks a re in o rder.Firstly in loga rithmi theo ries t ypi ally This is not the ase fo r some free �eld realizations of WZW mo dels; otherwise one should treat su h o rrelato rs mo re fo rmally � − λ � � as a means to ompute OPE's. d 2 w : J e ¯ � J a ( z 1 , ¯ z 1 ) J b ( z 2 , ¯ z 2 ) � λ = � J a ( z 1 ) J b ( z 2 ) exp J r : κ re � 0 kπ Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels � 1 � = 0

Se ondly , the p erturbation series integrals have divergen es. The integrands (fo r �nite sepa ration) a re p rop o rtional to Ea h o rrelato r is given b y the sum of singula r terms in the OPE ontra tions and thus lo oks lik e a rational fun tion times an inva riant tenso r onstru ted from κ ab and f abc . Fixing the metho d of subtra tion is equivalent to de�ning the omp osite op erato r : J e ¯ , i.e. w e de�ne its o rrelation fun tions as distributions. Conta t term ambiguities in these o rrelato rs a re related b y oupling onstant rede�nitions. There a re � J a ( z 1 ) J b ( z 2 ) J e 1 ( w 1 ) . . . J e n ( w n ) � 0 � ¯ J e 1 ( ¯ w 1 ) . . . J e n ( ¯ w n ) � 0 no infra red divergen es. Anatoly K one hny Non- hiral urrent algeb ras fo r defo rmed sup ergroup WZW mo dels J r : κ re