Yangian Superalgebras in Conformal Field Theory

of the global symmetry vanishes identically. These theories are argued to be conformal[1, 2, 3], a recent article is [4]. The problem is that generically these sigma models areonly known to have Virasoro and f**in**ite-dimensional Lie superalgebra symmetry, which is**in** most cases not sufficient to determ**in**e physical quantities as spectra and correlationfunctions. It is thus crucial to f**in**d more symmetry and to learn how to use it.The state of the art of sigma models on target superspace is that WZW models on supergroupsare fairly well understood due to their Kac-Moody symmetry, see [5, 6, 7, 8, 9,10, 11, 12, 13] for bulk and [13, 14, 15, 16, 17, 18, 19] for boundary models. The situationchanges when Kac-Moody symmetry ceases to be present. Then perturbative computationsstart**in**g from the WZW model led to the computation of specific boundary spectra[20, 21, 22, 23] and current correlation functions [24, 25, 26, 27, 28]. In both cases thevanish**in**g of the Kill**in**g form simplified the perturbative treatment. Non-perturbativelyone can s**in**gle out some subsectors of the theory us**in**g the global symmetry and computecorrelation functions of fields **in** these sectors **in** simpler models [29].The motivation to study target superspace sigma models are its appearance **in** str**in**gtheory and condensed matter physics.Str**in**g theory on AdS d × S d is described by superspace sigma models [30, 31, 32, 33, 34].These theories are important as they are conjectured to be dual to some gauge theories[36]. They areclassical **in**tegrable [33,37, 38] 1 . Quantum**in**tegrability istightly connectedwith **Yangian** symmetry at the quantum level.In condensed matter physics **in** the case of the **in**teger quantum Hall effect as well asfor disordered systems psl(N|N) symmetry and its central extension gl(N|N) are present[39, 40]. Boundary spectra were also computed perturbatively **in** this context [41].Animportantquantity**in**thetheoryoford**in**ary**Yangian**sistheeigenvalueofthequadraticCasimir element of the Lie subalgebra **in** the adjo**in**t representation. If the Kill**in**g form isdegenerate then this operator acts non-**in**vertible **in** the adjo**in**t 2 . In this case the natureof the **Yangian** algebra is expected to be rather unusual. Hav**in**g a natural constructionof the algebra as we provide it **in** this article should help understand**in**g these special**Yangian** superalgebras. Reviews on ord**in**ary **Yangian**s are [42, 43].The present work is **in**spired by the follow**in**g two developments. Lüscher found non-localcharges **in** massive two-dimensional quantum field theory [44], which were then identifiedas **Yangian** algebras by Bernard [45]. These methods were extended to WZW models oncompact Lie groups **in** [46]. As conformal fields do not vanish at **in**f**in**ity the proceduredepends on a po**in**t of reference P.1 In the AdS 4 ×CP 4 case classical **in**tegrability of the non-sigma model part of the str**in**g theory wasfound **in** [35].2 This also applies to the superalgebras of type gl(N|N) where the Casimir acts non-zero but nilpotent**in** the adjo**in**t.2

The second development is twistor str**in**g theory. It is argued to be dual to N=4 superYang-Mills theory [47, 48]. Amplitudes **in** super Yang-Mills are computed to be **Yangian****in**variant [49, 50]. The world-sheet theory of the open twistor str**in**g is essentially a freefield representation of the psl(4|4) current algebra at level one. Us**in**g the methods of[46] one can then establish **Yangian** symmetry [51]. The procedure simplifies considerablydue to the vanish**in**g of the Kill**in**g form of psl(4|4), e.g. the dependence on the po**in**t ofreference P eventually vanishes. The ma**in** goal is to generalize this result and to establish**Yangian**symmetry **in**manyconformalfieldtheoriesbasedonsuperalgebraswithvanish**in**gKill**in**g form and not necessarily possess**in**g Kac-Moody current symmetry.Our strategy is as follows. We consider Lie superalgebras whose Kill**in**g form vanishesidentically and which possess a non-degenerate **in**variant bil**in**ear form. This applies tothe algebras of the groups PSL(N|N), OSP(2N+2|2N) and D(2,1;α). The first step is totake a conformal field theory with current symmetry of such a Lie superalgebra. Thesecurrents are then used to construct non-local currents and charges, which depend on apo**in**t of reference P. Their properties are computed **in** complete analogy to the psl(4|4)level one example of the twistor str**in**g [51]. The first result is that a special subsector ofthe conformal field theory carries a representation of the **Yangian**, given by the action ofthesenon-localcharges. Moreoverthisisasymmetryandgives**in**terest**in**gWardidentities.These identities are **in**dependent of the po**in**t of reference P. This leads us to def**in**e newcharges which also provide a **Yangian** symmetry of this special subsector of the model.The new charges allow us to **in**vestigate truly marg**in**al perturbations that preserve the**Yangian** symmetry. We then list three examples. The ma**in** one are sigma models onthe supergroups PSL(N|N), OSP(2N+2|2N) and D(2,1;α). Further, if we take the discas world-sheet, we also have OSP(2N+2|2N) Gross-Neveu models which are conjecturedto be dual to supersphere S 2N+1|2N sigma models. F**in**ally aga**in** for the disc, we haveperturbations of supersymmetric ghost-systems. We show that these are equivalent tosigma models on super projective space CP N−1|N . There are many more examples, as e.g.massive deformations, which we do not consider.2 **Superalgebras**We present the superalgebras that appear **in** the conformal field theories.2.1 Lie superalgebrasWe start with some properties of Lie superalgebras. They were first studied by VictorKac [54]. A Lie superalgebra is def**in**ed as follows.Def**in**ition 2.1 Let g be a Z 2 graded algebra g = g¯0 ⊕g¯1 with product [ , ] : g×g → g3

one part as∆(t a 0) = t a 0 ⊗ 1 + 1 ⊗ t a 0(2.13)∆(t a 1) = t a 1 ⊗ 1 + 1 ⊗ t a 1 + αf a bct c 0 ⊗ t b 0is an algebra homomorphism. In addition the co-multiplication has to be co-associative(∆ ⊗ 1) ◦ ∆ = (1 ⊗ ∆) ◦ ∆. (2.14)Further there is a co-unit ǫ : Y(g) → Rǫ(t a i ) = 0 for i = 0,1 (2.15)and an anti-pode S : Y(g) → Y(g)S(t a i ) = −ta i for i = 0,1 (2.16)This def**in**es a Hopf superalgebra. Note that the anti-pode axiom has such a simple formbecause of the vanish**in**g of the Kill**in**g form.The co-multiplication be**in**g a homomorphism restricts the algebra bracket of level oneelements by some algebraic relations, the Serre relations. More precisely one can writethe bracket as[t a 1 , tb 1 ] = fab ct c 2 + Xab . (2.17)The Serre relations then fix X ab uniquely as follows (see [42] for a similar argument).Consider u ab satisfy**in**gu ab = −(−1) ab u ba and u ab f ab c = 0 (2.18)this means that we can identify u ab with a closed element u = u ab t a 0 ∧ t b 0 **in** ∧ 2 g ⊗ C.For homology of Lie superalgebras see [52]. We def**in**eY ab = (−1) de f a dcf b gef dg ht c 0 te 0 th 0 . (2.19)Vanish**in**g of the Kill**in**g form implies that Y ab is totally supersymmetric, i.e. we can viewit as an element of Sym 3 (g). Requir**in**g that ∆ is a homomorphism leads to the Serrerelationsu ab [t a 1, t b 1] = u ab X ab α 2= u ab3 Y ab . (2.20)If the second homology of ∧ ∗ g with coefficient **in** the trivial representation were zero,6

then these relation would be similar to the standard Serre relations of **Yangian** algebras:−f kl i[t i 1, t j 1] + (−1) lj f kj i[t i 1, t j 1]−(−1) k(l+j) f lj i[t i 1, t k 1] == α 2 (−1) ih+gj f gid f ld hf ki cf jg et c 0t h 0 t e 0.(2.21)This is the case for g ∈ {osp(2N +2|2N),d(2,1;α)}, but not for psl(N|N) [52].2.3 RepresentationsA representation of the **Yangian** is also one of its Lie subalgebra. But not every Liesuperalgebra representation can be lifted to the **Yangian**. The simplest lift is the trivialone, i.e. let ρ : g :→ End(V) be a representation of g, then we def**in**e ˜ρ : Y(g) :→ End(V)by˜ρ(t a 0) = ρ(t a 0) and ˜ρ(t a 1) = 0. (2.22)Look**in**g back to the Serre relations (2.20), we see that this becomes a representation ofthe **Yangian** if and only if ρ(Y ab ) = 0. It is not known **in** general when this condition issatisfied, however there are some criteria [55, 53]. These representations are important tous.2.4 Current algebraThe conformal field theories we start to lookat we require to possess currents J a (z) tak**in**gvalues **in** our Lie superalgebra g and satisfy**in**g the operator product expansionJ a (z)J b (w) ∼kκ ab(z −w) + fab cJ c (w). (2.23)2 (z −w)These currents have the operator valued Laurent expansionJ a (z) = ∑ nJ a n z−n−1 (2.24)and the modes satisfy the relations of the aff**in**ization ĝ of g[J a n ,Jb m ] = fab cJ c n+m + kκab δ n+m,0 , (2.25)where the central element takes the fixed value k, the level. Especially the zero modesform a copy of g. The action of these modes on the vacuum state isJ a n|0〉 = 0 = 〈0|J a −n for n ≥ 0. (2.26)7

Important fields of the conformal field theory are the primary fields φ(z,¯z). Let ρ : g →End(V) be a representation of g. Then to a vector φ **in** V, we associate the primary fieldφ(z,¯z) and this field has the follow**in**g operator product expansion with the currentsJ a (z)φ(w, ¯w) ∼ ρ(ta )φ(w, ¯w)(z −w). (2.27)The field φ(z,¯z) is bosonic/fermionic if the vector φ is bosonic/fermionic and we denoteits parity by |φ|. F**in**ally, we def**in**e the subsector of primary fields that transform **in**representations that can be lifted trivially to representations of the **Yangian**Y = {φ(z,¯z)|φ **in** V and ρ(Y ab ) = 0 for ρ : g → End(V)}. (2.28)This term**in**ates our preparations.3 Non-local ChargesMost of this section is **in** analogy of section four **in** [51] 4 , which handled the special caseof psl(4|4) current algebra at level one. We start with a conformal field theory hav**in**g acurrent algebra of g, and g has vanish**in**g Kill**in**g form as before. The zero modes of thecurrents form a copy of g. They are the charges of level zeroQ a 0 = Ja 0 = 1 ∮2πidzJ a (z). (3.1)The ma**in** goal of this section is to construct level one charges and to work out theirproperties. These charges correspond to non-local currents. Symmetries from non-localcharges were considered **in** [57].We **in**troduce the non-local field χ a (z;P),χ a (z;P) =∫ zPdwJ a (w) (3.2)and the non-local currentY a (z;P) = f a bc∫ zPdwJ c (w) J b (z) = f a bc χ c (z;P) J b (z). (3.3)Note that there is no normal order**in**g necessary, s**in**ce vanish**in**g of the Kill**in**g form plusantisupersymmetry of the structure constants imply4 Which is based on [46].f a bc J c (w) J b (z) ∼ 0. (3.4)8

We now **in**vestigate the correspond**in**g charges and their action on primary fields.3.1 Co-algebraThe co-algebra is determ**in**ed by the action of the charges on a product of fields.The operator product with a primary φ(w, ¯w) isY a (z;P) φ(w, ¯w) ∼ f a bc( ×× χc (w;P)ρ(t b )φ(w, ¯w) × ×+(z −w)( z −w) )×−ln×P −wJc (z) ρ(t b )φ(w, ¯w) × .×(3.5)The action of the non-local charge Q a 1(P) correspond**in**g to the current Y(z;P) on theprimary field φ(w, ¯w) isQ a 1 (P)( φ(w, ¯w) ) =∮C wdzY a (z;P) φ(w, ¯w) = 2 f a bc × × χc (w;P) ρ(t b )φ(w, ¯w) × × , (3.6)where the cut for the logarithm extends from w pass**in**g through the po**in**t P, and for thecontour C w see Figure 1. The computation of the **in**tegral is then as **in** [51]. The actionC wwPFigure 1: The contour C w starts just above P, circles around w and stops just below P.of the non-local charge on two fields is)Q a 1(φ (P) 1 (w 1 , ¯w 1 )φ 2 (w 2 , ¯w 2 )=∮C w1 ,w 2dzY a (z;P)φ 1 (w 1 , ¯w 1 )φ 2 (w 2 , ¯w 2 ))= Q a 1(φ (P) 1 (w 1 , ¯w 1 ) φ 2 (w 2 , ¯w 2 )+)+(−1) a|φ1| φ 1 (w 1 , ¯w 1 ) Q a 1(φ (P) 2 (w 2 , ¯w 2 ) +(3.7)−2πi f a bc Q c 0( ) ( )φ 1 (w 1 , ¯w 1 ) Q b 0 φ 2 (w 2 , ¯w 2 ) .For the contour C w1 ,w 2see Figure 2. Aga**in** the computation is exactly as **in** [51]. The9

w 1 w 2C w1 ,w 2PFigure 2: The contour C w1 ,w 2starts at P above both cuts (cut from w 1 to P, and cut fromw 2 to P), encircles both w 1 ,w 2 and stops below both cuts at P.action on two fields can be written **in** terms of the co-product of the **Yangian** (2.13)∆(Q a 1(P)) = Q a 1(P)⊗1+1⊗Q a 1(P)−2πi f a bc Q c 0 ⊗Q b 0. (3.8)3.2 AlgebraThe algebra axiom is a straightforward computation. We computeQ a 0 Y b (v;P) = 1 ∮2πi= f b cddzJ a (z)f b cd∫ vPdvJ d (w)J c (v)()f ad eχ e (v;P)J c (v)+(−1) ad f ac eχ d (v;P)J e (v)(3.9)= f ab cY c (v;P).The last equality follows from the Jacobi identity. Hence, the charges transform **in** theadjo**in**t[Q a 0,Q b 1(P)] = f ab cQ c 1(P). (3.10)3.3 Serre relationsRecall that the co-multiplication is only well def**in**ed if the Serre relations (2.20) aresatisfied. If these relations fail to hold, the action of the level one **Yangian** charges onproducts of fields are preposterous.We show that the left-hand side of the Serre relations vanish act**in**g on one field. Thismeans that not the full space of fields of our conformal field theory forms a representationof the **Yangian**, but only the subspace Y (2.28) spanned by those fields transform**in**g **in** arepresentation that can be lifted trivially to a representation of the **Yangian**.The left-hand side of the Serre relations (2.20) for the level one charges Q 1 (P) vanishes10

act**in**g on a primary field φ(w, ¯w), if[Q a 1 (P),Qb 1 (P)]φ(w, ¯w) = fab cX c (w, ¯w;P) (3.11)holds for some X c (w, ¯w;P). We show that the charges act **in**deed **in** this form.We start with Q 1 (P) act**in**g on χ(w;P). We computef a bcJ c (w)J b (z)J d (v) ∼ f a bc( kκ bd J c (w)(z −v) 2 + (−1)bc kκ cd J b (z)(w −v) 2 ++ fbd e × × Jc (w)J e (v) × ×(z −v)+ (−1)bc f cd e × × Jb (z)J e (v) ) ××.(w −v)(3.12)Us**in**g the **in**tegrals **in** appendix A, we getQ a 1(P)(χ d (w;P)) = f a bcf bd eχ c (w;P)χ e (w;P)−2f a cbkκ cd χ b (w;P). (3.13)This together with (3.6) impliesQ a 1(P)(Q b 1(P)(φ(w, ¯w))) = 2f b cdQ a 1( )χ d (w;P)ρ(t c )φ(w, ¯w)= 2f b cdQ a 1( )χ d (w;P) ρ(t c )φ(w, ¯w))+(3.14)+(−1) ad 2f b cdχ d (w;P)Q a 1( )ρ(t c )φ(w, ¯w) .We comb**in**e this toQ a 1 (P)(Qb 1 (P)(φ(w, ¯w)))−(−1)ab Q b 1 (P)(Qa 1 (P)(φ(w, ¯w))) = Aab +B ab (3.15)withA ab = 4f b cdf a ef(−1) ef kκ fd (χ e ρ(t c )φ)(w, ¯w;P)+B ab =−4(−1) ab f a cdf b ef(−1) ef kκ fd (χ e ρ(t c )φ)(w, ¯w;P)(f b cdf a eff ed g +(−1) af f b dff a egf ed c+and(3.16)−(−1) ab f a cdf b eff ed g −(−1) ab+bf f a dff b egf ed c)2(χ f χ g ρ(t c )φ)(w, ¯w;P).11

The Jacobi identity impliesA ab = −4f ab df d ce(χ e ρ(t c )φ)(w, ¯w;P)B ab = 2f ab ef dc gf e df(χ f χ g ρ(t c )φ)(w, ¯w;P).(3.17)We have thus shown (3.11), and hence the left-hand side of the Serre relations vanishesact**in**g on one primary field φ(w, ¯w). This means that our charges Q a 0 and Qb 1 (P) forma representation of the **Yangian** if and only if the representation ρ of the primary fieldcan be lifted trivially to a representation of the **Yangian** (recall subsection 2.3). Thus, werestrict to fields φ(z,¯z) **in** Y .3.4 Co-unitBefore we turn to Ward identities, we have to show thatQ a 0|0〉 = Q a 1(P)|0〉 = 0. (3.18)This means that the vacuum is a co-unit and implies Ward identities for these charges.Recall that the vacuum satisfiesJn a |0〉 = 0 for n ≥ 0. (3.19)HenceQ a 0|0〉 = 0, (3.20)and furtherQ a 1(P)|0〉 = 1 ∮2πidz∫ zPdwf a bcJ c (w)J b (z)|0〉= 1 ∮2πidz∫ zdwf a bcP∑Jn c w−n−1 Jm b z−m−1 |0〉m,n(3.21)= 0.The last l**in**e follows from the vanish**in**g of the Kill**in**g form plus supersymmetry of the**in**variant bil**in**ear form. In complete analogy one can show that〈0|Q a 0 = 〈0|Q a 1(P) = 0. (3.22)12

3.5 Ward identitiesIn this section, we show that the non-local charges are **in**deed symmetries, that is theyannihilate correlation functions. But, we stress aga**in** that the action of the non-localcharges on products of fields is only well def**in**ed if these fields transform **in** representationthat allow to be lifted trivially to a **Yangian** representation. Hence, the Ward identitiesonly hold for correlation functions conta**in****in**g these fields. Let φ i (z i ,¯z i ) be such fields.From last section, we get the Ward identities〈0|Q a i φ 1(z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉 = 0. (3.23)For the level zero charges this takes the form0 =n∑ρ i (t a )〈0|φ 1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉. (3.24)i=1Here and **in** the follow**in**g we use the notationρ i (t a )〈0|φ 1 (z 1 ,¯z 1 )...φ i (z i ,¯z i )...φ n (z n ,¯z n )|0〉 == 〈0|φ 1 (z 1 ,¯z 1 )...ρ i (t a )φ i (z i ,¯z i )...φ n (z n ,¯z n )|0〉.(3.25)Us**in**g co-multiplication (3.8) and also (3.6) the level-one charges annihilat**in**g the vacuumimply0 = 〈0|Q a 1(P)φ 1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉= 2f a bc∑i

We choose the phase 5 ln (z i −z j )(z j −z i )= ln(−1) = πi. (3.27)Us**in**g ord**in**ary global symmetry (3.24) of the correlation functions, antisupersymmetryof the structure constants and the vanish**in**g of the Kill**in**g form, we getf a bc∑i

co-associativity. This is only consistent with the Serre relations, if the right-hand side ofthe Serre relations acts trivially on a s**in**gle field. We thus cont**in**ue to restrict to fieldsφ(z,¯z) **in** Y. The crucial observation now is, that this representation of the **Yangian** isalso a symmetry of correlation functions∆ n( Q1) a 〈0|φ1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉 == f a bc∑ρ i (t c ) ρ j (t b ) 〈0|φ 1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉 = 0.(3.32)i

Let φ i (z i ,¯z i ) **in** Y as before. A correlation function can be computed perturbatively as〈0|φ 1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉 λ == ∑ m〈0|φ 1 (z 1 ,¯z 1 )...φ n (z n ,¯z n ) 1 m!( ∫ λ2π) m|0〉. (3.35)d 2 zO(z,¯z)The right-hand side is to be computed **in** the unperturbed theory. The trivial representationcan obviously be lifted to the trivial representation of the **Yangian**. S**in**ce theperturb**in**g field transforms by assumption **in** this representation, our Ward identities(3.29) still hold **in** the perturbed theory0 = ∑ m∆ n+m( Q a i)〈0|φ1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )m!( 1 ∫ λ2πd 2 zO(z,¯z)) m|0〉= ∑ (〈0| ∆ n( ) 1( ∫ λm|0〉Q a i )φ 1(z 1 ,¯z 1 )...φ n (z n ,¯z n ) d zO(z,¯z)) 2 (3.36)m! 2πm= ∆ n( )Q a i 〈0|φ1 (z 1 ,¯z 1 )...φ n (z n ,¯z n )|0〉 λfor i = 0,1 and all a. But this says that the perturbed theory still satisfies the nice Wardidentity∆ n( Q a 1)〈0|V1 (z 1 )V 2 (z 2 )...V n (z n )|0〉 λ =2πif a bc∑ρ i (t c ) ρ j (t b ) 〈0|V 1 (z 1 )V 2 (z 2 )...V n (z n )|0〉 λ = 0.(3.37)i

supergroup valued field. The supergroup sigma model action isS k,f [g] = 12πf 2 ∫d 2 z str ( g −1 ∂gg −1¯∂g ∫) k −12πd 2 z d −1 str ( (g −1 dg) 3) (4.1)For ord**in**ary Lie groups and supergroups this model is only quantum conformal for theWess-Zum**in**o-Witten model, i.e. for k = f −2 . In our case they are conformal at anycoupl**in**g and at the WZW po**in**t we have additional Lie superalgebra current symmetry.We thus view the sigma model as a perturbation from the WZW modelS k,f [g] = S WZW [g] − λ ∫2πd 2 z str ( g −1 ∂gg −1¯∂g ), (4.2)where λ = f −2 −k. The pr**in**cipal chiral field O PC = str ( g −1 ∂gg −1¯∂g )satisfiesQ a 0 O PC(z,¯z) = 0. (4.3)Hence at any sigma model on these supergroups possesses the additional symmetry. Especiallylet us mention aga**in** that the Ward identities (3.37) hold at any po**in**t λ.Remark, that we could have chosen a perturbation by the Wess-Zum**in**o term equally wellas the total action is **in**variant under the global symmetry.4.2 OSP(2N+2|2N) Gross-Neveu modelsOrthosymplectic Gross-Neveu models are supersymmetric generalizations of Gross-Neveumodelsfororthogonalgroups. Inthecaseoford**in**aryGross-Neveu modelstheO(2)-modelis special, as it is quantum conformal at any coupl**in**g. The reason for this is that O(2)is abelian. In the supergroup case despite be**in**g non-abelian the OSP(2N+2|2N) Gross-Neveu models are still quantum conformal. These theories are conjectured to be dualto sigma models on the superpheres S 2N+1|2N [21, 23]. The OSP(2N+2|2N) Gross-Neveumodel is a current-current perturbation of a free theory given by 2N+2 real dimension1/2 fermions plus N bosonic βγ−systems also of dimension 1/2,S λ = S + λ ∫2πd 2 zJ a κ ba ¯JbS = 1 ∫2πd 2 z(2N+2∑i=1+ 1 2ψ i¯∂ψi + ¯ψ i ∂ ¯ψ i)+( N∑a=1β a¯∂γa −γ a¯∂βa + ¯β a ∂¯γ a − ¯γ a ∂¯β a).(4.4)17

At λ = 0 the model has a holomorphic and a anti-holomorphic osp(2N+2|2N) currentsymmetry. The holomorphic o(2N+2) currents are generated by the dimension one currentsψ i ψ j , the sp(2N) currents are generated by the β a β b ,γ a β b and γ a γ b and the fermioniccurrents are of the form ψ i β a and ψ i γ a . This realisation is the aff**in**e analogue of an oszillatorrealisation for the horizontal subsuperalgebra for which the vanish**in**g of the righthandside of the Serre relations is understood [58]. The current-current perturbation fieldO GN = J a κ ¯Jb ba transforms **in** the adjo**in**t representation under both holomorphic andanti-holomorphic currents. It is thus not an operator that preserves **Yangian** symmetry.The situation changes if we take the world-sheet to have a boundary, i.e. the disc orequivalently the upper half plane. In this case, we have to demand glu**in**g conditions forthe fields at the boundary. We require thatψ i (z) = ¯ψ i (¯z) , β a (z) = ¯β a (¯z) and γ a (z) = ¯γ a (¯z) for z = ¯z. (4.5)This implies that the currents satisfy trivial boundary conditions J a (z) = ¯J a (¯z) for z = ¯z.Now, we view the anti-holomorphic coord**in**ates as holomorphic coord**in**ates on the lowerhalf plane. We can then analytically cont**in**ue the currents J a (z) on the entire planeJ a (z) ={ J a (z) z **in** upper half plane¯J a (z)z **in** lower half plane(4.6)Now, the holomorphic currents as well as the anti-holomorphic currents transform **in** theadjo**in**t representation of the charges Q a 0 correspond**in**g to the currents that are def**in**edon the full plane. It follows that the current-current perturbation transforms trivially,Q a 0 Jb (z)κ ¯Jc cb (¯z) = ( )f ab dκ cb +(−1) ad f ab cκ bd J d (z)¯J c (¯z) = 0. (4.7)Hence, the boundary Gross-Neveu model also has a subsector with **Yangian** symmetry.Aga**in** especially the Ward identity (3.37) holds.4.3 Ghost-systems and CP N−1|N sigma modelsOur next example is given by N fermionic ghost systems, each of central charge c = −2,and N bosonic ghost systems each of central charge c = 2. Such a system has a gl(N|N)current symmetry. Denote the bosonic ghosts by β i ,γ i , i = 1,...,N and the fermionic18

ones by β i ,γ i , i = N +1,...,2N. Then the action isS λ = S − λ ∫2πS = 1 ∫2πd 2 zO βγd 2 zβ i (¯∂ +Ā)γi + ¯β i (∂ +A)¯γ i(4.8)O βγ = O β O γ, O β = ∑ iβ i¯βi, O γ = ∑ iγ i¯γ i .The gauge field A,Ā ensures that β iγ i (often called U(1) R current) acts as zero and thusthere is only sl(N|N) symmetry. Further the total central charge is c = −2. The superalgebrasl(N|N) has a central element E, restrict**in**g to those representations that possessE-eigenvalue zero effectively reduces the symmetry to psl(N|N). The perturbation is acurrent-current perturbation correspond**in**g to the quadratic Casimir. **Yangian** symmetryis preserved **in** the boundary case by the same arguments as **in** the Gross-Neveu model.Interest**in**gly all vertex operators that only depend on the γ i and ¯γ i obey the criterionof [53] as argued **in** the case N = 4 [51], i.e. they allow for a trivial lift to a **Yangian**representation.We are look**in**g for a geometric **in**terpretation of this theory. Recently path **in**tegralarguments ledto correspondences [11, 61, 62] anddualities [59, 60] **in**volv**in**g sigma modelson non-compact (super) target spaces. In our case such arguments turn out to be verysimple.We restrict to correlation functions of fields V i (z i ) = V i (γ(z i ),¯γ(z i )) only depend**in**g onthe γ i ,¯γ i as for example gluon vertex operators **in** the twistor str**in**g. Then perform**in**gthe path **in**tegral first for the β i , ¯β i gives〈V 1 (z 1 )...V n (z n )〉 λ =∫d 2 γd 2 βd 2 −SAV 1 (z 1 )...V n (z n )eλ[β,¯β,γ,¯γ,A,Ā]==∫∫d 2 γd¯βd 2 AV 1 (z 1 )...V n (z n )δ(λ¯β i O γ +(¯∂ +Ā)γ −Sλ[0,¯β,γ,¯γ,A,Ā]i)ed 2 γd 2 AV 1 (z 1 )...V n (z n )e −S1 λ [γ,¯γ,A,Ā] (4.9)where S λ[γ,¯γ,A,Ā] 1 = 1 ∫2πλd 2 z 1 ∑(¯∂O +Ā)γi (∂ +A)¯γ i .γi19

Further **in**tegrat**in**g the gauge fields and redef**in****in**g(z i ,¯z i ) = 1 O γ(γ i ,¯γ i ) (4.10)yields the action of the CP N−1|N sigma model with coupl**in**gs θ = 2 λ and g2 = πλ2[22].In summary, the complete (not only a subsector) boundary CP N−1|N sigma model atcoupl**in**gs θ = 2 λ and g2 = πλ2has **Yangian** symmetry.5 OutlookWe have found **Yangian** symmetry **in** a subsector of many conformal field theories withglobal superalgebra symmetry. The underly**in**g Lie superalgebra has the crucial propertythat its Kill**in**g form vanishes identically. Our derivation was divided **in**to several steps.Thestart**in**gpo**in**twasaconformalfieldtheorythatpossesses Kac-Moodycurrent symmetry.Non-local charges depend**in**g on a po**in**t P were constructed **in** terms of these currentsand their properties were computed. The crucial result was that correlation functions ofprimaries **in** the subsector Y, that allows for trivial lifts to **Yangian** representation, obeythe Ward identity (3.37). This identity is **in**dependent of P. This observation led usto def**in**e new charges that are aga**in** a symmetry for the subsector Y due to the Wardidentity (3.37). Then perturbations that preserve this **Yangian** but not the Kac-Moodysymmetry were considered. F**in**ally, we gave some examples. We also derived a first orderformulation for CP N−1|N sigma models.There are several open questions.The apparent problem is to understand which representation of the Lie superalgebra canbe lifted trivially to a representation of the **Yangian**, i.e. to identify the subsector of theconformal field theory hav**in**g **Yangian** symmetry. Once know**in**g these representation oneshould search for solutions to the Ward identities and classify them. We hope that theyconstra**in** correlation functions severely.The list of examples can certa**in**ly be extended, e.g. one could search for massive deformations,one can apply it to world-sheet supersymmetric supergroup WZW models[63]. Also our method applies to the current-current deformations recently studied **in**[28] **in** this case for the disc as world-sheet. Here it would be **in**terest**in**g to understandhow the **Yangian** symmetry is preserved by current-current perturbations accord**in**g tothe quadratic Casimir for the sphere as world-sheet.20

AcknowledgementI am very grateful to Louise Dolan for regular discussions and helpful comments. I alsowould like to thank Brendan McLellan for discussions on **Yangian** superalgebras and theirrepresentations, and Peter Rønne, Till Bargheer, Dmitri Sorok**in**, Cosmas Zachos and JanPlefka for their comments.A Useful **in**tegralsLet f(z) be an analytic function **in**side and on the contour C x . For us f is either J or χ.Note, that the contour is not go**in**g around the po**in**t P. Then [51, 46] have the follow**in**g**in**tegrals∮1( z −x)dzf(z)ln2πi C xP −x=∫ P∮ ∫1 vdzf(z) dv g(v) ∫ x2πi C x P (z −v) =∮1dzf(z)2πi C x∫ xPdvxP1(z −v) 2 = f(x)∮ ∫1 z ∫ x1dzf(z) dw dv =2πi C x P P (w−v) ′ 2∮C xdz∫ zPdw∫ xdv g(z)f(w)P (w −v)′∫ P ′x= 0.dzf(z)dzf(z)g(z)dvf(v)(A.1)References[1] M. Bershadsky, S. Zhukov and A. Va**in**trob, Nucl. Phys. B 559 (1999) 205[arXiv:hep-th/9902180].[2] D. Kaganand C. A. S. Young, Nucl. Phys. B 745 (2006) 109 [arXiv:hep-th/0512250].[3] A. Babichenko, Phys. Lett. B 648 (2007) 254 [arXiv:hep-th/0611214].[4] K. Zarembo, JHEP 1005 (2010) 002. [arXiv:1003.0465 [hep-th]].[5] L. Rozansky, H. Saleur, Nucl. Phys. B389 (1993) 365-423. [hep-th/9203069].[6] L. Rozansky, H. Saleur, Nucl. Phys. B376 (1992) 461-509.[7] V. Schomerus, H. Saleur, Nucl. Phys. B734 (2006) 221-245. [hep-th/0510032].21

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