S-duality transformation of N =4 SYM theory at the operator level
SS-duality transformation of N = 4 SYM theory at theoperator level
Shan Hu
Department of Physics, Faculty of Physics and Electronic Sciences, Hubei University,Wuhan 430062, P. R. China [email protected]
We consider the S-duality transformation of gauge invariant operators and states in N = 4 SYM theory. The transformation is realized through an operator S which is the SL (2 , Z )canonical transformation in loop space with the gauge invariant electric and the magneticflux operators composing the canonical variables. Based on S , S-duals for all of the physicaloperators and states can be defined. The criterion for the theory to be S-duality invariant isthat the superconformal charges and their S-duals differ by a U (1) Y phase. The verificationcan be done by checking the S transformation for supersymmetry and special supersymmetryvariations of the loop operators. The fact that supercharges preserved by BPS Wilsonoperators and the S-dual BPS ’t Hooft operators differ by a 4 d chiral rotation could in somesense serve as a proof. 1 a r X i v : . [ h e p - t h ] M a y . INTRODUCTION N = 4 super Yang-Mills (SYM) theory is believed to realize an SL (2 , Z ) duality [1, 2].The action of the duality group on the gauge coupling τ is τ = τ + iτ = θ π + i πg → aτ + bcτ + d , a bc d ∈ SL (2 , Z ) . (1)Theories with τ ’s related by the SL (2 , Z ) transformations are physically equivalent. Espe-cially, when θ = 0, a = d = 0, (1) reduces to g → π/g , the strong-weak duality of Montonenand Olive [3–5].In Coulomb phase, the action of SL (2 , Z ) on BPS states is well known [1]. BPS massspectrum is invariant under (1). The perturbative massive vector multiplets are mapped intothe vector multiplets arising from the quantization of the monopoles or dyons. In conformalphase, when the theory exhibits the P SU (2 , |
4) superconformal symmetry, the observablesof interest are gauge invariant operators and their correlation functions. The SL (2 , Z ) trans-formation will make a gauge invariant operator O mapped into a gauge invariant operator O (cid:48) . Correlation functions of O (cid:48) in theory with the coupling constant τ equal the correlationfunctions of O in the same theory with the coupling constant ( aτ + b ) / ( cτ + d ): (cid:104) (cid:89) i O (cid:48) i (cid:105) τ = (cid:104) (cid:89) i O i (cid:105) aτ + bcτ + d . (2)Explicit conjectures have been made for the S-duality actions on the local operators [6–8],line operators [9, 10], surface operators [11, 12] and domain walls [13, 14]. Calculations ofthe correlation functions are made in [6, 7, 15, 16]. N = 4 SYM theory is dual to the type IIB string theory on AdS × S which is also SL (2 , Z ) invariant [17–19]. The duality transformation of the latter offers clues for theduality transformation of the former. In type IIB, components of the supergravity multipletare assigned with the definite U (1) Y charges, and the S-duality transformation can be seenas a U (1) Y transformation [20–23]. The corresponding operators in SYM theory shouldhave the opposite U (1) Y charges [24], which then leads to the conjecture that correlationfunctions of 1 / N = 4 SYM theory should exhibit the U (1) Y symmetry[6, 7]. In [8], the action of the SL (2 , Z ) duality on the local gauge invariant operators isstudied based on the conformal weight. It was suggested that for operator with the modularinvariant conformal weight, such as the BPS operator, SL (2 , Z ) transformation will map itinto itself up to a possible multiplicative factor if there is no degeneracy. Otherwise, theoperator will transform as part of a finite or infinite dimensional SL (2 , Z ) multiplet, whichis the situation for the Konishi operator. Generically, S-duality makes N = 4 SYM theory with the gauge group G mapped into the one with thedual group L G [3]. Here, we only consider U ( N ), which is self-dual.
2n this paper, we will study the S-duality transformation for operators and states of N = 4 SYM theory in more detail. Based on the commutation relation between the Wilson and’t Hooft operators in canonical quantization formalism [25], we propose a canonical commu-tation relation in loop space with the gauge invariant electric and magnetic fluxes composingthe canonical variables. The canonical transformation in loop space is the SL (2 , Z ) dual-ity transformation of the theory. The transformation is realized through an operator S ,based on which, the S-duals for all of gauge invariant operators and states can be defined.The criterion for the theory to be S-duality invariant is that superconformal charges andtheir S-duals differ by a U (1) Y phase, or equivalently, supercharges and special superchargestransform with a U (1) Y phase. The fact that supercharges preserved by BPS Wilson oper-ators and those by BPS ’t Hooft operators differ by a 4 d chiral transformation [26] wouldindicate the operator S making the Wilson operator transformed into the ’t Hooft operatorwill also make the supercharges transform with a U (1) Y phase thus could serve as a prooffor the duality invariance.With the criterion met, correlation functions of the dual operators are equal to the cor-relation functions of the original operators in a theory with the dual coupling constant as isin (2). The P SU (2 , |
4) irreducible state with the energy E ( τ ) is mapped into a state in thesame theory with the energy E ( aτ + bcτ + d ) and the identical SO (3) × SU (4) quantum number.Although the correlation functions of 1 / U (1) Y phase under the SL (2 , Z ) transformation according to AdS/CFT[6], we show that not all of the 1 / U (1) gauge theory; in section 3, we review the S-duality transformationin type IIB string theory; in section 4, we study the S-duality transformation of N = 4 SYMtheory; the discussion is in section 5. II. S-DUALITY TRANSFORMATION IN U (1) GAUGE THEORY
For Maxwell theory, S-duality transformation is implemented as E i → B i , B i → − E i both classically and quantum mechanically, i = 1 , ,
3. In temporal gauge, the canonicaloperators are ( A i , E i ), and the gauge invariant physical Hilbert space H ph is obtained byimposing the Gauss constraint ∂ i E i = 0 in H . ∂ i B i = 0 always holds but ∂ i E i = 0 isonly valid in H ph . Both E i and B i are physical operators. S-duality transformation can berealized via a unitary operator S in H ph : S ˆ E i S − = ˆ B i , S ˆ B i S − = − ˆ E i , (1)where ˆ represents a projection in H ph . There is no unitary operator relating E i and B i in H , since ∂ i E i = 0 only in H ph .The situation for the U (1) Born-Infeld theory with the couping g is also similar [27–30].3n temporal gauge, the canonical operators are ( A i , D i ) and H ph is obtained by imposing ∂ i D i = 0 in H . ∂ A i = E i (cid:54) = D i = δLδE i = D i ( g , A i , E i ). D i and B i are physical operators.S-duality transformation is still realized via a unitary operator S : S ˆ D i S − = ˆ B i , S ˆ B i S − = − ˆ D i . (2)The Hamiltonian is h = (cid:114) g D i D i + 1 g B i B i + ( (cid:15) ijk D j B k )( (cid:15) imn D m B n ) − , (3) S ˆ h ( g ) S − = ˆ h (1 /g ). Hamiltonians with the coupling constants g and 1 /g are related bythe unitary transformation S .Generically, for the arbitrary U (1) gauge theory in temporal gauge, the canonical opera-tors are ( A i , Π i ) with ∂ i Π i = 0 in H ph . The operator S is constructed as S ˆΠ i S − = ˆ B i , S ˆ B i S − = − ˆΠ i . (4)In canonical quantization formalism, the definition of the S-duality transformation onlydepends on the field content and has nothing to do with the Hamiltonian. We can studythe effect of the transformation on the Hamiltonian h ( g , B i , Π i ). If S ˆ h ( g , B i , Π i ) S − = ˆ h ( g , − Π i , B i ) = ˆ h ( 1 g , B i , Π i ) , (5)the theory is S-duality invariant. Otherwise, the original theory is dual to a different theorywith the coupling constant g and the different Hamiltonian.In (4), S is defined through its action on the local gauge invariant operators in theordinary space, but it could also be equivalently defined in loop space. For U (1) gaugetheory, the Wilson operator is W ( C ) = e iw ( C ) with the magnetic flux operator w ( C ) = (cid:73) C A i ( x ) dx i = (cid:90) (cid:90) Σ C B i ( x ) dσ i . (6)The projection of w ( C ) in H ph is denoted as ˆ w ( C ). The U (1) version of the ’t Hooft operatoris ˆ T ( C ) = e i ˆ t ( C ) with the electric flux operatorˆ t ( C ) = (cid:90) (cid:90) Σ C ˆΠ i ( x ) dσ i . (7)4he direct calculation shows [ˆ t ( C (cid:48) ) , ˆ w ( C )] = 2 πin , (8)where C and C (cid:48) are arbitrary two spatial loops at the equal time with the linking number n . (8) could be taken as the canonical commutation relation in loop space and ( ˆ w, ˆ t ) arecanonical variables with the suitable constraints imposed to eliminate the unphysical con-figurations that could not be obtained from the spacetime gauge potential. From (8), the U (1) version of the commutation relation for Wilson and ’t Hooft operators [25]ˆ W ( C ) ˆ T ( C (cid:48) ) = ˆ T ( C (cid:48) ) ˆ W ( C ) exp(2 πin ) (9)can be obtained.From (4), we have S ˆ t ( C ) S − = ˆ w ( C ) , S ˆ w ( C ) S − = − ˆ t ( C ) . (10)Conversely, if (10) is satisfied for the arbitrary spatial loops, (4) will hold. (4) and (10) aretwo equivalent ways to define S . In nonabelian theory, the definition (4) does not work sinceS-duality transformation can only be definitely determined for operators in H ph but B i andΠ i are not gauge invariant any more. On the other hand, loop operators are always gaugeinvariant, so (10) is still valid. Later, we will try to extend (10) into the N = 4 SYM theoryto construct a S-duality transformation operator S . III. S-DUALITY TRANSFORMATION IN TYPE IIB STRING THEORY
Type IIB string theory is SL (2 , Z ) invariant. The 32 supercharges of Type IIB theorycould be combined into two left handed Majorana-Weyl spinors Q L and Q R , satisfying boththe Majorana condition Q TL C = Q L , Q TR C = Q R , (1)and the Weyl condition Γ Q L = Q L , Γ Q R = Q R . (2)It is convenient to combine two Majorana-Weyl spinors into complex Weyl spinors Q ± = Q L ± iQ R . (3) We make a rescaling here to make [ A i ( x ) , Π j ( y )] = − πiδ ij δ ( x − y ) since eg = 2 π according to the Diracquantization condition. Here, L and R refer to the chirality in string worldsheet. Q + → ( cτ + dc ¯ τ + d ) Q + , Q − → ( cτ + dc ¯ τ + d ) − Q − , (4)where a bc d ∈ SL (2 , Z ) . (5) τ is the complex scalar field built out of the dilaton and the the axion. Type IIB supergravityfields could be organized into a supermultiplet with the definite U (1) Y charge [20–23]. Theaction of SL (2 , Z ) will make the fields with the U (1) Y charge q transform as F → ( cτ + dc ¯ τ + d ) q F . (6)S-duality transformation makes F-string mapped into the ( a, b ) string, and the first quan-tization of them gives the string fields f (1 , m,n and f ( a,b ) m,n that should also be mapped into eachother. As the ( m, n ) excitation of the ( a, b ) string, f ( a,b ) m,n is non-perturbative in (1 ,
0) F-stringframe. However, the massless excitations for all of the ( a, b ) strings are the same supergrav-ity multiplet related by a U (1) Y rotation. In this sense, supergravity multiplet is universalin all of the S-frame. IV. S-DUALITY TRANSFORMATION IN N = 4 SYM THEORY
S-duality transformation in U (1) gauge theory is realized through a unitary operator S in physical Hilbert space. We will construct the similar S for N = 4 SYM theory with thegauge group U ( N ). A. Definition of the S-duality transformation
The Lagrangian of the N = 4 SYM theory with the coupling constant τ = τ + iτ is L = τ π tr {− F µν F µν + τ τ F µν ∗ F µν − i ¯Ψ a ¯ σ µ D µ Ψ a − D µ X I D µ X I + 12 C abI Ψ a [ X I , Ψ b ] + 12 ¯ C Iab ¯Ψ a [ X I , ¯Ψ b ] + 14 [ X I , X J ] } . (1) D µ f = ∂ µ f − [ A µ , f ]. In temporal gauge, A = 0, canonical fields are ( X I , Ψ a , A i ) with theconjugate momentum (Π I , Π a , Π i ). I = 1 , , · · · , a = 1 , , , i = 1 , ,
3. LetΩ A = ∂ i Π Ai − if ABC A i B Π Ci + if ABC X I B Π CI − if ABC Ψ a B Π Ca , (2)6 = 1 , , · · · , N . The physical Hilbert space H ph is composed by states satisfying Ω A | ψ (cid:105) =0. Ω A and H ph are τ -independent.Hilbert space H could be decomposed into the direct sum of H ph and its complementaryspace: H = ¯ H ph ⊕ H ph . A generic operator O could be correspondingly decomposed as O = O O O O . (3)Ω A and the physical operators take the form ofΩ A = Ω A
00 0 O ph = O
00 ˆ O ph , (4)where ˆ O ph is the projection of O ph in H ph .Extending (10) into the U ( N ) theory requires the construction of the corresponding fluxoperators. For U ( N ) YM theory, the Wilson operator isˆ W ( C ) = 1 N ˆ tr (cid:20) P exp { i (cid:73) C A i ( x ) dx i } (cid:21) (5)in fundamental representation. ˆ tr represents a projection in H ph . W ( C ) for the arbitraryspatial loops compose the complete gauge invariant observables of the theory. For the YMtheory in temporal gauge, the fundamental field is A i . Polyakov has shown that the gaugeequivalent class of A i can be extracted from the element of the holonomy groupΦ( C, x (0)) = P exp { i (cid:73) C A i ( x ) dx i } , (6)where all loops begin and end at the same fixed point x (0) [34]. W ( C ) = 1 N tr [Φ(
C, x (0))] . (7)Suppose C n is a loop beginning and ending at x (0), winding C n times, then W ( C n ) = 1 N tr [Φ n ( C, x (0))] . (8)With W ( C n ) for n = 1 , , · · · , N given, Φ( C, x (0)) can be determined up to a gauge trans-formation, so W ( C ) contains complete degrees of freedom of the theory. In N = 4 SYMtheory, the complete gauge invariant observables should be the super Wilson operator built7rom the N = 4 vector multiplet. In [35–37], super Wilson operators are constructed andcould be taken as the Wilson operators for 10 d YM theory in superspace.With the super Wilson operator ˆ W ( C ) given, the magnetic flux operator ˆ w ( C ) is obtainedvia ˆ W ( C ) = exp { i ˆ w ( C ) } (9)following the definition in [31]. ˆ w ( C ) could act as the canonical coordinate in loop spacewith the conjugate momentum ˆ t ( C ). The equal time canonical commutation relation is[ˆ t ( C (cid:48) ) , ˆ w ( C )] = 2 πinN (10)and [ ˆ w ( C ) , ˆ w ( C (cid:48) )] = [ˆ t ( C ) , ˆ t ( C (cid:48) )] = 0 (11)with n the linking number between C (cid:48) and C . Forˆ T ( C ) = exp { i ˆ t ( C ) } , (12)from (10), ˆ W ( C ) ˆ T ( C (cid:48) ) = ˆ T ( C (cid:48) ) ˆ W ( C ) exp(2 πin/N ) , (13)which is the commutation relation for the Wilson and ’t Hooft operators [25], so ˆ T ( C ) canbe taken as the ’t Hooft operator.Consider the canonical transformation in loop space: S [ a bc d ; τ ] ˆ w ( C )ˆ t ( C ) S − [ a bc d ; τ ] = a cb d ˆ w ( C )ˆ t ( C ) . (14)To preserve (10)-(11), ad − bc = 1. Moreover, ( W ( C )) N = ( T ( C )) N = 1, ˆ w ( C ) and ˆ t ( C ) arequantized in units 2 π/N [31], which also breaks SL (2 , R ) to SL (2 , Z ). The correspondingloop operators transform as S − ˆ W ( C ) S = ˆ T − c ( C ) ˆ W d ( C ) S − ˆ T ( C ) S = ˆ T a ( C ) ˆ W − b ( C ) . This is the S-duality transformation rule for Wilson and ’t Hooft operators in N = 4 SYMtheory [9, 32, 33], so the canonical transformation operator S generates the SL (2 , Z ) dualitytransformation.The unitary operator S in H ph can also be extended into a unitary physical operator U H : U = V S , (15)where V is an arbitrary unitary operator acting on ¯ H ph . In this way, although not unique,the S-duality transformation of canonical fields can also be defined: U − XU = ˜ X U − Ψ U = ˜Ψ U − AU = ˜ AU − Π X U = ˜Π X U − Π Ψ U = ˜Π Ψ U − Π A U = ˜Π A . (16)(16) could be regarded as a canonical transformation in ordinary space. In contrast to loopspace, where S has a simple representation as an SL (2 , Z ) transformation, for the nonabeliantheory in ordinary space, it is difficult to tell the exact relation between the S-dual fieldsand the original fields. B. Criterion for the S-duality invariance
At this stage, the action of the S-duality transformation has been defined. The defini-tion is based on the field content with no dynamical information involved. The dynamicalinformation of N = 4 SYM theory is encoded in superconformal charges.For N = 4 SYM theory with the coupling constant τ , suppose G is a P SU (2 , |
4) gener-ator with the U (1) Y charge q , G ( τ ) = G [ X, Ψ , A ; Π X , Π Ψ , Π A | τ ] . (1)The generator constructed from the S-dual canonical fields with the coupling constant aτ + bcτ + d is ˜ G ( aτ + bcτ + d ) = G [ ˜ X, ˜Ψ , ˜ A ; ˜Π X , ˜Π Ψ , ˜Π A | aτ + bcτ + d ]= U − G ( aτ + bcτ + d ) U = G (cid:48) [ X, Ψ , A ; Π X , Π Ψ , Π A | τ ] = G (cid:48) ( τ ) . (2)The projection of G ( τ ) and G (cid:48) ( τ ) in H ph is ˆ G ( τ ) and ˆ G (cid:48) ( τ ). Then the criterion for the9heory to be S-duality invariant is ˆ G (cid:48) ( τ ) = ˆ˜ G ( aτ + bcτ + d ) = S − ˆ G ( aτ + bcτ + d ) S = ( cτ + dc ¯ τ + d ) − q ˆ G ( τ ) . (4)Especially, when G is the Hamiltonian,ˆ H (cid:48) ( τ ) = ˆ˜ H ( aτ + bcτ + d ) = S − ˆ H ( aτ + bcτ + d ) S = ˆ H ( τ ) . (5)If (5) is not satisfied, the original theory will be dual to another theory with a differentHamiltonian and the coupling constant aτ + bcτ + d . N = 4 SYM theory is the low energy effective theory on D D
3, we may select two sets of canonical fields ( X, Ψ , A ) and ( ˜ X, ˜Ψ , ˜ A ) comingfrom the quantization of the open F-string and the open ( a, b )-string, and then constructtwo theories with the coupling constants τ and aτ + bcτ + d . Two theories have the same physicalHilbert space H ph with the superconformal charges related via (4). The dynamics of D S defined through (14) could make (4) satisfied. Ac-cording to (4), for the supercharge Q and the special supercharge S ,( cτ + dc ¯ τ + d ) − ˆ Q aα ( τ ) = S − ˆ Q aα ( aτ + bcτ + d ) S , ( cτ + dc ¯ τ + d ) ˆ¯ Q a ˙ α ( τ ) = S − ˆ¯ Q a ˙ α ( aτ + bcτ + d ) S , ( cτ + dc ¯ τ + d ) ˆ S a ˙ α ( τ ) = S − ˆ S a ˙ α ( aτ + bcτ + d ) S , ( cτ + dc ¯ τ + d ) − ˆ¯ S aα ( τ ) = S − ˆ¯ S aα ( aτ + bcτ + d ) S . (6)Based on the superconformal algebra, { ˆ Q aα , ˆ¯ Q b ˙ β } = 2 σ µα ˙ β ˆ P µ δ ab , { ˆ S aα , ˆ¯ S b ˙ β } = 2 σ µα ˙ β ˆ K µ δ ba , (7) { ˆ Q aα , ˆ S bβ } = (cid:15) αβ ( δ ab ˆ D + ˆ R ab ) + 12 δ ab σ µναβ ˆ M µν , (8)(6) also givesˆ P µ ( τ ) = S − ˆ P µ ( aτ + bcτ + d ) S , ˆ M µν ( τ ) = S − ˆ M µν ( aτ + bcτ + d ) S , ˆ R ab ( τ ) = S − ˆ R ab ( aτ + bcτ + d ) S (9) In (4), it is necessary to make a projection in H ph to get ˆ G from G . For example, when G is the supercharge Q , since { Q aα , Q bβ } = − ig (cid:90) d x (cid:15) αβ tr { Ω X ab } { ¯ Q a ˙ α , ¯ Q b ˙ β } = − ig (cid:90) d x (cid:15) ˙ α ˙ β (cid:15) abcd tr { Ω X cd } , (3)there is no unitary transformation U making U QU − = e iθ Q and U ¯ QU − = e − iθ ¯ Q . On the other hand, in H ph , Ω = 0, so there can be a unitary transformation S making S ˆ QS − = e iθ ˆ Q and S ˆ¯ QS − = e − iθ ˆ¯ Q . In Coulomb phase, the perturbative degrees of freedom are vector multiplet, from which, the Hilbert spaceis constructed. The configuration space also contains monopoles and dyons, and the quantization of themgives a set of vector multiplets labelled by ( a, b ) in the same Hilbert space. The action of SL (2 , Z ) is alsodefined as a transformation from one set of the vector multiplet into another. K µ ( τ ) = S − ˆ K µ ( aτ + bcτ + d ) S , ˆ D ( τ ) = S − ˆ D ( aτ + bcτ + d ) S , (10)so it is sufficient to prove (6).Since { W ( C ) , T ( C ) } compose the complete physical operators, it is enough to checkthe action of Q, ¯ Q, S, ¯ S on W and T under the S transformation. In [26], supersym-metries preserved by various BPS Wilson-’t Hooft operators are studied. It was shownthat supercharges preserved by BPS Wilson loops and their magnetic counterparts are re-lated by a four dimensional chiral transformation. This could serve as a proof for theS-duality invariance of the theory. Let { A k | k = 1 , , · · · , n } represent a subset of super-charges and H { A k | k =1 , , ··· ,n } denote the space of BPS Wilson operators annihilated by them. S − H { A k | k =1 , , ··· ,n } S will be the space of the dual BPS ’t Hooft operators annihilated by { S − A k S | k = 1 , , · · · , n } . If S − H { A k | k =1 , , ··· ,n } S is also annihilated by { B k | k = 1 , , · · · , n } which is related to { A k | k = 1 , , · · · , n } by a chiral rotation, there will be { S − A k S | k =1 , , · · · , n } = { B k | k = 1 , , · · · , n } since otherwise, S − H { A k | k =1 , , ··· ,n } S would have thelower dimension. We arrive at the conclusion that the operator S making the Wilson oper-ator transformed into the ’t Hooft operator will also make the supercharges and the specialsupercharges transform with a U (1) Y phase so the theory is duality invariant. C. S-duality transformation of gauge invariant operators
In this section, we will study the S-duality transformation of gauge invariant operatorsand especially, the 1 /
1. Generic
In theory with the coupling constant τ , the generic on-shell gauge invariant operator O could be constructed from the canonical fields: O ( τ ) = O [ X, Ψ , A ; Π X , Π Ψ , Π A | τ ] . (1)The S-dual of O ( τ ) is constructed from the dual canonical fields and the dual couplingconstant: ˜ O ( aτ + bcτ + d ) = O [ ˜ X, ˜Ψ , ˜ A ; ˜Π X , ˜Π Ψ , ˜Π A | aτ + bcτ + d ]= U − O ( aτ + bcτ + d ) U = O (cid:48) [ X, Ψ , A ; Π X , Π Ψ , Π A | τ ] = O (cid:48) ( τ ) . (2)11 (cid:48) ( τ ) is still an operator in theory with the coupling constant τ and may be highly nonper-turbative when written in terms of the original fields. The projection in H ph isˆ˜ O ( aτ + bcτ + d ) = S − ˆ O ( aτ + bcτ + d ) S = ˆ O (cid:48) ( τ ) . (3)The correlation functions of O (cid:48) ( τ ) and O ( τ ) in theory with the coupling constant τ are (cid:104) | O (cid:48) ( x , t | τ ) O (cid:48) ( x , t | τ ) | (cid:105) = (cid:104) | O (cid:48) ( x , | τ ) e i ˆ H [ τ ]( t − t ) O (cid:48) ( x , | τ ) | (cid:105) = (cid:104) (cid:48) | O ( x , | aτ + bcτ + d ) S (0) e i ˆ H [ τ ]( t − t ) S − (0) O ( x , | aτ + bcτ + d ) | (cid:48) (cid:105) = (cid:104) (cid:48) | O ( x , | aτ + bcτ + d ) e i ˆ H [ aτ + bcτ + d ]( t − t ) O ( x , | aτ + bcτ + d ) | (cid:48) (cid:105) (4)and (cid:104) | O ( x , t | τ ) O ( x , t | τ ) | (cid:105) = (cid:104) | O ( x , | τ ) e i ˆ H [ τ ]( t − t ) O ( x , | τ ) | (cid:105) . (5)The correlation function of O (cid:48) in theory with the coupling constant τ is equal to the correla-tion function of O in the same theory with the coupling constant aτ + bcτ + d as is required in (2).To get (4), we used (5), which, if is not satisfied, will make the correlation function of O (cid:48) mapped into the correlation function of O in a different theory with the coupling constant aτ + bcτ + d .In special case, if S − ˆ O ( x | aτ + bcτ + d ) S = ˆ O (cid:48) ( x | τ ) = ( cτ + dc ¯ τ + d ) − q ˆ O ( x | τ ) (6)with q the U (1) Y charge of O , then according to (4), (cid:104) (cid:48) | O ( x , | aτ + bcτ + d ) e i ˆ H [ aτ + bcτ + d ]( t − t ) O ( x , | aτ + bcτ + d ) | (cid:48) (cid:105) = ( cτ + dc ¯ τ + d ) − q (cid:104) | O ( x , | τ ) e i ˆ H [ τ ]( t − t ) O ( x , | τ ) | (cid:105) . (7)The correlation functions of O in theories with the coupling constants aτ + bcτ + d and τ differ bya U (1) Y phase. Generically, (cid:104) (cid:89) i O ( q i ) i ( x i ) (cid:105) aτ + bcτ + d = ( cτ + dc ¯ τ + d ) − (cid:80) i q i (cid:104) (cid:89) i O ( q i ) i ( x i ) (cid:105) τ , (8)where q i is the U (1) Y charge of O ( q i ) i . This is the U (1) Y transformation rule for the correlationfunction of 1 / . / BPS operators / O ( τ ) = τ k/ ˆ tr ( X { I · · · X I k } ). ˆ O ( m,n ) ∼ δ m ¯ δ n ˆ O has the U (1) Y charge q = m − n . According to AdS/CFT, ˆ O ( m,n ) is mapped into the supergravity field F ( m,n ) .Under the SL (2 , Z ) transformation in type IIB string theory, F ( m,n ) → ( cτ + dc ¯ τ + d ) q F ( m,n ) . (9)The low energy effective action of type IIB string theory is expected to be SL (2 , Z ) invariantand so the correlation functions of 1 / O (cid:48) ( m,n ) ( x | τ ) = ( cτ + dc ¯ τ + d ) − q ˆ O ( m,n ) ( x | τ ) , (10)(8) can be automatically satisfied. Moreover, current multiplet is also 1 / U (1) Y phase thus could ensurethe S-duality invariance of the theory. In the following, we will check the validity of (10).Suppose the gauge group is SU (2) and for simplicity, let τ = iτ and consider the trans-formation with τ → − /τ . For O [ ab ][ cd ] ( τ ) = τ tr (2 X ab X cd + X ac X bd − X ad X bc ) (11) O a [ cd ] α ( τ ) = τ tr (2Ψ aα X cd + Ψ cα X ad − Ψ dα X ac ) (12) O ( ab ) ( τ ) = τ tr ( − Ψ αa Ψ bα + t ( ab ) cdefgh X cd X ef X gh ) (13)with the U (1) Y charges 0, 1 and 2, where O [ ab ][ cd ] ∼ O is the chiral primary operator, O a [ cd ] α ∼ δO and O ( ab ) ∼ δ O are descendants,ˆ O [ ab ][ cd ] ( τ ) → ˆ O [ ab ][ cd ] ( − /τ ) (14)ˆ O a [ cd ] α ( τ ) → e iπ ˆ O a [ cd ] α ( − /τ ) (15)ˆ O ( ab ) ( τ ) → e iπ ˆ O ( ab ) ( − /τ ) (16)should be realized. For SU (2) group, at the classical level, (14) can only be possible if X ab → X ab /τ up to a gauge transformation. The multiplication of e iπ for O a [ cd ] α requiresΨ aα → e iπ Ψ aα /τ . And then (16) cannot be satisfied unless the gauge group is U (1). Thesituation is the same with fields replaced by operators. The analysis does not rely on S justindicating O , δO and δ O cannot transform as in (14)-(16) simultaneously. So, at least forsome 1 / S should make (6) satisfied, so S cannot make the chiral primary operatorsˆ O remain invariant, otherwise, (10) will always hold. In fact, the invariance of ˆ O is a strongconstraint which requires ˜ X I = | cτ + d | X I in (16) up to a gauge transformation.13t remains to determine how the operator S making Wilson operators transformed intothe ’t Hooft operators will act on ˆ O . With S ˆ OS − given, X can be fixed up to a gaugetransformation and then, for an arbitrary physical operator K ( X ) composed by X , includ-ing all of the superconformal primary operators, S ˆ K ( X ) S − is also determined. With (4)satisfied, the successive action of superconformal charges will then give the dual operatorsfor the whole superconformal multiplet. D. S-duality transformation of physical states
The physical Hilbert space H ph forms a reducible representation of the superconformalgroup and could be decomposed into the direct sum of the irreducible subspaces: H ph = ⊕ H ( i ) ir ( τ ). The decomposition is τ -dependent and the states in H ph are required to benormalizable. The superconformal generators G ( τ ) are then the block diagonal matrices inthis representation: G ( τ ) = ⊕ G ( i ) ir ( τ ).The global time Hamiltonian is h = ( K + P ) = ( K + H ), S ˆ h ( τ ) S − = ˆ h ( aτ + bcτ + d ) . (1)ˆ h has the normalizable eigenstates {| E i (cid:105)} with the discrete eigenvalues { E i } ,ˆ h ( τ ) | E i ( τ ) (cid:105) = E i ( τ ) | E i ( τ ) (cid:105) . (2)Since ˆ h ( aτ + bcτ + d ) | E i ( aτ + bcτ + d ) (cid:105) = E i ( aτ + bcτ + d ) | E i ( aτ + bcτ + d ) (cid:105) , (3)we have ˆ h ( τ ) S − | E i ( aτ + bcτ + d ) (cid:105) = E i ( aτ + bcτ + d ) S − | E i ( aτ + bcτ + d ) (cid:105) . (4)If | E i ( τ ) (cid:105) is a normalizable eigenstate of ˆ h ( τ ) with the eigenvalue E i ( τ ), S − | E i ( aτ + bcτ + d ) (cid:105) willbe a normalizable eigenstate of ˆ h ( τ ) with the eigenvalue E i ( aτ + bcτ + d ). The spectrum of ˆ h ( τ ) is SL (2 , Z ) invariant: { E i ( τ ) } = { E i ( aτ + bcτ + d ) } .Generically, for a state | f ( τ ) (cid:105) , the dual state is S − | f ( aτ + bcτ + d ) (cid:105) . In each irreduciblerepresentation, if | f ( τ ) (cid:105) is the superconformal primary state with the energy E ( τ ), G ( τ ) · · · G ( τ ) | f ( τ ) (cid:105) will give the whole multiplet. The S-dual of | f ( τ ) (cid:105) is the super-conformal primary state S − | f ( aτ + bcτ + d ) (cid:105) with the energy E ( aτ + bcτ + d ), while the S-dual of G ( τ ) · · · G ( τ ) | f ( τ ) (cid:105) is S − G ( aτ + bcτ + d ) · · · G ( aτ + bcτ + d ) | f ( aτ + bcτ + d ) (cid:105) ∼ G ( τ ) · · · G ( τ ) S − | f ( aτ + bcτ + d ) (cid:105) . Sothe duality maps one irreducible representation into another. State and the dual state arein the same SO (3) × SU (4) representation with the energies E ( τ ) and E ( aτ + bcτ + d ), respectively.A normalizable state | O ( τ ) (cid:105) in H ph corresponds to a renormalized operator O r ( τ ) with14he finite two point function [38]. | O ( τ ) (cid:105) = e π ( H − K ) O r (0 | τ ) | (cid:105) . (5) O r can be expanded in terms of the bases { O Ir } with O Ir ( τ ) = Z IJ ( τ ) O J ( τ ). O J ( τ ) is the bareoperator and Z IJ ( τ ) is the renormalization matrix. The τ -dependence of O r ( τ ) also comesfrom the renormalization. If ˆ h ( τ ) | O ( τ ) (cid:105) = ∆( τ ) | O ( τ ) (cid:105) , (6)then [ ˆ D ( τ ) , ˆ O r ( τ )] = i ∆( τ ) ˆ O r ( τ ) . (7)The dual operator of ˆ O r ( τ ) is ˆ O (cid:48) r ( τ ) = S − ˆ O r ( aτ + bcτ + d ) S related with the normalizable state S − | O ( aτ + bcτ + d ) (cid:105) and so, are still properly renormalized with the finite two point function.[ ˆ D ( aτ + bcτ + d ) , ˆ O r ( aτ + bcτ + d )] = i ∆( aτ + bcτ + d ) ˆ O r ( aτ + bcτ + d ) . (8)Since S ˆ D ( τ ) S − = ˆ D ( aτ + bcτ + d ), [ ˆ D ( τ ) , ˆ O (cid:48) r ( τ )] = i ∆( aτ + bcτ + d ) ˆ O (cid:48) r ( τ ) . (9)The spectrum of conformal dimension, which is the same as the spectrum of ˆ h , is SL (2 , Z )invariant: { ∆ i ( τ ) } = { ∆ i ( aτ + bcτ + d ) } .Now consider the chiral primary operator ˆ O ( τ ) = τ k/ ˆ tr ( X { I · · · X I k } ) and the corre-sponding state | O ( τ ) (cid:105) , ˆ O (cid:48) ( τ ) (cid:54) = ˆ O ( τ ) according to the previous discussion. However, as the SO (3) invariant state with the energy k in (0 , k,
0) representation of SU (4), | O ( τ ) (cid:105) is unique,so | O ( τ ) (cid:105) = | O (cid:48) ( τ ) (cid:105) based on (4). We have to assume[ ˆ O (cid:48) ( τ ) − ˆ O ( τ )] | (cid:105) = 0 . (10)The difference between ˆ O (cid:48) and ˆ O annihilates the vacuum. Successive action of the super-charges gives [ ˆ O (cid:48) ( m,n ) ( τ ) − ( cτ + dc ¯ τ + d ) − q ˆ O ( m,n ) ( τ )] | (cid:105) = 0 . (11)With (11) satisfied, correlation functions of 1 / U (1) Y transformation rule. In fact, it is also emphasized in [7] that U (1) Y ruleonly applies for the separated operators.The coincident 1 / O ( τ ) ˆ O ( τ ) | (cid:105) andˆ O (cid:48) ( τ ) ˆ O (cid:48) ( τ ) | (cid:105) , when properly renormalized, will correspond to the two particle bound stateswith the coupling constant dependent energy. State and its S-dual have the different ener-gies thus could not be the same any more. The difference between ˆ O (cid:48) and ˆ O may have themanifestation here. E. θ structure and the large gauge transformation It is necessary to consider the impact of the S-duality transformation on the global struc-ture of the gauge theory. In temporal gauge, time-independent gauge transformation isrealized through the unitary operator Ξ( ω ( x i )) generating a U ( N ) gauge transformation u ( ω ( x i )) = e it A ω A . When | (cid:126)x | → ∞ , u ( ω ( x i )) → I . Π ( U ( N )) ∼ = Z , u ( ω ) is classified by itswinding number n , n = 124 π (cid:90) d x (cid:15) ijk tr [( u − ∂ i u )( u − ∂ j u )( u − ∂ k u )] . (1)Ξ and u with the winding number n are denoted as Ξ ( n ) and u ( n ) . Ξ ( n ) Ξ ( m ) = Ξ ( n + m ) , u ( n ) u ( m ) = u ( n + m ) [39]. H ph could be decomposed into the “direct integral” of the subspaces H ph θ . θ ∼ = θ + 2 π . ∀ | ψ (cid:105) θ ∈ H ph θ , [40] Ξ ( n ) | ψ (cid:105) θ = e inθ | ψ (cid:105) θ . (2)Especially, Ξ (0) | ψ (cid:105) = | ψ (cid:105) , ∀ | ψ (cid:105) ∈ H ph . For a theory with the coupling constant τ = θ π + i πg ,the true physical Hilbert space is H ph θ . Suppose | (cid:105) θ is the vacuum in H ph θ , states in H ph θ can be constructed as ˆ O | (cid:105) θ with ˆ O the gauge invariant operator, [Ξ ( n ) , ˆ O ] = 0, ∀ n .Under the S-duality transformation, τ → τ (cid:48) = ( aτ + b ) / ( cτ + d ) = θ (cid:48) π + i πg (cid:48) , we may have S [ a bc d ; τ ] | (cid:105) θ = | (cid:105) θ (cid:48) . (3)Accordingly, in theory with the coupling constant τ , the S-dual of | (cid:105) θ should be | (cid:105) (cid:48) θ = S − | (cid:105) θ (cid:48) = | (cid:105) θ (4)which is the same as | (cid:105) θ as is required.Generically, ∀ | ψ ( τ ) (cid:105) θ ∈ H ph θ , the dual state is | ψ ( τ ) (cid:105) (cid:48) θ = S − | ψ ( τ (cid:48) ) (cid:105) θ (cid:48) ∈ H ph θ . SinceΞ ( n ) | ψ ( τ ) (cid:105) θ = e inθ | ψ ( τ ) (cid:105) θ Ξ ( n ) | ψ ( τ (cid:48) ) (cid:105) θ (cid:48) = e inθ (cid:48) | ψ ( τ (cid:48) ) (cid:105) θ (cid:48) , (5)16here will be S − Ξ ( n ) S | ψ ( τ ) (cid:105) (cid:48) θ = e inθ (cid:48) | ψ ( τ ) (cid:105) (cid:48) θ = e in ( θ (cid:48) − θ ) Ξ ( n ) | ψ ( τ ) (cid:105) (cid:48) θ . (6)We may expect S − [ a bc d ; τ ]Ξ ( n ) S [ a bc d ; τ ] = e in ( θ (cid:48) − θ ) Ξ ( n ) , (7)or equivalently, Ξ ( n ) S [ a bc d ; τ ]Ξ ( n ) − = e in ( θ (cid:48) − θ ) S [ a bc d ; τ ] . (8)Although S commutes with Ξ (0) , it does not necessarily commute with Ξ ( n ) for n ≥ τ → τ + φ π with U ( φ )Π Ai U − ( φ ) = Π Ai + φ π B Ai U ( φ ) A Ai U − ( φ ) = A Ai . (9)The unitary operator U ( φ ) realizing such transformation is explicitly constructed [41]: U ( φ ) = e iφq , (10)where q = (cid:90) d x K ( x ) (11)is the topological charge. K is the time component of the current K µ = 18 π (cid:15) µνλσ ( A Aν F Aλσ − f ABC A Aν A Bλ A Cσ ) . (12)For the θ vacuum | (cid:105) θ ∼ (cid:88) n e inθ | n (cid:105) (13)with | n (cid:105) representing the pure gauge state with the winding number n , U ( φ ) | (cid:105) θ = | (cid:105) θ + φ . (14)Since Ξ ( n ) q Ξ ( n ) − = q + n , Ξ ( n ) e iφq Ξ ( n ) − = e iφn e iφq , (15)17or S [ φ π ; τ ] = e iφq , (16)there will be Ξ ( n ) S [ φ π ; τ ] Ξ ( n ) − = e iφn S [ φ π ; τ ] . (17) V. DISCUSSION
S-duality transformation in loop space can be explicitly realized as an SL (2 , Z ) canonicaltransformation. However, there is a gap between the loop space and the ordinary space, andthe question is how the same S will act on the local gauge invariant operators. Since localoperators also appear in the OPE of loop operators [16, 42, 43]: W ( L R ) = (cid:104) W ( L R ) (cid:105) (1 + (cid:88) i c i a ∆ i O i ) , (1) T ( L R ) = (cid:104) T ( L R ) (cid:105) (1 + (cid:88) i b (cid:48) i a ∆ (cid:48) i O (cid:48) i ) , (2)by studying the OPE of Wilson and ’t Hooft operators, the S-dual local operators can beobtained. For Wilson operator, the right hand side of (1) can be written explicitly [43], butthe challenge is to get O (cid:48) i from T ( L R ) which has a quantum mechanical definition in pathintegral formalism [15].The commutation relation between the Wilson and ’t Hooft operators leads to a naturalcanonical commutation relation in loop space with the linking number n ( C, C (cid:48) ) playing therole of δ ( (cid:126)x − (cid:126)x (cid:48) ). It seems that S-duality transformation rule always takes a more simpleform for loop operators. It is necessary to study the loop space formulation of N = 4 SYMtheory in more detail, especially, with the super Wilson operators [35–37].The operator S is defined through its action on operators which are both local and globalgauge invariant. The vacuum is also selected to be global gauge invariant, so our discussionis actually restricted to the global gauge invariant Hilbert space H g,ph which is a subspaceof H ph , H g,ph ⊂ H ph . This is consistent with AdS/CFT, where the bulk theory is dualto the colorless sector of the gauge theory. However, on gauge theory side, states in H ph only need to be local gauge invariant and could form the representation of the global gaugetransformation. The global gauge transformation operator is physical, and the S-dual ofwhich should also be constructed. In conformal phase, we do not have too many clues todefine it. Maybe it is better to address the problem in Coulomb phase, where the electriccharge and the magnetic charge can be obtained as the central charge of the superalgebra[44].Type IIB string theory in AdS × S is dual to the N = 4 SYM theory. With the type18IB Hilbert space in one-to-one correspondence with the SYM physical Hilbert space, N = 4 SYM theory gives a definition for the quantum Type IIB theory. We may expect theS-duality transformation in Type IIB theory can also be realized via an operator S , underwhich, the P SU (2 , |
4) charges transform with the U (1) Y phase and the AdS fields f (1 , m,n related with the (1 ,
0) string are mapped into the fields f ( a,b ) m,n for ( a, b ) string. ACKNOWLEDGMENTS
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