Master symmetry in the AdS 5 × S 5 pure spinor string

Journal of High Energy Physics, Jan 2017

We lift the set of classical non-local symmetries recently studied by Klose, Loebbert, and Münkler in the context of ℤ 2 cosets to the pure spinor description of the superstring in the AdS 5 × S 5 background.

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Master symmetry in the AdS 5 × S 5 pure spinor string

Received: August Master symmetry in the Open Access 0 1 2 4 c The Authors. 0 1 2 4 0 Texas A&M University 1 Diagonal Las Torres 2640, Pen ̃alol ́en , Santiago , Chile 2 Facultad de Ingenier ́ıa y Ciencias, Universidad Adolfo Ib ́an ̃ez 3 Departamento de Ciencias F ́ısicas, Universidad Andres Bello 4 Sazie 2212 , Santiago , Chile We lift the set of classical non-local symmetries recently studied by Klose, Loebbert, and Mu¨nkler in the context of Z2 cosets to the pure spinor description of the superstring in the AdS5 × S5 background. AdS-CFT Correspondence; Conformal Field Models in String Theory - pure spinor string 1 Introduction 2 3 4 Pure spinor string in AdS5 × S The flat current Master symmetry Conclusions and prospects Finding and studying integrable structures in the context of the AdS/CFT correspondence has been one of the most active areas of research in high energy physics. The theories on both sides of the conjecture enjoy a large number of symmetries that make it possible to obtain impressive results and checks of the conjecture. Although it lacks more recent updates, a good review with an extensive list of references is [1]. A more recent development not covered in [1] is the the Quantum Spectral Curve method [2, 3]. For some of its applications, including higher loop computations, see [4–9]. In the famous work of Bena, Polchinski, and Roiban [10], it was shown that the GreenSchwarz superstring in AdS5 × S5 [11] has an infinite set of classical conserved currents. The existence of an analogous set of currents in the context of the AdS5 × S5 pure spinor superstring was demonstrated in reference [12]. Since this string is a generalization of the usual Z2 coset to a super-coset with Z4 symmetry, the ability to lift this symmetry to the super-coset is non-trivial. In this note, we go one step further and show that the pure spinor Loebbert, and Mu¨nkler [13]. This symmetry complements the Yangian symmetry, acting as a raising operator on the classical Yangian charges. The master symmetry is not essentially new, however it provides a unifying picture containing all local and non-local symmetries of a coset model. In particular, it is interesting that the conserved charge associated with to be of practical use, for example, in applications to the study of supersymmetry Wilson loops in the AdS5 × S5 super-coset. The work presented here extends this structure to its super-analogue, specifically, to the Z4 super-coset description of the AdS5 × S5 pure spinor string. In a sense the ghosts present in the pure spinor string make the Z4 symmetry manifest with the ghosts’ Lorentz current playing the role of a gauge covariant current with vanishing Z4 charge. The classical and quantum integrability of the string in this background has been explored much more for the GS string (see e.g. [14]) than for the pure spinor version. Some interesting results concerning the classical and quantum integrability in the pure spinor formalism are given in references [15–18]. A possible application of integrability techniques to the quantum pure spinor string is to study its worldsheet dilatation operator [19]. It has been shown that semi-classical computations in the pure spinor string give the same results as the GS string for a set of classical solutions [20, 21], but very little is known about solutions dual to Wilson loops. This is an interesting line of research to which the results presented in this work may have suitable applications. This paper is organized as follows: in section 2, we give a short review of the pure spinor string in AdS5 × S5 including its flat current using a notation that will be useful in the subsequent sections. In section 3, we extend the master symmetry discussed in [13] to the pure spinor string. In section 4, we derive how the existence of the first Yangian charge is a consequence of the master symmetry and the global psu(2, 2|4) symmetry. We then give a general derivation of all higher non-local and non-abelian charges the superstring has. We conclude the paper and discuss directions for future research in section 5. Pure spinor string in AdS5 × S The pure spinor string in the AdS5 × S5 background is described in terms of the superg = Li3=0 gi with the projections satisfying h := g0 m := M gi. J = g−1dg = K + A, dJ + J ∧ J = 0 ∇ = d + [A, · ]. The Maurer-Cartan identity will also decompose into four independent identities along each gi. current and its dz component can be understood from context. The Killing form Str(·) also respects this symmetry in the sense that [gi, gj ] ⊂ gi+j mod 4. Str(gigj ) 6= 0 iff i + j = 0 mod 4. This is a Z4 generalization of the usual Z2 symmetry present in any symmetric space and, in particular, in the bosonic coset construction. For comparison and general convenience, we will define They suffer the gauge transformations where A and B are any two local bosonic elements of g2. We will also define which are the Lorentz generators for the ghosts. Note that they have zero Z4 charge. The pure spinor condition implies In addition to the geometric part, the pure spinor string is defined with pure spinor ghosts and their conjugate momenta. These are invariant under global PSU(2, 2|4) transformations. The ghosts are fermionic elements of the algebra This is the coset generalization of the pure spinor condition in flat space. The momenta conjugate to the pure spinor variables are denoted Having all the ingredients, we can write the pure spinor action [22–24] S = 1 Z d2z Str K1K¯3 + 2K2K¯2 + 3K3K¯1 − 4N N¯ + 4ω∇¯ λ + 4ω¯∇λ¯ . The geometric part of this action is the standard kinetic term of a coset model plus a SWZ = − 1 Z d2z Str K1K¯3 − K3K¯1 . This particular coefficient of the Wess-Zumino term is fundamental for BRST symmetry and integrability [12, 16]. By construction, the action has global PSU(2, 2|4) invariance and local SO(1, 4) × SO(5) invariance. Global transformations act on g by left multiplication and the local transformations act on g by right multiplication. The current J is invariant under the global symmetry. On the other hand, K tranforms in the adjoint representation of h if momenta transform in the adjoint representation of h as well. The next fundamental symmetry is BRST invariance defined by2 δg = g(λ + λ¯), δλ = 0, δλ¯ = 0, δω = −K3, δω¯ = −K¯1. 2These transformations are nilpotent only up to local SO(1, 4) × SO(5) transformations and equations of motion. There are ways to fix both these issues [25–28], however, they will not be needed here. The conserved current associated with BRST symmetry is given by the charges defined by after using the equations of motion which will be discussed below. This fact means that also generate symmetries for any two independent holomorphic and anti-holomorphic functions ǫ(z) and ǫ¯(z¯). In this case the BRST transformations above generalize to3 We now compute the current associated with the global PSU(2, 2|4) symmetry. The Noether method. The left invariant currents transform as When inserting this transformation into the action, we can drop the restriction on the subspaces since the transformations will always come together with a dual algebra element inside a supertrace. The action transforms as Qǫ = Qǫ¯ = 1 Z d2z Str g−1∂ΩgK¯3 + g−1∂¯ΩgK1 + · · · + 4g−1∂ΩgN¯ + 4g−1∂¯ΩgN , from which we read off the Noether current These equations are calculated by varying the action (2.12) with respect to a variation 4 decomposition, the Maurer-Cartan identity for J , and ∇N¯ − ∇¯ N − 2[N, N¯ ] = 0, 3It may seem surprising that the BRST invariance in the pure spinor superstring implies a much larger symmetry than the usual BRST symmetry in field theory. However, we should remember that the pure spinor BRST should also imply Virasoro symmetry which is an infinite-dimensional chiral symmetry. we can separate (2.22) into eight equations of motion. To derive this last equation we use the equations of motion for the ghosts coming from (2.12): Then, the action can be written as and the components of the Noether current can be written as S = 1 K¯ Σ(K) + ω∇¯ λ + ω¯∇λ¯ − N N¯ , jz = gΣ(K + N )g−1, jz¯ = gΣ(K¯ + N¯ )g−1. For the supertrace, we have it does not preserve the Lie bracket). The flat current In contrast to the Noether current of the bosonic cosets, the conserved current (2.21) of the Z4 super-coset is not flat. Instead, it was shown in reference [12] that the pure spinor string in AdS5 × S5 has a family of flat currents depending on a complex parameter µ : Lµ = lµ dz + ¯lµ dz¯, The current is flat as a consequence of the equations of motion. dLµ + Lµ ∧ Lµ = 0 and lµ = g e−µ Σ − 1 (J¯ + N¯ )g−1. The existence of this current is remarkable given that, as just mentioned, the conserved current of a Z4 super-coset is generally not flat. In particular, there is no value of µ for which the flat current (2.30) reduces to (2.21). However, note that L′0 = ∗j, the first charge generated by the flat current Lµ is the conserved charge of the psu(2, 2|4) algebra. This can be seen from the monodromy matrix4 The charge of the psu(2, 2|4) algebra M (µ ) = P exp Z ∞ Qpsu = Z ∞ is the coefficient of the first power of µ in the expansion of M (µ ). Master symmetry Maurer-Cartan current by Following Klose, Loebbert, and Mu¨nkler [13], we can define a flat deformation of the + A¯ + e−3µ K¯1 + e−2µ K¯2 + e−µ K¯3 + (e−4µ − 1)N¯ dz¯. (3.1) can be written as Lµ = eµ Σ(J + N ) − N dz + e−µ Σ(J¯ + N¯ ) − N¯ dz¯. (It is actually more straightforward to verify that Lµ satifies a flatness condition.) A deformation gµ of the coset element g can be defined by the differential equadgµ (z, z¯) = gµ (z, z¯)Lµ with gµ (z0, z¯0) = g(z0, z¯0). Here, (z0, z¯0) is any reference point on the worldsheet needed to fix an “initial condition”. This equation is well-defined since Lµ is flat. Consequently, this deformation of g is only defined on-shell. An ansatz to solve it is 4For simplicity and to avoid global subtleties, we will assume the worldsheet is infinite and open. This is not essential to any of the results discussed below which are classical. It would be interesting to check that finite size effects do not spoil our conclusions, but we expect they do not as Berkovits has shown that the Yangian symmetries are preserved even when such quantum corrections are taken into account [16]. Again, this equation is well defined since Lµ is flat. We can expand this differential equation of motion are satisfied. The solution is given by With this, we are finally in the position to define the “master symmetry” [13] (dzjz − dz¯jz¯). This is a non-local transformation acting on the currents as δˆJ = g−1dχ(1)g = δˆK2 + δˆK1 + δˆK3 + δˆA = g−1(jzdz − jz¯dz¯)g = = (K1 + 2K2 + 3K3 + 4N ) dz − K¯3 + 2K¯2 + 3K¯1 + 4N¯ dz¯ Up until this point, the master symmetry has been discussed at the level of geometry. Since the structure under discussion is on-shell, we will consider N and N¯ as fundamental will define the extension of the master symmetry to act on them as With these transformations, it is immediate to verify that (3.9) is a symmetry of the equations of motion (2.22), turning it into a Maurer-Cartan identity together with symmetries of a Z2 coset. The analogous statement for the pure spinor string for any value of µ will be proved at the end of this paper. The main difference here, apart from the presence of fermionic terms, is that the gauge field A transforms into the ghost current N , thereby mixing matter and ghosts. Since this symmetry is only defined on-shell, discussing it at the level of the action is potentially meaningless. Nevertheless, we follow reference [13] and try to use Noether procedure to calculate the current associated to the master symmetry anyway. We include a local parameter ǫ(z, z¯) in the transformation of g The currents transform as δˆJ = dǫg−1χ(1)g + ǫΣ (K + N ) dz − ǫΣ K¯ + N¯ dz¯. Inserting this transformation into the action and only collecting terms depending on dǫ, 1 Z d2z Str ∂ǫχ(1)jz¯ + ∂¯ǫχ(1)jz . Using the same normalization as (2.21), we read off the conserved current Note that by (3.6), Using this, we can perform the integral to find the conserved charge use that the psu(2, 2|4) charge is given by ∗ J (0) = C(0) = ∗J (0) = C(0) = Str QpsuQpsu) is the Casimir of the psu(2, 2|4) algebra, as in [13]. In principle we could find the higher scalar charges associated with higher powers of in (3.14) suggests that the scalar current containing all higher master symmetry charges element of the algebra, it is not conserved, and its µ 2 coefficient does not match the Casimir with respect to µ of the flatness condition (2.33). Then, the complete tower of nonlocal charges can be defined by Z ∞ Cµ = ∗Jµ = Z ∞ The zeroth power of µ in the expansion gives the Casimir C(0), and it is easy to show that δˆj = j(1) = g(K1 + 4K2 + 9K3 + 16N )g−1dz which is the first non-local current given by the monodromy matrix (2.35) constructed from the flat current (2.30). Schematically, this current generates a transformation on the use the Noether method again to see if we can obtain the non-local current as the Noether current associated with this transformation. However, carrying this out, we obtain only the last term in (4.1). As mentioned previously, it is not surprising that this time we could not obtain the desired result since this is an on-shell symmetry. In the case of the principal chiral model it is possible to interpret these non-local currents as Noether currents [32, 33], but it is not clear that we can use the same method here. We could obtain the higher non-local currents by successive applications of the master Non-local current The non-local current associated with the global symmetry and all higher Yangian charges is calculated by replacing g → gµ (3.3), (3.4) in the definition of the Noether current (2.21) and defining a finite µ deformation of ghost currents as N → Nµ = e4µ N, N¯ → N¯µ = e−4µ N¯ , from which we can see that the master symmetry (3.10) corresponds to the first power of the µ deformation. With these definitions we have Jµ = gµ ((Kµ )1 + 2(Kµ )2 + 3(Kµ )3 + 4Nµ )gµ−1dz + gµ (3(K¯µ )1 + 2(K¯µ )2 + (K¯µ )3 + 4N¯µ )gµ−1dz¯ = χµ g(eµ K1 + 2e2µ K2 + 3e3µ K3 + 4e4µ N )g−1χu−1dz + χµ g(3e−3µ K¯1 + 2e−2µ K¯2 + e−µ K¯3 + 4e−4µ N¯ )g−1χu−1dz¯, identify this current as − ∂¯lµ′ + lµ , −¯lµ′ ∂¯Jµ = χµ g[∇¯ + K¯ , 2e2µ K2 + eµ K1 + 3e3µ K3 + 4e4µ N ]g−1 + [¯lµ , g(2e2µ K2 + eµ K1 + 3e3µ K3 + 4e4µ N )g−1] χµ−1 ∂J¯µ = χµ g[∇ + K, 2e−2µ K¯2 + 3e−3µ K¯1 + e−µ K¯3 + 4e−4µ N¯ ]g−1 + [lµ , g(2e−2µ K¯2 + 3e−3µ K¯1 + e−µ K¯3 + 4e−4µ N¯ )g−1] χµ−1 So, the conservation of Jµ is simply the first derivative with respect to µ of the flatness condition (2.33) of Lµ : ∂¯Jµ + ∂J¯µ = χµ ∂¯lµ′ − ∂¯lµ′ + ¯lµ , lµ′ This relation proves that if g is a solution, then the deformation gµ is also a solution. The current Jµ contains a whole tower of non-local conserved currents of the model, starting with the global psu(2, 2|4) current (2.4). It is easily checked that where j(1) is the first Yangian current. To prove the higher µ powers are all higher Yangian Jµ = j + µ j(1) + · · · , After some manipulations, one can show that To get to final result let us now compute the derivative with respect to µ of Jµ χµ−1δˆJµ χµ = ∗Lµ′′ + [χµ−1δˆχµ , ∗Lµ′ ] + [χ(1), Lµ′ ]. χµ−1Jµ′ χµ = ∗Lµ′′ + [χµ−1χµ′ , ∗Lµ′ ]. If we subtract both equations we have χµ−1 δˆJµ − Jµ′ χµ = [χµ−1δˆχµ − χµ−1χµ′ , ∗Lµ′ ] + [χ(1), ∗Lµ′ ]. Let us call φµ := χµ−1δˆχµ − χµ−1χµ′ and note that φ0 = −χ(1). Using that it can be shown that the non-local Casimir (3.20) satisfies Conclusions and prospects We have shown that the classical pure spinor string in the AdS5 × S5 background has the full set of classical non-local symmetries extending those recently studied by Klose, Loebbert and Mu¨nkler in the context of Z2 cosets [13]. We find that the inclusion of ghosts in a sense makes the Z4 symmetry manifest, and all non-local symmetries can be lifted to the super-coset PSU(2, 2|4)/SO(1, 4) × SO(5). An immediate extension of the results of this paper is to derive the analog for the GreenSchwarz superstring. That can be done by erasing the ghosts and imposing an appropriate gauge choice. Classical solutions of the pure spinor string should preserve BRST symmetry which means the BRST charge should vanish when evaluated on the solution. If we do not set the ghosts to zero, this means that the currents K3 and K¯1 should vanish. In reference [12], it was shown that the pure spinor flat current is equivalent to the one in the Green-Schwarz formalism [10] in this gauge. We expect that the GS string enjoys all of the symmetries discussed in the present work. It would be interesting to apply the results of this paper to supersymmetric Wilson loops in AdS5 as in reference [13]. However, it is as yet not known how to study such classical solutions in the pure spinor formalism. There is hope such a task can be done, since it was shown by explicit computations in references [20, 21] that the semi-classical quantization of the pure spinor string is equivalent to the Green-Schwarz string in a certain class of solutions. In [34] it was argued that the equivalence holds for any physical solution. With these results in mind, it is likely that one can extend the results of, for example, references [35, 36] to the pure spinor string. A more speculative line of research is the relevance of the master symmetry in the quantum theory. Since the Yangian currents are still conserved at quantum level [16] it is possible that there is some quantum version of the master symmetry. However we cannot say if it will provide any additional help in achieving an exact solution of the model. The work of Oc and Bcv is partially supported by FONDECYT grant number 1151409. Bcv also has partial support from CONICYT grant number DPI20140115. 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Osvaldo Chandía, William Divine Linch III, Brenno Carlini Vallilo. Master symmetry in the AdS 5 × S 5 pure spinor string, Journal of High Energy Physics, 2017, 24, DOI: 10.1007/JHEP01(2017)024