Half-BPS vertex operators of the AdS 5 × S 5 superstring

: Using the pure spinor formalism for the superstring in an AdS 5 × S 5 background, a simple expression is found for half-BPS vertex operators. At large radius, these vertex operators reduce to the usual supergravity vertex operators in a ﬂat background. And at small radius, there is a natural conjecture for generalizing these vertex operators to non-BPS states.


Introduction
Although the computation of superstring scattering amplitudes in an AdS 5 ×S 5 background is complicated by the nonlinear form of the worldsheet action, the presence of maximal supersymmetry and the duality with d = 4 N = 4 super-Yang-Mills gives reasons to be optimistic that progress will be made. Since the RNS formalism can only be used to describe infinitesimal Ramond-Ramond backgrounds [1,2], one needs to use either the Green-Schwarz or pure spinor formalisms to fully describe AdS 5 × S 5 . The Green-Schwarz light-cone formalism is convenient for computing the physical spectrum of "long" strings [3], but amplitude computations using this formalism are complicated even in a flat background.
The pure spinor formalism in an AdS 5 × S 5 background has the advantage over the Green-Schwarz formalism of allowing manifestly PSU(2, 2|4)-covariant quantization [4]. Although less studied, this formalism was used to derive the quantum structure of the infinite set of nonlocal conserved currents in [5] and to compute the physical spectrum of "long" strings in [6]. And in a flat background, the pure spinor formalism has been used for computing multiloop superstring amplitudes [7] that have not yet been computed using either the RNS or Green-Schwarz formalisms.
To generalize these amplitude computations to an AdS 5 ×S 5 background, the first step is to explicitly construct the superstring vertex operators for half-BPS states. Although the behavior of half-BPS vertex operators near the AdS 5 × S 5 boundary was computed in [8], the complete BRST-invariant vertex operator was only previously known for some special states [9] such as the moduli for the AdS radius [10] and for the β-deformation [11].
In this paper, simple expressions will be constructed for general half-BPS vertex operators in an AdS 5 ×S 5 background using the pure spinor formalism. These simple expressions JHEP07(2019)084 were very recently proposed in [12] and will be shown here to be manifestly BRST-invariant and to closely resemble the vertex operators for Type IIB supergravity states in a flat background. Hopefully, these simple expressions for vertex operators will soon be used for computing superstring scattering amplitudes in an AdS 5 × S 5 background.
In section 2, the BRST-invariant vertex operator for Type IIB supergravity states in a flat background will be constructed in terms of the chiral supergravity superfield whose lowest components are the dilaton and axion. In section 3, this vertex operator will be expressed in a simple form using picture-changing operators. And in section 4, this simple expression for the Type IIB supergravity vertex operator in a flat background will be generalized to half-BPS vertex operators in an AdS 5 × S 5 background. Finally, section 5 will discuss the recent conjecture of [12] for generalizing this construction to non-BPS states in an AdS 5 × S 5 background at small radius.

Supergravity vertex operators
In any Type IIB supergravity background, the massless closed superstring vertex operator in unintegrated form in the pure spinor formalism is [13] where A αβ are bispinor superfields depending on the N = 2B d = 10 superspace variables (x m , θ α L , θ α R ), α = 1 to 16 are Majorana-Weyl spinor indices, and λ α L and λ α R are left and right-moving pure spinor variables satisfying λ L γ m λ L = λ R γ m λ R = 0 for m = 0 to 9. The onshell equations of motion and gauge invariances are implied by QV = 0 and δV = QΩ where and ∇ Lα and ∇ Rα are the 32 fermionic covariant derivatives in the supergravity background. These equations of motion and gauge invariances imply that A αβ satisfies where Ω Lα and Ω Rα satisfy γ αβ abcde ∇ Lα Ω Lβ = γ αβ abcde ∇ Rα Ω Rβ = 0.

Flat background
To construct solutions to (2.3) in a flat background, it is convenient to choose a reference frame where the momentum is only in the k + = k 0 + k 9 direction so that the covariant fermionic derivatives reduce to where a,ȧ are SO(8) chiral and antichiral spinor indices and

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Since k + is nonzero, (2.3) implies one can gauge-fix A aḃ = A ab = Aȧ b = 0, so that In the gauge of (2.6), where σ j aȧ are the SO(8) Pauli matrices. One method of solving (2.7) is to take the left-right product of the open superstring solutions of [14], but it will be useful to describe another method which can be easily generalized to the AdS 5 × S 5 background. This method is based on the SO (8) is a linear combination of the left and right-moving fermionic derivatives. In terms of (x + , θ a L , θ a R ), The superfield Φ will be defined to satisfy the reality condition ef gh Φ, and the 2 8 components of Φ describe the Type IIB supergravity multiplet where, at zeroth order in θ − , the real part of Φ is the Type IIB dilaton and the imaginary part of Φ is the Type IIB axion.
To construct the vertex operator of (2.6) for this multiplet, first consider the vertex operator Using the relation λ a L λȧ L = − 1 4 (σ jk λ L ) a (σ jk λ L )ȧ and λ a R λȧ R = − 1 4 (σ jk λ R ) a (σ jk λ R )ȧ, one finds that where λ a ± = λ a L ± iλ a R . Now consider the vertex operator Continuing this argument, one finds that

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So the BRST-invariant vertex operator with momentum k + in this gauge is and one can easily verify that at θ a L = θ a R = 0, Aȧ˙b is the bispinor Ramond-Ramond field in light-cone gauge Aȧ˙b = δȧ˙ba + σ jk aḃ a jk + σ jklṁ aḃ a jklm .
(2.15) It will be useful to note that one would end up with the same expression of (2.14) for V if one had instead started with the superfield Φ 1234 which is annihilated by

Picture-changing
To generalize this construction to an AdS 5 ×S 5 background, it will be useful to first consider the vertex operator V for the lowest component of can be reduced to just one term by writing it in a different "picture" as where P is the "picture-lowering" operator and λ a + = λ a L + i(σ 1234 λ R ) a . Note that the 8 λ a + 's in P are all independent so that 8 a=1 δ(λ a + ) is well-defined. Also note that P is BRST-invariant and is super-Poincaré invariant up to a BRST-trivial quantity. For example, under the supersymmetry transformation generated by q 1 , The original vertex operator V of (2.14) is related to V −1 of (3.1) by picture-raising as

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is the picture-raising operator and Q(ξ a ) is a formal expression whose action on V −1 is defined through the following procedure: using the notation of Friedan-Martinec-Shenker for picture-changing operators, δ(γ) = e −φ and ξδ(γ) = ξe −φ = 1 γ where (γ, β) are chiral bosons which have been fermionized as γ = ηe φ and β = ∂ξe −φ . Although λ a + and its conjugate w + a are not chiral bosons, one can formally define (3.6) Using this definition, CV −1 can be computed by using (3.6) to convert the factors of δ(λ a + ) in V −1 into factors of 1 λ a + . Furthermore, the BRST invariance of V −1 guarantees that CV −1 has no poles when λ a + = 0 and can be expressed in the form of (2.1) ) has no poles when λ a + = 0. Also note that if F has (left,right)-moving ghost number equal to (g L , g R ), then Q( F λ a + ) also has (left,right) ghost number (g L , g R ). This is easy to see since terms in QF must either carry ghost number (g L + 1, g R ) or (g L , g R + 1). So QF = Eλ a + for some E implies that E must carry ghost number (g L , g R ).

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contributes a factor of k + θ + . Continuing with this procedure of converting ξ a δ(λ a + ) into (λ a + ) −1 to compute the product with Q(ξ a ), it is expected that CV −1 will reproduce V of (2.14). To generalize this construction for half-BPS states in an AdS 5 × S 5 background, parameterize AdS 5 × S 5 using the supercoset g ∈ PSU(2,2|4) SO(4,1)×SO(5) as whereR = 1 to 4 is an SO(4, 1) spinor index,J = 1 to 4 is an SO(5) spinor index, and (λ L )R J and (λ R )R J are the left and right-moving pure spinors. Note that SO(4, 1) and SO (5) spinor indices can be raised and lowered using the matrices σRS 6 and σJK 6 which commute with SO(4, 1) and SO(5) rotations. The cosets G(X) and H(Y ) are defined up to local SO(4, 1) × SO(5) gauge transformations parameterized by Ω ∈ SO(4, 1) andΩ ∈ SO(5) as where the left and right-moving pure spinors λ L and λ R transform as SO(4, 1) × SO (5) spinors. More explicitly, G R R and H J J are 4 × 4 matrices which transform under the gauge transformations as and the AdS 5 coordinate X RS = −X SR and S 5 coordinate Y JK = −Y KJ are defined in terms of G R R and H J J by

Half-BPS vertex operator
To construct the vertex operator for a half-BPS state in an AdS 5 ×S 5 background, consider the state dual to the super-Yang-Mills gauge-invariant operator where Φ JK (x) are the six scalars located at the position x m on the AdS 5 boundary and y JK 0 is a fixed null six-vector satisfying ǫ JKLM y JK 0 y LM 0 = 0. It will be convenient to define the null six-vector where R = (A,Ȧ) with A,Ȧ = 1 to 2.  (4), and J will be defined to be the charge with respect to this U(1). Similarly, the choice of x RS 0 breaks SO(4, 2) conformal symmetry to SO(1, 1) × SO(3, 1), and ∆ will be defined to be the charge with respect to the SO(1, 1). The half-BPS state of (4.6) carries J = n and ∆ = n and is preserved by the 24 spacetime supersymmetries which carry J − ∆ ≥ 0.
In analogy with the construction of the vertex operator of V −1 in a flat background, it will now be argued that the BRST-invariant vertex operator for the state (4.6) is [12] where the picture-lowering operator P is defined as and θ a + are the 8 θ's which carry charge J − ∆ = 1. In terms of x RS 0 and y JK 0 , where only 8 of the 32 components of (x 0 ) RS (y 0 ) JK θ K S and (x 0 ) RS (y 0 ) JK θ S K are independent since (x 0 ) RS (x 0 ) ST = (y 0 ) JK (y 0 ) KL = 0.
To show that V −1 of (4.8) carries the same charges and is invariant under the same 24 supersymmetries as (4.6), note that Y · y 0 carries J = 1 and X · x 0 carries ∆ = −1 so that V −1 carries J = ∆ = n. Furthermore, both Y · y 0 and X · Similarly, under the BRST transformation of (4.2), Y ·y 0 X·x 0 transforms into terms containing products of Q(θ) with θ's where either Q(θ) carries J − ∆ = 1 or at least one of the θ's carries J − ∆ = 1. In both cases, the BRST transformation is killed by P = 8 a=1 θ a + δ(Q(θ a + )) of (4.9). And since P and (λ L )JR(λ R )R J are also BRST-invariant, it has been shown that V −1 of (4.8) is BRST-invariant.
Using the picture-lowering operator P = 8 a=1 θ a + δ(Q(θ a + )), the vertex operator of (4.8) is where θ a + are the 8 θ's which carry charge J − ∆ = 1 and (λ a + , λ a − , λȧ L , λȧ R ) for a,ȧ = 1 to 8 are defined by λ a + ≡ e and we have used that (λ L )JR(λ R )R J = λȧ L (σ 1234 )ȧ˙bλ˙b R when λ a + = 0. In the large radius limit where the AdS 5 × S 5 background approaches flat space, one can easily verify that V −1 of (4.12) approaches the flat space vertex operator V −1 of (3.1) where k + = n and ix + is identified with w − z. And the vertex operator for all other half-BPS states in an AdS 5 × S 5 background are obtained from (4.12) by acting with the appropriate PSU(2, 2|4) transformations, and reduce in the flat space limit to the vertex operators of other supergravity states in the muitiplet of (3.1).
Finally, one can relate V −1 of (4.12) to the supergravity vertex operator V = λ α L λ β R · A αβ (x, θ) of (2.1) by defining V = CV −1 (4.14) where C = 8 a=1 Q(ξ a ) and the 8 λ a + 's of (4.13) have been fermionized as in (3.5). Using the same procedure as in (3.7), this construction will produce an AdS 5 ×S 5 vertex operator of the form V = λ α L λ β R A αβ (θ + , z − w) where, as in a flat background, the potential poles coming from ξ a δ(λ a + ) = 1 λ a + are absent because of the BRST invariance of V −1 .

Summary
In this paper, a simple BRST-invariant vertex operator was constructed for half-BPS states in an AdS 5 × S 5 background. One possible application of this paper is to use these vertex operators to compute scattering amplitudes. Much is known about scattering amplitudes of half-BPS states in AdS 5 ×S 5 , and it would be very interesting to show how to compute these amplitudes using superstring vertex operators even for the simplest 3-point amplitude.
Another possible application of this paper is to construct AdS 5 × S 5 vertex operators for non-BPS states. As discussed in [12], the half-BPS vertex operator can be expressed as if one adds (n − 1) picture-raising operators C and (n − 1) picture-lowering operators P to V = CV −1 of (4.14). Since all states at zero 't Hooft coupling can be described as "spin chains" constructed from n super-Yang-Mills fields, it is natural to express the half-BPS vertex operator of (5.1) as where E ≡ P Y ·y 0 X·x 0 corresponds to the Yang-Mills field y JK 0 φ JK (x 0 ) on the spin chain. Therefore, a natural conjecture for general non-BPS vertex operators is where E 1 . . . E n describe n different super-Yang-Mills fields on the spin chain and are obtained from P Y ·y 0 X·x 0 by performing the appropriate PSU(2, 2|4) transformation. Since E and C are independently BRST-invariant, the vertex operator of (5.3) is BRST-invariant where : : denotes a normal-ordering prescription which is defined to be invariant under cyclic permuations of the E's. It would be very interesting to find evidence for this conjecture by using the topological description of [12] to study the AdS 5 × S 5 superstring at small radius.