Half-BPS Vertex Operators of the $AdS_5\times S^5$ Superstring

Nathan Berkovits (ICTP-SAIFR/IFT-UNESP; S\~ao Paulo)

arXiv:1904.06564·hep-th·September 4, 2019

Half-BPS Vertex Operators of the $AdS_5\times S^5$ Superstring

Nathan Berkovits (ICTP-SAIFR/IFT-UNESP, S\~ao Paulo)

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TL;DR

This paper derives simple half-BPS vertex operators for the superstring in $AdS_5\times S^5$ using pure spinor formalism, connecting to supergravity operators at large radius and proposing extensions to non-BPS states at small radius.

Contribution

It provides a new, simplified expression for half-BPS vertex operators in $AdS_5\times S^5$ superstring theory using pure spinor formalism.

Findings

01

Vertex operators reduce to supergravity operators at large radius.

02

Proposes a conjecture for non-BPS vertex operators at small radius.

03

Simplifies the understanding of BPS states in AdS/CFT correspondence.

Abstract

Using the pure spinor formalism for the superstring in an $A d S_{5} \times S^{5}$ 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.

Equations102

V = λ_{L}^{α} λ_{R}^{β} A_{α β} (x, θ_{L}, θ_{R})

V = λ_{L}^{α} λ_{R}^{β} A_{α β} (x, θ_{L}, θ_{R})

Q = λ_{L}^{α} \nabla_{L α} + λ_{R}^{α} \nabla_{R α}

Q = λ_{L}^{α} \nabla_{L α} + λ_{R}^{α} \nabla_{R α}

γ_{ab c d e}^{α γ} \nabla_{L α} A_{γ β} = γ_{ab c d e}^{α β} \nabla_{R α} A_{γ β} = 0, δ A_{α β} = \nabla_{L α} Ω_{R β} + \nabla_{R β} Ω_{L α},

γ_{ab c d e}^{α γ} \nabla_{L α} A_{γ β} = γ_{ab c d e}^{α β} \nabla_{R α} A_{γ β} = 0, δ A_{α β} = \nabla_{L α} Ω_{R β} + \nabla_{R β} Ω_{L α},

\nabla_{L a} \equiv (γ^{-} \nabla_{L})_{a} = \frac{\partial}{\partial θ _{L}^{a}} + θ_{L a} \partial_{+}, \nabla_{L \overset{a}{˙}} \equiv (γ^{+} \nabla_{L})_{\overset{a}{˙}} = \frac{\partial}{\partial θ _{L}^{\overset{a}{˙}}},

\nabla_{L a} \equiv (γ^{-} \nabla_{L})_{a} = \frac{\partial}{\partial θ _{L}^{a}} + θ_{L a} \partial_{+}, \nabla_{L \overset{a}{˙}} \equiv (γ^{+} \nabla_{L})_{\overset{a}{˙}} = \frac{\partial}{\partial θ _{L}^{\overset{a}{˙}}},

\nabla_{R a} \equiv (γ^{-} \nabla_{R})_{a} = \frac{\partial}{\partial θ _{R}^{a}} + θ_{R a} \partial_{+}, \nabla_{R \overset{a}{˙}} \equiv (γ^{+} \nabla_{R})_{\overset{a}{˙}} = \frac{\partial}{\partial θ _{R}^{\overset{a}{˙}}},

\nabla_{R a} \equiv (γ^{-} \nabla_{R})_{a} = \frac{\partial}{\partial θ _{R}^{a}} + θ_{R a} \partial_{+}, \nabla_{R \overset{a}{˙}} \equiv (γ^{+} \nabla_{R})_{\overset{a}{˙}} = \frac{\partial}{\partial θ _{R}^{\overset{a}{˙}}},

θ_{L a} = (γ^{+} θ_{L})_{a}, \overline{θ}_{L \overset{a}{˙}} = (γ^{-} θ_{L})_{\overset{a}{˙}}, θ_{R a} = (γ^{+} θ_{R})_{a}, \overline{θ}_{R \overset{a}{˙}} = (γ^{-} θ_{R})_{\overset{a}{˙}} .

θ_{L a} = (γ^{+} θ_{L})_{a}, \overline{θ}_{L \overset{a}{˙}} = (γ^{-} θ_{L})_{\overset{a}{˙}}, θ_{R a} = (γ^{+} θ_{R})_{a}, \overline{θ}_{R \overset{a}{˙}} = (γ^{-} θ_{R})_{\overset{a}{˙}} .

V = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\dot{b}} A_{\overset{a}{˙} \dot{b}} (x, θ_{L}, θ_{R})

V = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\dot{b}} A_{\overset{a}{˙} \dot{b}} (x, θ_{L}, θ_{R})

\frac{\partial}{\partial θ _{L}^{\overset{a}{˙}}} A_{\dot{b} \overset{c}{˙}} = \frac{\partial}{\partial θ _{R}^{\overset{a}{˙}}} A_{\dot{b} \overset{c}{˙}} = 0, \nabla_{L a} A_{\dot{b} \overset{c}{˙}} = \frac{1}{8} σ_{a \dot{b}}^{j} σ_{j}^{c \dot{d}} \nabla_{L c} A_{\dot{d} \overset{c}{˙}}, \nabla_{R a} A_{\dot{b} \overset{c}{˙}} = \frac{1}{8} σ_{a \overset{c}{˙}}^{j} σ_{j}^{c \dot{d}} \nabla_{R c} A_{\dot{b} \dot{d}}

\frac{\partial}{\partial θ _{L}^{\overset{a}{˙}}} A_{\dot{b} \overset{c}{˙}} = \frac{\partial}{\partial θ _{R}^{\overset{a}{˙}}} A_{\dot{b} \overset{c}{˙}} = 0, \nabla_{L a} A_{\dot{b} \overset{c}{˙}} = \frac{1}{8} σ_{a \dot{b}}^{j} σ_{j}^{c \dot{d}} \nabla_{L c} A_{\dot{d} \overset{c}{˙}}, \nabla_{R a} A_{\dot{b} \overset{c}{˙}} = \frac{1}{8} σ_{a \overset{c}{˙}}^{j} σ_{j}^{c \dot{d}} \nabla_{R c} A_{\dot{b} \dot{d}}

Φ (x^{+}, θ_{L}^{a}, θ_{R}^{a}) = e^{i k_{+} (x^{+} + i θ_{L}^{a} θ_{R}^{a})} f (θ_{-})

Φ (x^{+}, θ_{L}^{a}, θ_{R}^{a}) = e^{i k_{+} (x^{+} + i θ_{L}^{a} θ_{R}^{a})} f (θ_{-})

V_{0} = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\overset{a}{˙}} Φ.

V_{0} = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\overset{a}{˙}} Φ.

Q V_{0} = (λ_{-} \nabla_{+} + λ_{+} \nabla_{-}) (\overline{λ}_{L} \overline{λ}_{R}) Φ = (λ_{-} \nabla_{+}) (\overline{λ}_{L} \overline{λ}_{R}) Φ = - \frac{1}{4} (λ_{+} σ^{j k} \nabla_{+}) (\overline{λ}_{L} σ_{j k} \overline{λ}_{R}) Φ

Q V_{0} = (λ_{-} \nabla_{+} + λ_{+} \nabla_{-}) (\overline{λ}_{L} \overline{λ}_{R}) Φ = (λ_{-} \nabla_{+}) (\overline{λ}_{L} \overline{λ}_{R}) Φ = - \frac{1}{4} (λ_{+} σ^{j k} \nabla_{+}) (\overline{λ}_{L} σ_{j k} \overline{λ}_{R}) Φ

V_{1} = \frac{1}{32 i k _{+}} (\overline{λ}_{L} σ_{j k} \overline{λ}_{R}) (\nabla_{+} σ^{j k} \nabla_{+}) Φ.

V_{1} = \frac{1}{32 i k _{+}} (\overline{λ}_{L} σ_{j k} \overline{λ}_{R}) (\nabla_{+} σ^{j k} \nabla_{+}) Φ.

V_{2} = - \frac{1}{2048 k _{+}^{2}} (\overline{λ}_{L} σ_{j k l m} \overline{λ}_{R}) (\nabla_{+} σ^{j k} \nabla_{+}) (\nabla_{+} σ^{l m} \nabla_{+}) Φ.

V_{2} = - \frac{1}{2048 k _{+}^{2}} (\overline{λ}_{L} σ_{j k l m} \overline{λ}_{R}) (\nabla_{+} σ^{j k} \nabla_{+}) (\nabla_{+} σ^{l m} \nabla_{+}) Φ.

V_{n} = \frac{1}{n ! ( 32 i k _{+} ) ^{n}} (\overline{λ}_{L} σ_{j_{1} k_{1} ... j_{n} k_{n}} \overline{λ}_{R}) (\nabla_{+} σ^{j_{1} k_{1}} \nabla_{+}) ... (\nabla_{+} σ^{j_{n} k_{n}} \nabla_{+}) Φ.

V_{n} = \frac{1}{n ! ( 32 i k _{+} ) ^{n}} (\overline{λ}_{L} σ_{j_{1} k_{1} ... j_{n} k_{n}} \overline{λ}_{R}) (\nabla_{+} σ^{j_{1} k_{1}} \nabla_{+}) ... (\nabla_{+} σ^{j_{n} k_{n}} \nabla_{+}) Φ.

V = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\dot{b}} e^{i k_{+} x^{+}} A_{\overset{a}{˙} \dot{b}} (θ_{L}, θ_{R}) = V_{0} + V_{1} + V_{2} + V_{3} + V_{4},

V = \overline{λ}_{L}^{\overset{a}{˙}} \overline{λ}_{R}^{\dot{b}} e^{i k_{+} x^{+}} A_{\overset{a}{˙} \dot{b}} (θ_{L}, θ_{R}) = V_{0} + V_{1} + V_{2} + V_{3} + V_{4},

A_{\overset{a}{˙} \dot{b}} = δ_{\overset{a}{˙} \dot{b}} a + σ_{\overset{a}{˙} \dot{b}}^{j k} a_{j k} + σ_{\overset{a}{˙} \dot{b}}^{j k l m} a_{j k l m} .

A_{\overset{a}{˙} \dot{b}} = δ_{\overset{a}{˙} \dot{b}} a + σ_{\overset{a}{˙} \dot{b}}^{j k} a_{j k} + σ_{\overset{a}{˙} \dot{b}}^{j k l m} a_{j k l m} .

V_{n} = \frac{1}{n ! ( 32 i k _{+} ) ^{n}} (\overline{λ}_{L} σ_{j_{1} k_{1} ... j_{n} k_{n}} σ_{1234} \overline{λ}_{R}) (\nabla_{+} σ^{j_{1} k_{1}} \nabla_{+}) ... (\nabla_{+} σ^{j_{n} k_{n}} \nabla_{+}) Φ_{1234}

V_{n} = \frac{1}{n ! ( 32 i k _{+} ) ^{n}} (\overline{λ}_{L} σ_{j_{1} k_{1} ... j_{n} k_{n}} σ_{1234} \overline{λ}_{R}) (\nabla_{+} σ^{j_{1} k_{1}} \nabla_{+}) ... (\nabla_{+} σ^{j_{n} k_{n}} \nabla_{+}) Φ_{1234}

V_{- 1} = P V = (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}} a = 1 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})

V_{- 1} = P V = (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}} a = 1 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})

P = a = 1 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})

P = a = 1 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})

q_{1} P = δ (λ_{+}^{1}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) = Q [- θ_{+}^{1} δ^{'} (λ_{+}^{1}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})] .

q_{1} P = δ (λ_{+}^{1}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) = Q [- θ_{+}^{1} δ^{'} (λ_{+}^{1}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a})] .

C = a = 1 \prod 8 Q (ξ_{a})

C = a = 1 \prod 8 Q (ξ_{a})

λ_{+}^{a} = η^{a} e^{ϕ_{a}}, w_{a}^{+} = \partial ξ_{a} e^{- ϕ_{a}}

λ_{+}^{a} = η^{a} e^{ϕ_{a}}, w_{a}^{+} = \partial ξ_{a} e^{- ϕ_{a}}

ξ_{a} δ (λ_{+}^{a}) = ξ_{a} e^{- ϕ_{a}} = \frac{1}{λ _{+}^{a}} .

ξ_{a} δ (λ_{+}^{a}) = ξ_{a} e^{- ϕ_{a}} = \frac{1}{λ _{+}^{a}} .

C V_{- 1} = b = 2 \prod 8 Q (ξ_{b}) Q (ξ_{1}) V_{- 1} = b = 2 \prod 8 Q (ξ_{b}) Q (ξ_{1} V_{- 1})

C V_{- 1} = b = 2 \prod 8 Q (ξ_{b}) Q (ξ_{1}) V_{- 1} = b = 2 \prod 8 Q (ξ_{b}) Q (ξ_{1} V_{- 1})

= - b = 2 \prod 8 Q (ξ_{b}) Q (\frac{θ _{+}^{1}}{λ _{+}^{1}} a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= - b = 2 \prod 8 Q (ξ_{b}) Q (\frac{θ _{+}^{1}}{λ _{+}^{1}} a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= - b = 2 \prod 8 Q (ξ_{b}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}}

= - b = 2 \prod 8 Q (ξ_{b}) a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}}

= - b = 3 \prod 8 Q (ξ_{b}) Q (ξ_{2} a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= - b = 3 \prod 8 Q (ξ_{b}) Q (ξ_{2} a = 2 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= b = 3 \prod 8 Q (ξ_{b}) Q (\frac{θ _{+}^{2}}{λ _{+}^{2}} a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= b = 3 \prod 8 Q (ξ_{b}) Q (\frac{θ _{+}^{2}}{λ _{+}^{2}} a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}})

= b = 3 \prod 8 Q (ξ_{b}) (1 + 2 i k_{+} (λ_{-}^{1} θ_{+}^{1}) \frac{θ _{+}^{2}}{λ _{+}^{2}}) a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}}

= b = 3 \prod 8 Q (ξ_{b}) (1 + 2 i k_{+} (λ_{-}^{1} θ_{+}^{1}) \frac{θ _{+}^{2}}{λ _{+}^{2}}) a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) (\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) e^{i k_{+} \overset{x}{^}^{+}}

= b = 3 \prod 8 Q (ξ_{b}) ((\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) + \frac{i}{2} k_{+} θ_{+}^{1} θ_{+}^{2} σ_{12}^{j k} (\overline{λ}_{L} σ^{j k} σ_{1234} \overline{λ}_{R})) a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) e^{i k_{+} \overset{x}{^}^{+}}

= b = 3 \prod 8 Q (ξ_{b}) ((\overline{λ}_{L} σ_{1234} \overline{λ}_{R}) + \frac{i}{2} k_{+} θ_{+}^{1} θ_{+}^{2} σ_{12}^{j k} (\overline{λ}_{L} σ^{j k} σ_{1234} \overline{λ}_{R})) a = 3 \prod 8 θ_{+}^{a} δ (λ_{+}^{a}) e^{i k_{+} \overset{x}{^}^{+}}

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Full text

Half-BPS Vertex Operators

of the $AdS_{5}\times S^{5}$ Superstring

Nathan Berkovits††1 e-mail: [email protected]

ICTP South American Institute for Fundamental Research

Instituto de Física Teórica, UNESP - Univ. Estadual Paulista

Rua Dr. Bento T. Ferraz 271, 01140-070, São Paulo, SP, Brasil

Using the pure spinor formalism for the superstring in an $AdS_{5}\times S^{5}$ 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.

April 2019

1. Introduction

Although the computation of superstring scattering amplitudes in an $AdS_{5}\times S^{5}$ 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 $AdS_{5}\times S^{5}$ . 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 $AdS_{5}\times S^{5}$ 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 $AdS_{5}\times S^{5}$ 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 $AdS_{5}\times S^{5}$ 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 ${\beta}$ -deformation [11].

In this paper, simple expressions will be obtained for general half-BPS vertex operators in an $AdS_{5}\times S^{5}$ background using the pure spinor formalism. These expressions will be manifestly BRST-invariant and will 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 $AdS_{5}\times S^{5}$ 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 $AdS_{5}\times S^{5}$ background. Finally, section 6 will discuss the recent conjecture of [12] for generalizing this construction to non-BPS states in an $AdS_{5}\times S^{5}$ background at small radius.

2. 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]

[TABLE]

where $A_{{\alpha}{\beta}}$ are bispinor superfields depending on the N=2B d=10 superspace variables $(x^{m},{\theta}_{L}^{\alpha},{\theta}_{R}^{\alpha})$ , ${\alpha}=1$ to 16 are Majorana-Weyl spinor indices, and ${\lambda}_{L}^{\alpha}$ and ${\lambda}_{R}^{\alpha}$ are left and right-moving pure spinor variables satisfying ${\lambda}_{L}{\gamma}^{m}{\lambda}_{L}={\lambda}_{R}{\gamma}^{m}{\lambda}_{R}=0$ for $m=0$ to 9. The onshell equations of motion and gauge invariances are implied by $QV=0$ and ${\delta}V=Q{\Omega}$ where

[TABLE]

and $\nabla_{L{\alpha}}$ and ${\nabla}_{R{\alpha}}$ are the 32 fermionic covariant derivatives in the supergravity background. These equations of motion and gauge invariances imply that $A_{{\alpha}{\beta}}$ satisfies

[TABLE]

where ${\Omega}_{L{\alpha}}$ and ${\Omega}_{R{\alpha}}$ satisfy ${\gamma}_{abcde}^{{\alpha}{\beta}}\nabla_{L{\alpha}}{\Omega}_{L{\beta}}={\gamma}_{abcde}^{{\alpha}{\beta}}{\nabla}_{R{\alpha}}{\Omega}_{R{\beta}}=0$ .

2.1. Flat background

To construct solutions to (2.1) 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

[TABLE]

where $a,{\dot{a}}$ are SO(8) chiral and antichiral spinor indices and

[TABLE]

Since $k_{+}$ is nonzero, (2.1) implies one can gauge-fix $A_{a{\dot{b}}}=A_{ab}=A_{{\dot{a}}b}=0$ , so that

[TABLE]

where ${\lambda}_{L}^{a}=({\gamma}^{+}{\lambda}_{L})^{a}$ , ${\overline{\lambda}}_{L}^{\dot{a}}=({\gamma}^{-}{\lambda}_{L})^{\dot{a}}$ , ${\lambda}_{R}^{a}=({\gamma}^{+}{\lambda}_{R})^{a}$ , ${\overline{\lambda}}_{R}^{\dot{a}}=({\gamma}^{-}{\lambda}_{R})^{\dot{a}}$ . In the gauge of (2.1), $QV=0$ together with ${\overline{\lambda}}_{L}^{\dot{a}}{\lambda}_{L}^{a}{\sigma}^{j}_{a{\dot{a}}}={\overline{\lambda}}_{R}^{\dot{a}}{\lambda}_{R}^{a}{\sigma}^{j}_{a{\dot{a}}}=0$ implies that

[TABLE]

where ${\sigma}^{j}_{a{\dot{a}}}$ are the SO(8) Pauli matrices.

One method of solving (2.1) 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 $AdS_{5}\times S^{5}$ background. This method is based on the SO(8) chiral superfield $\Phi$ satisfying ${\nabla}_{-}^{a}\Phi=0$ where ${\nabla}_{\pm}^{a}\equiv{\nabla}_{L}^{a}\pm i{\nabla}_{R}^{a}$ is a linear combination of the left and right-moving fermionic derivatives. In terms of $(x^{+},{\theta}_{L}^{a},{\theta}_{R}^{a})$ ,

[TABLE]

where ${\theta}_{-}^{a}={\theta}_{L}^{a}-i{\theta}_{R}^{a}$ . The superfield $\Phi$ will be defined to satisfy the reality condition $({\nabla}_{+})^{4}_{abcd}\Phi={1\over{24}}{\epsilon}_{abcdefgh}({\nabla}_{-})_{efgh}^{4}\overline{\Phi}$ , and the $2^{8}$ components of $\Phi$ describe the Type IIB supergravity multiplet where, at zeroth order in ${\theta}_{-}$ , the real part of $\Phi$ is the Type IIB dilaton and the imaginary part of $\Phi$ is the Type IIB axion.

To construct the vertex operator of (2.1) for this multiplet, first consider the vertex operator

[TABLE]

Using the relation ${\lambda}_{L}^{a}{\overline{\lambda}}_{L}^{\dot{a}}=-{1\over 4}({\sigma}^{jk}{\lambda}_{L})^{a}({\sigma}_{jk}{\overline{\lambda}}_{L})^{\dot{a}}$ and ${\lambda}_{R}^{a}{\overline{\lambda}}_{R}^{\dot{a}}=-{1\over 4}({\sigma}^{jk}{\lambda}_{R})^{a}({\sigma}_{jk}{\overline{\lambda}}_{R})^{\dot{a}}$ , one finds that

[TABLE]

where ${\lambda}_{\pm}^{a}={\lambda}_{L}^{a}\pm i{\lambda}_{R}^{a}$ . Now consider the vertex operator

[TABLE]

Since $\{{\nabla}_{-},{\nabla}_{+}\}=4{\partial}_{+}$ , (2.1) implies that $QV_{0}=-({\lambda}_{+}{\nabla}_{-})V_{1}$ . Furthermore, a similar argument implies that $({\lambda}_{-}{\nabla}_{+})V_{1}=-({\lambda}_{+}{\nabla}_{-})V_{2}$ where

[TABLE]

Continuing this argument, one finds that $QV=0$ where $V=V_{0}+V_{1}+V_{2}+V_{3}+V_{4}$ and

[TABLE]

Note that $({\lambda}_{-}{\nabla}_{+})V_{4}=0$ since $({\nabla}_{+})^{9}\Phi=0$ .

So the BRST-invariant vertex operator with momentum $k_{+}$ in this gauge is

[TABLE]

and one can easily verify that at ${\theta}_{L}^{a}={\theta}_{R}^{a}=0$ , $A_{{\dot{a}}{\dot{b}}}$ is the bispinor Ramond-Ramond field in light-cone gauge

[TABLE]

It will be useful to note that one would end up with the same expression of (2.1) for $V$ if one had instead started with the superfield $\Phi_{1234}$ which is annihilated by $\nabla_{-}^{a}\equiv{\nabla}_{L}^{a}-i({\sigma}_{1234}{\nabla}_{R})^{a}$ . In this case, $V_{0}=({\overline{\lambda}}_{L}{\sigma}_{1234}{\overline{\lambda}}_{R})\Phi_{1234}$ and

[TABLE]

where ${\nabla}_{+}^{a}\equiv{\nabla}_{L}^{a}+i({\sigma}_{1234}{\nabla}_{R})^{a}$ .

3. Picture-Changing

To generalize this construction to an $AdS_{5}\times S^{5}$ background, it will be useful to first consider the vertex operator $V$ for the lowest component of $\Phi_{1234}$ in (2.1), i.e. $\Phi_{1234}=\exp(ik_{+}\hat{x}^{+})$ where $\hat{x}^{+}\equiv x^{+}+i{\theta}_{L}{\sigma}_{1234}{\theta}_{R}$ . Although this vertex operator of (2.1) has various terms $V_{0}...V_{4}$ with different powers of ${\theta}_{+}^{a}={\theta}_{L}^{a}+i({\sigma}_{1234}{\theta}_{R})^{a}$ , it can be reduced to just one term by writing it in a different “picture” as

[TABLE]

where $P$ is the “picture-lowering” operator

[TABLE]

and ${\lambda}_{+}^{a}={\lambda}_{L}^{a}+i({\sigma}_{1234}{\lambda}_{R})^{a}$ . Note that the 8 ${\lambda}_{+}^{a}$ ’s in $P$ are all independent so that $\prod_{a=1}^{8}{\delta}({\lambda}^{a}_{+})$ 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 $q_{1}$ ,

[TABLE]

The original vertex operator $V$ of (2.1) is related to $V_{-1}$ of (3.1) by picture-raising as $V=CV_{-1}$ where

[TABLE]

is the picture-raising operator and $Q(\xi_{a})$ 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, ${\delta}(\gamma)=e^{-\phi}$ and $\xi{\delta}({\gamma})=\xi e^{-\phi}={1\over{\gamma}}$ where $({\gamma},{\beta})$ are chiral bosons which have been fermionized as ${\gamma}=\eta e^{\phi}$ and $\beta={\partial}\xi e^{-\phi}$ . Although ${\lambda}_{+}^{a}$ and its conjugate $w_{a}^{+}$ are not chiral bosons, one can formally define

[TABLE]

so that

[TABLE]

Using this definition, $CV_{-1}$ can be computed by using (3.1) to convert the factors of ${\delta}({\lambda}_{+}^{a})$ in $V_{-1}$ into factors of ${1\over{{\lambda}_{+}^{a}}}$ . Furthermore, the BRST invariance of $V_{-1}$ guarantees that $CV_{-1}$ has no poles when ${\lambda}_{+}^{a}=0$ and can be expressed in the form of (2.1) as $V={\lambda}_{L}^{\alpha}{\lambda}_{R}^{\beta}A_{{\alpha}{\beta}}(x,{\theta}_{L},{\theta}_{R})$ . To see why, note that $Q(F\delta({\lambda}_{+}^{a}))=0$ implies that $Q(F)$ is proportional to ${\lambda}_{+}^{a}$ . So $Q({F\over{{\lambda}_{+}^{a}}})$ has no poles when ${\lambda}_{+}^{a}=0$ . Also note that if $F$ has (left,right)-moving ghost number equal to $(g_{L},g_{R})$ , then $Q({F\over{{\lambda}_{+}^{a}}})$ 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 $QF=E{\lambda}_{+}^{a}$ for some $E$ implies that $E$ must carry ghost number $(g_{L},g_{R})$ .

One can explicitly compute $CV_{-1}$ for the vertex operator of (3.1) as

[TABLE]

where we have used that ${\lambda}_{-}^{1}({\overline{\lambda}}_{L}{\sigma}_{1234}{\overline{\lambda}}_{R})={1\over 4}({\sigma}^{jk}{\lambda}_{+})^{1}({\overline{\lambda}}_{L}{\sigma}^{jk}{\sigma}_{1234}{\overline{\lambda}}_{R})$ . Continuing with this procedure of converting $\xi_{a}{\delta}({\lambda}_{+}^{a})$ into $({\lambda}_{+}^{a})^{-1}$ to compute the product with $Q(\xi_{a})$ , it is expected that $CV_{-1}$ will reproduce $V$ of (2.1).

4. $AdS_{5}\times S^{5}$ Vertex Operators

4.1. Parameterization of $AdS_{5}\times S^{5}$

To generalize this construction for half-BPS states in an $AdS_{5}\times S^{5}$ background, parameterize $AdS_{5}\times S^{5}$ using the supercoset $g\in{{PSU(2,2|4)}\over{SO(4,1)\times SO(5)}}$ as

[TABLE]

where $F({\theta})=\exp({\theta}^{J}_{R}q_{J}^{R}+{\theta}^{R}_{J}q^{J}_{R})$ is a fermionic ${{PSU(2,2|4)}\over{SO(4,2)\times SO(6)}}$ coset, $(q^{R}_{J},q^{J}_{R})$ are the 32 fermionic generators of $PSU(2,2|4)$ , $R=1$ to 4 are $SO(4,2)$ spinor indices, $J=1$ to 4 are $SO(6)$ spinor indices, $G(X)$ is an ${{SO(4,2)}\over{SO(4,1)}}$ coset for $AdS_{5}$ and $H(Y)$ is an ${{SO(6)}\over{SO(5)}}$ coset for $S^{5}$ . Under global $PSU(2,2|4)$ transformations, ${\delta}g=\Sigma g$ where $\Sigma\in PSU(2,2|4)$ , and under BRST transformations,

[TABLE]

where $\tilde{R}=1$ to 4 is an $SO(4,1)$ spinor index, $\tilde{J}=1$ to 4 is an $SO(5)$ spinor index, and $({\lambda}_{L})^{\tilde{R}}_{\tilde{J}}$ and $({\lambda}_{R})^{\tilde{R}}_{\tilde{J}}$ 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 ${\sigma}_{6}^{\tilde{R}\tilde{S}}$ and ${\sigma}_{6}^{\tilde{J}\tilde{K}}$ which commute with $SO(4,1)$ and $SO(5)$ rotations.

The cosets $G(X)$ and $H(Y)$ are defined up to local $SO(4,1)\times SO(5)$ gauge transformations parameterized by $\Omega\in SO(4,1)$ and $\hat{\Omega}\in SO(5)$ as

[TABLE]

where the left and right-moving pure spinors ${\lambda}_{L}$ and ${\lambda}_{R}$ transform as $SO(4,1)\times SO(5)$ spinors. More explicitly, $G^{R}_{\tilde{R}}$ and $H^{J}_{\tilde{J}}$ are $4\times 4$ matrices which transform under the gauge transformations as

[TABLE]

and the $AdS_{5}$ coordinate $X^{RS}=-X^{SR}$ and $S^{5}$ coordinate $Y^{JK}=-Y^{KJ}$ are defined in terms of $G^{R}_{\tilde{R}}$ and $H^{J}_{\tilde{J}}$ by

[TABLE]

Defining $X_{RS}={1\over 2}{\epsilon}_{RSTU}X^{TU}$ and $Y_{JK}={1\over 2}{\epsilon}_{JKLM}Y^{LM}$ , (4.1) implies $X^{RS}X_{RS}=4$ and $Y^{JK}Y_{JK}=4$ .

4.2. Half-BPS vertex operator

To construct the vertex operator for a half-BPS state in an $AdS_{5}\times S^{5}$ background, consider the state dual to the super-Yang-Mills gauge-invariant operator

[TABLE]

where $\Phi_{JK}(x)$ are the six scalars located at the position $x^{m}$ on the $AdS_{5}$ boundary and $y_{0}^{JK}$ is a fixed null six-vector satisfying ${\epsilon}_{JKLM}y_{0}^{JK}y_{0}^{LM}=0$ . It will be convenient to define the null six-vector

[TABLE]

where $R=(A,\dot{A})$ with $A,\dot{A}=1$ to 2. $x_{0}^{RS}$ transforms covariantly under $SO(4,2)$ conformal transformations of the $AdS_{5}$ boundary and satisfies ${\epsilon}_{RSTU}x_{0}^{RS}x_{0}^{TU}=0$ .

The choice of $y_{0}^{JK}$ breaks $SO(6)$ $R$ -symmetry to $U(1)\times SO(4)$ , and $J$ will be defined to be the charge with respect to this $U(1)$ . Similarly, the choice of $x_{0}^{RS}$ breaks $SO(4,2)$ conformal symmetry to $SO(1,1)\times SO(3,1)$ , and $\Delta$ will be defined to be the charge with respect to the $SO(1,1)$ . The half-BPS state of (4.1) carries $J=n$ and $\Delta=n$ and is preserved by the 24 spacetime supersymmetries which carry $J-\Delta\geq 0$ .

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 (4.1) is

[TABLE]

where the picture-lowering operator $P$ is defined as

[TABLE]

and ${\theta}_{+}^{a}$ are the 8 ${\theta}$ ’s which carry charge $J-\Delta=1$ . In terms of $x_{0}^{RS}$ and $y_{0}^{JK}$ ,

[TABLE]

where only 8 of the 32 components of $(x_{0})^{RS}(y_{0})_{JK}{\theta}^{K}_{S}$ and $(x_{0})_{RS}(y_{0})^{JK}{\theta}_{K}^{S}$ are independent since $(x_{0})^{RS}(x_{0})_{ST}=(y_{0})^{JK}(y_{0})_{KL}=0$ .

To show that $V_{-1}$ of (4.1) carries the same charges and is invariant under the same 24 supersymmetries as (4.1), note that $Y\cdot y_{0}$ carries $J=1$ and $X\cdot x_{0}$ carries $\Delta=-1$ so that $V_{-1}$ carries $J=\Delta=n$ . Furthermore, both $Y\cdot y_{0}$ and $X\cdot x_{0}$ are invariant under the 8 supersymmetries with $J-\Delta=1$ . And under the 16 supersymmetries with $J-\Delta=0$ , ${{Y\cdot y_{0}}\over{X\cdot x_{0}}}$ transforms into terms which contain at least one ${\theta}$ with $J-\Delta=1$ . However, all 8 $\theta$ ’s with $J-\Delta=1$ are contained in the picture-lowering operator $P$ of (4.1). So $V_{-1}$ is invariant under all 24 supersymmetries which carry $J-\Delta\geq 0$ .

Similarly, under the BRST transformation of (4.1), ${{Y\cdot y_{0}}\over{X\cdot x_{0}}}$ transforms into terms containing products of $Q({\theta})$ with ${\theta}$ ’s where either $Q({\theta})$ carries $J-\Delta=1$ or at least one of the ${\theta}$ ’s carries $J-\Delta=1$ . In both cases, the BRST transformation is killed by $P=\prod_{a=1}^{8}{\theta}_{+}^{a}{\delta}(Q({\theta}_{+}^{a}))$ of (4.1). And since $P$ and $({\lambda}_{L})^{\tilde{J}}_{\tilde{R}}({\lambda}_{R})^{\tilde{R}}_{\tilde{J}}$ are also BRST-invariant, it has been shown that $V_{-1}$ of (4.1) is BRST-invariant.

4.3. Explicit example

For example, consider the state corresponding to $Tr[(\Phi_{12}(0))^{n}]$ which carries $\Delta=J=n$ where $\Delta$ is the dilatation charge and $J$ is the $U(1)$ charge. To simplify the vertex operator, parameterize the supercoset $g\in{{PSU(2,2|4)}\over{SO(4,1)\times SO(5)}}$ as

[TABLE]

where $(q_{+},q_{0},q_{-})$ are the $(8,16,8)$ fermionic isometries with $(+1,0,-1)$ charge with respect to $J-\Delta$ , and $K$ and $R$ are the four conformal boosts and four $R$ -symmetries with charge $J-\Delta=1$ . Since the vertex operator $V$ is annihilated by $(q_{+},q_{0},K,R,\Delta-J)$ , the parameterization of (4.1) implies that $V$ is independent of $({\theta}_{-},{\theta}_{0},x,y,w+z)$ and only depends on $({\theta}_{+},z-w)$ and the pure spinor ghosts.

Using the picture-lowering operator $P=\prod_{a=1}^{8}{\theta}_{+}^{a}{\delta}(Q({\theta}_{+}^{a}))$ , the vertex operator of (4.1) is

[TABLE]

where $({\lambda}_{+}^{a},{\lambda}_{-}^{a},{\overline{\lambda}}_{L}^{\dot{a}},{\overline{\lambda}}_{R}^{\dot{a}})$ for $a,\dot{a}=1$ to 8 are defined by

[TABLE]

and we have used that $({\lambda}_{L})^{\tilde{J}}_{\tilde{R}}({\lambda}_{R})^{\tilde{R}}_{\tilde{J}}={\overline{\lambda}}_{L}^{\dot{a}}({\sigma}_{1234})_{\dot{a}\dot{b}}{\overline{\lambda}}_{R}^{\dot{b}}$ when ${\lambda}_{+}^{a}=0$ .

In the large radius limit where the $AdS_{5}\times S^{5}$ background approaches flat space, one can easily verify that $V_{-1}$ of (4.1) 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 $AdS_{5}\times S^{5}$ background are obtained from (4.1) 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 (4.1) to the supergravity vertex operator $V={\lambda}_{L}^{\alpha}{\lambda}_{R}^{\beta}A_{{\alpha}{\beta}}(x,{\theta})$ of (2.1) by defining

[TABLE]

where $C=\prod_{a=1}^{8}Q(\xi_{a})$ and the 8 ${\lambda}_{+}^{a}$ ’s of (4.1) have been fermionized as in (3.1). Using the same procedure as in (3.1), this construction will produce an $AdS_{5}\times S^{5}$ vertex operator of the form $V={\lambda}_{L}^{\alpha}{\lambda}_{R}^{\beta}A_{{\alpha}{\beta}}({\theta}_{+},z-w)$ where, as in a flat background, the potential poles coming from $\xi^{a}\delta({\lambda}_{+}^{a})={1\over{{\lambda}_{+}^{a}}}$ are absent because of the BRST invariance of $V_{-1}$ .

5. Summary

In this paper, a simple BRST-invariant vertex operator was constructed for half-BPS states in an $AdS_{5}\times S^{5}$ 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 $AdS_{5}\times S^{5}$ , 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 $AdS_{5}\times S^{5}$ vertex operators for non-BPS states. As discussed in [12], the half-BPS vertex operator can be expressed as

[TABLE]

if one adds $(n-1)$ picture-raising operators $C$ and $(n-1)$ picture-lowering operators $P$ to $V=CV_{-1}$ of (4.1). 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

[TABLE]

where $E\equiv P{{Y\cdot y_{0}}\over{X\cdot x_{0}}}$ corresponds to the Yang-Mills field $y_{0}^{JK}\phi_{JK}(x_{0})$ on the spin chain. Therefore, a natural conjecture for general non-BPS vertex operators is

[TABLE]

where $E_{1}...E_{n}$ describe $n$ different super-Yang-Mills fields on the spin chain and are obtained from $P{{Y\cdot y_{0}}\over{X\cdot x_{0}}}$ by performing the appropriate $PSU(2,2|4)$ transformation. Since $E$ and $C$ are independently BRST-invariant, the vertex operator of (5.1) 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 $AdS_{5}\times S^{5}$ superstring at small radius.

Acknowledgements: I would like to thank Thales Agricola, Thiago Fleury, Juan Maldacena, Andrei Mikhailov and Pedro Vieira for useful discussions, and CNPq grant 300256/94-9 and FAPESP grants 2016/01343-7 and 2014/18634-9 for partial financial support.

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