Superstrings on Penrose limits of AdS$_7$ solutions

We consider the Penrose limit of AdS$_7$ solutions of massive Type IIA supergravity. The resulting pp-wave geometry is supported by RR and NSNS fields. We quantize the Green--Schwarz superstring on the obtained pp-wave background, in the light-cone gauge.


Introduction
The presence of Ramond-Ramond fields poses a technical challenge to the quantization of superstrings.An exception are pp-wave spacetimes which provide curved backgrounds with Ramond-Ramond fields where superstrings can be quantized.String theory on pp-wave spacetimes is of interest in the context of the AdS/CFT correspondence, as pp-wave spacetimes arise in the Penrose limit of anti-de Sitter solutions.In [1] it was proposed that the Penrose limit of the Type IIB AdS 5 × S 5 solution corresponds to singling out a subset of operators in the dual N = 4 supersymmetric U(N ) Yang-Mills theory, where both the conformal dimension ∆ and a U(1) R-charge J are taken to be of order √ N with the difference ∆ − J finite as N → ∞.
In this paper we consider the Penrose limit of AdS 7 solutions of (massive) Type IIA supergravity, which are supported by RR and NSNS fluxes.These are dual to six-dimensional N = (1, 0) superconformal field theories which admit a description in terms of linear quivers.
Upon obtaining the pp-wave spacetime in the Penrose limit, we quantize the Green-Schwarz closed superstring in the light-cone gauge.The Hamiltonian has a "harmonic oscillator" form.
The identification of the operators of the dual six-dimensional field theories which correspond to the string excitations is under consideration.
The rest of the paper is organised as follows: in section 2 we review the family of AdS 7 solutions and select two of them on which we focus on the remainder of the paper.In section 3 we obtain the pp-wave background in the Penrose limit and analyze its supersymmetry.
In section 4 we perform the quantization of the Green-Schwarz closed superstring in the light-cone gauge.

AdS 7 solutions
The AdS 7 solutions of massive Type IIA supergravity [2,3], in the form introduced in [4], are They are determined by a cubic in z function α(z) and its derivatives (a dot (˙) denotes derivation with respect to z) which is subject to: ...
B is the NSNS two-form potential, ϕ the dilaton, F 2 the RR two-form field strength and F 0 the Romans mass.
We will endow the seven-dimensional anti-de Sitter spacetime AdS 7 and the two-sphere S 2 with the metrics where ds 2 (S 5 ) is the unit-radius metric on the five-sphere S 5 .Also, in (2.1), vol(S . . .
More general CFT6 from F-theory . . .• F-theory allows to include more general gauge groups • The D8's should be dual in F-theory to an object called "T-brane" We will consider two representative solutions (a) and (b).Solution (a) is determined by where n 0 ∈ Z are the quanta of the Romans mass and N ∈ Z the NSNS flux quanta.z ∈ [0, N ]; at z = 0 the geometry is regular while at z = N there is a D6-brane type singularity.The dual field theory is described by the quiver in Figure 1(a).
Solution (b) has zero Romans mass and is determined by where k ∈ Z are the RR flux quanta and N ∈ Z the NSNS flux quanta.z ∈ [0, N ], and at the endpoints of the interval there are D6-brane type singularities.The dual field theory is described by the quiver in Figure 1(b).

The Penrose limit
In order to define the Penrose limit, we consider null geodesic motion in the ψ (isometry) direction with afffine parameter λ.Furthermore, we will take the geodesic located at the center of AdS 7 at ρ = 0 and at the equator of the S 2 at θ = π/2.Thus, the U(1) ⊂ SU(2) isometry of the two-sphere is preserved, and similarly for the isometries of the five-sphere in AdS 7 .
The geodesic equation yields while imposing g µν (x µ ) ′ (x ν ) ′ = 0 (null condition) yields Substituting in the last equation coming from the geodesic equation we have which yields α α( α2 We conclude that z is fixed at a critical value z c for which (3.5) is satisfied.For the specific solutions we will study we will see that z c is such that α(z c ) = 0.
For solution (a) we have and, as mentioned, α(z c ) = 0.
We introduce and expand the metric near ζ = 0: where We now make the coordinate transformation The Penrose limit is L → ∞ i.e.N → ∞.The metric becomes: (3.12) In the limit under consideration, the dilaton is In order to have a finite dilaton we need to take Following the same procedure for the rest of the fields we obtain For solution (b) we have: We introduce again and expand the metric near ζ = 0: where We now make the coordinate transformation The Penrose limit is L → ∞ i.e.N → ∞.The metric becomes that of (3.11).
In the limit under consideration, the dilaton is In order to have a finite dilaton we need to take k → ∞ such that N 1/2 k stays finite as N → ∞.For the rest of the fields we find the same expressions as (3.14).
The pp-wave backgound for both solutions (a) and (b) is the same except for the constant value of the dilaton.It is straightforward to check that the equations of motion are satisfied.
For instance, the non-trivial component of the Einstein equations is: 2 where R ++ is the only non-zero component of the Ricci tensor.Given that We can express the light-cone momentum in terms of the conformal dimension ∆ and 2 Given a p-form C, we use the notation: the R-charge J of a dual operator [1]: (3.24b)
The equations coming from setting the gravitini supersymmetry variations to zero are: (3.30) ϵ 1 and ϵ 2 are Majorana-Weyl spinors of positive and negative chirality respectively.The equations coming from setting the dilatini supersymmetry variations to zero are: We now substitute the pp-wave background of the previous section.The equations coming from the dilatini variations are: The equations coming from the gravitini variations are: where we have used that the non-zero components of the spin connection are We find that are solved by where χ 1 , χ 2 are constant Majorana-Weyl spinors of opposite chirality and in addition, ϵ 1 , ϵ 2 are subject to In total sixteen supersymmetries are preserved, as for the AdS 7 solutions.

Bosonic part
The bosonic part of the Green-Schwarz action is We fix: It follows that X + satisfies the wave equation in two dimensions and so we can further impose the light-cone gauge The bosonic string action then becomes Introducing the center-of-mass variable we have where L b is the (bosonic) Lagrangian.Hence, We consider closed strings so impose the periodicity condition The equations of motion that follow from the action (4.5) are then: where i = 1, 2, . . ., 6.

.14)
For i = 7, 8 the solution is where Ãi n (τ, σ) := with and The canonical quantization condition where yields with the rest of the commutators vanishing.
The bosonic Hamiltonian is It has the following expression in terms of the creation and annihilation operators: where we have introduced and the normal-ordering constant

Fermionic part
The quadratic fermionic part of the Green-Schwarz action is where I, J , K = 1, 2, is the pullback to the worldsheet of the supercovariant connection that appears in the gravitini supersymmetry variation, θ 1 , θ 2 are Majorana-Weyl spinors of positive and negative chirality respectively and θI := (θ I ) † Γ 0 .
We fix h ab as in (4.2).In the light-cone gauge and the action becomes where We thus obtain The equations of motion are: We derive the solutions: where ω n := n 2 + (α ′ p + ) 2 and (4.38) The conjugate momenta are where we have rewritten Γ which follows from the light-cone gauge condition.
The canonical quantization conditions [5,6] where The projectors P ± reflect the fact that θ I are Weyl and subject to the light-cone gauge condition.
Upon using the equations of motion the Hamiltonian is given by We find where (4.45) In deriving the normal-ordering constant ε f we have used Tr(P ± ) = 8 and Tr(P ± Γ 7 ) = Tr(P ± Γ 78 ) = 0.

Light-cone Hamiltonian
Combining the bosonic (4.25) and fermionic (4.44) parts we obtain the Hamiltonian where the normal-ordering constant ε is the sum of the bosonic and fermionic ones, whose contributions cancel, up to a finite number of terms: