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Regularizing Cubic Open Neveu-Schwarz String Field Theory

After introducing non-minimal variables, the midpoint insertion of Y\bar Y in cubic open Neveu-Schwarz string field theory can be replaced with an operator N_\rho depending on a constant parameter \rho. As in cubic open superstring field theory using the pure spinor formalism, the operator N_\rho is invertible and is equal to 1 up to a BRST-trivial quantity. So unlike the linearized equation of motion Y\bar Y QV=0 which requires truncation of the Hilbert space in order to imply QV=0, the linearized equation N_\rho QV=0 directly implies QV=0.


Introduction
Open bosonic string field theory has recently been used to study classical solutions of string theory such as tachyon condensation that are difficult to analyze using firstquantized approaches. Some progress has been made in extending these techniques to open superstring field theory, and there are presently three versions of open superstring field theory available.
The first version is based on the cubic action [1][2] where V is the Neveu-Schwarz (NS) string field of zero picture and +1 ghost-number in the small Hilbert space, and YȲ is an operator of −2 picture inserted at the string midpoint.
The second version is based on the Wess-Zumino-Witten-like action [3] where φ is the NS string field of zero picture and zero ghost-number in the large Hilbert space. And the third version is based on the cubic action [4] where Φ is a superstring field of +1 ghost-number in the GSO(+) sector using the nonminimal pure spinor formalism, and N ρ = e −ρ{Q,χ} is a BRST-invariant regulator inserted at the midpoint which depends on a constant parameter ρ.
Each of these three versions has advantages and disadvantages. The first version of (1.1) has the advantage of being cubic, but has the disadvantage of being singular since the midpoint insertion YȲ is not invertible. So the linearized action YȲ QV = 0 does not imply QV = 0 unless one truncates out states in the kernel of YȲ . [5] The second version of (1.2) has the disadvantage of being non-polynomial, but has the advantage of being non-singular since there are no midpoint insertions. So the linearized equation of motion is ηQφ = 0, which implies QV = 0 where V ≡ ηφ is in the small Hilbert space.
Finally, the third version of (1.3) has the advantage of being cubic and non-singular since, unlike the operator YȲ in (1.1), the operator N ρ has no kernel and is invertible. So the linearized equation of motion N ρ QΦ = 0 implies QΦ = 0. The disadvantage of the third version is that Φ includes the GSO(+) NS and Ramond sectors of the superstring, but does not include the GSO(−) NS sector and cannot be used to describe tachyon condensation.
In this paper, we shall propose a fourth version of open superstring field theory which combines the advantages of the first and third versions and eliminates their disadvantages.
After adding a pair of non-minimal variables to the NS formalism, it will be possible to replace the singular operator YȲ of the first version with a non-singular invertible operator N ρ depending on the non-minimal variables and on a constant parameter ρ. The action will be where V is a NS string field in the zero picture in the small Hilbert space which is allowed to depend on the non-minimal variables.

Cubic open NS string field theory
In the absence of operators involving δ(γ), functional integration over the bosonic (β, γ) ghost zero modes produces infinities in the NS tree amplitude. These delta functions can be inserted in a BRST-invariant manner using the inverse picture-changing operator where we use the notation Note that the OPE's of e φ imply that δ(γ) = γ( δ(γ) γ ). Since γ has two zero modes on a disk, open string tree amplitudes require two inverse-picture-changing operators, and it is convenient to insert the operator YȲ at the midpoint interaction whereȲ =c∂ξe −2φ is constructed from the antiholomorphic ghosts.
Since YȲ has a non-trivial kernel, e.g. YȲ c = 0, it is clear that YȲ is not invertible unless one truncates out states from the Hilbert space. In order to replace YȲ with an invertible operator, the first step will be to write where r andr are fermionic variables and u andū are bosonic variables. Note that The next step will be to treat (r,r) and (u,ū) as non-minimal worldsheet variables with conjugate momenta (s,s) and (v,v) by adding them to the worldsheet action S = S RNS + d 2 z(−s∂r −s∂r + v∂u +v∂ū) (2.5) and to the BRST operator Using the standard quartet argument, the additional terms in Q imply that physical states in the cohomology of Q are independent of the non-minimal variables. It will also be convenient to perform the similarity transformation so that (u, v) and (r, s) each carry conformal weight (− 1 2 , 3 2 ). The final step is to define N ρ = e ρ{Q,χ} = e ρ[rc+rc+u(γ 2 + 3 2 c∂c)+ū(γ 2 + 3 2c∂c )] (2.8) where χ = uc +ūc, ρ is a nonzero constant, and we have used that χ has − 3 2 conformal weight to compute the uc∂c term in (2.8). Using a similar computation as in (2.3), it is easy to check that where the ρ dependence cancels out and the uc∂c term in (2.8) does not contribute because of the factor of cc in YȲ .
To define a cubic open NS string field theory using N ρ , one needs to allow the string field V to depend both on the original NS worldsheet variables (x m , ψ m ; b, c, β, γ) and on the new non-minimal variables (r, s, u, v). Note that bosonization is unnecessary since both V and N ρ can be expressed in terms of (β, γ) and (β,γ). The cubic string field theory action is where ... is defined as usual by functional integration over all the worldsheet variables.
Since u and r have two zero modes on the disk, their zero mode integration reproduces (2.9). So if one writes V = V 0 + V where V 0 is independent of the non-minimal variables, (2.9) implies that the terms in S which are independent of V are the same as in the original cubic action of (1.1). However, the terms in S which depend on V are necessary for guaranteeing that the linearized equation of motion is equivalent to QV = 0.
To see how V contributes to the action, note that rescaling removes the ρ dependence from N ρ and leaves invariant the worldsheet action and BRST operator. After this rescaling, the string field depends on ρ as where n counts the number of non-minimal fields in V n , i.e.
[ dz(uv + rs) + dz(ūv +rs) , V n ] = nV n . (2.13) Note that for string fields of finite conformal weight, n is bounded from below since (v, s,v,s) carry positive conformal weight.
Using (2.12), the action of (2.10) can be expressed as S = ∞ n=−∞ S n ρ −n where (2.14) As in the cubic action of (1.3) using the pure spinor formalism, (2.14) involves an infinite chain of auxiliary fields V n depending on the non-minimal variables. Since the non-minimal variables include bosons, (2.12) and (2.14) resemble the construction of superfields and actions in harmonic superspace. It should be noted that, unlike the non-minimal variables in the pure spinor formalism which all carry non-negative conformal weight, the nonminimal variables u and r carry − 1 2 conformal weight. So V n for n > 0 involves states of negative conformal weight which could complicate computations using level truncation. 1 Another possible difficulty, which is also a difficulty with all the other cubic superstring field theory actions, is gauge-fixing. Since the midpoint insertion of N ρ (like the insertion of YȲ ) involves the c ghost, fixing the b 0 = 0 gauge may be subtle. One could try to implement alternative gauge choices such as Schnabl gauge or the gauge choices of [1], but these also have subtleties. Note that in the cubic action using the pure spinor formalism, there are difficulties with gauge-fixing because of the (λλ) poles in the b ghost [4]. The only action that appears to be free of gauge-fixing difficulties is the Wess-Zumino-Witten-like action of (1.2) where one can easily choose the gauge-fixing conditions b 0 = ξ 0 = 0 as in [7].
It would be interesting to extend the action of this paper to include the Ramond sector. Although one can describe the GSO(+) Ramond sector using the pure spinor version of (1.3), there is no non-singular action which can covariantly describe both the GSO(+) and GSO(−) Ramond and NS sectors. It is intriguing that the non-minimal worldsheet fields (u, v, r, s) introduced here have the same statistics and conformal weights as the non-minimal worldsheet fields ( γ, β, ξ, µ) which were used in [8] to allow a more symmetric treatment of the NS and Ramond sectors.