HUTP–01/A061 PreprinttypesetinJHEPstyle. -HYPERVERSION hep-th/0112231 Towards Vacuum Superstring Field Theory: The Supersliver 2 0 0 2 n Marcos Mari˜no and Ricardo Schiappa a J Department of Physics, 3 Harvard University, 2 Cambridge, MA 02138, USA marcos, [email protected] 2 v 1 3 2 Abstract: We extend some aspects of vacuum string field theory to superstring field theory in 2 Berkovits’ formulation, andwestudythestaralgebrainthefermionicmattersector. Afterclarifying 1 1 the structure of the interaction vertex in the operator formalism of Gross and Jevicki, we provide an 0 algebraic construction of the supersliver state in terms of infinite–dimensional matrices. This state / h is an idempotent string field and solves the matter part of the equation of motion of superstring t - field theory with a pure ghost BRST operator. We determine the spectrum of eigenvalues and p e eigenvectors of the infinite–dimensional matrices of Neumann coefficients in the fermionic matter h sector. We then analyze coherent states based on the supersliver and use them in order to construct : v higher–rank projector solutions, as well as to construct closed subalgebras of the star algebra in i X the fermionic matter sector. Finally, we show that the geometric supersliver is a solution to the r superstring field theory equations of motion, including the (super)ghost sector, with the canonical a choice of vacuum BRST operator recently proposed by Gaiotto, Rastelli, Sen and Zwiebach. Keywords: String Field Theory, Supersymmetry, Sliver, Tachyon Condensation. Contents 1. Introduction and Summary 2 2. Berkovits’ Superstring Field Theory 4 2.1 A Short Review of Berkovits’ Superstring Field Theory 4 2.2 Superstring Field Theory Around a Classical Solution 5 3. Neumann Coefficients and Overlap Equations 9 3.1 The Identity 9 3.2 Interaction Vertex and Overlap Equations 11 4. The Supersliver 15 4.1 Algebraic Construction 16 4.2 Numerical Results and Comparison to the Geometric Sliver 19 4.3 Conservation Laws 21 5. Fermionic Star Algebra Spectroscopy 22 5.1 An Eigenvector of M and M 22 5.2 Diagonalizing K 23 1 5.3 Diagonalizing M and M f 24 6. Coherent States and Higher–Rank Projectors 28 f 6.1 Coherent States on the Supersliver 28 6.2 Higher–Rank Projectors 29 7. The Geometric Supersliver and the (Super)Ghost Sector 34 8. Conclusions and Future Directions 36 A. Appendix 39 1 1. Introduction and Summary In the last two years, the search for nonperturbative information in string field theory [1, 2] has experienced a renewed interest mainly due to a series of conjectures by Sen [3, 4, 5] (also see [6] for a review and a list of references). These conjectures have been tested numerically to a high degree of precision in level truncated cubic string field theory, and some of them have been proven in boundary string field theory (see, e.g., [7] for a review and a list of references). In the meantime, the elegant construction of Berkovits [8, 9, 10, 11] has emerged as a promising candidate for an open superstring field theory describing the NS sector: in here, Sen’s conjectures about the fate of the tachyon in the non–BPS D9–brane have been successfully tested by level truncation to a high level of accuracy [12, 13, 14, 15], and kink solutions have been found that describe lower–dimensional D–branes [16] (see, e.g., [17] for a review and a more complete list of references). So far, most of our understanding about tachyon condensation in both cubic string field theory and Berkovits’ superstring field theory is based on level–truncated computations and it would be of course desirable to have an analytical control over the problem. For the bosonic string, Rastelli, Sen and Zwiebach have proposed in a series of papers [18, 19, 20, 21, 22] a new approach to this problem called vacuum string field theory (VSFT). In VSFT, the form of the cubic string field theory action around the tachyonic vacuum is postulated by exploiting some of the expected properties it should have (like the absence of open string states). Then one can show that this theory has solutions that describe the perturbative vacuum and the various D–branes. In particular, the matter sector of the maximal D25–brane is described by a special state called the sliver. This state was first constructed geometrically by Rastelli and Zwiebach [23] and then algebraically by Kostelecky and Potting [24], and it is an idempotent state of the string field star algebra, in the matter sector. The construction of VSFT has been recently completed in [25], where Gaiotto, Rastelli, Sen and Zwiebach have proposed a canonical choice of the ghost BRST operator around the vacuum, with which they identified closed string states; and also in [26], where Rastelli, Sen and Zwiebach have found the eigenvalue and eigenvector spectrum of the Neumann matrices, which could allow for a proper definition of the string field space. The study of VSFT has also unveiled beautiful algebraic structures in cubic string field theory (for example, projectors of arbitrary rank in the star algebra have been constructed in detail in [20, 27, 28]). The main purpose of this paper is to give the first steps towards the construction of vacuum superstring field theory around the tachyonic vacua of the non–BPS maximal D9–brane in Type IIA superstring theory, and to explore the algebraic structure of the star algebra in the fermionic part of the matter sector. In section 2, we begin with a review of Berkovits’ open superstring field theory for the NS sector and discuss the general features of vacuum superstring field theory. We shall show in detail that, assuming a pure ghost BRST operator around the vacuum as in VSFT, Berkovits’ equation of motionforthe superstring field admits factorized solutions whose matter part is an idempotent state of the star algebra. In a sense, idempotency is even more useful in Berkovits’ theory since it drastically reduces the nonlinearity of the equation of motion. Idempotent string field solutions can be constructed in the GSO(+) sector or in both GSO( ) sectors. ± In order to construct idempotent states in superstring field theory, one first has to understand in detail the structure of the star algebra in the fermionic matter sector. To do that, we use the 2 operator construction of the interaction vertex for the superstring due to Gross and Jevicki [29], which extends their previous work on the bosonic string [30, 31] to the NSR superstring. In section 3 we review some of the relevant results and we further clarify the structure of the vertex. This allows us to write the Neumann coefficients in terms of two simple infinite–dimensional matrices which shall play a key rˆole in the constructions of this paper. Given any boundary conformal field theory (BCFT) one can construct geometrically a special state which is an idempotent of the star algebra [21]. When the BCFT is that of a D25–brane, this state is called the sliver1 [23, 18]. This geometric construction extends in a very natural way to the BCFT given by the NS sector of the open superstring which describes the unstable D9–brane. This yields an idempotent state that we call the supersliver. The matter part of the supersliver is a product of two squeezed states: one made of bosonic oscillators (the bosonic sliver previously considered in [24, 19]) and the other made of fermionic oscillators, that we shall call the fermionic sliver. Although the geometric construction gives a precise determination of the fermionic sliver, it is important for many purposes to have an algebraic construction as well. In section 4, and making use the results of section 3 for the interaction vertex, we find a simple expression for the fermionic sliver in terms of infinite–dimensional matrices, as in [24, 19], and we compare the result to the geometric construction. We also briefly address the supersliver conservation laws. In section 5 we use the techniques recently introduced in [26] to determine the eigenvalue spectrum and the eigenvectors of the various infinite–dimensional matrices involved in the fermionic star algebra, including the matrices of Neumann coefficients. Oncethefermionicsliver hasbeenconstructedalgebraically, onecantakeitasasortof“vacuum state” in order to build fermionic coherent states. This we do in section 6, where after constructing these fermionic coherent states on the fermionic sliver, we study their star algebras. As in [20], one can use these coherent states to construct higher–rank projectors of the fermionic star algebra. We shall show that one can also construct closed fermionic star subalgebras. These star subalgebras provide new idempotent states which yield new solutions to the vacuum superstring field theory equation of motion. However, some of them turn out to be related to the fermionic sliver by gauge transformations. In section 7 we consider the ghost/superghost sector, and we show that if one chooses the vacuum BRST operator to be the recent canonical choice of Gaiotto, Rastelli, Sen and Zwiebach, [25], then the geometrical sliver is a solution to Berkovits’ superstring field theory equations of motion, i.e., we solve the equations of motion in the (super)ghost sector. Finally, in section 8 we state some conclusions and open problems for the future. In the Appendix we give some of the details needed in the proof that the structure of the vertex found in section 3 agrees with the explicit expressions found by Gross and Jevicki in [29] using conformal mapping techniques. 1 Strictly speaking,the constructionof the sliverstate is purely geometricand is thus valid for arbitraryBCFT’s. However, in this paper, we shall use the denotation of “sliver” for the particular BCFT associated to the maximal brane in flat space. 3 2. Berkovits’ Superstring Field Theory 2.1 A Short Review of Berkovits’ Superstring Field Theory In this paper, we shall study the non–GSO projected open superstring in the NS sector. In the matter sector, there are two fermions ψ±(σ) with the mode expansion ψµ(σ) = e±irσψµ, (2.1) ± r r∈XZ+21 where the modes satisfy the anticommutation relations ψµ,ψν = ηµνδ . (2.2) { r s} r+s,0 We will therefore write ψ† = ψ for r > 0. The ghost/superghost sector includes the b,c, and the r −r β,γ, system and we bosonize the last one in the standard way [32]: β = ∂ξe−φ, γ = ηe−φ. (2.3) A superstring field theory describing the GSO–projected NS sector of the open superstring was proposed by Berkovits in [8] (recent reviews can be found in, e.g., [10, 17]). In this theory, the string field Φ is Grassmann even, has zero ghost number and zero picture number. The action has the structure of a WZW model: 1 1 S[Φ] = (e−ΦQ eΦ)(e−Φη eΦ) dt(e−tΦ∂ etΦ) (e−tΦQ etΦ),(e−tΦη etΦ) , (2.4) B 0 t B 0 2 − { } Z (cid:18) Z0 (cid:19) where Q is the BRST operator of the superstring and η the zero–mode of η (the bosonized B 0 superconformal ghost) [32]. In a WZW interpretation of this model, these operators play the role of a holomorphic and an anti–holomorphic derivatives, respectively. In this action, the integral and the star products are evaluated with Witten’s string field theory interaction [1]. The exponentiation of the string field Φ is defined by a series expansion with star products: eΦ = +Φ+ 1Φ⋆Φ+ , I 2 ··· where is the identity string field. As usual, we refer to the first term in (2.4) as the kinetic term I and to the second one as the Wess–Zumino term. It can be shown that the equation of motion derived from this action is [8]: η e−ΦQ eΦ = 0. (2.5) 0 B (cid:16) (cid:17) The action (2.4) has a gauge symmetry given by 4 δeΦ = Ξ eΦ +eΦΞ , (2.6) L R where the gauge parameters Ξ satisfy L,R Q Ξ = 0, η Ξ = 0. (2.7) B L 0 R One can include GSO( ) states by introducing Chan–Paton–like degrees of freedom [12, 13]. − The string field then reads, Φ = Φ 1+Φ σ , (2.8) + − 1 ⊗ ⊗ where Φ are respectively in the GSO( ) sectors, and σ is one of the Pauli matrices. The Q and ± 1 B ± η operators also have to be tensored with the appropriate matrices: 0 ˆ Q = Q σ , ηˆ = η σ . (2.9) B B 3 0 0 3 ⊗ ⊗ The action is again given by one–half times (2.4), where the bracket now includes a trace over the Chan–Paton–like matrices(the 1/2factorisincluded tocompensate forthe traceover thematrices). The gauge symmetry is given again by (2.6), where Ξ take values in both sectors as in (2.8). L,R It has been shown that Berkovits superstring field theory correctly reproduces the four–point tree amplitude in [11], and it can be used to computed the NS tachyon potential in level truncation (see [17] for a review), giving results which are compatible with Sen’s conjectures. 2.2 Superstring Field Theory Around a Classical Solution In the cubic theory of Witten, one can consider a particular solution of the classical equations of motion, Φ , and study fluctuations around it: Φ = Φ + Φ˜. It is easy to see that the action 0 0 governing the fluctuations Φ˜ has the structure of the original action for Φ, but with a different BRST operator, . Bosonic VSFT, as formulated in the series of papers [18, 19, 20, 21, 22], is Q based on two assumptions: 1) First, it is assumed that, when one expands around the tachyonic vacuum, the new BRST operator has vanishing cohomology and is made purely of ghost operators. Q 2) Second, it is assumed that all Dp–brane solutions of VSFT have the factorized form: Φ = Φ Φ , (2.10) g m ⊗ where Φ denote states containing only ghost and only matter modes, respectively. Since the star g,m product factorizes into the ghost and the matter sector, and since we have assumed that is pure Q ghost, the equations of motion split into: 5 Φ +Φ ⋆Φ = 0, (2.11) g g g Q and Φ ⋆Φ = Φ . (2.12) m m m The second equation says that the matter part is an idempotent of the star algebra (where the star product is now restricted to the matter sector). If these assumptions hold, the string field action evaluated at a solution of the form (2.10) is simply proportional to the BPZ norm of Φ , and this m | i allows one to compare in a simple way ratios of tensions of different D–branes [19, 21]. An interesting question is to which extent are these assumptions valid in Berkovits’ superstring field theory. In order to answer this question, the first step is to analyze the fluctuations around a solution to the equations of motion. This was first addressed by Kluson in [33], where it was shown that with an appropriate parameterization of the fluctuations, the equation of motion is identical to (2.5), albeit with a deformed Q operator. It was thus concluded (without proof) in [33] that the action for the fluctuation should have the form (2.4) with the deformed operator. We shall now derive the equation of motion in a slightly different way from the one presented in [33], and this will allow us to show that the action is indeed of the required form by direct computation. Let us define G = eΦ, the exponential of the string field that appears in Berkovits’ action. Let Φ be a solution to the classical equations of motion (2.5) and let us consider a fluctuation around 0 this solution parameterized as follows [33], G = G ⋆h, G = eΦ0, h = eφ. (2.13) 0 0 Since Berkovits’ action has the structure of a WZW theory, one should expect an analog of the Polyakov–Wiegmann equation [34] to be valid. In fact, it is easy to show (by using for example the geometric formulation of [33, 35]) that the action (2.4) satisfies S[G ⋆h] = S[G ]+S[h] (G−1Q G )(hη h−1), (2.14) 0 0 − 0 B 0 0 Z for arbitrary G and h. The effective action for the fluctuations is then 0 S [h] = S[h] (G−1Q G )(hη h−1). (2.15) eff − 0 B 0 0 Z Let us now obtain the equation of motion satisfied by h. Varying S[h], one obtains h−1δhη (h−1Q h), 0 B Z 6 and from the extra term in S [h] one gets eff h−1δhη (h−1Ah), 0 Z where we denoted A = G−1Q G and we have used the equation of motion η (A) = 0. Putting 0 B 0 0 both pieces together, one finds that η (h−1Q h+h−1Ah A) = 0. (2.16) 0 B − Therefore, the equation of motion is identical to (2.5) but with the deformed Q operator: Q (X) = Q (X)+AX ( 1)XXA. (2.17) A B − − One can moreover easily prove [33] that the new operator satisfies all the axioms of superstring field theory (it is a nilpotent derivation and it anticommutes with η ). 0 We shall now show that S [h] has in fact the structure of (2.4) but with the operator Q . For eff A that, we simply need to notice that 1 1 A(hη h−1) = (h−1Ah A)(h−1η h) dtA∂ (hˆη hˆ−1 hˆ−1η hˆ) , (2.18) 0 0 t 0 0 2 − − − Z Z (cid:18) Z0 (cid:19) where we haveused integrationby partswithrespect toη , andthe factthatΦ satisfies itsequation 0 0 of motion. We have also denoted hˆ = etφ. The first term in the RHS of (2.18) when added to the kinetic term in S[h] gives a kinetic term with the Q operator, while the second term when added A to the Wess–Zumino term in S[h] gives a Wess–Zumino term with Q . The conclusion of this A computation is that the action for the fluctuations is simply 1 1 S [h] = (e−φQ eφ)(e−φη eφ) dt(e−tφ∂ etφ) (e−tφQ etφ),(e−tφη etφ) , (2.19) eff A 0 t A 0 2 − { } Z (cid:18) Z0 (cid:19) as anticipated in [33]. Let us now consider the superstring field theory describing the non–BPS D9–brane, i.e., Berkovits’ superstring field theory including both the GSO( ) sectors. It has been shown in level ± truncation that this theory has two symmetric vacua where the tachyon condenses. According to Sen’s conjectures, at any of these two vacua there are no open superstring degrees of freedom. Let us then choose one of these vacua and study the action for fluctuations around it. As we have seen, the action for the fluctuations has the same form as the original one, but with a different BRST operator, that we shall now denote by . According to Sen’s conjectures, at this chosen vacuum Q there are no open string degrees of freedom and it is thus natural to assume, as in the VSFT for the bosonic string, that the new BRST operator has vanishing cohomology and is made purely of (super)ghost operators. In addition we will also assume that this operator annihilates the identity, 7 = 0. (2.20) QI This condition, although very natural, is strictly not necessary in order to preserve some of the basic features of bosonic VSFT. In the superstring case however, it is crucial. It was noticed in [18] that operators of the form = c + u c also have vanishing cohomology in the superstring Q 0 n n n case. In particular, the operator recently proposed in [25] for the bosonic VSFT is of this form Q P and annihilates the identity after some proper regularization, so that in principle it is a possible candidate for the superstring as well (where the superconformal ghost sector would be handled separately). We shall come back to this question in section 7. With these assumptions at hand, and given the fact that the action around the vacuum has the same form as the original one but with a pure (super)ghost operator , it is now easy to show Q that the ansatz (2.10) solves the superstring field theory equations of motion if Φ is idempotent m and Φ satisfies g η e−Φg eΦg = 0. (2.21) 0 Q (cid:16) (cid:17) In order to see this, notice that idempotency of Φ and factorization of the star product in matter m and ghost parts yields eΦ = eΦg Φ + Φ , (2.22) m m ⊗ I − and, since kills the identity and is pure ghost, one has Q eΦ = eΦg Φ . (2.23) m Q Q ⊗ Using again idempotency of Φ , the equation(cid:0)of mo(cid:1)tion becomes: m η e−Φg eΦg Φ = 0. (2.24) 0 m Q ⊗ (cid:18) (cid:19) (cid:16) (cid:17) Therefore, the above conditions are sufficient to solve the equations of motion. In the same way, one can show that in these circumstances the action factorizes as S = K Φ Φ , (2.25) m m h | i where K = S[Φ ]. (2.26) g 8 Let us now look at the gauge symmetry of the new action around the tachyon vacuum. We are particularly interested in transformations that preserve the structure of (2.22). Since both and Q η annihilate the identity, it is easy to see that the gauge transformation (2.6) with 0 Ξ = Ξ , Ξ = Ξ , (2.27) L m g R m g ⊗I − ⊗I preserves (2.22). This gauge transformation leaves Φ invariant and changes Φ as follows: g m δΦ = [Ξ ,Φ ] , (2.28) m m m ⋆ where [A,B] = A⋆B B⋆A is the commutator in the star algebra. Notice that this transformation ⋆ − preserves idempotency of Φ at linear order. The gauge symmetry (2.28) is precisely the one that m appears in bosonic VSFT when annihilates the identity [19, 21, 25]. Q The condition of idempotency of Φ in the non–GSO projected theory involves in fact two m different conditions. In general, a matter string field Φ has components in both GSO( ) sectors, m ± Φ = Φ+ 1+Φ− σ . (2.29) m m ⊗ m ⊗ 1 In this equation, Φ± is Grassmann even (odd), and idempotency of Φ is equivalent to the following m m equations Φ+ ⋆Φ+ +Φ− ⋆Φ− = Φ+, m m m m m Φ+ ⋆Φ− +Φ− ⋆Φ+ = Φ−. (2.30) m m m m m One particular solution is of course to take Φ+ as an idempotent state and Φ− = 0. The matter m m supersliver state that we will discuss later is an example of such a solution. Another possibility is to take Φ+ an idempotent and Φ− a nilpotent state satisfying the second equation in (2.30). In m m section 6 we will construct solutions with these characteristics, although we will also show that they are related to the supersliver solution by gauge transformations at the vacuum. 3. Neumann Coefficients and Overlap Equations In this section we review some of the results of [29] and we explain in detail the structure of the overlap equations involving the matter part of the fermionic sector. 3.1 The Identity As in bosonic string field theory, the simplest vertex in superstring field theory is the integration, which corresponds to folding the string and identifying the two halves [1] thus defining the identity string field I , | i 9