1. Surface States Global coordinates Local coordinate Surface state is a bra <Σ| associated to a Riemann surface Σ For any local operator Φ(ξ) ↔ | Φ>= lim Φ(ξ)|0> ξ→0 = D h s ○ Φ(0) = (s’ (0)) Φ(s(0)) m : ξ → u =s(ξ)

2. M M’

3. M

4. M

5. M

6. M

7. M

8. Wedge States Star Multiplication of Wedge States =

9. Conformal Sliver Conformal map Comparison with algebraic sliver

10. What we know Two sets of basis: i) related with spectrum of free string ii) related with "strong coupling" regime (may be suitable for study VSFT Identity and Sliver in Moyal Basis Identity Sliver SFT proposes a hard, but a surmountable way to get answers concerning non- perturbative phenomena

11. Open Problems More tests for checking validity of VSSFT Use the Moyal basis to construct the tachyon condensate and other solutions Use the Moyal basis to construct the tachyon condensate and other solutions Classification of projectors in open string field algebra and its physical meaning Closed string excitations in VSFT

12. References: I.Ya.~Aref'eva, D.M.~Belov, A.S.~Koshelev and P.B.~Medvedev,Tachyon Condensation in the Cubic Superstring Field Theory, Nucl.Phys., hep-th/0011117; Gauge invariance and tachyon condensation in cubic superstring field theory, Nucl.Phys., hep-th/010719 I.Ya.Aref'eva, A.A.Giryavets and A.S.Koshelev,NS Ghost Slivers, Phys.Lett., hep-th/0203227 I.Ya.Aref'eva, A.A.Giryavets and P.B.Medvedev,NS Matter Sliver, Phys.Lett.B 532 (2002) 291, hep-th/0112214; I.Ya.Aref'eva, A.A.Giryavets, Open Superstring Star as a Continuous Moyal Product Product, hep-th/0204164 D.M.Belov, Diagonal Representation of Open String Star and Moyal I.Ya.Aref'eva, D.M.Belov and A.A.Giryavets,Construction of the Vacuum String Field Theory on a Non-BPS Brane, hep-th/0201197 Noncommutative Field Theories and (Super)String Field Theories, hep-th/0111208