“Y-formalism & Curved Beta-Gamma Systems”

1. 28 Jul.- 1 Aug.2008, Yukawa Institute’s workshop “Y-formalism & Curved Beta-Gamma Systems” P.A. Grassi (Univ. of Piemonte Orientale) M. Tonin (Padova Univ.) I. O. (Univ. of the Ryukyus） N.P.B (in press)

2. Motivations of this study Covariant quantization of Green-Schwarz superstring action (1984) Pure spinor formalismby N. Berkovitz(2000) = CFT on a cone SO(10)/U(5) A simple question: “What kind of conformal field theory can be constructed on a given hypersurface?” Sigma models on a constrained surface Difficult to compute the spectrum and correlation functions Chiralmodel of beta-gamma systems Infinite radius limit plus holomorphy

3. Chiral model of beta-gamma systems • An infinite tower of states • Non-trivial partition function • Neither operator nor functional formalism • Some aspects are known: “Chiral de Rham Complex” • by F. Malikov et al., math.AG/9803041 • = N=2 superconformal field theory The most interesting case Bosonic pure spinor formalism One interesting approach: Cechcohomology construction by Nekrasov, hep-th/0511008 The procedure of gluing of free CFT on different patches Unpractical (!) since it works only if the path structure is known

4. Review of curved beta-gamma systems = World-sheet Riemann surface = Target-space complex manifold surface = Open covering of X = Local coordinates in = (1, 0)-form on Action of Beta-gamma system (Holomorphic sector):

5. Sigma model Local coordinates on X Hermitian components In conformal gauge, using first-order formalism Holomorphy Infinite radius limit Redefinition By construction, this action is a free, conformal fieldtheory.

6. Basic OPE Diffeomorphisms Current Anomaly term Witten, hep-th/0504078 Nekrasov, hep-th/0511008

7. Y-formalism M. Tonin & I. O. , P.L.B520(2001)398; N.P.B639(2002)182; P.L.B606(2005)218; N.P.B727(2005)176; N.P.B779(2007)63 We wish to use Y-formalism to study beta-gamma systems Quantization of a system with constraints (on hypersurface) Our strategy: A radically different way Impose constraints at each step of computation without solving the constraints! • It relies on the existence of patches but it does not use it • Easy to compute contact terms and anomalies in OPE’s • Easy to construct b-ghost

8. Y-formalism for beta-gamma models with quadratic constraint Target space manifold X = a hypersurface in n dimensions defined by constraints = Homogeneous function of degree h Gauge symmetry

9. Quadraticconstraint Pure spinorconstraint Conifold = singular CY space Basic OPE

10. = Constant vector Gauge symmetry

11. Gauge-invariant currents Ghost number current SO(N) generators Stress-energy tensor

12. Ghost number current SO(N) generators Stress-energy tensor Cf.

13. Current algebra

14. Adding other variables Purely bosonic beta-gamma systems • No BRST charge (needed for constructing physical states) • No conformal field theory with zero central charge Necessity for adding other variables! Bosonic variables Fermionic variables

15. BRST charge Stress-energy tensor b-ghost

16. Difficulty of treating constraints more than quadratic

17. Conclusion • Construction of Y-formalism on a given hypersurface • Derivation of algebra among currents • Construction of quantum b-ghost • Calculation of partition function • Construction of Y-formalism on a given super-hypersurface A remaining question: How to treat systems with non-quadratic constraints?