1 “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) 28 Jul.- 1 Aug.2008, Yukawa Institute’s workshop

Covariant quantization of Green-Schwarz superstring action (1984) Pure spinor formalism by 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 Chiral model of beta-gamma systems Motivations of this study 2 Infinite radius limit plus holomorphy

Chiral model of beta-gamma systems 3 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/ = N=2 superconformal field theory The most interesting case Bosonic pure spinor formalism One interesting approach: Cech cohomology construction by Nekrasov, hep-th/ The procedure of gluing of free CFT on different patches Unpractical (!) since it works only if the path structure is known

Review of curved beta-gamma systems 4 = 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 By construction, this action is a free, conformal field theory. HolomorphyInfinite radius limit Redefinition

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

Y-formalism 7 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 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 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!

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

9 Quadratic constraint Pure spinor constraint Conifold = singular CY space Basic OPE

10 = Constant vector Gauge symmetry

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

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

Current algebra 13

Adding other variables 14 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

Difficulty of treating constraints more than quadratic 16

17 Conclusion 1.Construction of Y-formalism on a given hypersurface 2.Derivation of algebra among currents 3.Construction of quantum b-ghost 4.Calculation of partition function 5.Construction of Y-formalism on a given super-hypersurface A remaining question: How to treat systems with non-quadratic constraints?