Ivancevic_Applied-Diff-Geom

1166 Applied Differential Geometry: A Modern Introductionwhere Q L is defined by integrating the BRST current j BRST (σ) with respectto σ over the left half regionQ L =∫ π/20dσj BRST (σ).Equation (6.274) is also represented by the product ⋆ asψ ⋆ (Qψ) = ψ ⋆ [(Q L I) ⋆ ψ + ψ ⋆ (Q L I)]. (6.275)Here, I stands for the identity element with respect to the ⋆−product,carrying the ghost number - 3 2, which corresponds to |I〉 in the operatorformulation. As is discussed by [Horowitz et. al. (1986)], in order to showthe relation (6.275) we need the formulasQ R I = −Q L I, (Q R ψ) ⋆ ξ = −(−1) n ψψ ⋆ (Q L ξ) (6.276)for arbitrary string fields ψ and ξ, where Q R is the integrated BRST currentover the right half region of σ. n ψ stands for the ghost number of thestring field ψ minus 1 2, and takes an integer value. The first formula meansthat the identity element is a physical quantity, also the second does theconservation of the BRST charge. By using these formulas, the first termin the bracket in r. h. s. of (6.275) becomes(Q L I) ⋆ ψ = −(Q R I) ⋆ ψ = I ⋆ (Q L ψ) = Q L ψ.Also, it turns out that the second term is equal to Q R ψ. Combining these,we can see that (6.275) holds.Further, we should remark that because the BRST current does notcontain the center of mass coordinate x j , it commute with the momentump i . From the continuity condition p i |I〉 = 0, it can be seen that p i Q L |I〉 = 0.Expanding the exponential in the expression of the unitary operator (6.272)and passing the momentum p 0,i to the right, we getUQ L |I〉 = Q L |I〉. (6.277)Now we can write down the result of the kinetic term. As a result of the