Ivancevic_Applied-Diff-Geom

1218 Applied Differential Geometry: A Modern IntroductionThus, we find that the left– and right–moving SEM charges satisfyT + = 2πiL 0 = 1 2 {Q +, ¯Q + }T − = 2πi¯L 0 = − 1 2 {Q −, ¯Q − }.To find G in the B−model, we should write these charges as commutatorswith respect to Q B = ¯Q + + ¯Q − , which givesT + = 1 2 {Q B, Q + }T − = − 1 2 {Q B, Q − },so we arrive at the conclusion that for the B−model,Now, we can rewrite (6.327) asG + = 1 2 Q +G − = − 1 2 Q −.(O (2)a ) +− = −{G + , [G − , O (0)a ]} = 1 4 {Q +, [Q − , O (0)a ]} (6.328)= 1 8 { ¯Q B , [(Q − − Q + ), O (0)a ]},which proves our claim that O a(2) is Q B −exact.An N = (2, 2) sigma model with a real action does, apart from the term(6.326), also contain a term∫tā Ō a (2) , (6.329)Σwhere tā is the complex conjugate of t a . It is not immediately clear that Ō(2) ais a physical operator: we have seen that physical operators in the B−modelcorrespond to forms that are ¯∂−closed, but the complex conjugate of sucha form is ∂−closed. However, by taking the complex conjugate of (6.328),we see that(Ō(2) a ) +− = 1 8 {Q B, [( ¯Q − − ¯Q + ), Ō(0) a ]},so not only is the operator Q B −closed, it is even Q B −exact. Thismeans that we can add terms of the form (6.329) to the action, and takingtā−derivatives inserts Q B −exact terms in the correlation functions.Naively, we would expect this to give a zero result, so all the physical quantitiesseem to be t−independent, and thus holomorphic in t. We will see ina moment that this naive expectation turns out to be almost right, but notquite.