Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1197that the equations (6.303) and (6.304) already contain some flavor of cohomology.Cohomological field theories are theories where this analogy canbe made exact.With this in mind, the first property of a cohomological field theory willnot come as a surprise: it should contain a fermionic symmetry operator Qwhich squares to zero, Q 2 = 0. This may seem like a strange requirementfor a field theory, but symmetries of this type occur for example when wehave a gauge symmetry and fix it by using the Faddeev–Popov procedure;the resulting theory will then have a global BRST symmetry, which satisfiesprecisely this constraint. Another example is found in supersymmetry,where one also encounters symmetry operators that square to zero, as wewill see in detail later on.The second property a cohomological field theory should have is reallya definition: we define the physical operators in this theory to be theoperators that are closed under the action of this Q−operator 28 :{Q, O i } = 0. (6.305)Again, this may seem to be a strange requirement for a physical theory,but again it naturally appears in BRST quantization, and for examplein conformal field theories, where we have a 1–1 correspondence betweenoperators and states. In such theories, the symmetry requirement (6.303)on the states translates into the requirement (6.305) on the operators.Thirdly, we want to have a theory in which the Q−symmetry is notspontaneously broken, so the vacuum is symmetric. Note that this impliesthe equivalenceO i ∼ O i + {Q, Λ}. (6.306)The reason for this is that the expectation value of an operator productinvolving a Q−exact operator {Q, Λ} takes the form〈0|O i1 · · · O ij {Q, Λ}O ij+1 · · · O in |0〉 = 〈0|O i1 · · · O ij (QΛ−ΛQ)O ij+1 · · · O in |0〉,and each term vanishes separately, e.g.,〈0|O i1 · · · O ij QΛO ij+1 · · · O in |0〉 = ±〈0|O i1 · · · QO ij ΛO ij+1 · · · O in |0〉 (6.307)= ±〈0|QO i1 · · · O ij ΛO ij+1 · · · O in |0〉 = 0,28 From now on, we denote both the commutator and the anti-commutator by curlybrackets, unless it is clear that one of the operators inside the brackets is bosonic.