New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

280 P. BouwknegtA = A M dx M = dx • + A μ dx μ . (53)We can decompose both the canonical metric on Y and the B-field, with respect tothe base, in terms of the connectiong = g μν dx μ dx ν + (dx • + A μ dx μ ) 2B = 1 2 B μνdx μ ∧ dx ν + B μ dx μ ∧ (dx • + A ν dx ν ), (54)where 1 2 B μνdx μ ∧dx ν and B μ dx μ are a 2-form and a 1-form on the base X, respectively.Then, by inserting (54) into the Buscher rules (52), we see that the Buscherrules essentially correspond to the interchange A μ ↔ B μ . To be precise, startingwithg MN =( )gμν + A μ A ν A μ, B MN =A ν 1the Buscher rules giveĝ MN =( )gμν + B μ B ν B μ, ̂B MN =B ν 1( )Bμν + (B μ A ν − A μ B ν ) B μ,−B ν 0(55)( )Bμν A μ. (56)−A ν 0Denoting the coordinate of the dual circle by ˆx • , we can interpret  = d ˆx • +B μ dx μ , locally, as a connection on a dual circle bundle ̂π : Ŷ → X. We deducefrom (56) that on the correspondence space Y × X Ŷ ={(y, ŷ) ∈ Y × Ŷ | π(y) =̂π(ŷ)}, with local coordinates (x μ , x • , ˆx • ),̂B = B + A ∧  − dx • ∧ d̂x • , (57)so thatĤ − H = d(A ∧ Â) = F ∧  − A ∧ ̂Fx, (58)where F = dA and ̂F = d  are the curvatures of A and Â, respectively, andare (globally) defined forms on M. Equation (58) actually makes sense globally onY × X Ŷ . Rewriting this equation asH − ̂F ∧ A = Ĥ − F ∧ Â, (59)we see that the left-hand side is a form on Y , while the right-hand side is a form onŶ . Thus, in order to have equality, we conclude that both have to equal a form H 3defined on X, i.e.H = H 3 + ̂F ∧ Ax,Ĥ = H 3 + F ∧ Âx. (60)