Ivancevic_Applied-Diff-Geom

1184 Applied Differential Geometry: A Modern Introductionsubstitution of this gauge choice into S[x, h] leaves the gauge–fixed actionS = T ∫d 2 ση αβ ∂ α x · ∂ β x. (6.300)2Quantum mechanically, the story is more subtle. Instead of eliminatingh via its classical field equations, one should perform a Feynman pathintegral, using standard machinery to deal with the local symmetries andgauge fixing. When this is done correctly, one finds that in general φ doesnot decouple from the answer. Only for the special case d = 26 does thequantum analysis reproduce the formula we have given based on classicalreasoning. Otherwise, there are correction terms whose presence can betraced to a conformal anomaly (i.e., a quantum–mechanical breakdown ofthe conformal invariance).The gauge–fixed action is quadratic in the x’s. Mathematically, it is thesame as a theory of d free scalar fields in two dimensions. The equations ofmotion obtained by varying x µ are free 2D wave equations:ẍ µ − x ′′µ = 0.However, this is not the whole story, because we must also take accountof the constraints T αβ = 0, which evaluated in the conformally flat gauge,readT 01 = T 10 = ẋ · x ′ = 0, T 00 = T 11 = 1 2 (ẋ2 + x ′2 ) = 0.Adding and subtracting givesBoundary Conditions(ẋ ± x ′ ) 2 = 0. (6.301)To go further, one needs to choose boundary conditions. There are threeimportant types. For a closed string one should impose periodicity in thespatial parameter σ. Choosing its range to be π (as is conventional)x µ (σ, τ) = x µ (σ + π, τ).For an open string (which has two ends), each end can be required to satisfyeither Neumann or Dirichlet boundary conditions for each value of µ,Neumann :Dirichlet :∂x µ= 0∂σat σ = 0 or π,∂x µ= 0∂τat σ = 0 or π.