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However, this expansion is useless for large t → 2 T/R , due to the imaginary part of the argument of sn the expansion will involve hyperbolic functions with arguments that can be large. Instead we can use the expansion of sn in terms of inverse sines ([11], 8.147.1), where these sines can be large, so only the first term is relevant: 

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Equation (34) is applicable for t → R/ 2 T from above and Eq. (35) is applicable when t → 2 T/R . 

The quadratic action, describing Gaussian fluctuations around t ∗ ( s ), is like (12) rewritten for the real-axis parametrization: 

The Gaussian approximations is justified for large KRT , when 

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To calculate the path integral over the quantum fluctuations around the minimizing function t ∗ ( s ), we need a mode expansion of the infinitesimal reparametrizing function β ( t ). To get rid of the projective symmetry, we keep fixed 3 points, e.g. − 1, 0, +1 or − √ µ , 0, √ µ fixed: β ( − 1) = 0, β (0) = 0, β (+1) = 0 or β ( − t ∗ ( − √ µ )) = 0, β (0) = 0, β ( t ∗ ( √ µ )) = 0. 

For each segment from t$_{i}$ to t$_{f}$ , we consider the mode expansion 

obeying the boundary condition β ( t$_{i}$ ) = β ( t$_{f}$ ) = 0. This set of sines forms a complete basis on the given interval. Actually we shall need the mode expansion for 4 segments attached to t = t ∗ ( ± √ µ ) ≈ ± R/ 2 T because the large contribution of 1 /µ will appear in A$_{2}$ only for those. 

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The appearance of the large factor 1 /µ is seen already from Eqs. (30), (33) because s ∗ in the denominator in Eq. (36) is proportional to √ µ . We need, however, to show that this 1 /µ is multiplied by a factor ∼ 1. 

Let us analyze the contribution to (36) that comes from − R/ 2 T < t$_{1}$, t$_{2}$ < + R/ 2 T , i.e. from the bottom side of the rectangle. Introducing the variable 

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