By Ohmori, K

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61), the action can be rewritten as 1 S˜ = − 2 go 1 1 I ◦ T (0)QB T (0) + f1 ◦ T (0)f2 ◦ T (0)f3 ◦ T (0) ′ 2α 3 1 1 I ◦ T (0)Q T (0) + f1 ◦ T (0)f2 ◦ T (0)f3 ◦ T (0) = −Vp+1 2π 2 α′3 τp B norm. 2α′ 3 norm. Generically, the correlation function includes the momentum conservation delta function (2π)p+1δ p+1 ( k), but since T ∈ H11 has no momentum dependence, this factor simply gives the volume Vp+1 of the Dp-brane. Accordingly, the correlation functions in the second line are normalized such that 1 matter = 1.

36). We proceed to the cubic term. 24), we get √ w1 ≡ f1 (0) = − 3, w2 ≡ f2 (0) = 0, √ w3 ≡ f3 (0) = 3, w1′ w2′ w3′ 1 8 2πi ≡ = e− 3 3 (g1 (0) + 1)2 2 ≡ f2′ (0) = , 3 8 ′ ≡ f3 (0) = . 39) Conformal transformations are fi ◦ Φ(0) = = dd k φ(k)fi ◦ (ceikX )(0) + √ ′ 2 −1 dd k (fi′ (0))α k +√ i Aµ (k)fi ◦ (c∂X µ eikX )(0) ′ 2α φ(k)ceikX (fi (0)) i ′ 2 (fi′ (0))α k Aµ (k)c∂X µ eikX (fi (0)) . 41) where we dropped φφA and AAA terms because of a twist symmetry [21, 24]. A ‘twist operation’ Ω is a combination of the world-sheet parity reversal and an SL(2, R) transformation.

The number of real solutions to a quadratic equation ax2 + bx + c = 0 depends on the sign of its discriminant b2 − 4ac. In the region where the effective potential seems to be a two-valued function of φ, discriminant is positive and two real solutions exist. At the point where two branches meet, discriminant vanishes and two solutions coincide. Even if we try to extend the branch beyond that point, no real solutions exist and we fail to do so. 100) of the effective potential around the perturbative vacuum φ = 0 has a finite radius of convergence.