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One-Variable Kähler Calculation of the No-Scale Cancellation

Example

Let $H = {ρ = x + i y : x, y \in R, y > 0} \subset C$ . Fix $m > 0$ and $w_{0} \in C$ . On $H$ , define the real function

K (ρ, \overset{ˉ}{ρ}) = - 3 m^{2} \log [- i (ρ - \overset{ˉ}{ρ})] = - 3 m^{2} \log (2 y),

the constant function $W (ρ) = w_{0}$ , the metric coefficient $g_{ρ \overset{ˉ}{ρ}} = \partial_{ρ} \partial_{\overset{ˉ}{ρ}} K$ , and the covariant derivative $D_{ρ} W = \partial_{ρ} W + (K_{ρ} / m^{2}) W$ . Let $g^{ρ \overset{ˉ}{ρ}} = 1 / g_{ρ \overset{ˉ}{ρ}}$ and define

V = e^{K / m^{2}} (g^{ρ \overset{ˉ}{ρ}} ∣ D_{ρ} W ∣^{2} - \frac{3 ∣ W ∣^{2}}{m^{2}}),

where $\log$ is the real logarithm on positive numbers and $∣ a + i b ∣^{2} = a^{2} + b^{2}$ . Then $g_{ρ \overset{ˉ}{ρ}} > 0$ and $V = 0$ for every $ρ \in H$ , while $D_{ρ} W \neq 0$ everywhere when $w_{0} \neq 0$ . This is a mathematical example about the stated functions and does not assert that any Type IIB flux compactification realizes them.

Example references

Giddings, Kachru, and Polchinski, §2.2.4, equations (2.37)–(2.43)

Graña, §5.2, equations (5.27)–(5.28)

Louis, §§8.1–8.2

Facts & Assumptions

Given: $x, y \in R$ , $y > 0$ , $m$ is a positive real constant, and $w_{0} \in C$ . For $ρ = x + i y$ define $\overset{ˉ}{ρ} = x - i y$ , and use Wirtinger derivatives $\partial_{ρ} = (\partial_{x} - i \partial_{y}) / 2$ and $\partial_{\overset{ˉ}{ρ}} = (\partial_{x} + i \partial_{y}) / 2$ .

[F1]

The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ defines $C$ as $R [x] / (x^{2} + 1)$ with $i^{2} = - 1$ and the embedded real numbers.

[F2]

$C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ proves that every element is uniquely $a + b i$ , the field operations are the displayed coordinate operations, and nonzero complex numbers have inverses.

[F3]

Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals, claims 1–3, constructs the oriented Riemann integral of a continuous curve in a complete normed real vector space, proves additivity, and proves that its variable-upper-endpoint integral has derivative equal to its integrand. We use $X = R$ with its absolute-value norm on compact intervals contained in $(0, \infty)$ .

[A1]

The Example defines $K, W, g, D_{ρ} W, V$ as the displayed mathematical functions. Conjugation is the real-linear map $a + i b \mapsto a - i b$ , well defined by the unique coordinates in [F2]; modulus squared is $∣ a + i b ∣^{2} = a^{2} + b^{2}$ . Use the real logarithm in its integral definition $\log t = \int_{1}^{t} d s / s$ for $t > 0$ , with oriented endpoints as in [F3]. No string-theory, flux, compactification, or empirical premise is assumed.

Verification

technique · direct real-coordinate differentiation and algebra

Given: [F1]–[F3] and [A1].

1.1givenF1F2F3A1algebra

The local conjugation convention in [A1] gives $\overset{ˉ}{ρ} = x - i y$ ; the field arithmetic in [F1]–[F2] then gives $- i (ρ - \overset{ˉ}{ρ}) = 2 y > 0$ . The integrand $s \mapsto 1 / s$ is continuous on each compact positive interval, so [F3] makes $\log t$ well defined and gives $(\log t)^{'} = 1 / t$ for every $t > 0$ . The difference quotient $((t + h)^{- 1} - t^{- 1}) / h = - 1 / [t (t + h)]$ gives $(\log t)^{''} = - 1 / t^{2}$ , continuously on $(0, \infty)$ . Thus $K = - 3 m^{2} \log (2 y)$ has continuous second real partial derivatives on $H$ .

2.1givenA1step 1.1algebra

Step 1.1 gives $\partial_{x} K = 0$ , $\partial_{y} K = - 3 m^{2} / y$ , and $\partial_{y}^{2} K = 3 m^{2} / y^{2}$ (rescale the logarithm difference quotient by $2$ ). Using the stated Wirtinger derivatives, $\partial_{ρ} K = 3 i m^{2} / (2 y) = - 3 m^{2} / (ρ - \overset{ˉ}{ρ})$ and

g_{ρ \overset{ˉ}{ρ}} = \partial_{ρ} \partial_{\overset{ˉ}{ρ}} K = - \frac{3 m^{2}}{(ρ - \overset{ˉ}{ρ})^{2}} = \frac{3 m^{2}}{4 y^{2}} > 0.

Therefore $g^{ρ \overset{ˉ}{ρ}} = 4 y^{2} / (3 m^{2})$ .

3.1givenA1step 2.1algebra

Since $W = w_{0}$ is constant, $D_{ρ} W = K_{ρ} w_{0} / m^{2} = - 3 w_{0} / (ρ - \overset{ˉ}{ρ})$ . Hence $g^{ρ \overset{ˉ}{ρ}} ∣ D_{ρ} W ∣^{2} = (4 y^{2} / (3 m^{2})) (9 ∣ w_{0} ∣^{2} / (4 y^{2})) = 3 ∣ w_{0} ∣^{2} / m^{2}$ .

4.1givenA1step 3.1algebra∎

The bracket defining $V$ is the difference of the equal terms computed in step 3.1, so $V = 0$ for every $y > 0$ . If $w_{0} \neq 0$ , then $ρ - \overset{ˉ}{ρ} = 2 i y \neq 0$ and step 3.1 gives $D_{ρ} W \neq 0$ ; if $w_{0} = 0$ , then $D_{ρ} W = 0$ as well. Thus zero $V$ does not force $D_{ρ} W = 0$ in this mathematical model, and $V$ is constant along the whole positive- $y$ direction.

Remarks

The displayed Kähler potential and cancellation are used in the cited flux-compactification literature as the one-modulus no-scale calculation. Here the example is fully verified from its mathematical definitions; the source citations motivate the choice of functions but are not proof premises. The relation to the physical theorem on the A page is nonlogical motivation only.

One-Variable Kähler Calculation of the No-Scale Cancellation

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources