Type IIB Five Form Self Duality Constraint

After varying Type IIB Two Derivative Supergravity Pseudoaction, separately impose ${\ast }_{g_s}{\tilde{G}}_5={\tilde{G}}_5$ on the gauge-invariant real five-form in the oriented Lorentzian ten-dimensional convention Lorentzian Hodge Star in Ten Dimensions. By Existence and Five-Form Properties of the Lorentzian Hodge Star, ${\ast }_{g_s}^2=1$ on five-forms, so the equation is a consistent real constraint. It is an adopted formulation postulate, not an equation obtained by varying the undoubled pseudoaction. Since the degree is odd, the constraint also gives ${\tilde{G}}_5\wedge {\tilde{G}}_5=0$ and hence $\mathrm{\mid }{\tilde{G}}_5{\mathrm{\mid }}^2=0$ pointwise. This scalar identity does not set ${\tilde{G}}_5$ to zero.

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Source Free Bianchi Identities

Type IIB Potential Gauge Transformations

Gauge Invariance of the Type IIB Tilded Field Strengths

Lorentzian Hodge Star in Ten Dimensions

Existence and Five-Form Properties of the Lorentzian Hodge Star

The wedge product is associative and graded commutative

Hamilton, §7.1, Definition 7.1

Sterckx, §3.2.3, equations (3.32)–(3.35)

For a real five-form the star is an involution and its $+1$ eigenspace has dimension $126$ in each fiber by the mathematical theorem above. The statement does not identify a global potential or establish global flux quantization.