Axiodilaton

Axiodilaton is the complex scalar field in type IIB string theory that combines the Ramond–Ramond axion χ and the dilaton φ into the complex parameter τ = χ + i e^−φ. It is the scalar of the type IIB supergravity multiplet and parametrizes the coset SL(2,R)/SO(2). The imaginary part Im τ = e^−φ is related to the string coupling via g_s = e^φ = 1/Im τ, while the real part χ is the RR scalar that shifts under certain dualities.

Classically, the axiodilaton field enjoys an SL(2,R) symmetry acting on τ by fractional linear transformations. In the quantum theory this continuous symmetry is broken to the discrete SL(2,Z) group, reflecting S-duality. Under τ → (aτ + b)/(cτ + d) with integers a,b,c,d and ad − bc = 1, strong and weak coupling are exchanged, and the axion mixes with the dilaton. This duality also rearranges charged states, relating fundamental strings to D-strings and, more generally, (p,q) string states.

Dynamically, the axiodilaton arises from a nonlinear sigma model on SL(2,R)/SO(2), with a kinetic term proportional to (∂μ τ ∂^μ τ̄)/(2 (Im τ)^2). Its SL(2,R) covariance governs couplings to other type IIB fields, including the NS-NS and RR 3-form field strengths.

In F-theory, τ is promoted to a spacetime-dependent complex structure parameter of an auxiliary torus, such that its variation over the compactification manifold geometrically encodes seven-branes and realizes S-duality as a geometric symmetry.