S-duality Clause Samples

S-duality. ‌ Type IIB supergravity in ten dimensions possesses a global SL(2, R) symmetry [4]. However, in the full quantum theory the SL(2, R) symmetry is broken to an SL(2, Z) subgroup, which is the S-duality group of type IIB string theory in ten dimensions [14, 59–64]. S-duality is a strong- weak coupling duality that relates type IIB string theory to itself in different regimes of the string coupling gs. Before reviewing the SL(2, Z) S-duality symmetry of the full theory we will rewrite the type IIB supergravity action with a global SL(2, R) symmetry in non-linearly realised form as a means to illustrate the methods involved in constructing non-linear realisations that are used throughout the rest of this thesis and described in detail in appendix B.3. The following type IIB action is written in standard SL(2, R) covariant form and produces the correct equations of motion when supplemented by a self-duality condition for the five-form field strength. In Einstein frame, the action may be expressed as S = 1 ∫ d10x√—g R — 1 HT MHµ1µ2µ3 1 µ 2κ2 1 ∫ 8κ2 d10x √ 1 5! µ1µ2µ3 1 3 4 F˜µ µ µ µ µ F˜µ1µ2µ3µ4µ5 + 5 ∂ M∂µM ϵabC4 ∧ Ha ∧ Hb 3 3 where a and b are SL(2, R) doublet indices, τ = χ + ie—φ is the axion-dilaton field and  |τ|2 —χ  M = eφ  —χ 1  . (3.82) The three form field strength H, with components Hµ1µ2µ3 , is defined by dB˜ where B˜ =  B  , (3.83)  C  and B, C are two-forms in the NS-NS and R-R sectors respectively. In terms of the SL(2, R) doublet indices, we have H1 = dB and H2 = dC. The five-form field strength F is written in the form ˜ 1 ˜i j F = F + 2 ϵijB ∧ H , (3.84) with components F˜µ1µ2µ3µ4µ5 . Note that F = dC4, where C4 is the R-R sector four-form. The self-duality condition, mentioned earlier, on the five-form field strength is F˜5 = ∗F˜5. The type IIB supergravity action in this form is invariant under the global SL(2, R) symmetry group with transformations aτ + b τ → cτ + d, B˜ → ΛB˜, M → Λ—1 T M Λ—1, (3.85) where a, b, c, d ∈ R and satisfy ad — bc = 1. The Einstein frame metric gˆ and R-R sector four-form C4 are SL(2, R) invariant. To rewrite the action in non-linearly realised form we first take a coset element g ∈ SL(2, R)/SO(2, R) parameterised by the type IIB scalar fields φ and χ in the form g = e 2 φ  e—φ —χ  0 1 The coset g ∈ SL(2, R)/SO(2, R) is related to M by  e—φ —χ   e—φ 0  ggT =eφ  0 1   —χ 1   |τ|2 —χ  (3.87) =eφ  =M, —χ 1  where T denotes the transpose. One may then show ∂µ M∂µM—1 = — 1 tr g—1∂ 4 µ = —tr (SµS...