<p>We implement axio-dilaton fields from spin geometry and indicate potential links between these structures and conformal spacetimes. Our approach is based on the complex generalization <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega \varepsilon _{AB}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <msub> <mi>ε</mi> <mrow> <mi mathvariant="italic">AB</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> of the metric spinor <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon _{AB}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mrow> <mi mathvariant="italic">AB</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, which simultaneously converts Maxwell electrodynamics in vacuum into axio-dilaton electrodynamics with modular coupling <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\tilde{\tau }=\textbf{i}\Omega ^{-2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">~</mo> </mover> <mo>=</mo> <mi mathvariant="bold">i</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and the spacetime metric into a conformally related metric rescaled by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((\Omega ^{*}\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mi mathvariant="normal">Ω</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By applying this complex rescaling to the Maxwell-Euler-Heisenberg theory, we demonstrate that this approach not only reproduces established axio-dilaton structures in the linear regime but also uncovers new mathematical structures within the nonlinear sector. Subsequently, by focusing on the duality group of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\tilde{\tau }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>, a set of SL<InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((2,\mathbb {Z})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dual metrics for the conformally equivalent spacetime is obtained through the Infeld-van der Waerden map. As a formal application of these dualities, we adapt our formulation to conformal cyclic cosmology and show how the pre- and post-conformal metrics can be mapped to each other via <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathcal {S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>-duality.</p>

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Spin-geometric Axio-dilatons, SL\((2,\mathbb {Z})\)-dual Metrics and Pre-post Duality for Conformal Cyclic Cosmology

  • André Martorano Kuerten

摘要

We implement axio-dilaton fields from spin geometry and indicate potential links between these structures and conformal spacetimes. Our approach is based on the complex generalization \(\Omega \varepsilon _{AB}\) Ω ε AB of the metric spinor \(\varepsilon _{AB}\) ε AB , which simultaneously converts Maxwell electrodynamics in vacuum into axio-dilaton electrodynamics with modular coupling \(\tilde{\tau }=\textbf{i}\Omega ^{-2}\) τ ~ = i Ω - 2 and the spacetime metric into a conformally related metric rescaled by \((\Omega ^{*}\Omega )\) ( Ω Ω ) . By applying this complex rescaling to the Maxwell-Euler-Heisenberg theory, we demonstrate that this approach not only reproduces established axio-dilaton structures in the linear regime but also uncovers new mathematical structures within the nonlinear sector. Subsequently, by focusing on the duality group of \(\tilde{\tau }\) τ ~ , a set of SL \((2,\mathbb {Z})\) ( 2 , Z ) -dual metrics for the conformally equivalent spacetime is obtained through the Infeld-van der Waerden map. As a formal application of these dualities, we adapt our formulation to conformal cyclic cosmology and show how the pre- and post-conformal metrics can be mapped to each other via \(\mathcal {S}\) S -duality.