[Paper Review] Simplicial and modular aspects of string dualities

Valeria Gili|ArXiv.org|May 4, 2006

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This thesis explores simplicial and modular structures in string dualities by analyzing random Regge triangulated surfaces (RRTs) as singular Euclidean structures and linking them to boundary conformal field theory (BCFT) via generalized Langlands boundary states. It introduces a novel gluing procedure along ribbon graphs, incorporates Chan-Paton factors for open strings, and constructs cosmological solutions in supergravity using the A₂ and E₈ algebras, revealing hidden symmetries via oxidation and billiard dynamics in de Sitter-like backgrounds.

ABSTRACT

This Ph.D. thesis collects results obtained investigating two different aspects of modern unifying theories. In the first part I summarized results achieved investigating simplicial aspects of string dualities. Exploiting Boundary Conformal Field Theory techniques, I investigated the coupling between random Regge triangulations and open string theory, discussing its implications in gauge/gravity correspondence. The second part reports results obtained in the paper hep-th/0309237, devoted to look for cosmological backgrounds of superstring theories.

Motivation & Objective

• To establish a correspondence between random Regge triangulated surfaces (RRTs) and singular Euclidean structures in string dualities.
• To investigate boundary conformal field theory (BCFT) on annuli using generalized Langlands boundary states and their quantum amplitudes.
• To extend the gluing procedure along ribbon graphs to include open string degrees of freedom via Chan-Paton factors.
• To construct cosmological solutions in supergravity using the A₂ toy model and its oxidation to E₈, revealing hidden dynamical symmetries.
• To analyze the role of infinite-dimensional Kac-Moody algebras, such as E₁₀(₁₀), in cosmological billiard dynamics and time-dependent backgrounds.

Proposed method

• Modeling random Regge triangulated surfaces (RRTs) as singular Euclidean structures using simplicial geometry and Regge calculus.
• Applying dual uniformization techniques to relate triangulated surfaces to Riemann surfaces via ribbon graphs on Regge polytopes.
• Formulating boundary insertion operators (BIOs) in BCFT and analyzing their algebraic structure at the T-duality self-dual radius.
• Introducing Chan-Paton factors on RRTs to incorporate open string gauge degrees of freedom and extending the gluing procedure to include chiral currents.
• Using the compensator method to generate new solutions in the A₂ model with specified Cartan and nilpotent field configurations.
• Oxidizing A₂ solutions to higher-dimensional E₈ models by embedding the A₂ algebra and verifying satisfaction of supergravity field equations.

Experimental results

Research questions

• RQ1 How can random Regge triangulated surfaces be uniformly mapped to Riemann surfaces via dual uniformization and ribbon graphs?
• RQ2 What is the role of generalized Langlands boundary states in determining the quantum amplitude on the annulus in BCFT?
• RQ3 How do boundary insertion operators transform under rotation and T-duality at the self-dual radius?
• RQ4 How can open string gauge degrees of freedom be consistently incorporated into the gluing procedure on RRTs?
• RQ5 How do the solutions of the A₂ model, when oxidized to E₈, satisfy supergravity field equations and what is their cosmological interpretation?

Key findings

• The dual uniformization of a random Regge triangulated surface is achieved via ribbon graphs embedded on Regge polytopes, establishing a correspondence between simplicial structures and Riemann surfaces.
• Generalized Langlands boundary states are constructed and shown to yield consistent quantum amplitudes on the annulus, with explicit dependence on boundary fields and glueing conditions.
• Boundary insertion operators (BIOs) at the T-duality self-dual radius close under rotation, forming a non-Abelian algebra that reflects modular invariance.
• The open string amplitude on a RRT is computed by extending the gluing procedure to include Chan-Paton factors, with chiral currents and boundary states contributing to the total amplitude.
• Solutions of the A₂ model with one or all three roots switched on are successfully oxidized to E₈, and the resulting field configurations satisfy the supergravity field equations.
• The cosmological billiard dynamics in the E₈ model reveal a structure consistent with a de Sitter-like evolution, with the solution having one root switched on exhibiting a well-defined time-dependent scalar profile.