MODULE I

Introduction to String Theory and the Swampland Program (12 hours) Goals:-Introduce the essential ideas of string theory needed to understand the Swampland Program -Motivate the emergence of the Landscape and the Swampland -Present the main Swampland conjectures and their physical implications Topics: 1. Why String Theory? Motivation from unification and quantum gravity Point particles vs. extended objects: ultraviolet behavior Critical dimensions and anomaly cancellation 2. Basic Ingredients of String Theory Strings and branes: open and closed strings, D-branes Worldsheet action and quantization: spectrum of string states T-duality and extra dimensions The role of supersymmetry 3. Compactifications and Effective Theories Kaluza–Klein reduction Calabi–Yau compactifications Moduli and fluxes Basics of moduli stabilization and the String Landscape 4. The Landscape and the Swampland What is the Swampland? Why effective field theories need not have UV completions Contrasting Landscape vacua with Swampland constraints 5. Main Swampland Conjectures The Swampland Distance Conjecture (SDC) The Weak Gravity Conjecture (WGC) The de Sitter Conjecture Other conjectures: No global symmetries, Completeness, Emergence 6. Examples and Heuristics Simple string compactification examples Infinite distance limits and towers of states Philosophical implications and criticisms.

MODULE II

Mathematical Aspects of the Swampland Program (12 hours) Goals: -Provide the mathematical foundation for understanding Swampland conjectures -Explore the geometry and structure of moduli spaces Study formal aspects of quantum gravity constraints Topics: 1. Moduli Spaces and Infinite Distance Limits Moduli spaces in string compactifications The metric on moduli space and geodesic distances Nilpotent Orbit Theorem and monodromy Application to the Swampland Distance Conjecture 2. The Weak Gravity Conjecture Charge-to-mass ratio bounds Generalizations to multiple U(1)’s and scalar fields Lattice WGC and convex hull condition Implications for UV cutoffs and consistency of EFTs 3. Towers and Emergence Kaluza–Klein towers, string excitations, wrapped branes Emergence of gravity and gauge couplings from integrating out towers Entropy and loop corrections 4. Global Symmetries and Cobordism No global symmetries in quantum gravity The Completeness Conjecture: all charges realized The Cobordism Conjecture and its implications 5. Mathematical Challenges and Open Problems Special geometry and mirror symmetry Arithmetic geometry and constraints on vacua Interplay with topological invariants and dualities.

MODULE III

Applications to Cosmology and Phenomenology (12 hours) Goals: -Explore how Swampland constraints impact models of the early and late Universe -Analyze implications for particle physics and observable phenomena -Assess the testability and predictive power of Swampland ideas Topics 1. Inflation and the Swampland Tension between slow-roll inflation and Swampland constraints Alternatives: k-inflation, DBI models, ekpyrotic cosmologies Trans-Planckian Censorship Conjecture (TCC) 2. Dark Energy and Quintessence Swampland bounds on scalar potentials Tracking fields, dynamical dark energy Observational signatures and constraints 3. Axions and the Axionic WGC Instantons, periodic fields, and axion monodromy Constraints on decay constants Applications to axion inflation and ALPs 4. Implications for Particle Physics U(1) extensions and Swampland constraints Neutrino masses and the seesaw mechanism Supersymmetry and the Swampland 5. Observational Windows and Prospects Current and future experimental probes: CMB, Euclid, DESI, LSST Can Swampland conjectures be falsified? Toward a data-driven Swampland perspective