On Marginal Deformations of Heterotic G2 Geometries Eirik Eik Svanes (KCL, ICTP) Based on work in collaboration with X. de la Ossa, M. Larfors, M.-A. Fiset, C. Quigley 1607.03473, 1704.08717, 1710.06865, etc. January 10th 2018, Imperial College London Moduli and Effective Theories – 1 Introduction Motivation and Overview General String Compactifications Why Heterotic? Steps in Understanding Moduli Compactifications to 3d The G Holonomy Case 2 Introduction The Heterotic G2 System World Sheet Moduli and Effective Theories – 2 Motivation and Overview Introduction The low energy theory of the heterotic string is ten dimensional supergravity coupled to Motivation and Overview General String Yang-Mills gauge theory. Compactifications Easy to obtain four-dimensional supersymmetric grand unified theories from Why Heterotic? Steps in Understanding compactifications on Calabi-Yau manifolds [Candelas et al 85, ..]. Moduli Compactifications to 3d The G2 Holonomy Case The Heterotic G2 System World Sheet Moduli and Effective Theories – 3 Motivation and Overview Introduction The low energy theory of the heterotic string is ten dimensional supergravity coupled to Motivation and Overview General String Yang-Mills gauge theory. Compactifications Easy to obtain four-dimensional supersymmetric grand unified theories from Why Heterotic? Steps in Understanding compactifications on Calabi-Yau manifolds [Candelas et al 85, ..]. Moduli Complications: Compactifications to 3d The G2 Holonomy Case Higher curvature corrections induce torsional (non Ricci-flat) geometries. The Heterotic G2 System Harder to understand geometries. Often loose toolbox of algebraic geometry and World Sheet Kahler¨ geometry. Harder to understand moduli (the deformation space). Moduli and Effective Theories – 3 Motivation and Overview Introduction The low energy theory of the heterotic string is ten dimensional supergravity coupled to Motivation and Overview General String Yang-Mills gauge theory. Compactifications Easy to obtain four-dimensional supersymmetric grand unified theories from Why Heterotic? Steps in Understanding compactifications on Calabi-Yau manifolds [Candelas et al 85, ..]. Moduli Complications: Compactifications to 3d The G2 Holonomy Case Higher curvature corrections induce torsional (non Ricci-flat) geometries. The Heterotic G2 System Harder to understand geometries. Often loose toolbox of algebraic geometry and World Sheet Kahler¨ geometry. Harder to understand moduli (the deformation space). This talk: Heterotic string on manifolds with reduced structure group G2. Review of heterotic G2 system. Review of infinitesimal moduli of heterotic G2 system (see talk by X. de la Ossa). The heterotic G2 sigma model and marginal deformations. Comments on work in progress.. Moduli and Effective Theories – 3 General String Compactifications Introduction Motivation and Overview General String Compactifications Why Heterotic? Steps in Understanding Moduli Compactifications to 3d The G2 Holonomy Case The Heterotic G2 System World Sheet Moduli and Effective Theories – 4 General String Compactifications Introduction String theory in the supergravity limit is ten-dimensional: Motivation and Overview General String M10 = Md × X10−d , Compactifications Why Heterotic? where Md is the d-dimensional external spacetime, and X is the internal (compact) Steps in Understanding Moduli geometry. Compactifications to 3d The G2 Holonomy Case The Heterotic G2 System World Sheet Moduli and Effective Theories – 4 General String Compactifications Introduction String theory in the supergravity limit is ten-dimensional: Motivation and Overview General String M10 = Md × X10−d , Compactifications Why Heterotic? where Md is the d-dimensional external spacetime, and X is the internal (compact) Steps in Understanding Moduli geometry. ′ Compactifications to 3d Supersymmetry: Puts conditions on X. Lowest order ⇔ α =0, mostly this talk. The G2 Holonomy Case The Heterotic G2 System d =4: To lowest order X is required to be a torsion-free Calabi-Yau manifold. World Sheet d =3: To lowest order X is a torsion-free G2-Holonomy manifold for Minkowski spacetimes. AdS3 geometries require torsional G2 structures. Moduli and Effective Theories – 4 General String Compactifications Introduction String theory in the supergravity limit is ten-dimensional: Motivation and Overview General String M10 = Md × X10−d , Compactifications Why Heterotic? where Md is the d-dimensional external spacetime, and X is the internal (compact) Steps in Understanding Moduli geometry. ′ Compactifications to 3d Supersymmetry: Puts conditions on X. Lowest order ⇔ α =0, mostly this talk. The G2 Holonomy Case The Heterotic G2 System d =4: To lowest order X is required to be a torsion-free Calabi-Yau manifold. World Sheet d =3: To lowest order X is a torsion-free G2-Holonomy manifold for Minkowski spacetimes. AdS3 geometries require torsional G2 structures. Deformations δX preserving conditions ⇔ Moduli fields in external spacetime. String Phenomenology: Find compact geometries whose set of moduli contains the Standard Model. Heterotic String: We do not yet understand moduli of generic compactifications. Special cases known (Standard Embedding [Candelas et al 85], etc). Moduli and Effective Theories – 4 Why Heterotic? Introduction Heterotic supergravity is a ten dimensional supergravity coupled to a Yang-Mills gauge Motivation and Overview General String theory. Compactifications Why Heterotic? Steps in Understanding Moduli Compactifications to 3d The G2 Holonomy Case The Heterotic G2 System World Sheet Moduli and Effective Theories – 5 Why Heterotic? Introduction Heterotic supergravity is a ten dimensional supergravity coupled to a Yang-Mills gauge Motivation and Overview General String theory. Compactifications Why Heterotic? Good for phenomenology and Particle Physics. Easy to obtain Standard Model like Steps in Understanding Moduli physics. Compactifications to 3d Useful for describing geometries with more fibration structures. Mathematically interesting: Natural generalisations of torsion free geometry with The G2 Holonomy Case bundles, when bundle can back-react on the base. The Heterotic G2 System World Sheet Moduli and Effective Theories – 5 Why Heterotic? Introduction Heterotic supergravity is a ten dimensional supergravity coupled to a Yang-Mills gauge Motivation and Overview General String theory. Compactifications Why Heterotic? Good for phenomenology and Particle Physics. Easy to obtain Standard Model like Steps in Understanding Moduli physics. Compactifications to 3d Useful for describing geometries with more fibration structures. Mathematically interesting: Natural generalisations of torsion free geometry with The G2 Holonomy Case bundles, when bundle can back-react on the base. The Heterotic G2 System World Sheet Complications: Torsional geometries not well understood. Few “non-trivial” examples [Dasgupta et al. 99, Becker et al 06, Halmagyi-Israel-EES 16,..]. Complicated equations to deal with, e.g. Bianchi Identity: α′ dH = 4 tr F ∧ F . Need a “nicer” description to deal with moduli [Anderson et al 10, Anderson et al 14, delaOssa-EES 14, Garcia-Fernandez et al 15 16, Candelas et al 16, McOrist 16, delaOssa-Larfors-EES 16 17, ..]. Moduli and Effective Theories – 5 Steps in Understanding Moduli Introduction 3 Steps in understanding moduli of a stringy geometry: Motivation and Overview General String Compactifications Why Heterotic? Steps in Understanding Moduli Compactifications to 3d The G2 Holonomy Case The Heterotic G2 System World Sheet Moduli and Effective Theories – 6 Steps in Understanding Moduli Introduction 3 Steps in understanding moduli of a stringy geometry: Motivation and Overview General String 2 Compactifications Step 1: Infinitesimal massless spectrum T M. Derive differential D with D =0. Why Heterotic? Tangent space of moduli space then given by cohomology Steps in Understanding Moduli 1 T M = HD(Q) . Compactifications to 3d The G2 Holonomy Case Moduli fields X usually one-forms with values in a bundle Q (or sheaf), naturally The Heterotic G2 System associated to the given moduli problem. World Sheet Moduli and Effective Theories – 6 Steps in Understanding Moduli Introduction 3 Steps in understanding moduli of a stringy geometry: Motivation and Overview General String 2 Compactifications Step 1: Infinitesimal massless spectrum T M. Derive differential D with D =0. Why Heterotic? Tangent space of moduli space then given by cohomology Steps in Understanding Moduli 1 T M = HD(Q) . Compactifications to 3d The G2 Holonomy Case Moduli fields X usually one-forms with values in a bundle Q (or sheaf), naturally The Heterotic G2 System associated to the given moduli problem. World Sheet Step 2: Understand geometry of M (Kahler¨ metric, etc). Higher order deformations, obstructions (Yukawa couplings). Maurer-Cartan elements, 1 DX + 2 [X , X ]=0 , and associated differentially graded Lie algebra (or L∞-algebra). Moduli and Effective Theories – 6 Steps in Understanding Moduli Introduction 3 Steps in understanding moduli of a stringy geometry: Motivation and Overview General String 2 Compactifications Step 1: Infinitesimal massless spectrum T M. Derive differential D with D =0. Why Heterotic? Tangent space of moduli space then given by cohomology Steps in Understanding Moduli 1 T M = HD(Q) . Compactifications to 3d The G2 Holonomy Case Moduli fields X usually one-forms with values in a bundle Q (or sheaf), naturally The Heterotic G2 System associated to the given moduli problem. World Sheet Step 2: Understand geometry of M (Kahler¨ metric,