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, 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).
Step 3: Understand quantum cohomology ring. Include non perturbative effects such as world-sheet instantons, and quantum corrections (higher genus effects). Construct topological theory of corresponding structures? Compute correlation functions in topological theory ⇒ define invariants for geometry?
Moduli and Effective Theories – 6 Introduction
Compactifications to 3d G2-structure compactifications The Canonical G2 Complex
The G2 Holonomy Case
The Heterotic G System 2 Compactifications to 3d World Sheet M10 = M3 × Y7
Moduli and Effective Theories – 7 G2-structure compactifications
Introduction Supersymmetry requires the internal geometry Y to have a G2-structure. Tangent bundle Compactifications to 3d has a reduced structure group G2 ⊂ SO(7). G2-structure compactifications G2 structure determined by a non-degenerate G2 invariant three-form ϕ, where ψ = ∗ϕ The Canonical G2 Complex and the Hodge dual is given by the metric defined by ϕ.
The G2 Holonomy Case Torsion classes (decomposed into irreduclible representations of G2) The Heterotic G2 System
World Sheet dϕ = τ1ψ +3 τ7 ∧ ϕ + ∗τ27
dψ =4 τ7 ∧ ψ + ∗τ14 .
LC Note that dϕ = dψ =0 ⇔ ∇ ϕ =0 (G2 holonomy).
Moduli and Effective Theories – 8 G2-structure compactifications
Introduction Supersymmetry requires the internal geometry Y to have a G2-structure. Tangent bundle Compactifications to 3d has a reduced structure group G2 ⊂ SO(7). G2-structure compactifications G2 structure determined by a non-degenerate G2 invariant three-form ϕ, where ψ = ∗ϕ The Canonical G2 Complex and the Hodge dual is given by the metric defined by ϕ.
The G2 Holonomy Case Torsion classes (decomposed into irreduclible representations of G2) The Heterotic G2 System
World Sheet dϕ = τ1ψ +3 τ7 ∧ ϕ + ∗τ27
dψ =4 τ7 ∧ ψ + ∗τ14 .
LC Note that dϕ = dψ =0 ⇔ ∇ ϕ =0 (G2 holonomy).
Supersymmetry requires [Papadopoulos et al 05, Lukas et al 10, ..],
1 M y τ1 = dvol 3 H(3) , τ7 = 2 dφ 1 y α′ τ14 =0 , τ27 = −H + 6 τ1ϕ − τ7 ψ , H = dB + 4 ωCS (A) .
Note, this is an integrable G2-structure. Gauge bundle: F ∧ ψ =0 ⇔ Curvature is an instanton.
Moduli and Effective Theories – 8 The Canonical G2 Complex
Introduction For integrable G2 structure (in particular G2 holonomy) we have the complex Compactifications to 3d [Reyes-Carrion 93, Fernandez et al 98] G2-structure compactifications 0 dˇ 1 dˇ 2 dˇ 3 The Canonical G2 . Complex 0 → Ω1 −→ Ω7 −→ Ω7 −→ Ω1 → 0
The G2 Holonomy Case Here dˇ = π ◦ d, where π is the projection onto the appropriate G2 representation. The Heterotic G2 System ˇ2 World Sheet This is a differential complex, i.e. d =0, iff the G2 structure is integrable (in close 2 analogy with integrable complex structure ⇔ ∂ =0).
Moduli and Effective Theories – 9 The Canonical G2 Complex
Introduction For integrable G2 structure (in particular G2 holonomy) we have the complex Compactifications to 3d [Reyes-Carrion 93, Fernandez et al 98] G2-structure compactifications 0 dˇ 1 dˇ 2 dˇ 3 The Canonical G2 . Complex 0 → Ω1 −→ Ω7 −→ Ω7 −→ Ω1 → 0
The G2 Holonomy Case Here dˇ = π ◦ d, where π is the projection onto the appropriate G2 representation. The Heterotic G2 System ˇ2 World Sheet This is a differential complex, i.e. d =0, iff the G2 structure is integrable (in close 2 analogy with integrable complex structure ⇔ ∂ =0).
This complex generalises to bundles (V, A)
ˇ ˇ ˇ 0 dA 1 dA 2 dA 3 0 → Ω1(V ) −−→ Ω7(V ) −−→ Ω7(V ) −−→ Ω1(V ) → 0 .
where dˇA = π ◦ dA. This is a differential complex iff F ∧ ψ =0.
These complexes are elliptic [Reyes-Carrion 93] ⇒ the corresponding cohomologies ∗ Hˇ (V ) are finite dimensional.
Moduli and Effective Theories – 9 Introduction
Compactifications to 3d
The G2 Holonomy Case Infinitesimal Geometric Moduli
Moduli of instanton bundles
The Heterotic G2 System World Sheet The G2 Holonomy Case
Moduli and Effective Theories – 10 Infinitesimal Geometric Moduli
Introduction Joyce: Infinitesimal geometric moduli counted by H3(Y ). Compactifications to 3d
The G2 Holonomy Case Infinitesimal Geometric Moduli
Moduli of instanton bundles
The Heterotic G2 System
World Sheet
Moduli and Effective Theories – 11 Infinitesimal Geometric Moduli
Introduction Joyce: Infinitesimal geometric moduli counted by H3(Y ). Compactifications to 3d Infinitesimal moduli are given by The G2 Holonomy Case Infinitesimal Geometric Moduli m np m m n ∂ ϕ 1 M ϕ dx , M M dx 1 T Y . Moduli of instanton bundles t = − 2 t ∧ mnp t = tn ∈ Ω ( )
The Heterotic G2 System where M is a symmetric matrix.
World Sheet Can show that dˇ∇M =0, where ∇ is the Levi-Civita connection (G2 holonomy). Modulo diffeomorphisms we find
M ∈ Hˇ 1(T Y ) . It follows that H3(Y ) embeds into Hˇ 1(T Y ), given by “the symmertric part” of Hˇ 1(T Y ).
Moduli and Effective Theories – 11 Infinitesimal Geometric Moduli
Introduction Joyce: Infinitesimal geometric moduli counted by H3(Y ). Compactifications to 3d Infinitesimal moduli are given by The G2 Holonomy Case Infinitesimal Geometric Moduli m np m m n ∂ ϕ 1 M ϕ dx , M M dx 1 T Y . Moduli of instanton bundles t = − 2 t ∧ mnp t = tn ∈ Ω ( )
The Heterotic G2 System where M is a symmetric matrix.
World Sheet Can show that dˇ∇M =0, where ∇ is the Levi-Civita connection (G2 holonomy). Modulo diffeomorphisms we find
M ∈ Hˇ 1(T Y ) . It follows that H3(Y ) embeds into Hˇ 1(T Y ), given by “the symmertric part” of Hˇ 1(T Y ).
The anti-symmetric part of Hˇ 1(T Y ) can be shown to correspond to closed two-forms. We can interpret this as deformations of the heterotic B-field.
In total we get
Hˇ 1(T Y ) =∼ H3(Y ) ⊕ H2(Y ) .
Moduli and Effective Theories – 11 Moduli of instanton bundles
Introduction Let us consider an infinitesimal deformation of the instanton condition ∂t (F ∧ ψ)=0, Compactifications to 3d which gives The G Holonomy Case 2 dˇA∂tA = Fˇ(Mt) , Infinitesimal Geometric Moduli where Moduli of instanton bundles n m Fˇ(Mt) = π7 (Fmndx ∧ Mt ) . The Heterotic G2 System
World Sheet
Moduli and Effective Theories – 12 Moduli of instanton bundles
Introduction Let us consider an infinitesimal deformation of the instanton condition ∂t (F ∧ ψ)=0, Compactifications to 3d which gives The G Holonomy Case 2 dˇA∂tA = Fˇ(Mt) , Infinitesimal Geometric Moduli where Moduli of instanton bundles n m Fˇ(Mt) = π7 (Fmndx ∧ Mt ) . The Heterotic G2 System Note the similarity with the Atiyah map for holomorphic bundles on complex manifolds. Fˇ World Sheet is a map in cohomology
p p Fˇ : Hˇ (T Y ) → Hˇ +1(End(V )) , by Bianchi identity for F together with the instanton condition.
Moduli and Effective Theories – 12 Moduli of instanton bundles
Introduction Let us consider an infinitesimal deformation of the instanton condition ∂t (F ∧ ψ)=0, Compactifications to 3d which gives The G Holonomy Case 2 dˇA∂tA = Fˇ(Mt) , Infinitesimal Geometric Moduli where Moduli of instanton bundles n m Fˇ(Mt) = π7 (Fmndx ∧ Mt ) . The Heterotic G2 System Note the similarity with the Atiyah map for holomorphic bundles on complex manifolds. Fˇ World Sheet is a map in cohomology
p p Fˇ : Hˇ (T Y ) → Hˇ +1(End(V )) , by Bianchi identity for F together with the instanton condition.
1 I.e. infinitesimal geometric moduli Mt ∈ ker(Fˇ). Note: elements of Hˇ (T Y ) corresponding to B-field deformations are in the kernel of Fˇ (representation theory argument). Theorem (delaOssa-Larfors-EES) The zeroth order infinitesimal heterotic moduli space on G2 manifolds to 3d Minkowski is
T M = Hˇ 1(End(V )) ⊕ ker(Fˇ) .
Moduli and Effective Theories – 12 Introduction
Compactifications to 3d
The G2 Holonomy Case
The Heterotic G2 System The Differential
The Differential
World Sheet The Heterotic G2 System
Moduli and Effective Theories – 13 The Differential
Introduction The heterotic G2 system naturally gives rise to a differential Compactifications to 3d dˇA Fˇ End(V ) End(V ) The G2 Holonomy Case Dˇ = : Ωˇ p → Ωˇ p+1 , ˇ ˇ + T Y T Y The Heterotic G2 System F d∇ The Differential The Differential where the upper right hand map is the G2 Atiyah map, and the map
World Sheet p p Fˇ : Ωˇ (End(V )) → Ωˇ +1(T Y )
is given by ˇ m α′ mn q F(α) = 4 π [ tr (g Fnqdx ∧ α)] , and π denotes the appropriate projection.
Moduli and Effective Theories – 14 The Differential
Introduction The heterotic G2 system naturally gives rise to a differential Compactifications to 3d dˇA Fˇ End(V ) End(V ) The G2 Holonomy Case Dˇ = : Ωˇ p → Ωˇ p+1 , ˇ ˇ + T Y T Y The Heterotic G2 System F d∇ The Differential The Differential where the upper right hand map is the G2 Atiyah map, and the map
World Sheet p p Fˇ : Ωˇ (End(V )) → Ωˇ +1(T Y )
is given by ˇ m α′ mn q F(α) = 4 π [ tr (g Fnqdx ∧ α)] , and π denotes the appropriate projection.
The connection d∇+ has conncetion symbols
+ p − p p 1 p Γmn =Γnm =Γnm − 2 Hnm , − where Γ denotes the unique metric connection with totally anti-symmetric torsion − preserving the G2 structure, ∇ ϕ =0. ′ NOTE: The connection Γ+ is NOT an instanton when α -corrections are included. However ′ R+ ∧ ψ = 0 + O(α ) . Moduli and Effective Theories – 14 The Differential
Introduction We compute the square of the differential Compactifications to 3d 2 2 dˇ + Fˇ Fˇdˇ∇+ + dˇAFˇ The G2 Holonomy Case ˇ2 A D = 2 2 =0 . dˇ∇+ Fˇ + FˇdˇA dˇ + + Fˇ The Heterotic G2 System ∇ The Differential
The Differential
World Sheet
Moduli and Effective Theories – 15 The Differential
Introduction We compute the square of the differential Compactifications to 3d 2 2 dˇ + Fˇ Fˇdˇ∇+ + dˇAFˇ The G2 Holonomy Case ˇ2 A D = 2 2 =0 . dˇ∇+ Fˇ + FˇdˇA dˇ + + Fˇ The Heterotic G2 System ∇ The Differential The Differential ˇ2 2 Instanton condition implies dA =0 ⇒ F ∈ Ω14(End(V )). 2 World Sheet Then Fˇ =0 follows from representation theory arguments.
Using dAF =0 and instanton condition, we also find
Fˇdˇ∇+ + dˇAFˇ = dˇ∇+ Fˇ + FˇdˇA =0 . ˇ2 ˇ2 Heterotic Bianchi identity ⇒ d∇+ + F =0.
Viewed as a connection on Q1 = End(V ) ⊕ T Y the connection D satisfies the instanton condition.
Moduli and Effective Theories – 15 The Differential
Introduction We compute the square of the differential Compactifications to 3d 2 2 dˇ + Fˇ Fˇdˇ∇+ + dˇAFˇ The G2 Holonomy Case ˇ2 A D = 2 2 =0 . dˇ∇+ Fˇ + FˇdˇA dˇ + + Fˇ The Heterotic G2 System ∇ The Differential The Differential ˇ2 2 Instanton condition implies dA =0 ⇒ F ∈ Ω14(End(V )). 2 World Sheet Then Fˇ =0 follows from representation theory arguments.
Using dAF =0 and instanton condition, we also find
Fˇdˇ∇+ + dˇAFˇ = dˇ∇+ Fˇ + FˇdˇA =0 . ˇ2 ˇ2 Heterotic Bianchi identity ⇒ d∇+ + F =0.
Viewed as a connection on Q1 = End(V ) ⊕ T Y the connection D satisfies the instanton condition.
Theorem [delaOssa-Larfors-EES ’17] The infinitesimal moduli space of the heterotic G2 system is 1 T M = HˇDˇ (Q1) .
See also [GarciaFernandez-Rubio-Tipler ’16] where the infinitesimal deformations where computed, and put in context of Generalised Geometry for G2 manifolds.
Moduli and Effective Theories – 15 Introduction
Compactifications to 3d
The G2 Holonomy Case
The Heterotic G2 System
World Sheet
The String World Sheet Conformal Invariance The World Sheet Perspective The Heterotic Sigma Model ′ Marginal Deformations (at O(α )=0) The BRST Operator
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 16 The String World Sheet
Introduction String theory is two dimensional. It is described by an action functional S(Ψ) on a Compactifications to 3d Riemann surface Σ, with local coordinates z, representing the string path through time. The G2 Holonomy Case We write The Heterotic G2 System S(Ψ) = L(Ψ) . World Sheet ZΣ The String World Sheet Conformal Invariance The fields collectively denoted Ψ correspond to coordinates in spacetime (or bundle). The Heterotic Sigma Model Space-time fields (metric, etc) are given by couplings between fields in S(Ψ). Marginal Deformations The BRST Operator Worldsheet supersymmetry + anomaly free conformal invariance ⇒ target space is Worldsheet vs Supergravity ten-dimensional. Outlook
Moduli and Effective Theories – 17 The String World Sheet
Introduction String theory is two dimensional. It is described by an action functional S(Ψ) on a Compactifications to 3d Riemann surface Σ, with local coordinates z, representing the string path through time. The G2 Holonomy Case We write The Heterotic G2 System S(Ψ) = L(Ψ) . World Sheet ZΣ The String World Sheet Conformal Invariance The fields collectively denoted Ψ correspond to coordinates in spacetime (or bundle). The Heterotic Sigma Model Space-time fields (metric, etc) are given by couplings between fields in S(Ψ). Marginal Deformations The BRST Operator Worldsheet supersymmetry + anomaly free conformal invariance ⇒ target space is Worldsheet vs Supergravity ten-dimensional. Outlook String theory has two expansions:
′ The α expansion: Perturbative quantum corrections to the effective action S(Ψ) on a given world sheet ⇒ higher curvature corrections on the supergravity side. The higher genus expansion: Corrections from putting the action S(Ψ) on higher genus surfaces ⇒ Quantum corrections for the space-time geometry.
Moduli and Effective Theories – 17 The String World Sheet
Introduction String theory is two dimensional. It is described by an action functional S(Ψ) on a Compactifications to 3d Riemann surface Σ, with local coordinates z, representing the string path through time. The G2 Holonomy Case We write The Heterotic G2 System S(Ψ) = L(Ψ) . World Sheet ZΣ The String World Sheet Conformal Invariance The fields collectively denoted Ψ correspond to coordinates in spacetime (or bundle). The Heterotic Sigma Model Space-time fields (metric, etc) are given by couplings between fields in S(Ψ). Marginal Deformations The BRST Operator Worldsheet supersymmetry + anomaly free conformal invariance ⇒ target space is Worldsheet vs Supergravity ten-dimensional. Outlook String theory has two expansions:
′ The α expansion: Perturbative quantum corrections to the effective action S(Ψ) on a given world sheet ⇒ higher curvature corrections on the supergravity side. The higher genus expansion: Corrections from putting the action S(Ψ) on higher genus surfaces ⇒ Quantum corrections for the space-time geometry.
The string partition function:
Z = DΨ exp − L(Ψ) . Z Z Xg Σg Moduli and Effective Theories – 17 The Heterotic Sigma Model
Introduction We are intetested in compactifications of the form Compactifications to 3d
The G2 Holonomy Case M(9,1) = M(2,1) × Y7 ,
The Heterotic G2 System where M(2,1) is Minkowski or AdS3, and Y7 admits an integrable G2 structure and is World Sheet equipped with a vector bundle V . The heterotic sigma model action restricted to Y then The String World Sheet reads Conformal Invariance 1 2 i j i j The Heterotic Sigma Model S = d z (Gij (φ)+ Bij (φ))∂z φ ∂z φ + Gij (φ)ψ Dz ψ 4π Z Marginal Deformations A A AB i j A B The BRST Operator i + λ Dz λ + 2 Fij ψ ψ λ λ , Worldsheet vs Supergravity
Outlook where the fermionic fields are
i ∗ A ∗ ψ ∈ Γ KΣ ⊗ φ (TY ) , λ ∈ Γ KΣ ⊗ φ (V ) . q p The fermionic fields (ψ,λ) are Grassmannian, i.e. they anti-commute.
Moduli and Effective Theories – 18 The Heterotic Sigma Model
Introduction We are intetested in compactifications of the form Compactifications to 3d
The G2 Holonomy Case M(9,1) = M(2,1) × Y7 ,
The Heterotic G2 System where M(2,1) is Minkowski or AdS3, and Y7 admits an integrable G2 structure and is World Sheet equipped with a vector bundle V . The heterotic sigma model action restricted to Y then The String World Sheet reads Conformal Invariance 1 2 i j i j The Heterotic Sigma Model S = d z (Gij (φ)+ Bij (φ))∂z φ ∂z φ + Gij (φ)ψ Dz ψ 4π Z Marginal Deformations A A AB i j A B The BRST Operator i + λ Dz λ + 2 Fij ψ ψ λ λ , Worldsheet vs Supergravity
Outlook where the fermionic fields are
i ∗ A ∗ ψ ∈ Γ KΣ ⊗ φ (TY ) , λ ∈ Γ KΣ ⊗ φ (V ) . q p The fermionic fields (ψ,λ) are Grassmannian, i.e. they anti-commute.
We also define the derivatives
i i − i j k Dz ψ = ∂z ψ +Γjk ∂z φ ψ A A AB i B Dzλ = ∂z λ − i A (φ)i∂zφ λ ,
− 1 − Note the appearance of Γ =Γ − 2 H, where ∇ ϕ =0.
Moduli and Effective Theories – 18 Conformal Invariance
Introduction Compute quantum corrections to the couplings (G, B, A), i.e. their expectation values, Compactifications to 3d e.g.
The G2 Holonomy Case hGij i = DΨ Gij (φ) exp − L(Ψ) . Z Z Xg Σg The Heterotic G2 System We then have World Sheet
The String World Sheet
Conformal Invariance hGij i = Gij + higher loop quantum corrections + higher genus corrections
The Heterotic Sigma Model Marginal Deformations Conformal Invariance: Want such corrections to vanish. Ignoring higher genus effects, the The BRST Operator first loop corrections vanish when A is Yang-Mills, and G is Ricci flat and H = dB =0 Worldsheet vs Supergravity for Minkowski spacetimes. Outlook
Moduli and Effective Theories – 19 Conformal Invariance
Introduction Compute quantum corrections to the couplings (G, B, A), i.e. their expectation values, Compactifications to 3d e.g.
The G2 Holonomy Case hGij i = DΨ Gij (φ) exp − L(Ψ) . Z Z Xg Σg The Heterotic G2 System We then have World Sheet
The String World Sheet
Conformal Invariance hGij i = Gij + higher loop quantum corrections + higher genus corrections
The Heterotic Sigma Model Marginal Deformations Conformal Invariance: Want such corrections to vanish. Ignoring higher genus effects, the The BRST Operator first loop corrections vanish when A is Yang-Mills, and G is Ricci flat and H = dB =0 Worldsheet vs Supergravity for Minkowski spacetimes. Outlook
Requiring these corrections to vanish give us the supergravity equations of motion.
1 kl ′ Rij − 4 HiklHj = 0 + O(α ) i ′ ∇ Hijk = 0 + O(α ) −i ′ ∇A Fij = 0 + O(α ) .
Note that the ”tree-level” supergravity equations of motion correspond to a one-loop computation on the worldsheet.
Moduli and Effective Theories – 19 The Heterotic Sigma Model
Introduction Introduce a bookkeeping Grassmann parameter θ, constant on Σ. Recall some basics Compactifications to 3d about Grassmann parameters
The G2 Holonomy Case 2 1 dθ = 0 , θ dθ = 1 , θ = 0 . The Heterotic G2 System Z Z
World Sheet The heterotic sigma model action restricted to Y can then be written in superspace as The String World Sheet 1 2 i j A A AB A i B Conformal Invariance S = d zdθ [Gij (Φ) + Bij (Φ)] ∂Φ DΦ − Λ DΛ + iA (Φ)Λ DΦ Λ , 4π Z i The Heterotic Sigma Model Marginal Deformations where the superfields Φ and Λ are given by The BRST Operator Φi = φi + i θψi Worldsheet vs Supergravity A A A Outlook Λ = λ + θl . We think of ψi as the right-moving superpartner of φi, while the fermions λA are left moving, and the fields lA is auxilary.
Moduli and Effective Theories – 20 The Heterotic Sigma Model
Introduction Introduce a bookkeeping Grassmann parameter θ, constant on Σ. Recall some basics Compactifications to 3d about Grassmann parameters
The G2 Holonomy Case 2 1 dθ = 0 , θ dθ = 1 , θ = 0 . The Heterotic G2 System Z Z
World Sheet The heterotic sigma model action restricted to Y can then be written in superspace as The String World Sheet 1 2 i j A A AB A i B Conformal Invariance S = d zdθ [Gij (Φ) + Bij (Φ)] ∂Φ DΦ − Λ DΛ + iA (Φ)Λ DΦ Λ , 4π Z i The Heterotic Sigma Model Marginal Deformations where the superfields Φ and Λ are given by The BRST Operator Φi = φi + i θψi Worldsheet vs Supergravity A A A Outlook Λ = λ + θl . We think of ψi as the right-moving superpartner of φi, while the fermions λA are left moving, and the fields lA is auxilary.
The superspace derivative is D = ∂θ + θ∂z . The action is further invariant under supersymmetry, Noether ⇒ there is a conservation law and a superchagre operator
Q = ∂θ − θ∂z . The equations of motion (δS =0) read
i − i j k i ij AB k A B D∂z Φ + Γjk ∂z Φ DΦ + 2 G Fjk DΦ Λ Λ = 0 A AB i B DΛ − iAi (Φ)DΦ Λ = 0 .
Moduli and Effective Theories – 20 Marginal Deformations
Introduction The moduli of the supergravity solution correspond on the world sheet side to deformations Compactifications to 3d of the couplings which preserve conformal invariance (supergravity equations of motion).
The G2 Holonomy Case These are referred to as marginal The Heterotic G2 System Two key results: World Sheet The String World Sheet A supersymmetric target space solution solves the supergravity equations of motion, Conformal Invariance thus preserves confromal invariance. The Heterotic Sigma Model Marginal Deformations One cannot deform away from supersymmetry without breaking equations of motion. The BRST Operator 1 Worldsheet vs Supergravity Hence, starting with a supersymmetric solution the cohomology HDˇ (Q) counts the Outlook marginal deformations modulo trivial deformations (corresponding to diffeomorphisms and gauge transformations on the target).
Moduli and Effective Theories – 21 Marginal Deformations
Introduction The moduli of the supergravity solution correspond on the world sheet side to deformations Compactifications to 3d of the couplings which preserve conformal invariance (supergravity equations of motion).
The G2 Holonomy Case These are referred to as marginal The Heterotic G2 System Two key results: World Sheet The String World Sheet A supersymmetric target space solution solves the supergravity equations of motion, Conformal Invariance thus preserves confromal invariance. The Heterotic Sigma Model Marginal Deformations One cannot deform away from supersymmetry without breaking equations of motion. The BRST Operator 1 Worldsheet vs Supergravity Hence, starting with a supersymmetric solution the cohomology HDˇ (Q) counts the Outlook marginal deformations modulo trivial deformations (corresponding to diffeomorphisms and gauge transformations on the target). It is a common feature that reduced structure on the target space implies more symmetry on the world sheet. E.g. an extra supersymmetry for even-dimensional targets gives rise to complex manifolds with holomorphic vector bundles.
Restricting to targets with a G2 structure satisfying the supergravity supersymmetry conditions enhances the world sheet superconformal symmetry to what we call the G2 algebra [Shatashvili-Vafa ’94].
As for supersymmetry, there is a nilpotent charge operator QBRST . Observables (moduli) are counted by its cohomology ⇒ expect the marginal deformations to be counted by the cohomology of the BRST operator of the G2 algebra. Moduli and Effective Theories – 21 The BRST Operator
Introduction A generic (infinitesimal) deformation of the superspace action reads Compactifications to 3d 1 2 i j AB A i B δS = d zdθ O , O = Mij (Φ)∂z Φ DΦ + iδAi (Φ)Λ DΦ Λ , The G2 Holonomy Case 4π Z
The Heterotic G2 System where the symmetric part of M corresponds to deformations of the metric, while the World Sheet anti-symmetric part of M corresponds to deformations of the B-field. The String World Sheet
Conformal Invariance
The Heterotic Sigma Model
Marginal Deformations
The BRST Operator
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 22 The BRST Operator
Introduction A generic (infinitesimal) deformation of the superspace action reads Compactifications to 3d 1 2 i j AB A i B δS = d zdθ O , O = Mij (Φ)∂z Φ DΦ + iδAi (Φ)Λ DΦ Λ , The G2 Holonomy Case 4π Z
The Heterotic G2 System where the symmetric part of M corresponds to deformations of the metric, while the World Sheet anti-symmetric part of M corresponds to deformations of the B-field. The String World Sheet
Conformal Invariance In the complex case, the BRST operator corresponds to a supercharge. In the G2 case, The Heterotic Sigma Model the BRST operator corresponds to the supercharge followed by an appropriate projection Marginal Deformations
The BRST Operator [Shatashvili-Vafa ’94, deBoer et al ’05].
Worldsheet vs Supergravity
Outlook In particular, a generic p-form Ξ is represented in superspace as p i1 i2 ip Ξ = Ξi1i2...ip (Φ)DΦ DΦ . . . DΦ , and the BRST operator is required to increase the degree of such form by one. Note that O corresponds to a one form.
Moduli and Effective Theories – 22 The BRST Operator
Introduction A generic (infinitesimal) deformation of the superspace action reads Compactifications to 3d 1 2 i j AB A i B δS = d zdθ O , O = Mij (Φ)∂z Φ DΦ + iδAi (Φ)Λ DΦ Λ , The G2 Holonomy Case 4π Z
The Heterotic G2 System where the symmetric part of M corresponds to deformations of the metric, while the World Sheet anti-symmetric part of M corresponds to deformations of the B-field. The String World Sheet
Conformal Invariance In the complex case, the BRST operator corresponds to a supercharge. In the G2 case, The Heterotic Sigma Model the BRST operator corresponds to the supercharge followed by an appropriate projection Marginal Deformations
The BRST Operator [Shatashvili-Vafa ’94, deBoer et al ’05].
Worldsheet vs Supergravity
Outlook In particular, a generic p-form Ξ is represented in superspace as p i1 i2 ip Ξ = Ξi1i2...ip (Φ)DΦ DΦ . . . DΦ , and the BRST operator is required to increase the degree of such form by one. Note that O corresponds to a one form.
The action of the supercharge reads i j k i j i j QO = Mij ∂z Φ ∂z Φ + Mi[j,k]DΦ ∂Φ DΦ + Mij D∂Φ DΦ A AB i B A AB i j B A AB i B + 2iDΛ δAi DΦ Λ − iΛ δA[j,i]DΦ DΦ Λ − iΛ δAi ∂z Φ Λ . The first and last terms correspond to zero forms and should hence be dropped.
Moduli and Effective Theories – 22 The Equations of Motion
′ Introduction Working at tree-level (α =0), we are free to impose the worldsheet equations of motion. Compactifications to 3d Doing so, one finds that the remaining terms correspond to two-forms on the target. The G2 Holonomy Case
The Heterotic G2 System
World Sheet
The String World Sheet
Conformal Invariance
The Heterotic Sigma Model
Marginal Deformations
The BRST Operator
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 23 The Equations of Motion
′ Introduction Working at tree-level (α =0), we are free to impose the worldsheet equations of motion. Compactifications to 3d Doing so, one finds that the remaining terms correspond to two-forms on the target. The G2 Holonomy Case The BRST operator further projects forms onto an appropriate G2 representation, the 7 The Heterotic G2 System representation for two forms. The BRST action on O thus becomes
World Sheet +l i QBRST O = Mk[j,i] − Γ[i|kMl|j] ∂Φ The String World Sheet h Conformal Invariance AB 1 AB k A B 2 ij m n − i D[iδAj] + 2 Fk[i M j] Λ Λ × (π7)mnDΦ DΦ , The Heterotic Sigma Model i Marginal Deformations 2 where π7 denotes the projection of two-forms onto the 7 representation of G2. The BRST Operator
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 23 The Equations of Motion
′ Introduction Working at tree-level (α =0), we are free to impose the worldsheet equations of motion. Compactifications to 3d Doing so, one finds that the remaining terms correspond to two-forms on the target. The G2 Holonomy Case The BRST operator further projects forms onto an appropriate G2 representation, the 7 The Heterotic G2 System representation for two forms. The BRST action on O thus becomes
World Sheet +l i QBRST O = Mk[j,i] − Γ[i|kMl|j] ∂Φ The String World Sheet h Conformal Invariance AB 1 AB k A B 2 ij m n − i D[iδAj] + 2 Fk[i M j] Λ Λ × (π7)mnDΦ DΦ , The Heterotic Sigma Model i Marginal Deformations 2 where π7 denotes the projection of two-forms onto the 7 representation of G2. The BRST Operator Worldsheet vs Supergravity We thus find that if QBRST O is to vanish for all field configurations, we must require Outlook ij +l ϕm Mk[j,i] − Γ[i|kMl|j] = 0 ij AB 1 AB k ϕm D[iδAj] + 2 Fk[i M j] = 0 . These are precicely the equations required for (M,δA) to be in the kernel of Dˇ ′ at α =0.
Similarly, one can check that QBRST exact deformations correspond to Dˇ exact terms on the supergravity side. Hence the marginal deformations are counted by
1 ∼ 1 ˇ ˇ 1 HDˇ (Q1) = Hdˇ (End(V )) ⊕ ker(F) , ker(F) ⊆ Hˇ (T Y ) . A d∇+
Moduli and Effective Theories – 23 Worldsheet vs Supergravity
Introduction We have identified the BRST operator on the worldsheet whose cohomology reproduces ′ Compactifications to 3d the moduli computation on the supergravity side at zeroth order in α .
The G2 Holonomy Case We have shown that the BRST cohomology counts the correct marginal deformations, i.e. The Heterotic G2 System 1 World Sheet the moduli HDˇ (Q1) found from supergravity.
The String World Sheet
Conformal Invariance
The Heterotic Sigma Model
Marginal Deformations
The BRST Operator
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 24 Worldsheet vs Supergravity
Introduction We have identified the BRST operator on the worldsheet whose cohomology reproduces ′ Compactifications to 3d the moduli computation on the supergravity side at zeroth order in α .
The G2 Holonomy Case We have shown that the BRST cohomology counts the correct marginal deformations, i.e. The Heterotic G2 System 1 World Sheet the moduli HDˇ (Q1) found from supergravity. We also recall the following
The String World Sheet Conformal Invariance Theorem (delaOssa-Larfors-EES): The operator Dˇ is nilpotent (and so D is an instanton The Heterotic Sigma Model connection) if and only if (Y,ϕ,B,A) satisfies the heterotic G2 system. Marginal Deformations The BRST Operator Similarly, from the worldsheet we have prooved: Worldsheet vs Supergravity Outlook Theorem (Fiset-Quigley-EES): The BRST operator QBRST is nilpotent if and only if (Y,ϕ,B,A) satisfies the (zeroth order) heterotic G2 system.
This is as expected since the worldsheet symmetry enhances to the G2 algebra precisely when supersymmetry is preserved in supergravity.
Moduli and Effective Theories – 24 Outlook
Introduction Outlook, and work in progress: Compactifications to 3d Apply to explicit examples, e.g. instanton bundles on G2 holonomy [Wapulski, SaEarp, The G2 Holonomy Case Menet-Nordstrm-SaEarp,..], or Nilmanifolds [Fernandez et al]. Can we compute the The Heterotic G2 System cohomologies? World Sheet The String World Sheet What are the exactly marginal deformations? Integrable deformations on the Conformal Invariance supergravity side. Need a better understanding of the deformation complex of the The Heterotic Sigma Model ∞ Marginal Deformations heterotic G2 system (L structure, etc). ′ The BRST Operator α corrections? Can we understand what the BRST operator is when these are Worldsheet vs Supergravity included? Insight from Dˇ ? Outlook What about non-perturbative effects, world sheet instantons, NS5-branes? Correct the Bianchi Identity
α′ 2 (2,2) dH + W5 = 4 tr F , [W5] ∈ H (X) .
Are there topological twists to perform for the heterotic sigma model ala Wittens A and B model for Calabi-Yau, or Shatashvili-Vafa for G2 without bundles? What about a topological theory for the target, like the Hitchin functional, or holomorphic Chern-Simons theory, or the heterotic superpotential for 6d compactifications? Use to define and compute invariants for heterotic geometries such as generalisations of Gromov-Witten and Donaldson-Thomas invariants, and relate to work by
Donaldson-Segal and Joyce? Moduli and Effective Theories – 25 Thank you!
Introduction
Compactifications to 3d
The G2 Holonomy Case
The Heterotic G2 System
World Sheet
The String World Sheet
Conformal Invariance
The Heterotic Sigma Model
Marginal Deformations
The BRST Operator Thank you for your attention!
Worldsheet vs Supergravity
Outlook
Moduli and Effective Theories – 26