Heterotic Flux Compactifications
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1 Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv: , and joint work with Rubio, Tipler, Shahbazi (Math. Annalen & ongoing). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
2 Heterotic Flux Compactifications... Again??? Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv: , and joint work with Rubio, Tipler, Shahbazi (Math. Annalen & ongoing). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
3 Outline This talk is about the internal geometry of four-dimensional N = 1 compactifications of heterotic string theory with flux H. Hull 86, Strominger 86 For Calabi-Yau compactifications (H = 0, φ constant), topology of internal space determines the field content of the 4D-supergravity. h 1,1 sizes of 2-cycles, h 2,1 sizes of 3-cycles. Candelas-Horowitz-Strominger-Witten 85 Furthermore, Yau s Theorem 76 allowed the application of powerful methods from algebraic and Kähler geometry leading to important advances. However, the many massless scalar fields arising from generic CY compactifcations called for stabilising mechanisms. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
4 Outline This talk is about the internal geometry of four-dimensional N = 1 compactifications of heterotic string theory with flux H. Hull 86, Strominger 86 For Calabi-Yau compactifications (H = 0, φ constant), topology of internal space determines the field content of the 4D-supergravity. h 1,1 sizes of 2-cycles, h 2,1 sizes of 3-cycles. Candelas-Horowitz-Strominger-Witten 85 Furthermore, Yau s Theorem 76 allowed the application of powerful methods from algebraic and Kähler geometry leading to important advances. However, the many massless scalar fields arising from generic CY compactifcations called for stabilising mechanisms. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
5 Outline This talk is about the internal geometry of four-dimensional N = 1 compactifications of heterotic string theory with flux H. Hull 86, Strominger 86 For Calabi-Yau compactifications (H = 0, φ constant), topology of internal space determines the field content of the 4D-supergravity. h 1,1 sizes of 2-cycles, h 2,1 sizes of 3-cycles. Candelas-Horowitz-Strominger-Witten 85 Furthermore, Yau s Theorem 76 allowed the application of powerful methods from algebraic and Kähler geometry leading to important advances. However, the many massless scalar fields arising from generic CY compactifcations called for stabilising mechanisms. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
6 Outline This talk is about the internal geometry of four-dimensional N = 1 compactifications of heterotic string theory with flux H. Hull 86, Strominger 86 For Calabi-Yau compactifications (H = 0, φ constant), topology of internal space determines the field content of the 4D-supergravity. h 1,1 sizes of 2-cycles, h 2,1 sizes of 3-cycles. Candelas-Horowitz-Strominger-Witten 85 Furthermore, Yau s Theorem 76 allowed the application of powerful methods from algebraic and Kähler geometry leading to important advances. However, the many massless scalar fields arising from generic CY compactifcations called for stabilising mechanisms. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
7 Outline In the presence of fluxes, the geometry is not Kähler and classical methods no longer apply. Furthermore, H 0 requires holomorphic bundles which make the analysis really complicated dh = α (tr R R tr F F ). Recent developments in the study of geometry of equations of motion (GF 13, Coimbra-Minasian-Triendl-Waldran 14) heterotic T-duality (Baraglia-Hekmati 13, GF 16) moduli (Melnikov-Sharpe 11; De la Ossa-Svanes 14; Anderson-Gray-Sharpe 14; GF-Rubio-Tipler 15) show that generalized geometry is an essential ingredient of the internal geometry in a heterotic flux compactification. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
8 Outline In the presence of fluxes, the geometry is not Kähler and classical methods no longer apply. Furthermore, H 0 requires holomorphic bundles which make the analysis really complicated dh = α (tr R R tr F F ). Recent developments in the study of geometry of equations of motion (GF 13, Coimbra-Minasian-Triendl-Waldran 14) heterotic T-duality (Baraglia-Hekmati 13, GF 16) moduli (Melnikov-Sharpe 11; De la Ossa-Svanes 14; Anderson-Gray-Sharpe 14; GF-Rubio-Tipler 15) show that generalized geometry is an essential ingredient of the internal geometry in a heterotic flux compactification. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
9 Outline In the presence of fluxes, the geometry is not Kähler and classical methods no longer apply. Furthermore, H 0 requires holomorphic bundles which make the analysis really complicated dh = α (tr R R tr F F ). Recent developments in the study of geometry of equations of motion (GF 13, Coimbra-Minasian-Triendl-Waldran 14) heterotic T-duality (Baraglia-Hekmati 13, GF 16) moduli (Melnikov-Sharpe 11; De la Ossa-Svanes 14; Anderson-Gray-Sharpe 14; GF-Rubio-Tipler 15) show that generalized geometry is an essential ingredient of the internal geometry in a heterotic flux compactification. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
10 Moduli MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
11 In 1986 Hull and Strominger characterized warped 4d compactifications of the heterotic string, with N = 1 supersymmetry, non-zero flux H 0, and non-constant dilaton φ. Given by an SU(3)-structure (ψ, ω) with almost complex structure J : TM 6 TM 6 and metric g, and a gauge field A with field strength F, such that dω = 0, g ij F ij = 0, F ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0, where Ω = e φ ψ, H = i( )ω, φ = log Ω In this talk, first order equations in α -expansion taken as exact (my apologies). Imposing the equations of motion, is equivalent to g ij R ij = 0, R ij = 0. Reminiscent of R = R in α -expansion. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
12 In 1986 Hull and Strominger characterized warped 4d compactifications of the heterotic string, with N = 1 supersymmetry, non-zero flux H 0, and non-constant dilaton φ. Given by an SU(3)-structure (ψ, ω) with almost complex structure J : TM 6 TM 6 and metric g, and a gauge field A with field strength F, such that dω = 0, g ij F ij = 0, F ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0, where Ω = e φ ψ, H = i( )ω, φ = log Ω In this talk, first order equations in α -expansion taken as exact (my apologies). Imposing the equations of motion, is equivalent to g ij R ij = 0, R ij = 0. Reminiscent of R = R in α -expansion. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
13 In 1986 Hull and Strominger characterized warped 4d compactifications of the heterotic string, with N = 1 supersymmetry, non-zero flux H 0, and non-constant dilaton φ. Given by an SU(3)-structure (ψ, ω) with almost complex structure J : TM 6 TM 6 and metric g, and a gauge field A with field strength F, such that dω = 0, g ij F ij = 0, F ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0, where Ω = e φ ψ, H = i( )ω, φ = log Ω In this talk, first order equations in α -expansion taken as exact (my apologies). Imposing the equations of motion, is equivalent to g ij R ij = 0, R ij = 0. Reminiscent of R = R in α -expansion. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
14 In 1986 Hull and Strominger characterized warped 4d compactifications of the heterotic string, with N = 1 supersymmetry, non-zero flux H 0, and non-constant dilaton φ. Given by an SU(3)-structure (ψ, ω) with almost complex structure J : TM 6 TM 6 and metric g, and a gauge field A with field strength F, such that dω = 0, g ij F ij = 0, F ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0, where Ω = e φ ψ, H = i( )ω, φ = log Ω In this talk, first order equations in α -expansion taken as exact (my apologies). Imposing the equations of motion, is equivalent to g ij R ij = 0, R ij = 0. Reminiscent of R = R in α -expansion. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
15 Theorem ( -Rubio-Tipler 15) The Hull-Strominger system (below) is elliptic. Therefore, the moduli space is finite-dimensional. dω = 0, g ij F ij = 0, F ij = 0, g ij R ij = 0, R ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0. Remark: does not take into account b-fields. Where are they? Canonically attached to M and the gauge bundle V, there is a set of string classes H 3 str (V, R) (Redden 11). Fixing (H 0, 0, A 0 ) a solution of the Bianchi identity, yields identification H 3 str (V, R) = H 3 (M, R) dh 0 = α tr R 0 R 0 α tr F A0 F A0. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
16 Theorem ( -Rubio-Tipler 15) The Hull-Strominger system (below) is elliptic. Therefore, the moduli space is finite-dimensional. dω = 0, g ij F ij = 0, F ij = 0, g ij R ij = 0, R ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0. Remark: does not take into account b-fields. Where are they? Canonically attached to M and the gauge bundle V, there is a set of string classes H 3 str (V, R) (Redden 11). Fixing (H 0, 0, A 0 ) a solution of the Bianchi identity, yields identification H 3 str (V, R) = H 3 (M, R) dh 0 = α tr R 0 R 0 α tr F A0 F A0. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
17 Theorem ( -Rubio-Tipler 15) The Hull-Strominger system (below) is elliptic. Therefore, the moduli space is finite-dimensional. dω = 0, g ij F ij = 0, F ij = 0, g ij R ij = 0, R ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0. Remark: does not take into account b-fields. Where are they? Canonically attached to M and the gauge bundle V, there is a set of string classes H 3 str (V, R) (Redden 11). Fixing (H 0, 0, A 0 ) a solution of the Bianchi identity, yields identification H 3 str (V, R) = H 3 (M, R) dh 0 = α tr R 0 R 0 α tr F A0 F A0. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
18 Theorem ( -Rubio-Tipler 15) The Hull-Strominger system (below) is elliptic. Therefore, the moduli space is finite-dimensional. dω = 0, g ij F ij = 0, F ij = 0, g ij R ij = 0, R ij = 0, d ω i( ) log Ω = 0, 2i ω α (tr R R tr F F ) = 0. Remark: does not take into account b-fields. Where are they? Canonically attached to M and the gauge bundle V, there is a set of string classes H 3 str (V, R) (Redden 11). Fixing (H 0, 0, A 0 ) a solution of the Bianchi identity, yields identification H 3 str (V, R) = H 3 (M, R) dh 0 = α tr R 0 R 0 α tr F A0 F A0. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
19 Flux map A choice of integral string class σ H 3 str (V, Z), corresponds to a string structure (up to homotopy) Killingback 87, Witten 87 choice of quantization scheme for heterotic Σ-model Freed 86, Waldorf 13. There is a well-defined map flux on moduli, so that (a first approximation to) moduli of heterotic compactifications is given by M σ HS = flux 1 (σ) M HS flux H 3 str (V, R). Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
20 Flux map A choice of integral string class σ H 3 str (V, Z), corresponds to a string structure (up to homotopy) Killingback 87, Witten 87 choice of quantization scheme for heterotic Σ-model Freed 86, Waldorf 13. There is a well-defined map flux on moduli, so that (a first approximation to) moduli of heterotic compactifications is given by M σ HS = flux 1 (σ) M HS flux H 3 str (V, R). Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
21 Flux map A choice of integral string class σ H 3 str (V, Z), corresponds to a string structure (up to homotopy) Killingback 87, Witten 87 choice of quantization scheme for heterotic Σ-model Freed 86, Waldorf 13. There is a well-defined map flux on moduli, so that (a first approximation to) moduli of heterotic compactifications is given by M σ HS = flux 1 (σ) M HS flux H 3 str (V, R). Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
22 Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). Now, δflux = 0, implies there exists a 2-form b on M such that db = δ(d c J ω) 2α tr R + 2α tr A F A. (ω, b,, A) determines generalized metric on E σ = T T End T End V Having added parameters b Ω 2 (M), need to add symmetries: B-field symmetries. Imposing equations are preserved by B-field symmetries: automorphism group of E σ. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
23 Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). Now, δflux = 0, implies there exists a 2-form b on M such that db = δ(d c J ω) 2α tr R + 2α tr A F A. (ω, b,, A) determines generalized metric on E σ = T T End T End V Having added parameters b Ω 2 (M), need to add symmetries: B-field symmetries. Imposing equations are preserved by B-field symmetries: automorphism group of E σ. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
24 Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). Now, δflux = 0, implies there exists a 2-form b on M such that db = δ(d c J ω) 2α tr R + 2α tr A F A. (ω, b,, A) determines generalized metric on E σ = T T End T End V Having added parameters b Ω 2 (M), need to add symmetries: B-field symmetries. Imposing equations are preserved by B-field symmetries: automorphism group of E σ. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
25 Theorem ( -Rubio-Tipler 15) MHS σ corresponds to a moduli space of natural Killing spinor equations in generalized geometry, on a Courant algebroid E σ determined by σ Hstr 3 (V, Z). Idea: infinitesimally δflux( Ω, ω,, A) = [δ(d c J ω) 2α tr R + 2α tr A F A ] H 3 (M, R). Now, δflux = 0, implies there exists a 2-form b on M such that db = δ(d c J ω) 2α tr R + 2α tr A F A. (ω, b,, A) determines generalized metric on E σ = T T End T End V Having added parameters b Ω 2 (M), need to add symmetries: B-field symmetries. Imposing equations are preserved by B-field symmetries: automorphism group of E σ. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
26 Kähler moduli Fixing the complex structure X on M, and consider the Aeppli cohomology group H 1,1 A (X ) = Ker i : Ω 1,1 Ω 2,2 Im : Ω 0,1 Ω 1,0 Ω 1,1. Fixing the holomorphic bundle structure on V and TX, the equation db = δ(d c J ω) 2α tr R + 2α tr A F A implies that there is a well-defined map [( Ω, ω,, A)] [ ω + ib 2α tr s 1 R + 2α tr s 2 F A ] H 1,1 A (X ) for suitable infinitesimal complex gauge transformations s 1, s 2. Kähler moduli of a heterotic flux compactification is H 1,1 A (X ) ( d-closed (1, 1)-currents analytic 2-cycles) Remark: assuming -Lemma holds, De la Ossa-Svanes 14 and Anderson-Gray-Sharpe 14 propose H 1,1 (X ) as Kähler moduli. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
27 Kähler moduli Fixing the complex structure X on M, and consider the Aeppli cohomology group H 1,1 A (X ) = Ker i : Ω 1,1 Ω 2,2 Im : Ω 0,1 Ω 1,0 Ω 1,1. Fixing the holomorphic bundle structure on V and TX, the equation db = δ(d c J ω) 2α tr R + 2α tr A F A implies that there is a well-defined map [( Ω, ω,, A)] [ ω + ib 2α tr s 1 R + 2α tr s 2 F A ] H 1,1 A (X ) for suitable infinitesimal complex gauge transformations s 1, s 2. Kähler moduli of a heterotic flux compactification is H 1,1 A (X ) ( d-closed (1, 1)-currents analytic 2-cycles) Remark: assuming -Lemma holds, De la Ossa-Svanes 14 and Anderson-Gray-Sharpe 14 propose H 1,1 (X ) as Kähler moduli. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
28 Kähler moduli Fixing the complex structure X on M, and consider the Aeppli cohomology group H 1,1 A (X ) = Ker i : Ω 1,1 Ω 2,2 Im : Ω 0,1 Ω 1,0 Ω 1,1. Fixing the holomorphic bundle structure on V and TX, the equation db = δ(d c J ω) 2α tr R + 2α tr A F A implies that there is a well-defined map [( Ω, ω,, A)] [ ω + ib 2α tr s 1 R + 2α tr s 2 F A ] H 1,1 A (X ) for suitable infinitesimal complex gauge transformations s 1, s 2. Kähler moduli of a heterotic flux compactification is H 1,1 A (X ) ( d-closed (1, 1)-currents analytic 2-cycles) Remark: assuming -Lemma holds, De la Ossa-Svanes 14 and Anderson-Gray-Sharpe 14 propose H 1,1 (X ) as Kähler moduli. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
29 Kähler moduli Fixing the complex structure X on M, and consider the Aeppli cohomology group H 1,1 A (X ) = Ker i : Ω 1,1 Ω 2,2 Im : Ω 0,1 Ω 1,0 Ω 1,1. Fixing the holomorphic bundle structure on V and TX, the equation db = δ(d c J ω) 2α tr R + 2α tr A F A implies that there is a well-defined map [( Ω, ω,, A)] [ ω + ib 2α tr s 1 R + 2α tr s 2 F A ] H 1,1 A (X ) for suitable infinitesimal complex gauge transformations s 1, s 2. Kähler moduli of a heterotic flux compactification is H 1,1 A (X ) ( d-closed (1, 1)-currents analytic 2-cycles) Remark: assuming -Lemma holds, De la Ossa-Svanes 14 and Anderson-Gray-Sharpe 14 propose H 1,1 (X ) as Kähler moduli. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
30 Twisted Heterotic Compactifications MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
31 A key ingredient for the description of Hull-Strominger system using generalized geometry in arxiv: is the treatment of the dilaton field φ using Dirac generating operators (Ševera, Alekseev-Xu, unpublished 01). In this approach, φ may be only locally defined, with field-strength given by a closed 1-form in the internal manifold M. Assuming H 1 (M) is trivial, this implies... a globally defined scalar field φ. If H 1 (M) is not trivial, there are interesting new possibilities [30]. A. Strominger, Superstrings with torsion 86. [30] K.S. Narain, Rutherford preprint (1985) E. Witten, private communication to K. S. Narain. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
32 A key ingredient for the description of Hull-Strominger system using generalized geometry in arxiv: is the treatment of the dilaton field φ using Dirac generating operators (Ševera, Alekseev-Xu, unpublished 01). In this approach, φ may be only locally defined, with field-strength given by a closed 1-form in the internal manifold M. Assuming H 1 (M) is trivial, this implies... a globally defined scalar field φ. If H 1 (M) is not trivial, there are interesting new possibilities [30]. A. Strominger, Superstrings with torsion 86. [30] K.S. Narain, Rutherford preprint (1985) E. Witten, private communication to K. S. Narain. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
33 A key ingredient for the description of Hull-Strominger system using generalized geometry in arxiv: is the treatment of the dilaton field φ using Dirac generating operators (Ševera, Alekseev-Xu, unpublished 01). In this approach, φ may be only locally defined, with field-strength given by a closed 1-form in the internal manifold M. Assuming H 1 (M) is trivial, this implies... a globally defined scalar field φ. If H 1 (M) is not trivial, there are interesting new possibilities [30]. A. Strominger, Superstrings with torsion 86. [30] K.S. Narain, Rutherford preprint (1985) E. Witten, private communication to K. S. Narain. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
34 Common sector: A twisted heterotic compactification to 10 2n dimensions is given by a SU(n)-structure (ω, Ψ) on M 2n satisfying dψ θ Ψ = 0, dθ = 0, dd c ω = 0. (1) where θ = Jd ω is the Lee form of the Hermitian structure. Mild class of non-geometric backgrounds, which violate Maldacena-Nuñez Theorem. Very small moduli space. Remark: If M is compact and θ exact, then H = 0, θ = 0 and g is a Calabi-Yau metric. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
35 Common sector: A twisted heterotic compactification to 10 2n dimensions is given by a SU(n)-structure (ω, Ψ) on M 2n satisfying dψ θ Ψ = 0, dθ = 0, dd c ω = 0. (1) where θ = Jd ω is the Lee form of the Hermitian structure. Mild class of non-geometric backgrounds, which violate Maldacena-Nuñez Theorem. Very small moduli space. Remark: If M is compact and θ exact, then H = 0, θ = 0 and g is a Calabi-Yau metric. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
36 Common sector: A twisted heterotic compactification to 10 2n dimensions is given by a SU(n)-structure (ω, Ψ) on M 2n satisfying dψ θ Ψ = 0, dθ = 0, dd c ω = 0. (1) where θ = Jd ω is the Lee form of the Hermitian structure. Mild class of non-geometric backgrounds, which violate Maldacena-Nuñez Theorem. Very small moduli space. Remark: If M is compact and θ exact, then H = 0, θ = 0 and g is a Calabi-Yau metric. MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
37 Example: Twisted compactification to 6d on M 4 = S 3 S 1. Identify M = SU(2) U(1), with Lie algebra where su(2) R = e 1, e 2, e 3, e 4, de 1 = e 2 e 3, de 2 = e 3 e 1, de 3 = e 1 e 2, de 4 = 0. Using T-duality with parameters, obtain all homogeneous solutions on M, given by (β, γ, l, τ) R >0 R 3. g = e 1 e 1 + e 2 e 2 + e 3 e 3 + γ2 β 2 e4 e 4 H = βe 1 de 1, Ψ = (e 1 + i γ β e4 ) (e 2 + ie 3 ) θ = γ β e4, b = τe 1 e 4 where e 1 = e 1 + l β e4. Remark: These are homogeneous primary Hopf surfaces. Moduli of homogeneous complex structures M cx = C (Hasegawa-Kamishima). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
38 Example: Twisted compactification to 6d on M 4 = S 3 S 1. Identify M = SU(2) U(1), with Lie algebra where su(2) R = e 1, e 2, e 3, e 4, de 1 = e 2 e 3, de 2 = e 3 e 1, de 3 = e 1 e 2, de 4 = 0. Using T-duality with parameters, obtain all homogeneous solutions on M, given by (β, γ, l, τ) R >0 R 3. g = e 1 e 1 + e 2 e 2 + e 3 e 3 + γ2 β 2 e4 e 4 H = βe 1 de 1, Ψ = (e 1 + i γ β e4 ) (e 2 + ie 3 ) θ = γ β e4, b = τe 1 e 4 where e 1 = e 1 + l β e4. Remark: These are homogeneous primary Hopf surfaces. Moduli of homogeneous complex structures M cx = C (Hasegawa-Kamishima). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
39 Moduli M (twisted common sector) (β, γ, l, τ) M flux H 3 (M, R) = R β ( γ l + i β l ) M cx = C H 1,1 A Remarks: (X ) = C γ + iτ. -Lemma does not hold (in fact, H 1,1 (X ) = 0). Kähler moduli given by non-topological, transcendental quantities Standard compactifications on K 3 have (real) 82-dimensional moduli. Fixing Aeppli class and complex structure unique solution! (Analogue of Yau s Theorem). Bundle moduli studied by Moraru (explicit description as abelian fibration over P 2c 2+1 ). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
40 Moduli M (twisted common sector) (β, γ, l, τ) M flux H 3 (M, R) = R β ( γ l + i β l ) M cx = C H 1,1 A Remarks: (X ) = C γ + iτ. -Lemma does not hold (in fact, H 1,1 (X ) = 0). Kähler moduli given by non-topological, transcendental quantities Standard compactifications on K 3 have (real) 82-dimensional moduli. Fixing Aeppli class and complex structure unique solution! (Analogue of Yau s Theorem). Bundle moduli studied by Moraru (explicit description as abelian fibration over P 2c 2+1 ). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
41 Moduli M (twisted common sector) (β, γ, l, τ) M flux H 3 (M, R) = R β ( γ l + i β l ) M cx = C H 1,1 A Remarks: (X ) = C γ + iτ. -Lemma does not hold (in fact, H 1,1 (X ) = 0). Kähler moduli given by non-topological, transcendental quantities Standard compactifications on K 3 have (real) 82-dimensional moduli. Fixing Aeppli class and complex structure unique solution! (Analogue of Yau s Theorem). Bundle moduli studied by Moraru (explicit description as abelian fibration over P 2c 2+1 ). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
42 Moduli M (twisted common sector) (β, γ, l, τ) M flux H 3 (M, R) = R β ( γ l + i β l ) M cx = C H 1,1 A Remarks: (X ) = C γ + iτ. -Lemma does not hold (in fact, H 1,1 (X ) = 0). Kähler moduli given by non-topological, transcendental quantities Standard compactifications on K 3 have (real) 82-dimensional moduli. Fixing Aeppli class and complex structure unique solution! (Analogue of Yau s Theorem). Bundle moduli studied by Moraru (explicit description as abelian fibration over P 2c 2+1 ). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
43 Moduli M (twisted common sector) (β, γ, l, τ) M flux H 3 (M, R) = R β ( γ l + i β l ) M cx = C H 1,1 A Remarks: (X ) = C γ + iτ. -Lemma does not hold (in fact, H 1,1 (X ) = 0). Kähler moduli given by non-topological, transcendental quantities Standard compactifications on K 3 have (real) 82-dimensional moduli. Fixing Aeppli class and complex structure unique solution! (Analogue of Yau s Theorem). Bundle moduli studied by Moraru (explicit description as abelian fibration over P 2c 2+1 ). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
44 Heterotic T-duality MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
45 Fix a heterotic flux compactification (Ω, ω,, A) with gauge group G, let P = P g M P V (a Spin(6, R) G-bundle). Definition: The string class of the heterotic flux compactification (Ω, ω,, A) is [Ĥ] = [p d c ω α CS( ) + α CS(A)] H 3 (P, R), where CS(A) = tr(a F A + 1 A [A A]) 6 Assuming that M and V have torus action T k, can consider T k -invariant string classes: σ H 3 str (V, R) T k H 3 (P, R) MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
46 Fix a heterotic flux compactification (Ω, ω,, A) with gauge group G, let P = P g M P V (a Spin(6, R) G-bundle). Definition: The string class of the heterotic flux compactification (Ω, ω,, A) is [Ĥ] = [p d c ω α CS( ) + α CS(A)] H 3 (P, R), where CS(A) = tr(a F A + 1 A [A A]) 6 Assuming that M and V have torus action T k, can consider T k -invariant string classes: σ H 3 str (V, R) T k H 3 (P, R) MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
47 Let (P, σ), (P, σ) a pair as before, with commuting diagram (B = M/T k = M/T k and P 0 = P/T k = P/T k ) P P0 P P p P 0 p P M B M Definition (Baraglia-Hekmati 15): (P, σ) is T-dual to (P, σ) if exists an invariant 2-form F on P P0 P which is non-degenerate on the fibres of P P0 P P 0 and representants Ĥ σ and Ĥ σ, such that p Ĥ p Ĥ = df. Theorem (GF 16) Solutions of the Hull-Strominger system are preserved by Heterotic T-duality. Remarks: Existence of T -dual requires flux quantization σ H 3 str (V, Z). Previous partial check by (Evslin-Minasian 08). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
48 Let (P, σ), (P, σ) a pair as before, with commuting diagram (B = M/T k = M/T k and P 0 = P/T k = P/T k ) P P0 P P p P 0 p P M B M Definition (Baraglia-Hekmati 15): (P, σ) is T-dual to (P, σ) if exists an invariant 2-form F on P P0 P which is non-degenerate on the fibres of P P0 P P 0 and representants Ĥ σ and Ĥ σ, such that p Ĥ p Ĥ = df. Theorem (GF 16) Solutions of the Hull-Strominger system are preserved by Heterotic T-duality. Remarks: Existence of T -dual requires flux quantization σ H 3 str (V, Z). Previous partial check by (Evslin-Minasian 08). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
49 Let (P, σ), (P, σ) a pair as before, with commuting diagram (B = M/T k = M/T k and P 0 = P/T k = P/T k ) P P0 P P p P 0 p P M B M Definition (Baraglia-Hekmati 15): (P, σ) is T-dual to (P, σ) if exists an invariant 2-form F on P P0 P which is non-degenerate on the fibres of P P0 P P 0 and representants Ĥ σ and Ĥ σ, such that p Ĥ p Ĥ = df. Theorem (GF 16) Solutions of the Hull-Strominger system are preserved by Heterotic T-duality. Remarks: Existence of T -dual requires flux quantization σ H 3 str (V, Z). Previous partial check by (Evslin-Minasian 08). MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
50 Thank you! MGF (ICMAT) Heterotic Flux Compactifications String Pheno / 19
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