Hermitian Yang-Mills equation • At string tree level, the connection of the vector bundle has to satisfy the hermitian Yang-Mills equations g ab F ab = ⋆ [ J ∧ J ∧ F ] = 0 . F ab = F ab = 0 , F has to be a holomorphic vector bundle. • A necessary condition is the so-called Donaldson-Uhlenbeck-Yau (DUY) condition, � � J ∧ J ∧ c 1 ( V N i ) = 0 , J ∧ J ∧ c 1 ( L m i ) = 0 , X X to be satisfied for all n i , m . If so, a theorem by Uhlenbeck-Yau guarantees a unique solution provided each term is µ - stable . Florence, 7. June 2006 – p.12/31
One-loop DUY equation Florence, 7. June 2006 – p.13/31
One-loop DUY equation Computing the FI-terms, reveals a one-loop correction to the DUY equation in the presence of M5-branes, which leads to the conjecture. Florence, 7. June 2006 – p.13/31
One-loop DUY equation Computing the FI-terms, reveals a one-loop correction to the DUY equation in the presence of M5-branes, which leads to the conjecture. There exists a corresponding stringy one-loop correction to the HYM equation of the form � ℓ 4 � 4(2 π ) 2 e 2 φ 10 F ab J ∧ J ∧ F ab s − ∧ tr E 8 i ( F i ∧ F i ) − ⋆ 6 i i � 2 � � � 1 1 + ℓ 4 s e 2 φ 10 � F ab 2tr( R ∧ R ) 2 ∓ λ a ∧ γ a + N a i a (non − pert . terms) = 0 .. Florence, 7. June 2006 – p.13/31
One-loop DUY equation Florence, 7. June 2006 – p.14/31
One-loop DUY equation There exists a unique solution, once the bundle satisfies the corresponding integrability condition and the bundle is Λ -stable with respect to the slope �� 1 � J ∧ J ∧ c 1 ( F ) − ℓ 4 s g 2 Λ( F ) = c 1 ( F ) ∧ s rk( F ) X X M i � ch 2 ( V N i ) + 1 1 ( L n i ) + 1 � � c 2 2 c 2 ( T ) + (npt) . 2 n i =1 Florence, 7. June 2006 – p.14/31
One-loop DUY equation There exists a unique solution, once the bundle satisfies the corresponding integrability condition and the bundle is Λ -stable with respect to the slope �� 1 � J ∧ J ∧ c 1 ( F ) − ℓ 4 s g 2 Λ( F ) = c 1 ( F ) ∧ s rk( F ) X X M i � ch 2 ( V N i ) + 1 1 ( L n i ) + 1 � � c 2 2 c 2 ( T ) + (npt) . 2 n i =1 If, as for SU ( N ) Bundles λ ( V ) = µ ( V ) , we can immediately conclude that a µ -stable bundle is also λ -stable for sufficiently small string coupling g s . Florence, 7. June 2006 – p.14/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.15/31
Flipped SU (5) vacua Consider heterotic string on a Calabi-Yau manifold X with bundle W = V ⊕ L with structure group G = SU (4) × U (1) . Florence, 7. June 2006 – p.15/31
Flipped SU (5) vacua Consider heterotic string on a Calabi-Yau manifold X with bundle W = V ⊕ L with structure group G = SU (4) × U (1) . reps. Cohomology H ∗ ( M , V ⊗ L − 1 ) 10 − 1 H ∗ ( M , L 4 ) 10 4 H ∗ ( M , V ⊗ L 3 ) 5 3 H ∗ ( M , � 2 V ⊗ L − 2 ) 5 − 2 H ∗ ( M , V ⊗ L − 5 ) 1 − 5 Florence, 7. June 2006 – p.15/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.16/31
Flipped SU (5) vacua Consider heterotic string on a Calabi-Yau manifold X with bundle W = V ⊕ L with structure group G = SU (4) × U (1) . Florence, 7. June 2006 – p.16/31
Flipped SU (5) vacua Consider heterotic string on a Calabi-Yau manifold X with bundle W = V ⊕ L with structure group G = SU (4) × U (1) . reps. Cohomology H ∗ ( M , V ⊗ L − 1 ) 10 − 1 H ∗ ( M , L 4 ) 10 4 H ∗ ( M , V ⊗ L 3 ) 5 3 H ∗ ( M , � 2 V ⊗ L − 2 ) 5 − 2 H ∗ ( M , V ⊗ L − 5 ) 1 − 5 Florence, 7. June 2006 – p.16/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.17/31
Flipped SU (5) vacua • If this really is flipped SU (5) , then GUT breaking via Higgs in 10 . Florence, 7. June 2006 – p.17/31
Flipped SU (5) vacua • If this really is flipped SU (5) , then GUT breaking via Higgs in 10 . • However, for c 1 ( L ) � = 0 the U (1) receives a mass via the GS mechanism → standard SU (5) GUT with extra exotics + GUT breaking via discrete Wilson lines (Tatar, Watari, hep-th/0602238), (Andreas, Curio, hep-th/0602247) Florence, 7. June 2006 – p.17/31
Flipped SU (5) vacua • If this really is flipped SU (5) , then GUT breaking via Higgs in 10 . • However, for c 1 ( L ) � = 0 the U (1) receives a mass via the GS mechanism → standard SU (5) GUT with extra exotics + GUT breaking via discrete Wilson lines (Tatar, Watari, hep-th/0602238), (Andreas, Curio, hep-th/0602247) • Embed a second line bundle into the other E 8 , such that a linear combination of the two observable U (1) ’s remains massless . Florence, 7. June 2006 – p.17/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.18/31
Flipped SU (5) vacua • Concretely, we embed the line bundle L also in the second E 8 , where it leads to the breaking E 8 → E 7 × U (1) 2 and the decomposition E 7 × U (1) � � − → ( 133 ) 0 + ( 1 ) 0 + ( 56 ) 1 + ( 1 ) 2 + c.c. . 248 Florence, 7. June 2006 – p.18/31
Flipped SU (5) vacua • Concretely, we embed the line bundle L also in the second E 8 , where it leads to the breaking E 8 → E 7 × U (1) 2 and the decomposition E 7 × U (1) � � − → ( 133 ) 0 + ( 1 ) 0 + ( 56 ) 1 + ( 1 ) 2 + c.c. . 248 • The resulting massless spectrum is E 7 × U (1) 2 bundle L − 1 56 1 L − 2 1 2 Florence, 7. June 2006 – p.18/31
Flipped SU (5) vacua • Concretely, we embed the line bundle L also in the second E 8 , where it leads to the breaking E 8 → E 7 × U (1) 2 and the decomposition E 7 × U (1) � � − → ( 133 ) 0 + ( 1 ) 0 + ( 56 ) 1 + ( 1 ) 2 + c.c. . 248 • The resulting massless spectrum is E 7 × U (1) 2 bundle L − 1 56 1 L − 2 1 2 • More general breakings are possible. Florence, 7. June 2006 – p.18/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.19/31
Flipped SU (5) vacua • Tadpole cancellation condition � ch 2 ( V ) + 3 ch 2 ( L ) − N a γ a = − c 2 ( T ) . a Florence, 7. June 2006 – p.19/31
Flipped SU (5) vacua • Tadpole cancellation condition � ch 2 ( V ) + 3 ch 2 ( L ) − N a γ a = − c 2 ( T ) . a • The linear combination U (1) X = − 1 � U (1) 1 − 5 � 2 U (1) 2 2 remains massless if the following conditions are satisfied � � c 1 ( L ) ∧ c 2 ( V ) = 0 , c 1 ( L ) = 0 for all M5 branes . Γ a X Florence, 7. June 2006 – p.19/31
Flipped SU (5) vacua: spectrum Florence, 7. June 2006 – p.20/31
Flipped SU (5) vacua: spectrum reps. bundle SM part. ( q L , d c R , ν c ( 10 , 1 ) 1 χ ( V ) = g R ) + [ H 10 ] 2 χ ( L − 1 ) = 0 − ( 10 , 1 ) − 2 χ ( V ⊗ L − 1 ) = g ( u c ( 5 , 1 ) − 3 R , l L ) 2 χ ( � 2 V ) = 0 ( 5 , 1 ) 1 [( h 3 , h 2 ) + ( h 3 , h 2 )] χ ( V ⊗ L ) + χ ( L − 2 ) = g e c ( 1 , 1 ) 5 R 2 χ ( L − 1 ) = 0 ( 1 , 56 ) 5 − 4 Table 2: Massless spectrum of H = SU (5) × U (1) X × E 7 models with g = 1 � X c 3 ( V ) . 2 Florence, 7. June 2006 – p.20/31
Flipped SU (5) vacua Florence, 7. June 2006 – p.21/31
Flipped SU (5) vacua • One gets precisely g generations of flipped SU (5) matter. Florence, 7. June 2006 – p.21/31
Flipped SU (5) vacua • One gets precisely g generations of flipped SU (5) matter. • Right handed leptons from the second E 8 are absent if � c 3 1 ( L ) = 0 . X Florence, 7. June 2006 – p.21/31
Flipped SU (5) vacua • One gets precisely g generations of flipped SU (5) matter. • Right handed leptons from the second E 8 are absent if � c 3 1 ( L ) = 0 . X • The generalised DUY condition for the bundle L simplifies to � J ∧ J ∧ c 1 ( V ) = 0 , λ ( V ) = µ ( V ) = X Florence, 7. June 2006 – p.21/31
Flipped SU (5) vacua: couplings Florence, 7. June 2006 – p.22/31
Flipped SU (5) vacua: couplings • GUT breaking via H 10 + H 10 leads to a natural solution of the doublet-triplet splitting problem via a missing partner mechanism in the superpotential coupling 10 H 2 10 H 2 5 − 1 . 1 1 Florence, 7. June 2006 – p.22/31
Flipped SU (5) vacua: couplings • GUT breaking via H 10 + H 10 leads to a natural solution of the doublet-triplet splitting problem via a missing partner mechanism in the superpotential coupling 10 H 2 10 H 2 5 − 1 . 1 1 • Gauge invariant Yukawa couplings 2 5 j 5 i 2 10 j 2 1 j 10 i 10 i 2 5 − 1 , 2 5 1 , 2 5 − 1 , − 3 1 1 − 3 1 5 lead to Dirac mass-terms for the d , ( u, ν ) and e quarks and leptons after electroweak symmetry breaking. Florence, 7. June 2006 – p.22/31
Flipped SU (5) vacua: couplings Florence, 7. June 2006 – p.23/31
Flipped SU (5) vacua: couplings • Since the electroweak Higgs carries different quantum numbers than the lepton doublet, the dangerous dimension-four proton decay operators 5 i 2 5 k 2 5 k 2 1 j 2 10 j 10 i ∈ ∈ l l e 2 , q d l , u d d − 3 − 3 − 3 1 5 1 2 are not gauge invariant. Florence, 7. June 2006 – p.23/31
Flipped SU (5) vacua: gauge coupl. Florence, 7. June 2006 – p.24/31
Flipped SU (5) vacua: gauge coupl. • Breaking a stringy SU (5) or SO (10) GUT model via discrete Wilson lines, the Standard Model tree level gauge couplings satisfy α 3 = α 2 = 5 3 α Y = α GUT at the string scale. Florence, 7. June 2006 – p.24/31
Flipped SU (5) vacua: gauge coupl. • Breaking a stringy SU (5) or SO (10) GUT model via discrete Wilson lines, the Standard Model tree level gauge couplings satisfy α 3 = α 2 = 5 3 α Y = α GUT at the string scale. • Since the U (1) X has a contribution from the second E 8 , this relation gets modified to α 3 = α 2 = 8 3 α Y = α GUT Florence, 7. June 2006 – p.24/31
Bundles on elliptically fibered CYs Florence, 7. June 2006 – p.25/31
Bundles on elliptically fibered CYs Elliptically fibered Calabi-Yau manifold X π : X → B with the property that the fiber over each point is an elliptic curve E b and that there exist a section σ . Florence, 7. June 2006 – p.25/31
Bundles on elliptically fibered CYs Elliptically fibered Calabi-Yau manifold X π : X → B with the property that the fiber over each point is an elliptic curve E b and that there exist a section σ . • If the base is smooth and preserves only N = 1 supersymmetry in four dimensions, it is restricted to a del Pezzo surface, a Hirzebruch surface, an Enriques surface or a blow up of a Hirzebruch surface. Florence, 7. June 2006 – p.25/31
Bundles on elliptically fibered CYs Elliptically fibered Calabi-Yau manifold X π : X → B with the property that the fiber over each point is an elliptic curve E b and that there exist a section σ . • If the base is smooth and preserves only N = 1 supersymmetry in four dimensions, it is restricted to a del Pezzo surface, a Hirzebruch surface, an Enriques surface or a blow up of a Hirzebruch surface. • Friedman, Morgan and Witten have defined stable SU ( N ) bundles on such spaces via the so-called spectral cover construction. (Friedman, Morgan, Witten, hep-th/9701162) Florence, 7. June 2006 – p.25/31
Fourier-Mukai transform Florence, 7. June 2006 – p.26/31
Fourier-Mukai transform The idea is to use a simple description of SU ( n ) bundles over the elliptic fibers and then globally glue them together to define bundles over X . Florence, 7. June 2006 – p.26/31
Fourier-Mukai transform The idea is to use a simple description of SU ( n ) bundles over the elliptic fibers and then globally glue them together to define bundles over X . Mathematically, such a prescription is realized by the Fourier-Mukai transform V = π 1 ∗ ( π ∗ 2 N ⊗ P B ) with � � X × B C, P B ⊗ π ∗ 2 N π 1 π 2 � � � � C, N X, V Florence, 7. June 2006 – p.26/31
Fourier-Mukai transform The idea is to use a simple description of SU ( n ) bundles over the elliptic fibers and then globally glue them together to define bundles over X . Mathematically, such a prescription is realized by the Fourier-Mukai transform V = π 1 ∗ ( π ∗ 2 N ⊗ P B ) with � � X × B C, P B ⊗ π ∗ 2 N π 1 π 2 � � � � C, N X, V Florence, 7. June 2006 – p.26/31
Cohomology classes (R.B, Moster, Reinbacher, Weigand, to appear) Florence, 7. June 2006 – p.27/31
Cohomology classes (R.B, Moster, Reinbacher, Weigand, to appear) • The Leray spectral sequence for π 2 implies the following intriguing result H 0 ( X, V a ⊗ V b ) = 0 , H 1 ( X, V a ⊗ V b ) = H 0 ( C a ∩ C b , N a ⊗ N b ⊗ K B ) , H 2 ( X, V a ⊗ V b ) = H 1 ( C a ∩ C b , N a ⊗ N b ⊗ K B ) , H 3 ( X, V a ⊗ V b ) = 0 . For the special case V a = O X and C a = σ , one finds C b = σ 2 . (Donagi, He, Ovrut, Reinbacher, hep-th/0405014) Florence, 7. June 2006 – p.27/31
Cohomology classes (R.B, Moster, Reinbacher, Weigand, to appear) • The Leray spectral sequence for π 2 implies the following intriguing result H 0 ( X, V a ⊗ V b ) = 0 , H 1 ( X, V a ⊗ V b ) = H 0 ( C a ∩ C b , N a ⊗ N b ⊗ K B ) , H 2 ( X, V a ⊗ V b ) = H 1 ( C a ∩ C b , N a ⊗ N b ⊗ K B ) , H 3 ( X, V a ⊗ V b ) = 0 . For the special case V a = O X and C a = σ , one finds C b = σ 2 . (Donagi, He, Ovrut, Reinbacher, hep-th/0405014) • Determine cohomologies of line bundles over complete intersections of divisors in X → Koszul sequences allow one relate them eventually to line bundles on B . Florence, 7. June 2006 – p.27/31
Cohomology classes Florence, 7. June 2006 – p.28/31
Cohomology classes The cohomology classes of the anti-symmetric and symmetric tensor products are more involved but can be computed by similar methods. Florence, 7. June 2006 – p.28/31
Outlook Florence, 7. June 2006 – p.29/31
Outlook Using bundle extensions 0 → V 1 → V → V 2 → 0 we have so far found concrete flipped SU (5) models with just three generations of MSSM quarks and leptons plus one vector-like GUT Higgs, i.e. H i ( X, V ) = (0 , 1 , 4 , 0) . Florence, 7. June 2006 – p.29/31
Outlook Using bundle extensions 0 → V 1 → V → V 2 → 0 we have so far found concrete flipped SU (5) models with just three generations of MSSM quarks and leptons plus one vector-like GUT Higgs, i.e. H i ( X, V ) = (0 , 1 , 4 , 0) . The number of weak Higgses and the stability of these extensions are still under investigation. Florence, 7. June 2006 – p.29/31
Conclusions Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. • They do have multiple anomalous U (1) gauge symmetries, which are cancelled by a generalised Green-Schwarz mechanism. Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. • They do have multiple anomalous U (1) gauge symmetries, which are cancelled by a generalised Green-Schwarz mechanism. • There appears a one-loop correction to the DUY supersymmetry condition, motivating a new notion of stability of vector bundles. Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. • They do have multiple anomalous U (1) gauge symmetries, which are cancelled by a generalised Green-Schwarz mechanism. • There appears a one-loop correction to the DUY supersymmetry condition, motivating a new notion of stability of vector bundles. • Three generation flipped SU (5) and SM like vacua can be constructed on elliptically fibered CY manifolds. Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. • They do have multiple anomalous U (1) gauge symmetries, which are cancelled by a generalised Green-Schwarz mechanism. • There appears a one-loop correction to the DUY supersymmetry condition, motivating a new notion of stability of vector bundles. • Three generation flipped SU (5) and SM like vacua can be constructed on elliptically fibered CY manifolds. • Relation between heterotic orbifold constructions and the smooth Calabi-Yau description? (Buchm¨ uller, Hamaguchi, Lebedev, Ratz, hep-ph/0511035) Florence, 7. June 2006 – p.30/31
Conclusions • Heterotic string compactifications with U ( N ) bundles provide new prospects for string model building. • They do have multiple anomalous U (1) gauge symmetries, which are cancelled by a generalised Green-Schwarz mechanism. • There appears a one-loop correction to the DUY supersymmetry condition, motivating a new notion of stability of vector bundles. • Three generation flipped SU (5) and SM like vacua can be constructed on elliptically fibered CY manifolds. • Relation between heterotic orbifold constructions and the smooth Calabi-Yau description? (Buchm¨ uller, Hamaguchi, Lebedev, Ratz, hep-ph/0511035) • Heterotic Landscape? Florence, 7. June 2006 – p.30/31
Corrolar Sorry, but if our construction is correct then it follows for the LHC, Florence, 7. June 2006 – p.31/31
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