What’s new with G 2 ? Sakura Sch¨ afer-Nameki KEK Theory Workshop, December 2018 With Andreas Braun , Sebastjan Cizel, Max H¨ ubner appeared now
Related recent work on G 2 and Spin (7) : • with Andreas Braun (Oxford): 1708.07215 , construction of new G 2 manifolds • with Andreas Braun (Oxford): 1803.10755 , construction of new Spin(7) manifolds • with Braun, Del Zotto, Larfors, Halverson, Morrison: 1803.02343 and Acharya, Braun, Svanes, Valandro: 1812.04008 , identifying M2-instantons in G 2 • with Julius Eckhard (Oxford), Jin-Mann Wong (KIPMU): 1804.02368 on N=1 version of 3d 3d correspondence • with Julius Eckhard, Heeyeon Kim (Oxford): in progress on refinement of the N=1 3d 3d correspondence
Why G 2 ? Why now? M-theory on a 7d-manifold with G 2 -holonomy retains 4d N = 1 susy: SO (7) → G 2 8 → 7 ⊕ 1 . • Hype in 1990s/2000s on non-compact G 2 s for susy model building [Acharya, Witten, Atiyah, Maldacena, Vafa...] ⇒ Main challenge: construction of compact G 2 s with codim 4 and 7 singularities • In the meantime mathematicans have slowly, but steadily made progress, culminating recently with the largest class of compact G 2 manifolds (order 10 6 ): Twisted connected Sum G 2 manifolds ⇒ What’s the 4d physics? • With the resurgence of F-theory, new directions in geometric engineering have emerged. ⇒ precision matching of geometry and 4d physics ⇒ beyond susy-GUTs, e.g. superconformal field theories (SCFTs).
Some lessons from F-theory The framework of choice in recent years for geometric engineering, e.g. 4d N = 1 , is F-theory (i.e. Type IIB with varying axio-dilaton τ ) on elliptic Calabi-Yau four-folds (CY4). Lessons we learned there: • At the beginning there were ‘local’ models, i.e. Higgs bundles, encoding gauge sector of 7-branes on M 4 inside CY4 7-branes on M 4 × R 1 , 3 ≡ { ( φ,A ) : ω ∧ F A + i [ φ, ¯ φ ] = 0 ,Dφ = D ¯ φ = 0 } � φ � � = 0 breaks � G → G × G ⊥ . • Spectral cover description for [ φ, ¯ φ ] = 0 : The local ALE-fibration over M 4 is encoded in the eigenvalues of φ ∼ diag ( λ 1 , ··· ,λ n ) . • Most importantly: these spectral cover models opened up the systematic study of global F-theory compactifications.
G 2 vs. F-theory Engineering 4d N = 1 theories: • Pro F: Compact geometries (elliptic Calabi-Yau fourfolds) are very well understood by now, everything is holomorphic (great toolset) • Contra F: Models are not purely geometric, need G 4 flux to generate chiral matter. • Contra G 2 : very few compact examples, and differential geometry is much harder • Pro G 2 : Purely geometric, chirality from codim 7 singularities. Tandem of recent mathematical progress and recent emphasis on exploring gauge theories (decoupling gravity anyway) and interest in minimal susy SCFTs (and classification, such as in 6d), makes revisiting G 2 s a very exciting avenue to revisit.
G 2 • Lie group G 2 is defined as 14 dimensional subgroup of GL 7 R that leaves in variant the three-form ( dx ijk = dx i ∧ dx j ∧ dx k ) Φ 3 = dx 123 + dx 145 + dx 167 + dx 246 − dx 257 − dx 347 − dx 356 . • G 2 -holonomy manifolds are 7d admitting a Ricci-flat metric with holonomy G 2 . • Metric specified by a three-form, the G 2 -form, Φ d Φ = d ⋆ Φ = 0 . • Calibrated submanifolds are 3d associatives M 3 Φ | M 3 = vol ( M 3 ) . i.e. volume minimising in their homology class, or 4d co-associatives, which are calibrated by ⋆ Φ .
All known compact G 2 manifolds • First example: non-compact ( C 2 × S 3 ) / Γ ADE [Bryant, Salamon (1989)] • Compact: [Joyce (2000)] orbifolds T 7 / Γ . Order 10 examples, but far from fully classified • Compact: Calabi-Yau × S 1 with antiholomorphic involution [Joyce, Karigiannis (2017), some earlier work] • Compact: Twisted Connected sum: [Corti, Haskins, Nordstr¨ om, Pacini (2015)] . By now millions of examples... ... but they are very special and not quite what we need in M-theory.
Plan of Action 1. Gauge sector of G 2 -compactifications: Local Higgs bundles for G 2 s 2. Twisted Connected Sum (TCS) G 2 and Local Models 3. From TCS to chiral models.
4d N = 1 Gauge Theories from G 2 Holonomy
Gauge Sector of M-theory on G 2 Manifolds • M-theory on a singular, non-compact K3, i.e. C 2 / Γ ADE : C MNP KK-reduction and M2-branes gives 7d SYM with G =ADE. • ADE-singularity fibered over a three-manifold: C 2 / Γ ADE → M 3 This can be given a local G 2 -structure. • Adiabatic picture: 7d SYM on M 3 . SO (1 , 6) L × SU (2) R → SO (1 , 3) L × SO (3) M × SU (2) R M 3 has generic SO (3) holonomy. To retain susy in 4d, we need to topologically twist SO (3) M with SU (2) R-symmetry: ⇒ SO (3) twist = diag ( SO (3) M × SU (2) R ) . ⇒ 4 supercharges in 4d.
Higgs bundle on M 3 The supersymmetric field configurations on M 3 are characterized by the BPS equations � δψ � = 0 Background fields are one-forms 3 of SO (3) twist : • φ twisted scalars are adjoint valued one-forms, i.e. Ω 1 ( � M 3 ) ⊗ Ad ( G ⊥ ) • A gauge field components along M 3 0 = F A + i [ φ,φ ] , 0 = D A φ 0 = D † A φ. For [ φ,φ ] = 0 and φ regular, non-trivial solutions only exist for π 1 ( M 3 ) � = 0 .
Matter field zero-modes Zero-modes of 4d matter fields depend on background values of φ and A : χ α ∈ H 3 gauginos: D ( M 3 ) where D = d − [( φ + A ) ∧ · ] ψ α ∈ H 1 Wilson-line-inos: D ( M 3 ) Simplest class of solutions to BPS equations: dφ = d † φ = 0 A = 0 ⇒ ∃ f harmonic, with φ = d f For M 3 compact: no solutions. M 3 with boundaries or alternatively, Poisson equation with source ρ . ⇒ Morse theory for critical loci points [Pantev, Wijnholt] or Morse-Bott theory for more general critical loci [Braun, Cizel, Hubner, SSN] .
Setup that we will study: φ = d f , ∆ f = ρ, so that f = electrostatic potential ρ = charge density, supported on Γ ⊂ M 3 . In terms of the Higgs field, that is regular: excise tubular neighborhood T (Γ) , where ρ is supported on Γ : φ regular , φ = d f , ∆ f = 0 , ∂M 3 = T (Γ) � = ∅ T( ) M 3 = M 3 \T( ) M 3
Zero-Modes f breaks G → H × U (1) , then charge q states U (1) -valued Higgs field φ = d appearing in this decomposition, e.g. SU ( n + 1) → SU ( n ) × U (1) yields n q =1 and n − 1 , are counted by the cohomology of D f = d + q d f ∧ · . • Charge distribution: ρ support on Γ ⊂ M 3 . Either + or - charge Γ ± , with total charge distribution 0. • Boundary conditions: Excise tubular neighborhood of Γ ± and impose Neumann or Dirichlet b.c.: Dirichlet : α t = 0 , Neumann : ⋆α n = 0 .
Chiral zero modes: χ α ∈ H 3 α ∈ H 0 D f ( M 3 ) , χ ˙ ¯ D f ( M 3 ) ¯ ψ α ∈ H 1 α ∈ H 2 D f ( M 3 ) , ψ ˙ D f ( M 3 ) , Mathematiacally these can be computed in terms of relative cohomology of M 3 with respect to its boundary: H ∗ D f ( M 3 ) = H ∗ ( M 3 ,∂ − M 3 ) Chiral index: χ ( M 3 ,∂ − M 3 ) = b 2 ( M 3 ,∂ − M 3 ) − b 1 ( M 3 ,∂ − M 3 ) . The matter is localized at φ = d f = 0 ↔ generically a codim 3 condition in M 3 which is precisely the well-known statement that chiral matter localizes at codimension 7 (points) in the G 2 . Note: for each critical point of f there is one matter mode localized.
Higher Rank and Spectral Cover Picture Consider [ φ,φ ] = 0 , diagonalizable φ in U (1) n n n � � b n − i s i = b 0 C : 0 = det( φ − s ) = ( s − λ i ) i =0 i =1 φ = d f = 0 becomes λ i = 0 loci, i.e. when one of the covers intersects the zero-section M 3 . p M 3 If C factorizes over M 3 then can evaluate the cohomologies H ∗ D f ( M 3 ) . f is a Morse function, i.e. it has non-degenerate, isolated critical loci.
Couplings From the 7d SYM the following coupling decends: � ψ ( a,p 1 ) ∧ ϕ ( b,p 2 ) ∧ ψ ( c,p 3 ) , Y abc pqr = Q 1 + Q 2 + Q 3 = 0 M 3 p i are the points where matter is localized; a,b,c labels the modes. α 1 α 2 This localizes along gradient flows p 1 p 2 γ ( f ) dγ ( f ) i ( f 1 ) ( f 2 ) = pg ij ∂ j f ds which emanuate from each critical α 3 The S 2 s in ALE-fiber fibered point. over the gradient flow tree gives rise to a supersymmetric three-cycle ( f 3 ) ⇒ M2-instanton contribution. p 3
Building of Models G → G × U (1) n , t i generate U (1) s, and consider a charge configuration � � φ = t i d ρ = t i ρ i , i = 1 ,...,n : f i , ∆ f i = ρ i , ρ i = 0 . M 3 Then for Q = ( q 1 , ··· ,q n ) n n � � ρ Q = q i ρ i , f Q = q i f i i =1 i =1 At every point in M 3 where d f Q = 0 , there is a localized chiral multiplet transforming in R Q .
Given charge configuration ρ Q , the resulting massless spectrum can be described in terms of the numbers n Q ± of positively and negatively ± of loops. r Q is the number charged component, and the total number ℓ Q of negatively charged loops which are independent in homology when embedded into M 3 \ ρ + Q . [Pantev, Wijnholt; Braun, Cizel, Hubner, SSN] Evaluating the cohomologies using Morse/Morse-Bott: − − r Q − 1 , # Chiral Multiplets valued in R Q = ℓ Q + + n Q + − r Q − 1 , # Conjugate-Chiral Multiplets valued in R Q = ℓ Q − + n Q χ Q = ( l Q − − l Q + ) + ( n Q + − n Q − ) . Interactions between three 4d chiral fields localized at points p s transforming in R Q s can only arise if Q 1 + Q 2 + Q 3 = 0 and if there exists a trivalent gradient flow tree between them.
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