Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Higgs Bundles and a Colored Supersymmetric QM Abelian Higgs Backgrounds Summary and Outlook Probing Higgs Bundles for Local G2-Manifolds Max H¨ubner [email protected] Seminar Series on String Phenomenology 1 December 2020, arXiv:2009.07136 Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Higgs Bundles and a Colored Supersymmetric QM Abelian Higgs Backgrounds Summary and Outlook Overview 1 Introduction and Motivation 2 Local G2-Manifolds and 7d twisted SYM 3 Higgs Bundles and a Colored Supersymmetric QM 4 Abelian Higgs Backgrounds 5 Summary and Outlook Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Higgs Bundles and a Colored Supersymmetric QM Abelian Higgs Backgrounds Summary and Outlook Introduction and Motivation M-theory on local (i.e. non-compact) G2-manifolds engineers minimally supersymmetric gauge theories in 4d. [Acharya, 2000], [Witten, 2001], [Atiyah, Witten, 2003], [Pantev, Wijnholt, 2009], [Braun, Cizel, H, Schafer-Nameki, 2018], [Barbosa, Cvetiˇc,Heckman, Lawrie, Torres, Zoccarato, 2019], [Cvetiˇc,Heckman, Rochais, Torres, Zoccarato, 2020], [H, 2020], [Acharya, Kinsella, Svanes, 2020], [Acharya, Najjar, Foscolo, Svanes, 2020] The 4d gauge theories describe the localized degrees of freedom in compact G2-manifolds. [Acharya, 1998], [Halverson, Morrison, 2015], [Guio, Jockers, Klemm, Yeh, 2017], [Braun, Schafer-Nameki, 2017], [Braun, Del Zotto, 2017], [Braun, 2019], [Xu, 2020] Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Higgs Bundles and a Colored Supersymmetric QM Abelian Higgs Backgrounds Summary and Outlook F-theory methods relying on Higgs bundles and their spectral covers can be applied to study the physics of local G2-manifolds. [Beasley, Heckman, Vafa, 2009], [Hayashi, Kawano, Tatar, Watari, 2009], [Marsano, Saulina, Schafer-Nameki, 2009], [Marsano, Saulina, Schafer-Nameki, 2010], [Blumenhagen, Grimm, Jurke, Weigand, 2010], [Donagi, Wijnholt 2011], [Donagi, Wijnholt 2014] Supersymmetric sigma models probing the geometries give insight into non-perturbative effects and indices. [Alvarez-Gaum´e,Witten, 1981], [Witten, 1982], [Pantev, Wijnholt, 2009], [Braun, Cizel, H, Schafer-Nameki, 2018], [H, 2020] Question: What determines the 4d physics engineered by local G2-manifolds in M-theory? Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook ALE Fibered, Local G2-Manifolds Geometric data 2 Local G2-Manifold : C^=ΓADE ,−! X7 ! M3 2 Fibral 2-Spheres : σI 2 H2(C^=ΓADE; R) 2 2 Hyperk¨ahlerTriplet : (!1;!2;!3) 2 H (C^=ΓADE; R) The Higgs field collects the K¨ahlerperiods Z i 1 Higgs field : φI = !i dx 2 Ω (M3) σI where I = 1;:::; rank gADE. Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook Singularities and Supersymmetric 3-cycles Singularity Enhancement at x 2 M3 : φI (x) = 0 (isolated) The vanishing cycles trace out 3-spheres: ,I,J,K ALE Fiber: ,I ALE Fiber: S3 S3 M3 M3 & &I = 0 I = 0 &J = 0 & = 0 I &K = 0 Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook Questions What is the global structure of the network of supersymmetric 3-spheres and the M2-brane instantons? What do the M2-branes wrapped on the supersymmetric 3-spheres descend to in an effective 7d field theory description? How can their contribution to the 4d superpotential be computed from the 7d field theory description? Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook Previous Work M-theory on 1;3 R × X7 [Acharya, Witten, 2001] [Harvey, Moore, 1999] [BCHLTZ, 2019] [Barbosa, 2019] [CHRTZ, 2020] Twisted 7d SYM [H, 2020] 4d N = 1 1;3 on R × M3 [Pantev, Wijnholt, 2009] Field theory [Braun, Cizel, H, Schafer-Nameki, 2018] Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook Effective 7d Physics M-theory on the local G2-manifold X7 gives a 1;3 Partially twisted 7d SYM on R × M3 with gauge group GADE . 1 Complex bosonic 1-form on M3 : ' = φ + iA 2 Ω (M3; gADE) Supersymmetric vacua are solutions to the Hitchin system: i(FA)ij + [φi ; φj ] = 0 ; (dAφ)ij = 0 ; ∗ dA ∗ φ = 0 Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G -Manifolds and 7d twisted SYM 2 ALE Fibered, Local G -Manifolds Higgs Bundles and a Colored Supersymmetric QM 2 Twisted 7d Supersymmetric Yang-Mills Theory Abelian Higgs Backgrounds Summary and Outlook Zero modes along M3 are determined by 1 n o H = Q; Qy ; Q = d + ['; · ] 2 and counted by the cohomologies ∗ HQ (M3; gADE) : Alternatively, consider perturbative zero modes ∗ χa 2 Ω (M3; gADE) $ Codimension 7 Singularity The zero modes are recovered from Z Mass Matrix : Mab = hχb; Qχai M3 Z Yukawa Couplings : Yabc = hχc ; [χa; χb]i : M3 Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Relevant Structures Higgs Bundles and a Colored Supersymmetric QM Supersymmetric Colored Particles Abelian Higgs Backgrounds Perturbative ground states and Instantons Summary and Outlook Colored N = 2 SUSY Quantum Mechanics i Bosonic Coordinates on M3 : x ; i = 1; 2; 3 ∗ i Fermions in x (TM3): ; i = 1; 2; 3 ∗ α Color fermions in x (adGADE): λ ; α = 1;:::; dim gADE 1 1 Lagrangian : L = x_ i x_ − φi φ + i ¯i r + iλ¯αD λ 2 i 2 λ λ,i τ i τ α − D φ ¯i j + i (F ) ¯i j + ζ λ¯αλ − 1 (i j) λ ij λ α Hilbertspace : Hphys. = Λ(M3; gADE) Supercharge : Q = d + ['; · ] Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Relevant Structures Higgs Bundles and a Colored Supersymmetric QM Supersymmetric Colored Particles Abelian Higgs Backgrounds Perturbative ground states and Instantons Summary and Outlook M3 #i(!) $⍺ (!) $A $B xi(!) i xB Bosonic Coordinates on M3 : x ; i = 1; 2; 3 x A ∗ i Fermions in x (TM3): ; i = 1; 2; 3 ∗ α Color fermions in x (adGADE): λ ; α = 1;:::; dim gADE M3 #i(!) $⍺ (!) xi(!) Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Relevant Structures Higgs Bundles and a Colored Supersymmetric QM Supersymmetric Colored Particles Abelian Higgs Backgrounds Perturbative ground states and Instantons Summary and Outlook ¯ i i Supercharge : Q = i x_ − φλ Perturbative Groundstates : (x; λ) 2 M3 × gADE with φλ(x) = 0 1=2-BPS instantons are piecewise solutions to the flow equations i i i α i ¯β γ α Flow line instanton :x _ − φλ =x _ − ic βγ φαλ λ = Dτ λ = 0 : M3 M3 #i(!) $⍺ (!) $C $A $A $B xC x xi(!) B xA $B xB xA Colored instantons ! Euclidean M2-brane instantons M3 Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds #i(!) $⍺ (!) xi(!) Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Abelian Higgs Backgrounds (split) Higgs Bundles and a Colored Supersymmetric QM Yukawa Couplings Abelian Higgs Backgrounds Abelian Higgs Backgrounds (non-split) Summary and Outlook Abelian Higgs Backgrounds (split) I Abelian solutions to the BPS equations: dA = d ; φ = φI H . The 1-form Cartan components φI are harmonic up to sources dφI = 0 ; ∗ d ∗ φI = ρI ! φI = dfI ; ∆fI = ρI ; which are supported on codimension ≥ 2 subloci in M3. Witten SQM into M3 with Root α 2 gADE $ (α) I supercharge Q = d + α dfI Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Abelian Higgs Backgrounds (split) Higgs Bundles and a Colored Supersymmetric QM Yukawa Couplings Abelian Higgs Backgrounds Abelian Higgs Backgrounds (non-split) Summary and Outlook Yukawa Couplings Z (γ) (α) (β) Yabc = hχc ; [χa ; χb ]i M3 Z Z 3 ¯ h (α) ¯ (β) ¯ (γ) ¯ i = d x0 DxD D exp i S [xa; a; a] + S [xb; b; b] + S [xc ; c ; c ] M3 Z Z Z ! X I I I = (±)Γabc exp − α φI − β φI + γ φI Γa Γb Γc Γabc X h 3 i = (±) exp −Vol (S ) ! [Harvey, Moore, 1999] Γabc Γabc Γabc −iHT iHT (α) e Ψr e (α) χr = lim ≡ Ψr ; 0 < δ 1 ; T →−∞(1+iδ) −iE T (α) −∞ e 0;r hχr jΨr i ¯ Y ¯ ¯ Y ¯ DxD D = d x; ; a,τ d x; ; b,τ d x; ; c,τ −∞< τ <0 0< τ <1 xa;−∞ = xa xc;1 = xc xb;−∞ = xb Max H¨ubner Probing Higgs Bundles for Local G2-Manifolds Introduction and Motivation Local G2-Manifolds and 7d twisted SYM Abelian Higgs Backgrounds (split) Higgs Bundles and a Colored Supersymmetric QM Yukawa Couplings Abelian Higgs Backgrounds Abelian Higgs Backgrounds (non-split) Summary and Outlook Effective 4d Theory (split) We find: 1 Mathematics fixing effective 4d physics is Fukaya's Morse theory with multiple Morse functions, [Fukaya, 1997] 2 Chiral and conjugate chiral multiplets in 4d counted by 1 (α) 2 (α) H (M3;@− M3) ; H (M3;@− M3) respectively.