Whats new with G 2 ? Sakura Sch afer-Nameki KEK Theory Workshop, - - PowerPoint PPT Presentation

What’s new with G2?

Sakura Sch¨ afer-Nameki KEK Theory Workshop, December 2018 With Andreas Braun, Sebastjan Cizel, Max H¨ ubner appeared now

Related recent work on G2 and Spin(7):

• with Andreas Braun (Oxford): 1708.07215, construction of new G2

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 G2

• with Julius Eckhard (Oxford), Jin-Mann Wong (KIPMU): 1804.02368
• n 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 G2? Why now?

M-theory on a 7d-manifold with G2-holonomy retains 4d N = 1 susy: SO(7) → G2 8 → 7 ⊕ 1.

• Hype in 1990s/2000s on non-compact G2s for susy model building

[Acharya, Witten, Atiyah, Maldacena, Vafa...] ⇒ Main challenge: construction

• f compact G2s with codim 4 and 7 singularities
• In the meantime mathematicans have slowly, but steadily made

progress, culminating recently with the largest class of compact G2 manifolds (order 106): Twisted connected Sum G2 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 M4 inside CY4 7-branes on M4 × R1,3 ≡ {(φ,A) : ω ∧ FA + 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 M4 is encoded in the eigenvalues of φ ∼ diag(λ1,··· ,λn).

• Most importantly: these spectral cover models opened up the

systematic study of global F-theory compactifications.

G2 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 G4 flux to generate

chiral matter.

• Contra G2: very few compact examples, and differential geometry is

much harder

• Pro G2: 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 G2s a very exciting avenue to revisit.

G2

• Lie group G2 is defined as 14 dimensional subgroup of GL7R that

leaves in variant the three-form (dxijk = dxi ∧ dxj ∧ dxk) Φ3 = dx123 + dx145 + dx167 + dx246 − dx257 − dx347 − dx356 .

• G2-holonomy manifolds are 7d admitting a Ricci-flat metric with

holonomy G2.

• Metric specified by a three-form, the G2-form, Φ

dΦ = d ⋆ Φ = 0.

• Calibrated submanifolds are 3d associatives M3

Φ|M3 = vol(M3). i.e. volume minimising in their homology class, or 4d co-associatives, which are calibrated by ⋆Φ.

All known compact G2 manifolds

• First example: non-compact (C2 × S3)/ΓADE [Bryant, Salamon (1989)]
• Compact: [Joyce (2000)] orbifolds T 7/Γ. Order 10 examples, but far

from fully classified

• Compact: Calabi-Yau ×S1 with antiholomorphic involution [Joyce,

Karigiannis (2017), some earlier work]

• Compact: Twisted Connected sum: [Corti, Haskins, Nordstr¨
• m, 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 G2-compactifications:

Local Higgs bundles for G2s

• 2. Twisted Connected Sum (TCS) G2 and Local Models
• 3. From TCS to chiral models.

4d N = 1 Gauge Theories from G2 Holonomy

Gauge Sector of M-theory on G2 Manifolds

• M-theory on a singular, non-compact K3, i.e. C2/ΓADE:

CMNP KK-reduction and M2-branes gives 7d SYM with G=ADE.

• ADE-singularity fibered over a three-manifold:

C2/ΓADE → M3 This can be given a local G2-structure.

• Adiabatic picture: 7d SYM on M3.

SO(1,6)L × SU(2)R → SO(1,3)L × SO(3)M × SU(2)R M3 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 M3

The supersymmetric field configurations on M3 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. Ω1(

M3) ⊗ Ad(G⊥)

• A gauge field components along M3

0 = FA + i[φ,φ], 0 = DAφ 0 = D†

Aφ.

For [φ,φ] = 0 and φ regular, non-trivial solutions only exist for π1(M3) = 0.

Matter field zero-modes

Zero-modes of 4d matter fields depend on background values of φ and A: gauginos: χα ∈ H3

D(M3)

Wilson-line-inos: ψα ∈ H1

D(M3)

where D = d − [(φ + A) ∧ ·] Simplest class of solutions to BPS equations: A = 0 ⇒ dφ = d†φ = 0 ∃f harmonic, with φ = d f For M3 compact: no solutions. M3 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 ⊂ M3 . 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, ∂M3 = T(Γ) = ∅

M3 T( ) M3=M3\T( )

Zero-Modes

U(1)-valued Higgs field φ = d f breaks G → H × U(1), then charge q states appearing in this decomposition, e.g. SU(n + 1) → SU(n) × U(1) yields nq=1 and n−1, are counted by the cohomology of Df = d + q d f ∧ ·.

• Charge distribution: ρ support on Γ ⊂ M3. Either + or - charge Γ±,

with total charge distribution 0.

• Boundary conditions: Excise tubular neighborhood of Γ± and impose

Neumann or Dirichlet b.c.: Dirichlet : αt = 0, Neumann : ⋆αn = 0.

Chiral zero modes: χα ∈ H3

Df (M3),

¯ χ ˙

α ∈ H0 Df (M3)

ψα ∈ H1

Df (M3),

¯ ψ ˙

α ∈ H2 Df (M3),

Mathematiacally these can be computed in terms of relative cohomology

• f M3 with respect to its boundary:

H∗

Df (M3) = H∗(M3,∂−M3)

Chiral index: χ(M3,∂−M3) = b2(M3,∂−M3) − b1(M3,∂−M3). The matter is localized at φ = d f = 0 ↔ generically a codim 3 condition in M3 which is precisely the well-known statement that chiral matter localizes at codimension 7 (points) in the G2. 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 C : 0 = det(φ − s) =

n

• i=0

bn−isi = b0

n

• i=1

(s − λi) φ = d f = 0 becomes λi = 0 loci, i.e. when one of the covers intersects the zero-section M3.

M3 p

If C factorizes over M3 then can evaluate the cohomologies H∗

Df (M3).

f is a Morse function, i.e. it has non-degenerate, isolated critical loci.

Couplings

From the 7d SYM the following coupling decends: Y abc

pqr =

• M3

ψ(a,p1) ∧ ϕ(b,p2) ∧ ψ(c,p3) , Q1 + Q2 + Q3 = 0 pi are the points where matter is localized; a,b,c labels the modes.

p1 p2 p3 (f3) (f1) (f2) α1 α2 α3

This localizes along gradient flows γ(f) dγ(f)i ds = pgij∂jf which emanuate from each critical point. The S2s in ALE-fiber fibered

• ver the gradient flow tree gives rise to

a supersymmetric three-cycle ⇒ M2-instanton contribution.

Building of Models

• G → G × U(1)n, ti generate U(1)s, and consider a charge configuration

i = 1,...,n : φ = tid fi , ρ = tiρi , ∆fi = ρi ,

• M3

ρi = 0. Then for Q = (q1,··· ,qn) ρQ =

n

• i=1

qiρi , fQ =

n

• i=1

qifi At every point in M3 where d fQ = 0, there is a localized chiral multiplet transforming in RQ.

Given charge configuration ρQ, the resulting massless spectrum can be described in terms of the numbers nQ

± of positively and negatively

charged component, and the total number ℓQ

± of loops. rQ is the number

• f negatively charged loops which are independent in homology when

embedded into M3 \ ρ+

Q.

[Pantev, Wijnholt; Braun, Cizel, Hubner, SSN]

Evaluating the cohomologies using Morse/Morse-Bott: #Chiral Multiplets valued in RQ = ℓQ

+ + nQ − − rQ − 1,

#Conjugate-Chiral Multiplets valued in RQ = ℓQ

− + nQ + − rQ − 1,

χQ = (lQ

− − lQ +) + (nQ + − nQ −).

Interactions between three 4d chiral fields localized at points ps transforming in RQs can only arise if Q1 + Q2 + Q3 = 0 and if there exists a trivalent gradient flow tree between them.

Example: Top Yukawa

E6 → SU(5) × U(1)a × U(1)b , Let the matter be localized along the critical loci of the following Morse functions, i.e. f: 5−3,3 : f5 = −3fa + 3fb , 10−1,−3 : f (1)

10 = −fa − 3fb ,

104,0 : f (2)

10 = 4fa .

fQ1 fQ2 fQ3

M3

+

_ _

+

_ _ 5

• q1

10(1)

• q1

2q1

• q2

10(2)

• q2

2q2

Local Models for TCS G2-Manifolds

Twisted Connected Sums

S1 x Z+\S+ S- S1 x Z-\S-

HKR

S+ S- K3

Building blocks: Calabi-Yau three-folds = K3s S± over P1. Remove a fiber (S±

0 ), take a prod-

uct with S1 and glue S± with a hyper-K¨ ahler rotation (HKR) ω± ↔ ReΩ(2,0)

∓

, ImΩ(2,0) ↔ −ImΩ(2,0)

[Kovalev; Corti, Haskins, Nordstr¨

• m, Pacini]

S3 P1 E K3

Let S± be elliptically fibered K3 with sections, i.e. Weierstrass models over P1, and e.g. S+: smooth elliptic fibration S−: two II∗ singular fibers Singular K3-fibers result in non-abelian gauge groups, e.g. En

[Braun, SSN]

Field Theoretic Interpretation of TCS

S1 x Z+\S+ S- S1 x Z-\S-

HKR

S+ S- K3

• M-theory on Calabi-Yau Z± ×S1 preserves N = 2 in 4d.
• Central region: K3 × T 2×interval preserves N = 4 in 4d.
• HyperK¨

ahler rotation and gluing retains only a common N = 1 susy.

• Key: building block have algebraic models.
• TCS are globally K3 → S3. Apply M on K3/het on T 3 duality; and

even het/F-theory duality to e.g. understand instantons [Braun, SSN;

Braun, del Zotto, Halverson, Larfors, SSN; Acharya, Braun, Svanes, Valandro]

TCS Higgs-Bundle

Local Higgs bundle model for Calabi-Yau threefolds in each building block is a spectral cover model over P1 (with charge loci excised). Charges: circles (red/blue), and critical loci are circles (yellow).

S1 x Z+\S+

S-

S1 x Z-\S-

HKR

S+ S- K3

Due to product structure of each building block the critical loci of f, and so matter loci, are always 1d! Requires generalization to Morse-Bott

• theory. Upshot: Matter Spectrum is always non-chiral.

Singular Transitions in TCS G2-manifolds

Can TCS be deformed to yield chiral 4d theories? Deformation of concentric circular charge configurations to e.g. ellipses: gives 4 critical points with equal chiral and conjugate-chiral matter: Singular transitions in the local model that will generate chirality:

Spin(7)

Recent resurgence of insights in 3d N = 1 theories and dualities. Geometric engineering of these in M-theory: Spin(7) 8-manifold. [Alternatively: M5-branes on associative three-cycles in G2 [Eckhard, SSN,

Wong]]

Compact Spin(7) manifolds are equally sparse:

• [Joyce (2000)] orbifold T 8/Γ
• Calabi-Yau four-fold orientifold [Kovalev (2018?)]
• Inspired by TCS for G2 we developed a Generalized Connected Sum

construction.

[Braun, SSN (2018)]

Generalized Connected Sum Spin(7)-manifolds

Generalized Connected Sum (GCS):

[Braun, SSN (2018)]

Z+=CY4 Z-=G2 x S1 CY3

Field theoretic construction: Z± preserves 3d N = 2. Central region preserves 3d N = 4, but gluing retains only common 3d N = 1. Examples

• f new compact Spin(7) manifolds [Braun, SSN].

For CY3 is elliptic, there is an F-theory dual with 4d ‘N = 1/2’ [Vafa]. Generalized Connected Sums were used recently to build F-theory dark matter model [Heckman, Lawrie, Lin, Zoccarato]. Idea: standard M/F-duality lifts 3d N = 2 to 4d N = 1. M/Spin(7) uplift to 4d ‘N = 1/2′. Could realize cancellatin of zero point energy without supersymmetry [Witten].

Summary and Outlook

• G2 manifolds provide a purely geometric way of engineering gauge

theories in 4d with minimal susy.

• Local Higgs bundle model gives insights into the structure of the

gauge sector

• Future: using insights into deformations of TCS form local model, try

to construct compact G2 with codim 7 singularities

• Non-compact G2: revisit in light of SCFTs in 4d.