Toric ideas for special geometriesswann/talks/Uppsala-2020.pdfmetry’, J. Geom. Phys. 138: 144–53...

Toric ideas for special geometries

Andrew Swann

Department of Mathematics, and DIGIT,Aarhus [email protected]

Strings, geometry, and quantum elds zoominar17th June, 2020

Madsen, T. B. and Swann, A. F. (2019a), ‘Toric geometry ofG2-manifolds’,Geom. Topol.23 (7): 3459–500Madsen, T. B. and Swann, A. F. (2019b), ‘Toric geometry of Spin(7)-manifolds’, to bepublished in Int. Math. Res. Not. IMRN, arXiv: 1810.12962 [math.DG]Russo, G. and Swann, A. F. (2019), ‘Nearly Kähler six-manifolds with two-torus sym-metry’, J. Geom. Phys. 138: 144–53 DFF - 6108-00358

[email protected]

https://arxiv.org/abs/1810.12962

Symplectic Multi Spin(7) Nearly Kähler

Outline

1 Symplectic background

2 Multi-Hamiltonian actions

3 Spin(7)

4 Nearly Kähler

Andrew Swann Toric ideas for special geometries


Symplectic constructions

( symplectic form: degree 2, closed, non-degenerate

X symmetry: LX( = 0

moment map: X : M → ℝ, dX = ((X, · )

Flat spaceM = ℝ2m coordinates (w1, . . . ,wm, x1, . . . , xm)( = dw1 ∧ dx1 + · · · + dwm ∧ dxm

X = c1

(x1

)

)w1− w1

)

)x1

)+ · · · + cm

(xm

)

)wm− wm

)

)xm

)X = 1

2(c1(w21 + x21 ) + · · · + cm (w2m + x2m)

)+ c



Delzant construction

(M2m,() compact symplectic with Hamiltonian action of torus T m:invariant function : M → ℝm = Lie(T m)∗ with

d〈,X〉 = ((X, · ) ∀X ∈ ℝm = Lie(T m)

Delzant (1988)Compact symplectictoric manifolds

correspond to

Delzant polytopes

Polytopes in ℝm with normal vectors in ℤm and the normals atintersections of faces being part of a ℤ-basis for ℤm.



• b1(M2m) = 0 =⇒ every symplectic torus T j action isHamiltonian• then ( pulls back to 0 on each torus orbit, so j 6 m

The Delzant dimensions are such that

dim(M2m/T m) = m = dim(codomain)

Can be used for non-compactM, e.g. Karshon and Lerman (2015)

Toric Calabi-YauInclude symplectic quotients of ℝ2N = ℂN by subtori of T N whoseweights sum to zero.

ℂ4// diag(eis, eis, e−is, e−is) = −1(0)/T 1

=(O(−1) ⊕ O(−1) → ℂP(1)

)Tian and Yau (1991), . . . , Goto (2012)Futaki, Ono, and Wang (2009)



Ricci-flat special holonomy

Ricci-at geometries in the Berger holonomy classication 1955, . . .

Name Holonomy Dimension Form degrees

Calabi-Yau SU(m) 2m 2,m,mHyperKähler Sp(m) 4m 2, 2, 2G2 holonomy G2 7 3, 4Spin(7) holonomy Spin(7) 8 4

In these cases• the forms specify the geometry• holonomy reduction ⇐⇒ the forms are closed



Multi-Hamiltonian torus actions

(M,) manifold with closed ∈ Ωo (M) preserved by T j ismulti-Hamiltonian if there is an invariant

: M → Λo−1 (Lie(T j)∗) ℝN ,

d〈,X1 ∧ · · · ∧ Xo−1〉 = (X1, . . . ,Xo−1, · )

for all Xh ∈ Lie(T j).

• b1(M) = 0 =⇒ each T j-action preserving ismulti-Hamiltonian• then pulls-back to 0 on each T j-orbit

Geometry is toric if multi-Hamiltonian for T j and

dim(M/T j) = dim(codomain )



A Ricci-flat hierarchy

dimension 4 6 7 8holonomies Sp(1) SU(3) G2 Spin(7)closed forms (1,(2,(3 (,Ω+,Ω− %, ∗7% Φ

degrees 2, 2, 2 2, 3, 3 3, 4 4

Toric group S1 T 2 T 3 T 4

Extension M4 × S1 ×ℝ M6 × S1 M7 × S1

(1 = e45 + e67, (2 = e46 + e75, (3 = e47 + e56

( = e23 − (1, Ω+ = −e2 ∧ (2 − e3 ∧ (3, Ω− = −e3(2 + e2 ∧ (3% = e1 ∧ ( +Ω+ ∗7% = Ω− ∧ e1 + 1

2(2

Φ = e0 ∧ % + ∗7%



Toric Spin(7)

(M8,Φ) with toric T 4 generated by Uh, h = 0, 1, 2, 3, multi-momentmap = (0, 1, 2, 3)

dh = (−1)hΦ(Ui ∧Uj ∧Uℓ, · ) (hijℓ) = (0123)

On the open dense setM0 where T 4 acts freely

Φ = det(V)(Shijℓ

(−1)h (h ∧ dijℓ + ijℓ ∧ dh) + 12 (d

sV−1)2)

g = sV−1 + det(V)dsV−1d

for V = (g(Uh,Ui))−1, = (0, 1, 2, 3) connection one-forms



Smooth behaviour

TheoremHolonomy is contained in Spin(7) (i.e. dΦ = 0) if and only if

divV = 0 and L(V) + Q(dV) = 0

div(V)a =

3∑h=0

)Vha)h

L(V)ab =3∑

h,i=0Vhi

)2Vab

)h)i, Q(dV)ab = −

3∑h,i=0

)Vha

)i

)Vib

)h

Note

L(V) + Q(dV) =3∑

h,i=0

)2

)h)i(VhiVab − VhbVia )



Specialisations

Holonomy in G2, SU(3) and Sp(1) from V =( ∗ 00 1q

), q = 1, 2, 3

dim4, Sp(1), V =

(V00 00 13

)divV = 0 ⇐⇒ V00 = V00(1, 2, 3) =⇒ Q(dV) = 0 and L(V) = 0has only one scalar equation ∆ℝ3V33 = 0, V33 is harmonic ondomain(s) in ℝ3.

dim6, SU(3), V =

(V00 V01 0V01 V11 00 0 12

))V00)0+ )V01

)1= 0,

)V01)0+ )V11

)1= 0

)2

)h)idetV + (−1)h+i

()2

)22+ )2

)32

)V1−h,1−i = 0 h, i = 0, 1



Explicit full holonomy

Holonomy = Spin(7)V = diag(1, 2, 3, 0) > 0

g =

3∑h=0

1h+1

2h + h+2h+3hd2h dh = (−1)h+1h+2dh+2 ∧ dh+3

Holonomy = G2

V = diag(1, 0, 0, 0) > 0

g =10

3∑h=1

2h + 30d20 + 203∑h=1

d2h dh = di ∧ dj (hij) = (123)

Cf. Hein, Sun, Viaclovsky, and Zhang (2018)Andrew Swann Toric ideas for special geometries


Singular behaviour

Theorem = (0, 1, 2, 3) induces a local homeomorphismM/T 4 → ℝ4

Each stabiliser StabT 4 (o) is a connected subtorus of T 4 of rank 6 2Image under : M → ℝ4

• dim(StabT 4) = 1: straight lines with rational slopes• dim(StabT 4) = 2: pointswith three straight lines meeting at point, sum of primitivetangents is zero

Flat model

M = T 2 × ℂ3, T 4 6 T 2 × SU(3) planar graph



Li (2019) produced Taub-NUT family of complete metrics on ℂ3

with above orbit structureOther Calabi-Yau examples Mooney (2020)

Example

S1× (Bryant-Salamon G2 metric on S3 ×ℝ4):

non-planar, but lies in ℝ3 ⊂ ℝ4



ExampleS1× Foscolo, Haskins, and Nordström (2018) G2-example onMl,m:Ml,m a circle bundle over the canonical bundle of ℂP1 × ℂP1 withrst Chern class (l,−m) over the zero section, symmetry groupSU(2) × SU(2) × S1

Primitive directions(l − m, 0,m)(0,m −l,l)(m −l,l − m,−l − m)planar



Nearly Kähler in dimension 6

(N6, g, J,",'+,'−) is (strictly) nearly Kähler ifM7 = C (N6),- = s2ds ∧ " + s3'+ has holonomy in G2. Equivalently

d" = 3'+ and d'− = −2" ∧ "

Gray (1976) (N6, g) is positive EinsteinButruille (2005) the homogeneous examples are

S6 = G2/SU(3), ℂP(3) = Sp(2)/U(2),F1,2(ℂ3) = SU(3)/T 2 and S3 × S3 = SU(2)3/SU(2)

Foscolo and Haskins (2017) two new examples.All known examples have symmetry group of rank at least 2, so T 2symmetryConjecture (Moroianu and Nagy 2019): T 3 symmetry impliesN6 = S3 × S3 = SU(2)3/SU(2) cf. Dixon (2020)



Two-torus symmetry

T 2 generated by vector elds U,VMulti-moment map = "(U,V) d = 3'+(U,V, · )

1 0 is an interior point of the interval (N) ⊂ ℝ

2 the T 2 action is free on −1(r) whenever r is a regular value3 −1(r)/T 2 is then a smooth three manifold, parallelised by acoframe 0, 1, 2 satisfying

f d0 = 1 ∧ 2, d1 ∧ 0 = 0 = d2 ∧ 0

f = 4/(1 − r2/detG), G metric on T 2-bre4 can recover N via a geometric ow of the data 0, 1, 2,G



Singular behaviour

Stabilisers are either T 2, S1 or ℤj

T 2 and S1 stabilisers occur only in −1(0) and the quotient−1(0)/T 2 is homeomorphic to a smooth three-manifoldImages of points with S1 stabiliser give curves in thethree-manifoldImages of points with T 2 stabiliser give vertices at which threesuch curves meet.

S6 ℂP(3) F1,2(ℂ3)

Non-zero critical points lie on holomorphic tori.


References

References I

Berger, M. (1955), ‘Sur les groupes d’holonomie des variétés àconnexion ane et des variétés riemanniennes’, Bull. Soc. Math.France, 83: 279–330.

Butruille, J.-B. (2005), ‘Classication des variétésapproximativement kähleriennes homogènes’, Ann. GlobalAnal. Geom. 27 (3): 201–25.

Delzant, T. (1988), ‘Hamiltoniens périodiques et images convexesde l’application moment’, Bull. Soc. Math. France, 116 (3): 315–39.

Dixon, K. (2020), Global properties of toric nearly Kähler manifolds,11 Feb., arXiv: 2002.04534 [math.DG].

Foscolo, L. and Haskins, M. (2017), ‘New G2-holonomy cones andexotic nearly Kähler structures on S6 and S3 × S3’, Ann. of Math.(2), 185 (1): 59–130.



References

References II

Foscolo, L., Haskins, M., and Nordström, J. (2018), Innitely manynew families of complete cohomogeneity one G2-manifolds: G2analogues of the Taub-NUT and Eguchi-Hanson spaces, May,arXiv: 1805.02612 [math.DG].

Futaki, A., Ono, H., and Wang, G. (2009), ‘Transverse Kählergeometry of Sasaki manifolds and toric Sasaki-Einsteinmanifolds’, J. Dierential Geom. 83 (3): 585–635.

Goto, R. (2012), ‘Calabi-Yau structures and Einstein-Sasakianstructures on crepant resolutions of isolated singularities’, J.Math. Soc. Japan, 64 (3): 1005–52.

Gray, A. (1976), ‘The structure of nearly Kähler manifolds’,Math.Ann. 223: 233–48.



References

References III

Hein, H.-J., Sun, S., Viaclovsky, J., and Zhang, R. (2018), Nilpotentstructures and collapsing Ricci-at metrics on K3 surfaces, 24 July,arXiv: 1807.09367 [math.DG].

Karshon, Y. and Lerman, E. (2015), ‘Non-compact symplectic toricmanifolds’, SIGMA Symmetry Integrability Geom. Methods Appl.11, 055, Preprint version led.

Li, Y. (2019), SYZ geometry for Calabi-Yau 3-folds: Taub-NUT andOoguri-Vafa type metrics, 23 Feb., arXiv: 1902.08770 [math.DG].

Madsen, T. B. and Swann, A. F. (2019a), ‘Toric geometry ofG2-manifolds’, Geom. Topol. 23 (7): 3459–500.

Madsen, T. B. and Swann, A. F. (2019b), ‘Toric geometry ofSpin(7)-manifolds’, to be published in Int. Math. Res. Not.IMRN, arXiv: 1810.12962 [math.DG].





References

References IV

Mooney, C. (2020), Solutions to the Monge-Ampère equation withpolyhedral and Y-shaped singularities, 14 Apr., arXiv:2004.06696 [math.AP].

Moroianu, A. and Nagy, P.-A. (2019), ‘Toric nearly Kählermanifolds’, Ann. Global Anal. Geom. 55 (4): 703–17.

Russo, G. and Swann, A. F. (2019), ‘Nearly Kähler six-manifoldswith two-torus symmetry’, J. Geom. Phys. 138: 144–53.

Tian, G. and Yau, S.-T. (1991), ‘Complete Kähler manifolds withzero Ricci curvature. II’, Invent. Math. 106 (1): 27–60.



Toric ideas for special geometriesswann/talks/Uppsala-2020.pdfmetry’, J. Geom. Phys. 138: 144–53...

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