toric ideas for special geometriesswann/talks/uppsala-2020.pdfmetry’, j. geom. phys. 138: 144–53...
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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
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 Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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.
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
• 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)
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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 )
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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%
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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 )
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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)
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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
Andrew Swann Toric ideas for special geometries
Symplectic Multi Spin(7) Nearly Kähler
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.
Andrew Swann Toric ideas for special geometries
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.
Andrew Swann Toric ideas for special geometries
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.
Andrew Swann Toric ideas for special geometries
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].
Andrew Swann Toric ideas for special geometries
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.
Andrew Swann Toric ideas for special geometries