Geometric Duality - Aarhus Universitet · Outline 1 Geometry Metric geometry with torsion KT...

Transcript of Geometric Duality - Aarhus Universitet · Outline 1 Geometry Metric geometry with torsion KT...

Geometric Duality

Andrew Swann

IMADA / CP3-OriginsUniversity of Southern Denmark

[email protected]

November 2009 / Odense
Geometry Twists Superconformal Other

Outline

1 GeometryMetric geometry with torsionKT GeometryHKT Geometry

2 TwistsT-duality as a Twist Construction

3 Superconformal SymmetrySuperconformal Quantum MechanicsThe Superalgebras D(2, 1; α)Geometric StructureHKT ExamplesSummary

4 Other Examples via the Twist

Andrew Swann Geometric Duality

Outline






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Geometry Twists Superconformal Other Metric KT HKT

Geometry from supersymmetry

Classical geometries

Riemannian/Lorentzian metric g = (gij), has a uniquecovariant derivative ∇lc, Levi-Civita connection, that ismetric ∇lcg = 0 and torsion-free ∇lcX Y−∇lcY X = [X, Y]:

2g(∇lcX Y, Z) = Xg(Y, Z) + Yg(X, Z)− Zg(X, Y)+ g([X, Y], Z) + g([Z, X], Y) + g(X, [Z, Y]);

∇lc∂∂xi

∂∂xj = Γ

kij

∂∂xk , Γ

kij =

12 g

k`(g`i,j + gj`,i − gij,`) = Γkji.

Supersymmetry: usually parallel complex structures J = (Jij):

g(JX, JY) = g(X, Y), J2 = −1, ∇lcJ = 0;Ji`Jjkg`k = gij, JijJjk = −δki , Jik,j = Γ`ijJ`k.


One complex structureKähler geometry: Riemann surfaces, CP(n),projective varieties X =

⋂i(fi = 0) ⊂ CP(n),

Hermitian symmetric spaces,. . .Calabi-Yau manifolds, Kähler with c1 = 0:have Ric ≡ 0 so Einstein;X = (f = 0) ⊂ CP(n), deg f = n + 1; K3surface (x4 + y4 + z4 + w4 = 0) ⊂ CP(3).

Multiple complex structuresHyperKähler geometry I, J, K, IJ = K = −JI:are Calabi-Yau; K3 surfaces, T4k = R4k/Z4k;Hilbert schemes; instanton moduli;. . .

Holonomy classification (Berger,. . . ) essentially only getproducts of the above examples


Outline






Torsion Geometry

Metric geometry with torsionmetric g, connection ∇, torsionT∇(X, Y) = ∇XY−∇YX− [X, Y]∇g = 0

c(X, Y, Z) = g(T∇(X, Y), Z) athree-form

∇ ∂∂xi

∂∂xj = γ

kij

∂∂xk ,

T`ij = γk[ij],

cijk = c[ijk] = g`kT`ij,

∇ = ∇lc + 12 cγkij = Γ

kij +

12 cijk

Any c ∈ Ω3(M) willdo.∇, ∇lc have thesame geodesics(dynamics).The geometry isstrong if dc = 0.


Torsion Geometry

Metric geometry with torsionmetric g, connection ∇, torsionT∇(X, Y) = ∇XY−∇YX− [X, Y]∇g = 0c(X, Y, Z) = g(T∇(X, Y), Z) athree-form

∇ ∂∂xi

∂∂xj = γ

kij

∂∂xk ,

T`ij = γk[ij],


∇ = ∇lc + 12 cγkij = Γ

kij +

12 cijk



Torsion Geometry

Metric geometry with torsionmetric g, connection ∇, torsionT∇(X, Y) = ∇XY−∇YX− [X, Y]∇g = 0c(X, Y, Z) = g(T∇(X, Y), Z) athree-form

∇ ∂∂xi

∂∂xj = γ

kij

∂∂xk ,

T`ij = γk[ij],


∇ = ∇lc + 12 cγkij = Γ

kij +

12 cijk



Such geometries with extra structure from supersymmetryarise from:

Wess-Zumino terms in the Lagrangian, superstrings withtorsion, B-fields (Strominger, 1986)One-dimensional quantum mechanics with type Bsupersymmetry, blackhole dynamics and moduli(Michelson and Strominger, 2000; Coles andPapadopoulos, 1990; Hull, 1999; Gibbons et al., 1997)Constructions in supergravity (Grover et al., 2009)

Mathematically, one wishes to:clarify the basic definitions and relationships to knowngeometries,construct and classify examples in given categories.

In particular, we will be looking for compact simply-connectedtorsion geometries with compatible complex structures.


Outline






KT Geometry

g, ∇ = ∇lc + 12 c, c ∈ Λ3T∗M

KT geometryadditionally

I integrable complexstructureg(IX, IY) = g(X, Y)∇I = 0

Two form ωI(X, Y) = g(IX, Y)

∇ is unique

c = −IdωI

the Bismut connection

KT geometry = Hermitiangeometry + Bismutconnectionc = 0 is Kähler geometrystrong KT is ∂∂̄ωI = 0

ExampleM6 = S3 × S3 = SU(2)× SU(2)

Gauduchon (1991)

every compact Hermitian M4 isconformal to strong KT


KT Geometry

g, ∇ = ∇lc + 12 c, c ∈ Λ3T∗M




∇ is unique

c = −IdωI




Gauduchon (1991)



KT Geometry

g, ∇ = ∇lc + 12 c, c ∈ Λ3T∗M




∇ is unique

c = −IdωI




Gauduchon (1991)



KT Geometry

g, ∇ = ∇lc + 12 c, c ∈ Λ3T∗M




∇ is unique

c = −IdωI




Gauduchon (1991)



Outline






HKT Geometry

HKT structure(g,∇, I, J, K) with

(g,∇, A) KT, A = I, J, KIJ = K = −JI

c = −AdωA independent of A

Martín Cabrera andSwann (2008)

IdωI = JdωJ = KdωK

implies I, J, K integrable, soHKT.

ExamplesDim 4 T4, K3, S3 × S1 (Boyer,

1988)Dim 8 Hilbert schemes, SU(3),

nilmanifolds, vectorbundles over discretegroups (Verbitsky, 2003;Barberis and Fino, 2008)

Compact, simply-connectedexamples which are neitherhyperKähler norhomogeneous?


HKT Geometry


(g,∇, A) KT, A = I, J, KIJ = K = −JI










HKT Geometry


(g,∇, A) KT, A = I, J, KIJ = K = −JI










HKT Geometry


(g,∇, A) KT, A = I, J, KIJ = K = −JI









Geometry Twists Superconformal Other T-dual

Outline






Four-dimensional inspiration

HyperKähler M4

ds2 = V−1(dτ + ω)2

+ Vγijdxidxj

dV = ∗3dω

T duality

onX = ∂∂τ

Strong HKT W4

ds2 = V(d2τ + γijdxidxj)

c = −dτ ∧ dω

Gibbons, Papadopoulos, and Stelle, 1997Callan, Harvey, and Strominger, 1991Bergshoeff, Hull, and Ortı́n, 1995

For circle actions have:

R↔ 1/R and here W = (M/S1)× S1



HyperKähler M4

ds2 = V−1(dτ + ω)2

+ Vγijdxidxj

dV = ∗3dω

T duality

onX = ∂∂τ

Strong HKT W4


c = −dτ ∧ dω






HyperKähler M4

ds2 = V−1(dτ + ω)2

+ Vγijdxidxj

dV = ∗3dω

T duality

onX = ∂∂τ

Strong HKT W4


c = −dτ ∧ dω





T-duality as a Twist

Xp generating a n-torusaction on M(P, θ, Yq)

π−→ M aninvariant principalTn-bundle

X′p = X̃p + apqYq a lift of Xpgenerating a free torusaction, dapq = −Xp y Fθq

DefinitionA twist W of M with respect toXp is

W := P/〈X′p〉

Transverse locally free liftsalways exist for Xp y Fθqexact.W is at worst an orbifold.

M W

Pπ

YqπW

X′p

DuallyM is a twist of W with respectto XWq = (πW)∗Yq,θWp = (a−1)pqθq




π−→ M aninvariant principalTn-bundleX′p = X̃p + apqYq a lift of Xpgenerating a free torusaction, dapq = −Xp y Fθq


W := P/〈X′p〉


M W

Pπ

YqπW

X′p







W := P/〈X′p〉


M W

Pπ

YqπW

X′p







W := P/〈X′p〉


M W

Pπ

YqπW

X′p







W := P/〈X′p〉


M W

Pπ

YqπW

X′p



DefinitionTensors α on αW on M and W are H-related, αW ∼H α if theirpull-backs agree on H = ker θ

Move invariant geometry from M to W by using thecorresponding H-related tensors

gW ∼H g, ωWI ∼H ωI, etc.

For invariant forms

dαW ∼H dα− Fθq ∧ (a−1)pqXq y α

For the KT torsion form c = −IdωI:

cW ∼H c− (a−1)pqIFθq ∧Xp.


DefinitionTensors α on αW on M and W are H-related, αW ∼H α if theirpull-backs agree on H = ker θ

Move invariant geometry from M to W by using thecorresponding H-related tensors


For invariant forms

dαW ∼H dα− Fθq ∧ (a−1)pqXq y α

For the KT torsion form c = −IdωI:

cW ∼H c− (a−1)pqIFθq ∧Xp.

Geometry Twists Superconformal Other QM D(2, 1; α) Geometry HKT Summary

Outline






Superconformal Quantum Mechanics

N particles in 1 dimension

H =12

P∗a gabPb + V(x)

Standard quantisation

Pa ∼ −i∂

∂xa, a = 1, . . . , N

Michelson and Strominger (2000); Papadopoulos(2000)

operator D with [D, H] = 2iH ⇐⇒ vector field X withLXg = 2g & LXV = −2VK so span{iH, iD, iK} ∼= sl(2, R) ⇐⇒ X[ = g(X, ·) isclosedthen K = 12 g(X, X).

Choose a superalgebra containing sl(2, R) in its even part.


Superconformal Quantum Mechanics

N particles in 1 dimension

H =12

P∗a gabPb + V(x)

Standard quantisation

Pa ∼ −i∂

∂xa, a = 1, . . . , N

Michelson and Strominger (2000); Papadopoulos(2000)

operator D with [D, H] = 2iH ⇐⇒ vector field X withLXg = 2g & LXV = −2VK so span{iH, iD, iK} ∼= sl(2, R) ⇐⇒ X[ = g(X, ·) isclosedthen K = 12 g(X, X).

Choose a superalgebra containing sl(2, R) in its even part.


Outline






The Superalgebras D(2, 1; α)

The classification of simple Liesuperalgebras contains onecontinuous family

D(2, 1; α)

g = g0 + g1g0 =sl(2, C) + sl(2, C)+ + sl(2, C)−g1 = C2 ⊗C2+ ⊗C2− = C4Q + C4S[Sa, Qa] = D,[S1, Q2] = − 4α1+α R3+ −

41+α R

3−

Simple for α 6= −1, 0, ∞.

Over C,isomorphismsbetween the casesα±1, −(1 + α)±1,−(α/(1 + α))±1.Real formg0 = sl(2, R) +su(2)+ + su(2)−.Over R,isomorphisms forα±1




D(2, 1; α)

g = g0 + g1

g0 =sl(2, C) + sl(2, C)+ + sl(2, C)−g1 = C2 ⊗C2+ ⊗C2− = C4Q + C4S[Sa, Qa] = D,[S1, Q2] = − 4α1+α R3+ −

41+α R

3−






D(2, 1; α)

g = g0 + g1g0 =sl(2, C) + sl(2, C)+ + sl(2, C)−g1 = C2 ⊗C2+ ⊗C2− = C4Q + C4S

[Sa, Qa] = D,[S1, Q2] = − 4α1+α R3+ −

41+α R

3−






D(2, 1; α)


41+α R

3−






D(2, 1; α)


41+α R

3−






D(2, 1; α)


41+α R

3−


Over C,isomorphismsbetween the casesα±1, −(1 + α)±1,−(α/(1 + α))±1.

Real formg0 = sl(2, R) +su(2)+ + su(2)−.Over R,isomorphisms forα±1




D(2, 1; α)


41+α R

3−




Outline






Superconformal Geometry

N = 4B quantummechanicswith D(2, 1; α)superconformalsymmetry

↔

HKT manifold Mwith X a special homothety of type (a, b)

LXg = ag,LIXJ = bK,LXI = 0, LIXI = 0,. . .

α = ab − 1Action ofR× SU(2)rotating I, J, K

For a 6= 0

M is non-compact

µ = 2a(a−b)‖X‖2 is an HKT potential

ωI = 12 (ddI + dJdK)µ =12 (1− J)dIdµ.




↔




For a 6= 0

M is non-compact






↔




For a 6= 0

M is non-compact




Superconformal Geometry II

ExampleM = Hn+1 \ {0} →HP(n)a = 2, b = −2, α = −2.

Poon and Swann (2003)

a 6= 0 corresponds toQ = M/(R× SU(2)) =µ−1(1)/ SU(2) a QKT orbifold(of special type).

E.g. Q = k CP(2).For S 3-Sasaki, M = S×Rwarped product, ishyperKähler with specialhomothety α = −2

Get to a = 0, special isometry,by potential change

g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)

with g1 is HKT with specialisometry X

In this casedX[ = 0b1(M) > 1








g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)








E.g. Q = k CP(2).

For S 3-Sasaki, M = S×Rwarped product, ishyperKähler with specialhomothety α = −2


g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)










g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)










g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)










g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)










g1 =1µ

g− 12µ2

(dHµ)2

Discrete quotient

M = (µ−1(1)×R)/Z(ϕ, 2)




Outline






Twisting HKT

Twist by


Then

IdωWI ∼H IdωI + 1a X[ ∧ Iωθ

For HKT need

c = −IdωI = −JdωJ = −KdωKPropositionHKT twists to HKT via a circle ifand only if Fθ ∈ S2E =

⋂I Λ

1,1I ,

i.e., an instanton

X a special isometry, X y Fθ = 0twists to XW a special isometry

TheoremM HKT with special isometry(α = −1). Can

untwist locally to dX[ = 0on S× S1

change potential on S×R toa 6= 0, (α = −2)

Fθ = dX[ is an instanton

Many simply-connectedexamples when b2(S) > 1E.g., Q = k CP(2)


Twisting HKT

Twist by


Then


For HKT need

c = −IdωI = −JdωJ = −KdωK

PropositionHKT twists to HKT via a circle ifand only if Fθ ∈ S2E =

⋂I Λ

1,1I ,

i.e., an instanton








Twisting HKT

Twist by


Then


For HKT need


⋂I Λ

1,1I ,

i.e., an instanton








Twisting HKT

Twist by


Then


For HKT need


⋂I Λ

1,1I ,

i.e., an instanton








Twisting HKT

Twist by


Then


For HKT need


⋂I Λ

1,1I ,

i.e., an instanton








Twisting HKT

Twist by


Then


For HKT need


⋂I Λ

1,1I ,

i.e., an instanton








Twisting HKT

Twist by


Then


For HKT need


⋂I Λ

1,1I ,

i.e., an instanton








Outline






Summary

D(2, 1; α) superconformal symmetry realised by HKT withR× SU(2) action

α 6= −1 comes from R× SO(3) bundles over certain QKTorbifoldsα = −1 comes from previous examples via change ofpotential and twistconstruct non-homogeneous compact simply-connectedexamples with α = −1


Summary

D(2, 1; α) superconformal symmetry realised by HKT withR× SU(2) actionα 6= −1 comes from R× SO(3) bundles over certain QKTorbifolds

α = −1 comes from previous examples via change ofpotential and twistconstruct non-homogeneous compact simply-connectedexamples with α = −1


Summary

D(2, 1; α) superconformal symmetry realised by HKT withR× SU(2) actionα 6= −1 comes from R× SO(3) bundles over certain QKTorbifoldsα = −1 comes from previous examples via change ofpotential and twist

construct non-homogeneous compact simply-connectedexamples with α = −1


Summary

D(2, 1; α) superconformal symmetry realised by HKT withR× SU(2) actionα 6= −1 comes from R× SO(3) bundles over certain QKTorbifoldsα = −1 comes from previous examples via change ofpotential and twistconstruct non-homogeneous compact simply-connectedexamples with α = −1


General HKT with Torus Symmetry

M = N1 ×N2N2 with an HKT torus symmetry Xp[Fθq] ∈ H2(N1, Z), Fθq ∈ S2E instanton

Twists to N2 → W → N1 HKT with torus symmetry

ExampleN1 a K3 surfaceN2 = G compact Lie dim = 4k or N2 = S× S1, S 3-SasakiFθ self-dual, primitive

Generate:large number of simply-connected examples, includingnew examples with reduced holonomy;all examples on compact nilmanifolds, Ni tori.


Hypercomplex vs. HKT

Theorem (Swann (2008))

There is a simply-connected T4-bundle M over a K3 surface N thatadmits integrable I, J and K, but no compatible HKT metric.

This is constructed as a twist of T4 ×N using Fθq not ofinstanton type, but chosen so that integrability of the complexstructures is preserved.

References

References I

M. L. Barberis and A. Fino. New strong HKT manifoldsarising from quaternionic representations, May 2008. eprintarXiv:0805.2335[math.DG].

E. Bergshoeff, C. Hull, and T. Ortı́n. Duality in the type-IIsuperstring effective action. Nuclear Phys. B, 451(3):547–575,1995. ISSN 0550-3213.

C. P. Boyer. A note on hyperHermitian four-manifolds. Proc.Amer. Math. Soc., 102:157–164, 1988.

C. G. Callan, Jr., J. A. Harvey, and A. Strominger. Worldsheetapproach to heterotic instantons and solitons. Nuclear Phys.B, 359(2-3):611–634, 1991. ISSN 0550-3213.


arXiv:0805.2335 [math.DG]
References

References II

R. A. Coles and G. Papadopoulos. The geometry of theone-dimensional supersymmetric nonlinear sigma models.Classical Quantum Gravity, 7(3):427–438, 1990. ISSN0264-9381.

P. Gauduchon. Structures de Weyl et théorèmes d’annulationsur une varété conforme autoduale. Ann. Sc. Norm. Sup. Pisa,18:563–629, 1991.

G. W. Gibbons, G. Papadopoulos, and K. S. Stelle. HKT andOKT geometries on soliton black hole moduli spaces.Nuclear Phys. B, 508(3):623–658, 1997. ISSN 0550-3213.

Jai Grover, Jan B. Gutowski, Carlos A. R. Herdeiro, and WaficSabra. HKT geometry and de Sitter supergravity. NuclearPhys. B, 809(3):406–425, 2009. ISSN 0550-3213. doi:10.1016/j.nuclphysb.2008.08.024. URLhttp://dx.doi.org/10.1016/j.nuclphysb.2008.08.024.


http://dx.doi.org/10.1016/j.nuclphysb.2008.08.024
References

References III

C. M. Hull. The geometry of supersymmetric quantummechanics, October 1999. eprint arXiv:hep-th/9910028.

F. Martı́n Cabrera and A. F. Swann. The intrinsic torsion ofalmost quaternion-hermitian manifolds. Ann. Inst. Fourier,58(5):1455–1497, 2008. ISSN 0373-0956.

J. Michelson and A. Strominger. The geometry of (super)conformal quantum mechanics. Comm. Math. Phys., 213(1):1–17, 2000. ISSN 0010-3616.

G. Papadopoulos. Conformal and superconformal mechanics.Classical Quantum Gravity, 17(18):3715–3741, 2000. ISSN0264-9381.

Y. S. Poon and A. F. Swann. Superconformal symmetry andhyperKähler manifolds with torsion. Commun. Math. Phys.,241(1):177–189, 2003.


arXiv:hep-th/9910028
References

References IV

A. Strominger. Superstrings with torsion. Nuclear Phys. B, 274(2):253–284, 1986. ISSN 0550-3213.

A. F. Swann. Twisting Hermitian and hypercomplexgeometries. in preparation, 2008.

M. Verbitsky. Hyperkähler manifolds with torsion obtainedfrom hyperholomorphic bundles, March 2003. eprintarXiv:math.DG/0303129.


arXiv:math.DG/0303129

GeometryMetric geometry with torsionKT GeometryHKT Geometry

TwistsT-duality as a Twist Construction

Superconformal SymmetrySuperconformal Quantum MechanicsThe Superalgebras D(2,1;alpha)Geometric StructureHKT ExamplesSummary

Other Examples via the TwistAppendixReferences

Geometric Duality - Aarhus Universitet · Outline 1 Geometry Metric geometry with torsion KT...

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Transcript of Geometric Duality - Aarhus Universitet · Outline 1 Geometry Metric geometry with torsion KT...