Torsion Geometry, Superconformal Symmetry and T-duality

T orsion G eometry , S uperconformal S ymmetry and T- duality

Andrew Swann

University of Southern Denmark swann@imada.sdu.dk

May 2008 / Bilbao

Torsion Superconformal T-dual Summary

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure 3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure 3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Metric KT HKT

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Metric KT HKT

T orsion G eometry

Metric geometry with torsion metricg, connection∇, torsion T^∇(X,Y) =∇_XY− ∇_YX−[X,Y]

∇g=0

c(X,Y,Z) =g(T^∇(X,Y),Z)a three-form

∇=∇^LC+ ¹₂c anyc∈_Ω³(M)_will do

∇,∇^LCsame

geodesics/dynamics strongifdc=0 Studycompactsimply-connected

torsion geometries with

compatible complex structures and

small symmetry group

Torsion Superconformal T-dual Summary Metric KT HKT

T orsion G eometry

Metric geometry with torsion metricg, connection∇, torsion T^∇(X,Y) =∇_XY− ∇_YX−[X,Y]

∇g=0

c(X,Y,Z) =g(T^∇(X,Y),Z)a three-form

∇=∇^LC+ ¹₂c anyc∈_Ω³(M)_will do

∇,∇^LCsame

geodesics/dynamics strongifdc=0 Studycompactsimply-connected

torsion geometries with

compatible complex structures and

small symmetry group

Torsion Superconformal T-dual Summary Metric KT HKT

T orsion G eometry

Metric geometry with torsion metricg, connection∇, torsion T^∇(X,Y) =∇_XY− ∇_YX−[X,Y]

∇g=0

c(X,Y,Z) =g(T^∇(X,Y),Z)a three-form

∇=∇^LC+ ¹₂c anyc∈_Ω³(M)_will do

∇,∇^LCsame

geodesics/dynamics strongifdc=0

Studycompactsimply-connected torsion geometries with

compatible complex structures and

small symmetry group

Torsion Superconformal T-dual Summary Metric KT HKT

T orsion G eometry

Metric geometry with torsion metricg, connection∇, torsion T^∇(X,Y) =∇_XY− ∇_YX−[X,Y]

∇g=0

c(X,Y,Z) =g(T^∇(X,Y),Z)a three-form

∇=∇^LC+ ¹₂c anyc∈_Ω³(M)_will do

∇,∇^LCsame

geodesics/dynamics strongifdc=0 Studycompactsimply-connected

torsion geometries with

compatible complex structures and

small symmetry group

Torsion Superconformal T-dual Summary Metric KT HKT

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Metric KT HKT

KT G eometry

g,∇=∇^LC+¹₂c, c∈ _Λ³T^∗M KTgeometry

additionally

I integrable complex structure

g(IX,IY) =g(X,Y)

∇I =0

Two formF_I(X,Y) =g(IX,Y)

∇isunique

c=−IdF_I theBismut connection

KT geometry=Hermitian geometry+ Bismut connection

c=0 is K¨ahler geometry strong KT is∂∂F¯ _I =0 Example

M⁶=S³×S³ =SU(2)×SU(2) Gauduchon (1991)

every compact HermitianM⁴is conformal to strong KT

Torsion Superconformal T-dual Summary Metric KT HKT

KT G eometry

g,∇=∇^LC+¹₂c, c∈ _Λ³T^∗M KTgeometry

additionally

I integrable complex structure

g(IX,IY) =g(X,Y)

∇I =0

Two formF_I(X,Y) =g(IX,Y)

∇isunique

c= −IdF_I theBismut connection

KT geometry=Hermitian geometry+ Bismut connection

c=0 is K¨ahler geometry strong KT is∂∂F¯ _I =0 Example

M⁶=S³×S³ =SU(2)×SU(2) Gauduchon (1991)

every compact HermitianM⁴is conformal to strong KT

Torsion Superconformal T-dual Summary Metric KT HKT

KT G eometry

g,∇=∇^LC+¹₂c, c∈ _Λ³T^∗M KTgeometry

additionally

I integrable complex structure

g(IX,IY) =g(X,Y)

∇I =0

Two formF_I(X,Y) =g(IX,Y)

∇isunique

c= −IdF_I theBismut connection

KT geometry=Hermitian geometry+ Bismut connection

c=0 is K¨ahler geometry strong KT is∂∂F¯ _I =0

Example

M⁶=S³×S³ =SU(2)×SU(2) Gauduchon (1991)

every compact HermitianM⁴is conformal to strong KT

Torsion Superconformal T-dual Summary Metric KT HKT

KT G eometry

g,∇=∇^LC+¹₂c, c∈ _Λ³T^∗M KTgeometry

additionally

I integrable complex structure

g(IX,IY) =g(X,Y)

∇I =0

Two formF_I(X,Y) =g(IX,Y)

∇isunique

c= −IdF_I theBismut connection

KT geometry=Hermitian geometry+ Bismut connection

c=0 is K¨ahler geometry strong KT is∂∂F¯ _I =0 Example

M⁶=S³×S³ =SU(2)×SU(2) Gauduchon (1991)

every compact HermitianM⁴is conformal to strong KT

Torsion Superconformal T-dual Summary Metric KT HKT

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Metric KT HKT

HKT G eometry

HKTstructure (g,∇,I,J,K)with

(g,∇,A)KT, A=I,J,K IJ=K=−JI

c=−AdF_Ais independent ofA

Mart´in Cabrera and Swann(2007)

IdF_I =JdF_J=KdF_K impliesI,J,Kintegrable, so HKT.

Examples

Dim4 T⁴, K3,S³×S¹(Boyer, 1988)

Dim8 Hilbert schemes,SU(3), nilmanifolds, vector bundles over discrete groups (Verbitsky, 2003; Barberis and Fino, 2008) Compact, simply-connected examples which are neither hyperK¨ahler nor

homogeneous?

Torsion Superconformal T-dual Summary Metric KT HKT

HKT G eometry

HKTstructure (g,∇,I,J,K)with

(g,∇,A)KT, A=I,J,K IJ=K=−JI

c=−AdF_Ais independent ofA Mart´in Cabrera and Swann(2007)

IdF_I =JdF_J =KdF_K impliesI,J,Kintegrable, so HKT.

Examples

Dim4 T⁴, K3,S³×S¹(Boyer, 1988)

Dim8 Hilbert schemes,SU(3), nilmanifolds, vector bundles over discrete groups (Verbitsky, 2003; Barberis and Fino, 2008) Compact, simply-connected examples which are neither hyperK¨ahler nor

homogeneous?

Torsion Superconformal T-dual Summary Metric KT HKT

HKT G eometry

HKTstructure (g,∇,I,J,K)with

(g,∇,A)KT, A=I,J,K IJ=K=−JI

c=−AdF_Ais independent ofA Mart´in Cabrera and Swann(2007)

IdF_I =JdF_J =KdF_K impliesI,J,Kintegrable, so HKT.

Examples

Dim4 T⁴, K3,S³×S¹(Boyer, 1988)

Dim8 Hilbert schemes,SU(3), nilmanifolds, vector bundles over discrete groups (Verbitsky, 2003;

Barberis and Fino, 2008)

Compact, simply-connected examples which are neither hyperK¨ahler nor

homogeneous?

Torsion Superconformal T-dual Summary Metric KT HKT

HKT G eometry

HKTstructure (g,∇,I,J,K)with

(g,∇,A)KT, A=I,J,K IJ=K=−JI

c=−AdF_Ais independent ofA Mart´in Cabrera and Swann(2007)

IdF_I =JdF_J =KdF_K impliesI,J,Kintegrable, so HKT.

Examples

Dim4 T⁴, K3,S³×S¹(Boyer, 1988)

Dim8 Hilbert schemes,SU(3), nilmanifolds, vector bundles over discrete groups (Verbitsky, 2003;

Barberis and Fino, 2008) Compact, simply-connected examples which are neither hyperK¨ahler nor

homogeneous?

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal Q uantum M echanics

Nparticles in 1 dimension H= ¹

2P^∗_ag^abP_b+V(x)

Standard quantisation Pa∼ −i ∂

∂x^a, a=1, . . . ,N

Michelson andStrominger (2000); Papadopoulos (2000)

operatorDwith[D,H] =2iH ⇐⇒ vector fieldXwith L_Xg=2g&L_XV=−2V

Kso span{iH,iD,iK} ∼=sl(2,R) ⇐⇒ X^[=g(X,·)is closed

thenK= ¹₂g(X,X).

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

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal Q uantum M echanics

Nparticles in 1 dimension H= ¹

2P^∗_ag^abP_b+V(x)

Standard quantisation Pa∼ −i ∂

∂x^a, a=1, . . . ,N Michelson and Strominger (2000); Papadopoulos (2000)

operatorDwith [D,H] =2iH ⇐⇒ vector fieldXwith L_Xg=2g&L_XV=−2V

Kso span{iH,iD,iK} ∼=sl(2,R) ⇐⇒ X^[=g(X,·)is closed

thenK= ¹₂g(X,X).

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

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α})

g=g₀+g₁ g₀=

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁=_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁

g₀=

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁=_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁ g₀ =

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁ =_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S

[S^a,Q^a] =D, [S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁ g₀ =

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁ =_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋

Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁ g₀ =

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁ =_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁ g₀ =

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁ =_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1.

Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

T he S uperalgebras D ( 2, 1; α )

The classification of simple Lie superalgebras containsone continuous family

D(_{2, 1;α}) g=g₀+g₁ g₀ =

sl(2,C) +sl(2,C)₊+sl(2,C)₋ g₁ =_C²⊗_C²₊⊗_C²₋=_C⁴_Q+_C⁴_S [S^a,Q^a] =D,

[S¹,Q²] =−₁^4α₊

αR³₊− ₁₊⁴

αR³₋ Simple forα6= −1, 0,∞.

OverC, isomorphisms between the cases α^±1,−(1+α)^±1,

−(α/(1+α))^±1. Real form g₀ =sl(2,R) + su(2)++su(2)₋. OverR,

isomorphisms for α^±1

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure 3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry

N =4B quantum mechanics

withD(2, 1;α) superconformal symmetry

↔

HKTmanifoldM

withXaspecial homothety of type(a,b) L_Xg=ag,

L_IXJ=bK,

L_XI=0,L_IXI=0,. . .

α= ^a_b−1 Action of R×SU(2) rotatingI,J,K

For a6=0

Mis non-compact µ= ²

a(a−b)kXk²_{is an}HKT potential FI= ¹₂(_ddI+_dJdK)µ= ¹₂(₁−J)_dIdµ.

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry

N =4B quantum mechanics

withD(2, 1;α) superconformal symmetry

↔

HKTmanifoldM

withXaspecial homothety of type(a,b) L_Xg=ag,

L_IXJ=bK,

L_XI=0,L_IXI=0,. . . α= ^a_b−1

Action of R×SU(2) rotatingI,J,K

For a6=0

Mis non-compact µ= ²

a(a−b)kXk²_{is an}HKT potential FI= ¹₂(_ddI+_dJdK)µ= ¹₂(₁−J)_dIdµ.

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry

N =4B quantum mechanics

withD(2, 1;α) superconformal symmetry

↔

HKTmanifoldM

withXaspecial homothety of type(a,b) L_Xg=ag,

L_IXJ=bK,

L_XI=0,L_IXI=0,. . . α= ^a_b−1

Action of R×SU(2) rotatingI,J,K

For a6=0

Mis non-compact µ= ²

a(a−b)kXk²_{is an}HKT potential FI = ¹₂(ddI+dJdK)µ= ¹₂(1−J)dIdµ.

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)²

Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary QM D(2, 1;α) Geometry

S uperconformal G eometry II

Example

M=_Hⁿ⁺¹\ {0} →_HP(n) a=2,b=−2,α= −2.

Poon andSwann(2003) a6=0 corresponds to Q=M/(_R×SU(2)) =

µ⁻¹(₁)_/SU(₂)a QKT orbifold (of special type).

E.g. Q=kCP(2).

ForS3-Sasaki,M=S×_R warped product, is

hyperK¨ahler with special homothetyα=−2

Get toa=0, special isometry, by potential change

g₁= ¹ µg− ¹

2µ²(d^Hµ)² Discrete quotient

M= (µ⁻¹(1)×_R)_/Z(ϕ, 2) withg₁ is HKT with special isometryX

In this case dX^[ =0 b₁(M)>¹

Torsion Superconformal T-dual Summary Twist HKT Circle

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Twist HKT Circle

T- duality as a T wist

Xgenerating a circle action onM

(P,θ,Y)−→^π Man invariant principal S¹-bundle

X⁰ =X^˜ +aYa lift ofX generating a free circle action,da= −Xy^Fθ

Definition

Atwist Wof Mwith respect to Xis

W :=P/hX⁰i Transverse locally free lifts always exist forXy^Fθ exact.

M W

P

π Y

πW X⁰

Dually

Mis a twist ofW with respect toX_W = (π_W)_∗Y,θ_W = ¹_aθ Definition

Tensorsαon α_W onMandW areH-related,α_W ∼_Hαif their pull-backs agree onH=kerθ dα_W ∼_Hdα−F_θ∧¹_aXy^α if invariant

Torsion Superconformal T-dual Summary Twist HKT Circle

T- duality as a T wist

Xgenerating a circle action onM

(P,θ,Y)−→^π Man invariant principal S¹-bundle

X⁰ =X^˜ +aYa lift ofX generating a free circle action,da= −Xy^Fθ

Definition

Atwist Wof Mwith respect to Xis

W :=P/hX⁰i Transverse locally free lifts always exist forXy^Fθ exact.

M W

P

π Y

πW X⁰

Dually

Mis a twist ofW with respect toX_W = (π_W)_∗Y,θ_W = ¹_aθ Definition

Tensorsαon α_W onMandW areH-related,α_W ∼_Hαif their pull-backs agree onH=kerθ dα_W ∼_Hdα−F_θ∧¹_aXy^α if invariant

Torsion Superconformal T-dual Summary Twist HKT Circle

T- duality as a T wist

Xgenerating a circle action onM

(P,θ,Y)−→^π Man invariant principal S¹-bundle

X⁰ =X^˜ +aYa lift ofX generating a free circle action,da= −Xy^Fθ

Definition

Atwist WofM with respect to Xis

W :=P/hX⁰i Transverse locally free lifts always exist forXy^Fθ exact.

M W

P

π Y

πW X⁰

Dually

Mis a twist ofW with respect toX_W = (π_W)_∗Y,θ_W = ¹_aθ Definition

Tensorsαon α_W onMandW areH-related,α_W ∼_Hαif their pull-backs agree onH=kerθ dα_W ∼_Hdα−F_θ∧¹_aXy^α if invariant

Torsion Superconformal T-dual Summary Twist HKT Circle

T- duality as a T wist

Xgenerating a circle action onM

(P,θ,Y)−→^π Man invariant principal S¹-bundle

X⁰ =X^˜ +aYa lift ofX generating a free circle action,da= −Xy^Fθ

Definition

Atwist WofM with respect to Xis

W :=P/hX⁰i Transverse locally free lifts always exist forXy^Fθ exact.

M W

P

π Y

πW X⁰

Dually

M is a twist ofW with respect toX_W = (π_W)_∗Y,θ_W = ¹_aθ

Definition

Tensorsαon α_W onMandW areH-related,α_W ∼_Hαif their pull-backs agree onH=kerθ dα_W ∼_Hdα−F_θ∧¹_aXy^α if invariant

Torsion Superconformal T-dual Summary Twist HKT Circle

T- duality as a T wist

Xgenerating a circle action onM

(P,θ,Y)−→^π Man invariant principal S¹-bundle

X⁰ =X^˜ +aYa lift ofX generating a free circle action,da= −Xy^Fθ

Definition

Atwist WofM with respect to Xis

W :=P/hX⁰i Transverse locally free lifts always exist forXy^Fθ exact.

M W

P

π Y

πW X⁰

Dually

M is a twist ofW with respect toX_W = (π_W)_∗Y,θ_W = ¹_aθ Definition

Tensors αon α_W onMandW are H-related,α_W ∼_Hαif their pull-backs agree onH=kerθ dα_W ∼_Hdα−F_θ∧¹_aXy^α if invariant

Torsion Superconformal T-dual Summary Twist HKT Circle

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_{I ,} i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K

Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_{I ,} i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_I , i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_I , i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry

Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_I , i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_I , i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton

Many simply-connected examples whenb₂(S)>¹ E.g., Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

T wisting HKT

Twist by

g_W ∼_Hg, F^W_I ∼_HF_I, etc.

Then

IdF^W_I ∼_HIdF_I+ ¹_aX^[∧IF_θ For HKT need

c= −IdF_I =−JdF_J =−KdF_K Proposition

HKT twists to HKT via a circle if and only if Fθ ∈S²E= ^T_IΛ^1,1_I , i.e., an instanton

Xa special isometry,Xy^Fθ =0 twists toX_W a special isometry Theorem

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

untwist locally to dX^[=0 on S×S¹

change potential on S×_Rto a6=0, (α=−2)

F_θ =dX^[ isan instanton Many simply-connected examples whenb₂(S)>¹ E.g.,Q=kCP(2)

Torsion Superconformal T-dual Summary Twist HKT Circle

O utline

1 TorsionGeometry

Metric geometry with torsion KT Geometry

HKT Geometry

2 Superconformal Symmetry

Superconformal Quantum Mechanics The SuperalgebrasD(2, 1;α)

Geometric Structure

3 T-duality

T-duality as a Twist Construction HKT Examples

General HKT with Circle Symmetry

Torsion Superconformal T-dual Summary Twist HKT Circle

G eneral HKT with C ircle S ymmetry

M=N₁×N₂

N₂ with an HKT circle symmetryX

[F_θ]∈H²(N₁,Z),F_θ ∈S²E

Twists toN₂ →W→N₁HKT with circle symmetry

Generate simply-connected examples

Example N₁ a K3 surface F_θ self-dual, primitive Generalises to torus actions

HKT nilmanifoldM =_G/Γ g^∗ basise₁, . . . ,e_nwith

de_i+1 ∈_Λ²span{e₁, . . . ,e_i} Barberis, DottiMiatello, andVerbitsky(2007) I,J,Kare Abelian

de_i+1 ∈S²E∩_Λ²span{e₁, . . . ,e_i} Proposition

Every HKT nilmanifold may be obtained by successive twists of a torus T⁴ⁿ.

Torsion Superconformal T-dual Summary Twist HKT Circle

G eneral HKT with C ircle S ymmetry

M=N₁×N₂

N₂ with an HKT circle symmetryX

[F_θ]∈H²(N₁,Z),F_θ ∈S