SPINORS, MINIMAL SURFACES, CHIRALITY, HELICITY, TORSION, SPIN
Killing Spinors in Generalized Geometry -...
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Killing Spinors in Generalized Geometry
Mario Garcia-Fernandez
Instituto de Ciencias Matematicas, Madrid
Superstring Solutions, Supersymmetry and Geometry6 May 2016
Based on joint work with R. Rubio and C. Tipler (arXiv:1503.07562),and C. Shahbazi (in progress).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 1 / 31
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Outline
This talk is about the internal geometry of 4-dimensional N = 1supersymmetric compactifications of heterotic supergravity.
Classical ingredients of this geometry are complex manifolds (trivialcanonical bundle), Kahler-Ricci-flat metrics (∼ SU(3)-holonomy),(stable) holomorphic bundles (prescribed Chern classes), ... allshaken, not stirred.
In the presence of fluxes, the SU(3)-holonomy condition for themetric is replaced by: 1. a balanced condition for a hermitian metricand 2. the Bianchi identity, coupling the metric with the gauge field
ddcω = α′(tr R ∧ R − tr F ∧ F ).
Classical methods of algebraic and Kahler geometry, that lead toimportant advances in the understading of Calabi-Yau compactifications,
no longer apply.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 2 / 31
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Outline
This talk is about the internal geometry of 4-dimensional N = 1supersymmetric compactifications of heterotic supergravity.
Classical ingredients of this geometry are complex manifolds (trivialcanonical bundle), Kahler-Ricci-flat metrics (∼ SU(3)-holonomy),(stable) holomorphic bundles (prescribed Chern classes), ... allshaken, not stirred.
In the presence of fluxes, the SU(3)-holonomy condition for themetric is replaced by: 1. a balanced condition for a hermitian metricand 2. the Bianchi identity, coupling the metric with the gauge field
ddcω = α′(tr R ∧ R − tr F ∧ F ).
Classical methods of algebraic and Kahler geometry, that lead toimportant advances in the understading of Calabi-Yau compactifications,
no longer apply.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 2 / 31
![Page 4: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/4.jpg)
Outline
This talk is about the internal geometry of 4-dimensional N = 1supersymmetric compactifications of heterotic supergravity.
Classical ingredients of this geometry are complex manifolds (trivialcanonical bundle), Kahler-Ricci-flat metrics (∼ SU(3)-holonomy),(stable) holomorphic bundles (prescribed Chern classes), ... allshaken, not stirred.
In the presence of fluxes, the SU(3)-holonomy condition for themetric is replaced by: 1. a balanced condition for a hermitian metricand 2. the Bianchi identity, coupling the metric with the gauge field
ddcω = α′(tr R ∧ R − tr F ∧ F ).
Classical methods of algebraic and Kahler geometry, that lead toimportant advances in the understading of Calabi-Yau compactifications,
no longer apply.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 2 / 31
![Page 5: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/5.jpg)
Outline
This talk is about the internal geometry of 4-dimensional N = 1supersymmetric compactifications of heterotic supergravity.
Classical ingredients of this geometry are complex manifolds (trivialcanonical bundle), Kahler-Ricci-flat metrics (∼ SU(3)-holonomy),(stable) holomorphic bundles (prescribed Chern classes), ... allshaken, not stirred.
In the presence of fluxes, the SU(3)-holonomy condition for themetric is replaced by: 1. a balanced condition for a hermitian metricand 2. the Bianchi identity, coupling the metric with the gauge field
ddcω = α′(tr R ∧ R − tr F ∧ F ).
Classical methods of algebraic and Kahler geometry, that lead toimportant advances in the understading of Calabi-Yau compactifications,
no longer apply.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 2 / 31
![Page 6: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/6.jpg)
Outline
This talk is about the internal geometry of 4-dimensional N = 1supersymmetric compactifications of heterotic supergravity.
Classical ingredients of this geometry are complex manifolds (trivialcanonical bundle), Kahler-Ricci-flat metrics (∼ SU(3)-holonomy),(stable) holomorphic bundles (prescribed Chern classes), ... allshaken, not stirred.
In the presence of fluxes, the SU(3)-holonomy condition for themetric is replaced by: 1. a balanced condition for a hermitian metricand 2. the Bianchi identity, coupling the metric with the gauge field
ddcω = α′(tr R ∧ R − tr F ∧ F ).
Classical methods of algebraic and Kahler geometry, that lead toimportant advances in the understading of Calabi-Yau compactifications,
no longer apply.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 2 / 31
![Page 7: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/7.jpg)
Outline
Recent developments in the study of moduli of heterotic fluxcompactifications shows that generalized geometry is an essentialingredient of this new geometry (Melnikov-Sharpe ’11, de la Ossa-Svanes
’14, Anderson-Gray-Sharpe ’14, GF-Rubio-Tipler ’15).
Hope is that generalized geometry brings back some of the powerfulclassical methods, with a different incarnation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 3 / 31
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Outline
Recent developments in the study of moduli of heterotic fluxcompactifications shows that generalized geometry is an essentialingredient of this new geometry (Melnikov-Sharpe ’11, de la Ossa-Svanes
’14, Anderson-Gray-Sharpe ’14, GF-Rubio-Tipler ’15).
Hope is that generalized geometry brings back some of the powerfulclassical methods, with a different incarnation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 3 / 31
![Page 9: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/9.jpg)
Part I: Heterotic flux compactifications
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 4 / 31
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Heterotic supergravity
The (bosonic) field content of heterotic supergravity is a Lorentz metric g on
ten-dimensional space-time M10, dilaton φ ∈ C∞(M10), 3-form flux H and gauge
field A with field strength F
Ricgij +2∇gi ∇
gj φ−
1
4HiklH
klj − α′ tr FikF k
j + α′ tr RikRkj + O(α′2) = 0, EM
d∗(e−2φH) + O(α′2) = 0,
d∗A(e−2φF ) +e−2φ
2∗ (F ∧ ∗H) + O(α′2) = 0,
∇−ε+ O(α′2) = 0, SUSY
(2dφ− H) · ε+ O(α′2) = 0,
F · ε+ O(α′2) = 0
dH − α′ (tr R ∧ R − tr F ∧ F ) + O(α′2) = 0, Bianchi
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 5 / 31
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Heterotic supergravity
The (bosonic) field content of heterotic supergravity is a Lorentz metric g on
ten-dimensional space-time M10, dilaton φ ∈ C∞(M10), 3-form flux H and gauge
field A with field strength F
Ricgij +2∇gi ∇
gj φ−
1
4HiklH
klj − α′ tr FikF k
j + α′ tr RikRkj + O(α′2) = 0, EM
d∗(e−2φH) + O(α′2) = 0,
d∗A(e−2φF ) +e−2φ
2∗ (F ∧ ∗H) + O(α′2) = 0,
∇−ε+ O(α′2) = 0, SUSY
(2dφ− H) · ε+ O(α′2) = 0,
F · ε+ O(α′2) = 0
dH − α′ (tr R ∧ R − tr F ∧ F ) + O(α′2) = 0, Bianchi
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 5 / 31
![Page 12: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/12.jpg)
This slide may hurt your sensibilities
The (bosonic) field content of heterotic supergravity is a Lorentz metric g on
ten-dimensional space-time M10, dilaton φ ∈ C∞(M10), 3-form flux H and gauge
field A with field strength F
Ricgij +2∇gi ∇
gj φ−
1
4HiklH
klj − α′ tr FikF k
j + α′ tr RikRkj = 0, EM
d∗(e−2φH) = 0,
d∗A(e−2φF ) +e−2φ
2∗ (F ∧ ∗H) = 0,
∇−ε = 0, SUSY
(2dφ− H) · ε = 0,
F · ε = 0
dH − α′ (tr R ∧ R − tr F ∧ F ) = 0, Bianchi
In this talk, first order equations in α′-expansion taken as exact (my apologies).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 6 / 31
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This slide may hurt your sensibilities
The (bosonic) field content of heterotic supergravity is a Lorentz metric g on
ten-dimensional space-time M10, dilaton φ ∈ C∞(M10), 3-form flux H and gauge
field A with field strength F
Ricgij +2∇gi ∇
gj φ−
1
4HiklH
klj − α′ tr FikF k
j + α′ tr RikRkj = 0, EM
d∗(e−2φH) = 0,
d∗A(e−2φF ) +e−2φ
2∗ (F ∧ ∗H) = 0,
∇−ε = 0, SUSY
(2dφ− H) · ε = 0,
F · ε = 0
dH − α′ (tr R ∧ R − tr F ∧ F ) = 0, Bianchi
In this talk, first order equations in α′-expansion taken as exact (my apologies).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 6 / 31
![Page 14: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/14.jpg)
Calabi-Yau compactificationsIn 1985, Candelas, Horowitz, Strominger and Witten showed thatmetrics with SU(3) holonomy provide supersymmetric vacuum incompactifications of the heterotic string (zero flux, constant dilaton).
Hol(g) ⊂ SU(3)
Combined with Yau’s solution of the Calabi Conjecture in 1976 thisled to important advances in heterotic model building and modulispace.
Yau’s result relies in an important separation of parameters in killingspinor equations (complex vs metric parameters): reduces the problemto the complex Monge-Ampere equation for the Kahler potential:
log det ∂i∂jϕ = f
This ‘separation of variables’ produces complex and metric modulisplitting in moduli space ⇒ algebraic methods.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 7 / 31
![Page 15: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/15.jpg)
Calabi-Yau compactificationsIn 1985, Candelas, Horowitz, Strominger and Witten showed thatmetrics with SU(3) holonomy provide supersymmetric vacuum incompactifications of the heterotic string (zero flux, constant dilaton).
Hol(g) ⊂ SU(3)
Combined with Yau’s solution of the Calabi Conjecture in 1976 thisled to important advances in heterotic model building and modulispace.
Yau’s result relies in an important separation of parameters in killingspinor equations (complex vs metric parameters): reduces the problemto the complex Monge-Ampere equation for the Kahler potential:
log det ∂i∂jϕ = f
This ‘separation of variables’ produces complex and metric modulisplitting in moduli space ⇒ algebraic methods.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 7 / 31
![Page 16: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/16.jpg)
Calabi-Yau compactificationsIn 1985, Candelas, Horowitz, Strominger and Witten showed thatmetrics with SU(3) holonomy provide supersymmetric vacuum incompactifications of the heterotic string (zero flux, constant dilaton).
Hol(g) ⊂ SU(3)
Combined with Yau’s solution of the Calabi Conjecture in 1976 thisled to important advances in heterotic model building and modulispace.
Yau’s result relies in an important separation of parameters in killingspinor equations (complex vs metric parameters): reduces the problemto the complex Monge-Ampere equation for the Kahler potential:
log det ∂i∂jϕ = f
This ‘separation of variables’ produces complex and metric modulisplitting in moduli space ⇒ algebraic methods.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 7 / 31
![Page 17: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/17.jpg)
Calabi-Yau compactificationsIn 1985, Candelas, Horowitz, Strominger and Witten showed thatmetrics with SU(3) holonomy provide supersymmetric vacuum incompactifications of the heterotic string (zero flux, constant dilaton).
Hol(g) ⊂ SU(3)
Combined with Yau’s solution of the Calabi Conjecture in 1976 thisled to important advances in heterotic model building and modulispace.
Yau’s result relies in an important separation of parameters in killingspinor equations (complex vs metric parameters): reduces the problemto the complex Monge-Ampere equation for the Kahler potential:
log det ∂i∂jϕ = f
This ‘separation of variables’ produces complex and metric modulisplitting in moduli space ⇒ algebraic methods.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 7 / 31
![Page 18: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/18.jpg)
Hull-Strominger Geometry
In 1986 Hull and Strominger characterized warped 4d compactifications of the
heterotic string, with N = 1 supersymmetry and nonzero flux H 6= 0
M10 = R1,3 ×M6 g = e2f · (η ⊕ g) f ∈ C∞(M10)
where (M6, g) compact Riemannian
PG → M10 G ⊂ SO(32),E8 × E8
Imposing N = 1 supersymmetry, on M6 we obtain:
SU(3)-structure (ψ, ω) with metric g and almost complex structureJ : TM6 → TM6, ψ ∈ Λ3,0, ω ∈ Λ1,1,
φ = f ∈ C∞(M6),
3-form H,
gauge field A with field strength F .
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 8 / 31
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Hull-Strominger Geometry
In 1986 Hull and Strominger characterized warped 4d compactifications of the
heterotic string, with N = 1 supersymmetry and nonzero flux H 6= 0
M10 = R1,3 ×M6 g = e2f · (η ⊕ g) f ∈ C∞(M10)
where (M6, g) compact Riemannian
PG → M10 G ⊂ SO(32),E8 × E8
Imposing N = 1 supersymmetry, on M6 we obtain:
SU(3)-structure (ψ, ω) with metric g and almost complex structureJ : TM6 → TM6, ψ ∈ Λ3,0, ω ∈ Λ1,1,
φ = f ∈ C∞(M6),
3-form H,
gauge field A with field strength F .
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 8 / 31
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Hull-Strominger Geometry
Imposing N = 1 supersymmetry, on M6 we obtain: an SU(3)-structure (ψ, ω)
with almost complex structure J : TM6 → TM6 and metric g , φ ∈ C∞(M6),
3-form H and gauge field A with field strength F , such that
dΩ = 0,
g i jFi j = 0, Fij = 0,
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0,
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0,
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
Remark: The system is obtained taking Susy + Bianchi.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 9 / 31
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Hull-Strominger Geometry
Imposing N = 1 supersymmetry, on M6 we obtain: an SU(3)-structure (ψ, ω)
with almost complex structure J : TM6 → TM6 and metric g , φ ∈ C∞(M6),
3-form H and gauge field A with field strength F , such that
dΩ = 0,
g i jFi j = 0, Fij = 0,
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0,
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0,
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
Remark: The system is obtained taking Susy + Bianchi.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 9 / 31
![Page 22: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/22.jpg)
Hull-Strominger geometry
dΩ = 0,
g i jFi j = 0, Fij = 0,
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0,
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0,
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
Theorem (Fernandez-Ivanov-Ugarte-Villacampa ’08-’10)
EM + SUSY + Bianchi⇔ (⇑) and g i jRi j = 0, Rij = 0.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 10 / 31
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The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
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The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
![Page 25: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/25.jpg)
The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
![Page 26: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/26.jpg)
The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
![Page 27: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/27.jpg)
The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
![Page 28: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/28.jpg)
The Strominger system
Strominger System (ST)
dΩ = 0, (1)
Fij = 0, Rij = 0, (2)
g i jFi j = 0, g i jRi j = 0, (3)
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0, (4)
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0, (5)
where Ω = e2φψ, H = i(∂ − ∂)ω, φ = 18 log ‖Ω‖
First non-Kahler solutions of the Strominger system were found by Liand Yau in 2005, and in non-Kahlerian complex manifolds by Fu andYau in 2008.
Since then, a long list of people has been studying the existenceproblem for the Strominger system ... still no analogue of Yau’s Thm.
Remark: Crucial symmetry between curvature 2-form R of ∇ and F !MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 11 / 31
![Page 29: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/29.jpg)
Part II: Moduli
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 12 / 31
![Page 30: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/30.jpg)
Moduli
In recent work (arXiv:1503.07562, GF-Rubio-Tipler) it is proved thatthe Strominger system is an elliptic system of equations: an ellipticcomplex S∗ of (multidegree, real) differential operators is constructed,so that H1(S∗) is the infinitesimal moduli.
Previous work by de la Ossa-Svanes ’14, Anderson-Gray-Sharpe ’14
propose a (complex) vector space as infinitesimal moduli: the firstDolbeault cohomology H1(Q) of a holomorphic double extension Q.
Let me explain the relation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 13 / 31
![Page 31: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/31.jpg)
Moduli
In recent work (arXiv:1503.07562, GF-Rubio-Tipler) it is proved thatthe Strominger system is an elliptic system of equations: an ellipticcomplex S∗ of (multidegree, real) differential operators is constructed,so that H1(S∗) is the infinitesimal moduli.
Previous work by de la Ossa-Svanes ’14, Anderson-Gray-Sharpe ’14
propose a (complex) vector space as infinitesimal moduli: the firstDolbeault cohomology H1(Q) of a holomorphic double extension Q.
Let me explain the relation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 13 / 31
![Page 32: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/32.jpg)
Moduli
In recent work (arXiv:1503.07562, GF-Rubio-Tipler) it is proved thatthe Strominger system is an elliptic system of equations: an ellipticcomplex S∗ of (multidegree, real) differential operators is constructed,so that H1(S∗) is the infinitesimal moduli.
Previous work by de la Ossa-Svanes ’14, Anderson-Gray-Sharpe ’14
propose a (complex) vector space as infinitesimal moduli: the firstDolbeault cohomology H1(Q) of a holomorphic double extension Q.
Let me explain the relation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 13 / 31
![Page 33: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/33.jpg)
Moduli
In recent work (arXiv:1503.07562, GF-Rubio-Tipler) it is proved thatthe Strominger system is an elliptic system of equations: an ellipticcomplex S∗ of (multidegree, real) differential operators is constructed,so that H1(S∗) is the infinitesimal moduli.
Previous work by de la Ossa-Svanes ’14, Anderson-Gray-Sharpe ’14
propose a (complex) vector space as infinitesimal moduli: the firstDolbeault cohomology H1(Q) of a holomorphic double extension Q.
Let me explain the relation ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 13 / 31
![Page 34: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/34.jpg)
The flux map
In an ideal situation (obstructions!), given a solution of the Stromingersystem we obtain an open patch in the moduli space
H1(S∗) ⊃ U ⊂MST
Using the transgression formula for the Chern-Simons three-form: welldefined map given by flux charge
MST ⊃ U → H3(M,R)
Idea: (neglect ∇, take A abelian) given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 14 / 31
![Page 35: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/35.jpg)
The flux map
In an ideal situation (obstructions!), given a solution of the Stromingersystem we obtain an open patch in the moduli space
H1(S∗) ⊃ U ⊂MST
Using the transgression formula for the Chern-Simons three-form: welldefined map given by flux charge
MST ⊃ U → H3(M,R)
Idea: (neglect ∇, take A abelian) given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 14 / 31
![Page 36: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/36.jpg)
The flux map
In an ideal situation (obstructions!), given a solution of the Stromingersystem we obtain an open patch in the moduli space
H1(S∗) ⊃ U ⊂MST
Using the transgression formula for the Chern-Simons three-form: welldefined map given by flux charge
MST ⊃ U → H3(M,R)
Idea: (neglect ∇, take A abelian) given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 14 / 31
![Page 37: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/37.jpg)
The flux map
In an ideal situation (obstructions!), given a solution of the Stromingersystem we obtain an open patch in the moduli space
H1(S∗) ⊃ U ⊂MST
Using the transgression formula for the Chern-Simons three-form: welldefined map given by flux charge
MST ⊃ U → H3(M,R)
Idea: (neglect ∇, take A abelian) given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 14 / 31
![Page 38: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/38.jpg)
The flux map
MST ⊃ U → H3(M,R)
given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
The exterior derivative δ = dflux defines a closed H3(M,R)-valued 1-form
δ ∈ Ω1(MST ,H3(M,R)),
and hence natural foliation on the moduli space integrating Ker δ.
Fact: the leaf of the foliation passing trough a point in MST , can beidentified with a local moduli space of solutions of natural killing spinorequations in generalized geometry.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 15 / 31
![Page 39: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/39.jpg)
The flux map
MST ⊃ U → H3(M,R)
given by (A′ = A + a)
flux : (Ω′,A′, ω′) 7→ [dcJ′ω′ − dcω + 2α′a ∧ FA′ ]
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
The exterior derivative δ = dflux defines a closed H3(M,R)-valued 1-form
δ ∈ Ω1(MST ,H3(M,R)),
and hence natural foliation on the moduli space integrating Ker δ.
Fact: the leaf of the foliation passing trough a point in MST , can beidentified with a local moduli space of solutions of natural killing spinorequations in generalized geometry.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 15 / 31
![Page 40: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/40.jpg)
The flux map
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
Fact: the leave of the foliation passing trough p, can be identified with alocal moduli space MST of solutions of natural killing spinor equations ingeneralized geometry
MST//MST
δ // H3(X ,R).
Idea: flux : (Ω′,A′, ω′) = 0 implies
dcJ′ω′ − dcω + 2α′a ∧ FA′ = db
and (ω, b,A) determine a generalized metric on T ⊕ T ∗ ⊕ . . ..
Remark: flux quantization is a mechanism which ‘kills moduli’, asexpected in string theory folklore.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 16 / 31
![Page 41: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/41.jpg)
The flux map
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
Fact: the leave of the foliation passing trough p, can be identified with alocal moduli space MST of solutions of natural killing spinor equations ingeneralized geometry
MST//MST
δ // H3(X ,R).
Idea: flux : (Ω′,A′, ω′) = 0 implies
dcJ′ω′ − dcω + 2α′a ∧ FA′ = db
and (ω, b,A) determine a generalized metric on T ⊕ T ∗ ⊕ . . ..
Remark: flux quantization is a mechanism which ‘kills moduli’, asexpected in string theory folklore.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 16 / 31
![Page 42: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/42.jpg)
The flux map
Flux quantization: restricts to flux−1(H3(M,Z)) (string theory)
Fact: the leave of the foliation passing trough p, can be identified with alocal moduli space MST of solutions of natural killing spinor equations ingeneralized geometry
MST//MST
δ // H3(X ,R).
Idea: flux : (Ω′,A′, ω′) = 0 implies
dcJ′ω′ − dcω + 2α′a ∧ FA′ = db
and (ω, b,A) determine a generalized metric on T ⊕ T ∗ ⊕ . . ..
Remark: flux quantization is a mechanism which ‘kills moduli’, asexpected in string theory folklore.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 16 / 31
![Page 43: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/43.jpg)
Moduli of Killing spinors
The local moduli space MST of solutions of the killing spinor equations isconstructed using generalized diffeomorphisms. Restricting to the innergeneralized diffeomorphisms, obtain and H2(X ,R)-bundle MST over MST
H2(X ,R) // MST
MST
//MSTδ // H3(X ,R).
Key: MST is an even dimensional manifold.
Conjecture
The moduli MST carries a natural Kahler structure.
Evidence: there is a natural map TpMST → H1(Q) (complex).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 17 / 31
![Page 44: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/44.jpg)
Moduli of Killing spinors
The local moduli space MST of solutions of the killing spinor equations isconstructed using generalized diffeomorphisms. Restricting to the innergeneralized diffeomorphisms, obtain and H2(X ,R)-bundle MST over MST
H2(X ,R) // MST
MST
//MSTδ // H3(X ,R).
Key: MST is an even dimensional manifold.
Conjecture
The moduli MST carries a natural Kahler structure.
Evidence: there is a natural map TpMST → H1(Q) (complex).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 17 / 31
![Page 45: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/45.jpg)
Moduli of Killing spinors
The local moduli space MST of solutions of the killing spinor equations isconstructed using generalized diffeomorphisms. Restricting to the innergeneralized diffeomorphisms, obtain and H2(X ,R)-bundle MST over MST
H2(X ,R) // MST
MST
//MSTδ // H3(X ,R).
Key: MST is an even dimensional manifold.
Conjecture
The moduli MST carries a natural Kahler structure.
Evidence: there is a natural map TpMST → H1(Q) (complex).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 17 / 31
![Page 46: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/46.jpg)
Moduli of Killing spinors
The local moduli space MST of solutions of the killing spinor equations isconstructed using generalized diffeomorphisms. Restricting to the innergeneralized diffeomorphisms, obtain and H2(X ,R)-bundle MST over MST
H2(X ,R) // MST
MST
//MSTδ // H3(X ,R).
Key: MST is an even dimensional manifold.
Conjecture
The moduli MST carries a natural Kahler structure.
Evidence: there is a natural map TpMST → H1(Q) (complex).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 17 / 31
![Page 47: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/47.jpg)
Part III: Generalized geometry
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 18 / 31
![Page 48: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/48.jpg)
Basics on generalized geometry
Given a smooth manifold M, T ⊕ T ∗ has canonical pairing and bracket
〈X + ξ,X + ξ〉 = Xiξi , [X + ξ,Y + η] = [X ,Y ] + LXη − Yidξ[ij]
It has structure group O(n, n), and symmetries Ω2cl o Diff(M), with
B-fields acting byX + ξ 7→ X + ξ + XiB[ij]
Twisted version: an exact Courant algebroid
0→ T ∗ → E → T → 0.
is isomorphic to
(T + T ∗, 〈, 〉, [, ]H = [, ] + XiYjH[ijk])
for some H ∈ Ω3cl(M) (whose class [H] ∈ H3(M) parameterizes E ).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 19 / 31
![Page 49: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/49.jpg)
Basics on generalized geometry
Given a smooth manifold M, T ⊕ T ∗ has canonical pairing and bracket
〈X + ξ,X + ξ〉 = Xiξi , [X + ξ,Y + η] = [X ,Y ] + LXη − Yidξ[ij]
It has structure group O(n, n), and symmetries Ω2cl o Diff(M), with
B-fields acting byX + ξ 7→ X + ξ + XiB[ij]
Twisted version: an exact Courant algebroid
0→ T ∗ → E → T → 0.
is isomorphic to
(T + T ∗, 〈, 〉, [, ]H = [, ] + XiYjH[ijk])
for some H ∈ Ω3cl(M) (whose class [H] ∈ H3(M) parameterizes E ).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 19 / 31
![Page 50: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/50.jpg)
Basics on generalized geometry
Given a smooth manifold M, T ⊕ T ∗ has canonical pairing and bracket
〈X + ξ,X + ξ〉 = Xiξi , [X + ξ,Y + η] = [X ,Y ] + LXη − Yidξ[ij]
It has structure group O(n, n), and symmetries Ω2cl o Diff(M), with
B-fields acting byX + ξ 7→ X + ξ + XiB[ij]
Twisted version: an exact Courant algebroid
0→ T ∗ → E → T → 0.
is isomorphic to
(T + T ∗, 〈, 〉, [, ]H = [, ] + XiYjH[ijk])
for some H ∈ Ω3cl(M) (whose class [H] ∈ H3(M) parameterizes E ).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 19 / 31
![Page 51: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/51.jpg)
Basics on generalized geometry
Given a smooth manifold M, T ⊕ T ∗ has canonical pairing and bracket
〈X + ξ,X + ξ〉 = Xiξi , [X + ξ,Y + η] = [X ,Y ] + LXη − Yidξ[ij]
It has structure group O(n, n), and symmetries Ω2cl o Diff(M), with
B-fields acting byX + ξ 7→ X + ξ + XiB[ij]
Twisted version: an exact Courant algebroid
0→ T ∗ → E → T → 0.
is isomorphic to
(T + T ∗, 〈, 〉, [, ]H = [, ] + XiYjH[ijk])
for some H ∈ Ω3cl(M) (whose class [H] ∈ H3(M) parameterizes E ).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 19 / 31
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Generalized metrics
A metric is a reduction of the frame bundle from GL(n) to O(n).
A generalized metric is a reduction from O(n, n) to O(n)×O(n) ∼= a rankn positive-definite subbundle V+ ⊂ E .
A generalized metric on an exact Courant algebroid is actually equivalentto a usual metric g together with two-form b-field,
V+ = X + Xigij + Xibij.
It determines a closed 3-form H = H + db.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 20 / 31
![Page 53: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/53.jpg)
Generalized metrics
A metric is a reduction of the frame bundle from GL(n) to O(n).
A generalized metric is a reduction from O(n, n) to O(n)×O(n) ∼= a rankn positive-definite subbundle V+ ⊂ E .
A generalized metric on an exact Courant algebroid is actually equivalentto a usual metric g together with two-form b-field,
V+ = X + Xigij + Xibij.
It determines a closed 3-form H = H + db.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 20 / 31
![Page 54: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/54.jpg)
Generalized metrics
A metric is a reduction of the frame bundle from GL(n) to O(n).
A generalized metric is a reduction from O(n, n) to O(n)×O(n) ∼= a rankn positive-definite subbundle V+ ⊂ E .
A generalized metric on an exact Courant algebroid is actually equivalentto a usual metric g together with two-form b-field,
V+ = X + Xigij + Xibij.
It determines a closed 3-form H = H + db.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 20 / 31
![Page 55: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/55.jpg)
Generalized metrics
A metric is a reduction of the frame bundle from GL(n) to O(n).
A generalized metric is a reduction from O(n, n) to O(n)×O(n) ∼= a rankn positive-definite subbundle V+ ⊂ E .
A generalized metric on an exact Courant algebroid is actually equivalentto a usual metric g together with two-form b-field,
V+ = X + Xigij + Xibij.
It determines a closed 3-form H = H + db.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 20 / 31
![Page 56: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/56.jpg)
Generalized metrics
A metric is a reduction of the frame bundle from GL(n) to O(n).
A generalized metric is a reduction from O(n, n) to O(n)×O(n) ∼= a rankn positive-definite subbundle V+ ⊂ E .
A generalized metric on an exact Courant algebroid is actually equivalentto a usual metric g together with two-form b-field,
V+ = X + Xigij + Xibij.
It determines a closed 3-form H = H + db.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 20 / 31
![Page 57: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/57.jpg)
Generalized connections
A connection on E is a differential operator
D : Γ(E )→ Γ(T ∗ ⊗ E ),
satisfying the Leibniz rule (De fe′ = π(e)(f )e′ + fDee) and compatible with themetric (π(e)〈e′, e′′〉 = 〈Dee′, e′′〉+ 〈e′,Dee′′〉).
The space of connections is affine, modelled on Γ(E ∗ ⊗ o(E )).
There is a well-defined torsion TD ∈ Λ3E
TD(e1, e2, e3) = 〈De1e2 − De2e1 − [e1, e2], e3〉+ 〈De3e1, e2〉
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 21 / 31
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Generalized connections
A connection on E is a differential operator
D : Γ(E )→ Γ(T ∗ ⊗ E ),
satisfying the Leibniz rule (De fe′ = π(e)(f )e′ + fDee) and compatible with themetric (π(e)〈e′, e′′〉 = 〈Dee′, e′′〉+ 〈e′,Dee′′〉).
The space of connections is affine, modelled on Γ(E ∗ ⊗ o(E )).
There is a well-defined torsion TD ∈ Λ3E
TD(e1, e2, e3) = 〈De1e2 − De2e1 − [e1, e2], e3〉+ 〈De3e1, e2〉
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 21 / 31
![Page 59: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/59.jpg)
Generalized connections
A generalized connection on E is a differential operator
D : Γ(E )→ Γ(E ∗ ⊗ E ),
satisfying the Leibniz rule (De fe′ = π(e)(f )e′ + fDee) and compatible with themetric (π(e)〈e′, e′′〉 = 〈Dee′, e′′〉+ 〈e′,Dee′′〉).
The space of connections is affine, modelled on Γ(E ∗ ⊗ o(E )).
There is a well-defined torsion TD ∈ Λ3E
TD(e1, e2, e3) = 〈De1e2 − De2e1 − [e1, e2], e3〉+ 〈De3e1, e2〉
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 21 / 31
![Page 60: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/60.jpg)
Generalized connections
A generalized connection on E is a differential operator
D : Γ(E )→ Γ(E ∗ ⊗ E ),
satisfying the Leibniz rule (De fe′ = π(e)(f )e′ + fDee) and compatible with themetric (π(e)〈e′, e′′〉 = 〈Dee′, e′′〉+ 〈e′,Dee′′〉).
The space of connections is affine, modelled on Γ(E ∗ ⊗ o(E )).
There is a well-defined torsion TD ∈ Λ3E
TD(e1, e2, e3) = 〈De1e2 − De2e1 − [e1, e2], e3〉+ 〈De3e1, e2〉
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 21 / 31
![Page 61: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/61.jpg)
Generalized connections
A generalized connection on E is a differential operator
D : Γ(E )→ Γ(E ∗ ⊗ E ),
satisfying the Leibniz rule (De fe′ = π(e)(f )e′ + fDee) and compatible with themetric (π(e)〈e′, e′′〉 = 〈Dee′, e′′〉+ 〈e′,Dee′′〉).
The space of connections is affine, modelled on Γ(E ∗ ⊗ o(E )).
There is a well-defined torsion TD ∈ Λ3E
TD(e1, e2, e3) = 〈De1e2 − De2e1 − [e1, e2], e3〉+ 〈De3e1, e2〉
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 21 / 31
![Page 62: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/62.jpg)
An example: the Gualtieri-Bismut connection
Let V+ be a generalized metric (recall V± ∼= T ). Define, by projecting, amap C : E → E , C (V+) = V−, C (V−) = V+ and
DBe e ′ := [e−, e
′+]+ + [e+, e
′−]− + [Ce−, e
′−]− + [Ce+, e
′+]+,
The connection DB preserves V± and has totally skew torsion
TDB = π∗+H + π∗−H.
Projecting to T , DB encodes two metric connections with totally skewsymmetric torsion
∇± = ∇g ± 1
2g−1H
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 22 / 31
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An example: the Gualtieri-Bismut connection
Let V+ be a generalized metric (recall V± ∼= T ). Define, by projecting, amap C : E → E , C (V+) = V−, C (V−) = V+ and
DBe e ′ := [e−, e
′+]+ + [e+, e
′−]− + [Ce−, e
′−]− + [Ce+, e
′+]+,
The connection DB preserves V± and has totally skew torsion
TDB = π∗+H + π∗−H.
Projecting to T , DB encodes two metric connections with totally skewsymmetric torsion
∇± = ∇g ± 1
2g−1H
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 22 / 31
![Page 64: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/64.jpg)
An example: the Gualtieri-Bismut connection
Let V+ be a generalized metric (recall V± ∼= T ). Define, by projecting, amap C : E → E , C (V+) = V−, C (V−) = V+ and
DBe e ′ := [e−, e
′+]+ + [e+, e
′−]− + [Ce−, e
′−]− + [Ce+, e
′+]+,
The connection DB preserves V± and has totally skew torsion
TDB = π∗+H + π∗−H.
Projecting to T , DB encodes two metric connections with totally skewsymmetric torsion
∇± = ∇g ± 1
2g−1H
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 22 / 31
![Page 65: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/65.jpg)
An example: the Gualtieri-Bismut connection
Let V+ be a generalized metric (recall V± ∼= T ). Define, by projecting, amap C : E → E , C (V+) = V−, C (V−) = V+ and
DBe e ′ := [e−, e
′+]+ + [e+, e
′−]− + [Ce−, e
′−]− + [Ce+, e
′+]+,
The connection DB preserves V± and has totally skew torsion
TDB = π∗+H + π∗−H.
Projecting to T , DB encodes two metric connections with totally skewsymmetric torsion
∇± = ∇g ± 1
2g−1H
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 22 / 31
![Page 66: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/66.jpg)
The canonical Levi-Civita connection
Given a metric V+, we can define the Levi-Citiva connection
DLC = DB −1
3TDB
.
which encodes four different metric connections on M:
∇± = ∇g ± 1
2g−1H,
∇±13 = ∇g ± 1
6g−1H.
Remark: not unique torsion-free connection compatible with V+! (∼conformal generalized geometry).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 23 / 31
![Page 67: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/67.jpg)
The canonical Levi-Civita connection
Given a metric V+, we can define the Levi-Citiva connection
DLC = DB −1
3TDB
.
which encodes four different metric connections on M:
∇± = ∇g ± 1
2g−1H,
∇±13 = ∇g ± 1
6g−1H.
Remark: not unique torsion-free connection compatible with V+! (∼conformal generalized geometry).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 23 / 31
![Page 68: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/68.jpg)
The canonical Levi-Civita connection
Given a metric V+, we can define the Levi-Citiva connection
DLC = DB −1
3TDB
.
which encodes four different metric connections on M:
∇± = ∇g ± 1
2g−1H,
∇±13 = ∇g ± 1
6g−1H.
Remark: not unique torsion-free connection compatible with V+! (∼conformal generalized geometry).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 23 / 31
![Page 69: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/69.jpg)
Killing spinors and Calabi-Yau metricsIf M is spin and dim M = 2n, by V+
∼= T , we can talk about the spinorbundle S±(V+), so that the restrictions DLC
± : V+ → V+ ⊗ (V±)∗, extendto a differential operator on spinors
DLC± : S+(V+)→ S+(V+)⊗ (V±)∗,
with associated Dirac operator
/DLC+ : S+(V+)→ S−(V+).
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If (V+, η) is a solution to the Killing spinor eq. with η 6= 0 pure, thenH = 0 and g is a metric with holonomy contained in SU(n).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 24 / 31
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Killing spinors and Calabi-Yau metricsIf M is spin and dim M = 2n, by V+
∼= T , we can talk about the spinorbundle S±(V+), so that the restrictions DLC
± : V+ → V+ ⊗ (V±)∗, extendto a differential operator on spinors
DLC± : S+(V+)→ S+(V+)⊗ (V±)∗,
with associated Dirac operator
/DLC+ : S+(V+)→ S−(V+).
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If (V+, η) is a solution to the Killing spinor eq. with η 6= 0 pure, thenH = 0 and g is a metric with holonomy contained in SU(n).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 24 / 31
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Killing spinors and Calabi-Yau metricsIf M is spin and dim M = 2n, by V+
∼= T , we can talk about the spinorbundle S±(V+), so that the restrictions DLC
± : V+ → V+ ⊗ (V±)∗, extendto a differential operator on spinors
DLC± : S+(V+)→ S+(V+)⊗ (V±)∗,
with associated Dirac operator
/DLC+ : S+(V+)→ S−(V+).
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If (V+, η) is a solution to the Killing spinor eq. with η 6= 0 pure, thenH = 0 and g is a metric with holonomy contained in SU(n).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 24 / 31
![Page 72: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/72.jpg)
Killing spinors and Calabi-Yau metricsIf M is spin and dim M = 2n, by V+
∼= T , we can talk about the spinorbundle S±(V+), so that the restrictions DLC
± : V+ → V+ ⊗ (V±)∗, extendto a differential operator on spinors
DLC± : S+(V+)→ S+(V+)⊗ (V±)∗,
with associated Dirac operator
/DLC+ : S+(V+)→ S−(V+).
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If (V+, η) is a solution to the Killing spinor eq. with η 6= 0 pure, thenH = 0 and g is a metric with holonomy contained in SU(n).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 24 / 31
![Page 73: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/73.jpg)
Killing spinors and Calabi-Yau metricsIf M is spin and dim M = 2n, by V+
∼= T , we can talk about the spinorbundle S±(V+), so that the restrictions DLC
± : V+ → V+ ⊗ (V±)∗, extendto a differential operator on spinors
DLC± : S+(V+)→ S+(V+)⊗ (V±)∗,
with associated Dirac operator
/DLC+ : S+(V+)→ S−(V+).
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If (V+, η) is a solution to the Killing spinor eq. with η 6= 0 pure, thenH = 0 and g is a metric with holonomy contained in SU(n).
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 24 / 31
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Killing spinors and Calabi-Yau metrics
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If dim M = 2n and (V+, η) is a solution to the Killing spinor eq. with η 6= 0pure, then H = 0 and g is a metric with holonomy contained in SU(n).
Proof: Reduces to Ivanov-Papadopoulos No-Go Theorem.
Remark: embedded in generalized geometry, deformations of Calabi-Yaumetrics encode closed B-fields providing natural complexification Kahlercone.
V+ = X + Xigij + Xibij,
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 25 / 31
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Killing spinors and Calabi-Yau metrics
The Killing spinor equations for a spinor η ∈ S+(V+) are
DLC+ η = 0, /D
LC− η = 0.
Proposition ( ,Rubio,Tipler)
If dim M = 2n and (V+, η) is a solution to the Killing spinor eq. with η 6= 0pure, then H = 0 and g is a metric with holonomy contained in SU(n).
Proof: Reduces to Ivanov-Papadopoulos No-Go Theorem.
Remark: embedded in generalized geometry, deformations of Calabi-Yaumetrics encode closed B-fields providing natural complexification Kahlercone.
V+ = X + Xigij + Xibij,
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 25 / 31
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Killing spinors and the Strominger systemUsing the bundle of frames of the internal manifold M6, jointly with gaugebundle, construct a principal G -bundle P over the internal manifold withvanishing first Pontryagin class.
p1(P) = 0.
Choice of invariant class
[H] ∈ H3(P,R)G .
determines an equivariant (twisted) exact Courant algebroid
0→ T ∗P → E → TP → 0,
that can be reduced to a non-exact Courant algebroid E → M. As avector bundle, E ∼= T + ad P + T ∗, but not canonically.
Theorem ( ,Rubio,Tipler)
A solution to the Killing spinor eq. on E with η 6= 0 is equivalent to asolution of the Strominger system.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 26 / 31
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Killing spinors and the Strominger systemUsing the bundle of frames of the internal manifold M6, jointly with gaugebundle, construct a principal G -bundle P over the internal manifold withvanishing first Pontryagin class.
p1(P) = 0.
Choice of invariant class
[H] ∈ H3(P,R)G .
determines an equivariant (twisted) exact Courant algebroid
0→ T ∗P → E → TP → 0,
that can be reduced to a non-exact Courant algebroid E → M. As avector bundle, E ∼= T + ad P + T ∗, but not canonically.
Theorem ( ,Rubio,Tipler)
A solution to the Killing spinor eq. on E with η 6= 0 is equivalent to asolution of the Strominger system.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 26 / 31
![Page 78: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/78.jpg)
Killing spinors and the Strominger systemUsing the bundle of frames of the internal manifold M6, jointly with gaugebundle, construct a principal G -bundle P over the internal manifold withvanishing first Pontryagin class.
p1(P) = 0.
Choice of invariant class
[H] ∈ H3(P,R)G .
determines an equivariant (twisted) exact Courant algebroid
0→ T ∗P → E → TP → 0,
that can be reduced to a non-exact Courant algebroid E → M. As avector bundle, E ∼= T + ad P + T ∗, but not canonically.
Theorem ( ,Rubio,Tipler)
A solution to the Killing spinor eq. on E with η 6= 0 is equivalent to asolution of the Strominger system.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 26 / 31
![Page 79: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/79.jpg)
Killing spinors and the Strominger systemUsing the bundle of frames of the internal manifold M6, jointly with gaugebundle, construct a principal G -bundle P over the internal manifold withvanishing first Pontryagin class.
p1(P) = 0.
Choice of invariant class
[H] ∈ H3(P,R)G .
determines an equivariant (twisted) exact Courant algebroid
0→ T ∗P → E → TP → 0,
that can be reduced to a non-exact Courant algebroid E → M. As avector bundle, E ∼= T + ad P + T ∗, but not canonically.
Theorem ( ,Rubio,Tipler)
A solution to the Killing spinor eq. on E with η 6= 0 is equivalent to asolution of the Strominger system.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 26 / 31
![Page 80: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/80.jpg)
Killing spinors and the Strominger systemUsing the bundle of frames of the internal manifold M6, jointly with gaugebundle, construct a principal G -bundle P over the internal manifold withvanishing first Pontryagin class.
p1(P) = 0.
Choice of invariant class
[H] ∈ H3(P,R)G .
determines an equivariant (twisted) exact Courant algebroid
0→ T ∗P → E → TP → 0,
that can be reduced to a non-exact Courant algebroid E → M. As avector bundle, E ∼= T + ad P + T ∗, but not canonically.
Theorem ( ,Rubio,Tipler)
A solution to the Killing spinor eq. on E with η 6= 0 is equivalent to asolution of the Strominger system.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 26 / 31
![Page 81: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/81.jpg)
The Courant algebroid
More explicitely: assume for simplicity that gauge bundle isSU(r)-bundle, with associated hermitian vector bundle V . Then
E = TM ⊕ End TM ⊕ Endskw V ⊕ T ∗M
Pairing: for e = X + s + t + ξ
〈e, e〉 = Xiξi − α′ tr ss + α′ tr tt.
Recall: B-field transformations act on E by
Y → Y + YiB[ij].
Bracket: the B-field part of [e, ·] is
Bij = −∂[iξj] + Yk(dcω)[kij] − 2α′ tr(Rij , s) + 2α′ tr(Fij , t)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 27 / 31
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The Courant algebroid
More explicitely: assume for simplicity that gauge bundle isSU(r)-bundle, with associated hermitian vector bundle V . Then
E = TM ⊕ End TM ⊕ Endskw V ⊕ T ∗M
Pairing: for e = X + s + t + ξ
〈e, e〉 = Xiξi − α′ tr ss + α′ tr tt.
Recall: B-field transformations act on E by
Y → Y + YiB[ij].
Bracket: the B-field part of [e, ·] is
Bij = −∂[iξj] + Yk(dcω)[kij] − 2α′ tr(Rij , s) + 2α′ tr(Fij , t)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 27 / 31
![Page 83: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/83.jpg)
The Courant algebroid
More explicitely: assume for simplicity that gauge bundle isSU(r)-bundle, with associated hermitian vector bundle V . Then
E = TM ⊕ End TM ⊕ Endskw V ⊕ T ∗M
Pairing: for e = X + s + t + ξ
〈e, e〉 = Xiξi − α′ tr ss + α′ tr tt.
Recall: B-field transformations act on E by
Y → Y + YiB[ij].
Bracket: the B-field part of [e, ·] is
Bij = −∂[iξj] + Yk(dcω)[kij] − 2α′ tr(Rij , s) + 2α′ tr(Fij , t)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 27 / 31
![Page 84: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/84.jpg)
The Courant algebroid
More explicitely: assume for simplicity that gauge bundle isSU(r)-bundle, with associated hermitian vector bundle V . Then
E = TM ⊕ End TM ⊕ Endskw V ⊕ T ∗M
Pairing: for e = X + s + t + ξ
〈e, e〉 = Xiξi − α′ tr ss + α′ tr tt.
Recall: B-field transformations act on E by
Y → Y + YiB[ij].
Bracket: the B-field part of [e, ·] is
Bij = −∂[iξj] + Yk(dcω)[kij] − 2α′ tr(Rij , s) + 2α′ tr(Fij , t)
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 27 / 31
![Page 85: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/85.jpg)
A unifying framework
Generalized geometry is a unifying framework for the theory of theStrominger system and the well-stablished theory for metrics withSU(n)-holonomy.
dΩ = 0,
Fij = 0, Rij = 0,
g i jFi j = 0, g i jRi j = 0,
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0,
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0,
Remark: for simplicity, I have assumed that ‖Ω‖ = 1 (constant dilaton).General case requires conformal generalized geometry.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 28 / 31
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A unifying framework
Generalized geometry is a unifying framework for the theory of theStrominger system and the well-stablished theory for metrics withSU(n)-holonomy.
dΩ = 0,
Fij = 0, Rij = 0,
g i jFi j = 0, g i jRi j = 0,
d∗ω − i(∂ − ∂) log ‖Ω‖ = 0,
2i∂∂ω − α′(tr R ∧ R − tr F ∧ F ) = 0,
Remark: for simplicity, I have assumed that ‖Ω‖ = 1 (constant dilaton).General case requires conformal generalized geometry.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 28 / 31
![Page 87: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/87.jpg)
In the next episode ...
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 29 / 31
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Twisted heterotic supergravity
Conformal generalized geometry suggests a new class of heteroticcompactifications where the dilaton is only a locally defined functionon the internal manifold (GF-Shahbazi).
Present very interesting features:
compactifications to 6d with non-zero flux at zero order in α′ (∼Maldacena-Nunez)very small moduli spaceexplicit examples in Hopf surfaces (∼ WZW model).toy model for analysis of standard compactifications.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 30 / 31
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Twisted heterotic supergravity
Conformal generalized geometry suggests a new class of heteroticcompactifications where the dilaton is only a locally defined functionon the internal manifold (GF-Shahbazi).
Present very interesting features:
compactifications to 6d with non-zero flux at zero order in α′ (∼Maldacena-Nunez)very small moduli spaceexplicit examples in Hopf surfaces (∼ WZW model).toy model for analysis of standard compactifications.
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 30 / 31
![Page 90: Killing Spinors in Generalized Geometry - benasque.orgbenasque.org/2016ssg/talks_contr/053_BenasqueSugra2016-Garcia... · Killing Spinors in Generalized Geometry ... de la Ossa-Svanes](https://reader030.fdocuments.us/reader030/viewer/2022020121/5c670b5f09d3f2d0218d3a44/html5/thumbnails/90.jpg)
Thank you!
MGF (ICMAT) Killing spinors in generalized geometry Susy and Geom, Benasque 31 / 31