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Deformations of calibratedD-branes in flux generalized
complex manifoldshep-th/0610044 (with Luca Martucci)
Paul Koerber
Max-Planck-Institut fur Physik
Fohringer Ring 6
D-80805 Munchen
GermanyPaul Koerber, MPI – p.1/22
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Motivation
Generalized complex geometry � tailored todescribe susy
�� �� �
background sugra solutionswith fluxes
In the same way: supersymmetric D-branes �
generalized calibrations
Open string moduli � deformations of generalizedcalibrations
Paul Koerber, MPI – p.2/22
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Calibrations
A way to find minimal volumes surface in a curvedspace
Second-order equations � first-order equations
Analogous to self-duality solves Yang-Mills equations
Or more generally BPS equations solve equations ofmotion
Paul Koerber, MPI – p.3/22
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Calibrations
Calibration form
�
:
� � � �
(1)
Bound: � �� � � � �� � � (2)
(bound must be such that it can be saturated)
Calibrated submanifold
�
:
Saturates bound: � � � � � � � � � � � (3)
For
� � � �� �
� �Vol
� ��� �
��
�� � �
��
� � � ����
� � � ����
� � Vol
� � � �
Paul Koerber, MPI – p.4/22
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Calibrations
Calibration form
�
:
� � � �
(1)
Bound: � �� � � � �� � � (2)
(bound must be such that it can be saturated)
Calibrated submanifold
�
:
Saturates bound: � � � � � � � � � � � (3)
Calibration forms from invariant spinors: e.g.
��
�� � �
�
Paul Koerber, MPI – p.4/22
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Generalizedcalibrations
Introduce
bulk fields
�
and
�
RR
�
on the D-brane, where
� � � � � � � � �
such that
� � � �
Paul Koerber, MPI – p.5/22
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Generalizedcalibrations
Calibration polyform
�
:
� � � � � �
RR
� �
RR (1)
Bound:
� � � � � � � � � � � (2)
(bound must be such that it can be saturated)
Papadopoulos and Gutowski
Paul Koerber, MPI – p.6/22
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Generalizedcalibrations
Calibration polyform
�
:
� � � � � �
RR (1)
Bound:
� � � � � � � � � � � (2)
(bound must be such that it can be saturated)
Generalized geometry
Paul Koerber, MPI – p.6/22
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Generalizedcalibrations
Calibration polyform
�
:
� � � � � �
RR (1)
Bound:
� � � � � � � � � � � (2)
(bound must be such that it can be saturated)
Calibrated D-brane
� ��
�
:
Saturates bound:
� � � � � � � � � � �
For
� � � ��
� � � � �� �
��
�� �� �
��
E
� �� �
�� �
�
� � � �
� �� � ��
�� ��
� � � � � �
� � ��
�
� ��
� � � � � �� � �� E
� �� �
��
Paul Koerber, MPI – p.6/22
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Generalized calibrations
Correspond to supersymmetric D-branes
Calibration forms are the pure spinors
� � � � � � � � � � �� satisfying
� � � � � � � � ��
� � � � � � � ��
� � �� � ��
In the rest of the talk we will focus on space-fillingD-branes
Paul Koerber, MPI – p.7/22
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D-flatness and F-flatness conditions
Saturating bound consists of two parts
� � � � � � �� � � � � � , where �� �
varying phase� � ��
� �
is generalized complex submanifold withrespect to
��
This becomes an F-flatness condition in the4d-effective theory
� � � � � � � �
� � �
: analogous to the ‘special’ in speciallagrangianThis becomes a D-flatness condition in the4d-effective theory
We will study the deformations of these conditionsseparately!
Paul Koerber, MPI – p.8/22
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Some technology I
Decomposition of formsPure spinor: e.g.
��
��: Null space or also � �-eigenspace of �
Definition: forms in
� � � � � ��
can be written as
� � � � � �� �
�� �
with
� � � � � ���
. They have
�
-eigenvalue of �
Paul Koerber, MPI – p.9/22
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Some technology II
D-brane current
�� � � � � : generalization of the Poincaré dual:
�� � �
� �
�� ��
�� � � � ��
Explicitly:
�� � � � � � PD
� � � � �
�
Pure spinor and� � � � � � � � � �
Null space: generalized tangent bundle
�� � � � �
Paul Koerber, MPI – p.10/22
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Some technology III
Generalized normal bundle: � � � � � ��� � �� � � � ��
� �� � � �
Elements
� �
� � � � � look like
� � � �� �
�
: a normal vector to
� � geometric deformations
� � � � � � � � � � deformations gauge field
� � � � �
� � � � ��
work on�� � � � �
� � � � � � � ��� � � � �
Paul Koerber, MPI – p.11/22
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Some technology IV
Lie algebroid exterior derivativePure spinor
�
, null space
�
, natural� � ��
� �
metric
�
Isomorphism
� � � �
:
� � � � � � �� �
�
� � � � � � � �
�
can be viewed as element of � � � � � �
� �� � � � � � � � �
� � � � � � � �
��: Lie algebroid exterior derivative: for
� � � � � � �
�� � � � � � � � � �
�� � � � �
�
�
�� � � � � � � � � � � � � � � � �
��� � � � � �
���
��� �
�
�� � � � � � � � � � �� � � � �� � � � � � �
��� � � � � �
�� � � � � � �
���
Paul Koerber, MPI – p.12/22
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Deformations of gc submanifold
Generalized complex submanifold:
�� � � � � � � � ���
Deformation
� �
� � � � � :� � � � � � � ��� � � � �
, with
� � � � � �
�
� � �
Becomes
� �� � �� � � �� � � � �� � �
� �� � is a section of both
�� and � � � � � : it acts on
�� � � � � � ��� � � �� � � � �� �
�
deformation that transforms gc submanifold into gcsubmanifold:
�� �� � � �� �� � � �
Paul Koerber, MPI – p.13/22
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Cohomology
Gauge symmetry:
� � � �
generated by�� � � � �� � �
In fact: deformation equation
�� �� � � �� �� � � �
� enhanced gauge symmetry:
�� and � �
��
Divide out by
��� � �� � � �� ��
Deformations classified by
� � � �� � � � ��
Meaning: ‘imaginary’ gauge transformation:equivalent D-branes in topological string theory
Kapustin,Li
Paul Koerber, MPI – p.14/22
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Deformations of D-flatness
Second condition:
� � � � � � � �
� � �
(depends
� �)Deformations
�
that preserve this condition:
� � � � ��� � � � � � � � � � � �� � �
Provides gauge fixing ‘imaginary’ gaugetransformations
For calibration
� ��
� �
: natural metric on � � � � � :
� � � � ��� � � � �� � � � � ��
� �
��
��� � � � � � � �
�
� � define� �� �� � � �
Paul Koerber, MPI – p.15/22
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Deformations of D-flatness
So for deformations to preserve total calibrationcondition:
� �� �� � � �
� �� � � � �� �� � � �� �� � � �
The deformations are still classified by
� � � �� � � � ��
Depends only on the integrable
�� !
Paul Koerber, MPI – p.17/22
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Example I: deformations of SLag
McLean
� � � �
,
�� � �
� �
,
� � �
in type IIA
�� � � � � � � � ��� � � � � � � � � �
� � � � � � �
�� � � � � � � � �� �
� � � � � � � � � � � � �
�� � � � � � � � � �
,
�� �� � � �
� �Result:
� � � ��
� �Note: as opposed to McLean: also gaugedeformations
However, McLean also shows there are noobstructions
Paul Koerber, MPI – p.18/22
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Example II: B-branes with fluxes
� � � �� �
,
�� � �
,
� �� � � � �� � � �
in type IIB
�
complex �� � � � � � �� � � �
� � � �� � � � �� � � �
�� � � � � �
� ��� � �� � �
� � � �� ��
� ���� � � � � � � �
�
�� � � � � � � �� �� � � � � �
� only as vector space
� � � � �
�� � � � � �
� � � � � �
� � � � � �
�
Paul Koerber, MPI – p.19/22
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Example II: B-branes with fluxes
� � � �� � � � �
� � � � � �� �� �
� � �
�
�� �� � � �
� � � � ��
� � � � � � � � � � � �Kapustin
Marchesano,Gomis,Mateos
Paul Koerber, MPI – p.20/22
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Example II: B-branes with fluxes
� � � �� � � � �
� � � � � �� �� �
� � �
�
�� �� � � �
� � � � ��
� � � � � � � � � � � �
� � � �� � � � �� � � �� �� �
� � � � � ��� �� � � �
�
� � �
� �
� ��� �� � � �
�
� � �
� � � � � � � �� ��
� � � � �� ��
� � �
� � � � �� �� �
� � � �
Paul Koerber, MPI – p.20/22
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Example III: type-changing gcs
Gcs:
� � �� � � � �� � � � �� � � (type 1)
�� � � � �
at certain points � local complex structure
Susy D3-brane can only move on�� � � locus
Analysis shows: deformations off the locus lifted
Paul Koerber, MPI – p.21/22
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Future work
Find more examples (non-
� � � � �
-structure case):depends also on non-trivial
� � � � � � � � � � �
background examples
Calibrated D-branes on
� � ��� �
Instantons ?
Coinciding D-branes: hard problem!
The
end
� �
�
The end
�
�
�
T he end �
Paul Koerber, MPI – p.22/22