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Deformations of calibrated D-branes in flux generalized complex manifolds hep-th/0610044 (with Luca Martucci) Paul Koerber [email protected] Max-Planck-Institut f ¨ ur Physik F ¨ ohringer Ring 6 D-80805 M ¨ unchen Germany Paul Koerber, MPI – p.1/22
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Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Jul 22, 2020

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Page 1: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Deformations of calibratedD-branes in flux generalized

complex manifoldshep-th/0610044 (with Luca Martucci)

Paul Koerber

[email protected]

Max-Planck-Institut fur Physik

Fohringer Ring 6

D-80805 Munchen

GermanyPaul Koerber, MPI – p.1/22

Page 2: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 3: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 4: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 5: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 6: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Generalizedcalibrations

Introduce

bulk fields

and

RR

on the D-brane, where

� � � � � � � � �

such that

� � � �

Paul Koerber, MPI – p.5/22

Page 7: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 8: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Generalizedcalibrations

Calibration polyform

:

� � � � � �

RR (1)

Bound:

� � � � � � � � � � � (2)

(bound must be such that it can be saturated)

Generalized geometry

Paul Koerber, MPI – p.6/22

Page 9: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 10: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 11: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 12: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 13: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 14: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 15: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 16: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 17: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 18: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 19: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 20: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 21: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Example II: B-branes with fluxes

� � � �� �

,

�� � �

,

� �� � � � �� � � �

in type IIB

complex �� � � � � � �� � � �

� � � �� � � � �� � � �

�� � � � � �

� ��� � �� � �

� � � �� ��

� ���� � � � � � � �

�� � � � � � � �� �� � � � � �

� only as vector space

� � � � �

�� � � � � �

� � � � � �

� � � � � �

Paul Koerber, MPI – p.19/22

Page 22: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Example II: B-branes with fluxes

� � � �� � � � �

� � � � � �� �� �

� � �

�� �� � � �

� � � � ��

� � � � � � � � � � � �Kapustin

Marchesano,Gomis,Mateos

Paul Koerber, MPI – p.20/22

Page 23: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

Example II: B-branes with fluxes

� � � �� � � � �

� � � � � �� �� �

� � �

�� �� � � �

� � � � ��

� � � � � � � � � � � �

� � � �� � � � �� � � �� �� �

� � � � � ��� �� � � �

� � �

� �

� ��� �� � � �

� � �

� � � � � � � �� ��

� � � � �� ��

� � �

� � � � �� �� �

� � � �

Paul Koerber, MPI – p.20/22

Page 24: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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

Page 25: Deformations of calibrated D-branes in flux generalized ... · Calibrations Calibration form: (1) Bound: (2) (bound must be such that it can be saturated) Calibrated submanifold

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