Abelian and Nonabelian Dbrane Effective Actions

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Abelian and Non-abelian D-brane Effective Actions

• by Paul Koerber

Pythagoras There is geometry in the humming of
the strings.
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Contents

• Introduction
• Strings, D-branes, effective actions,...
• Explain the title
• Summary of most significant results

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Part I
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String Theory

• Two types of strings

Closed string
Open string

• Particles oscillation modes of the string
• In fact infinite number of modes infinite
  number of different particles
• Massless modes finite number

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Size and weight of strings

• Quantum fluctuations determine size
• Fundamental length a1/2 ¼ lp 1.6 10-33cm
• Planck mass lp-1 1.2 1019 GeV 25 mg
• LHC (CERN) 7 103 GeV
• Braneworld ?

Massless modes !
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Massless sector of two types of strings

• Gauge sector
• Gauge bosons
• Forces of the Standard Model

Open string

• Gravitational sector
• Graviton
• Antisymmetric tensor
• Dilaton
• Gravity !

Closed string
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Scattering amplitudes

• n-point amplitudes S-matrix

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Effective action I

• String scattering amplitudes
• External massless states
• Massive states can propagate internally

S-matrix

• Effective action
• Action of particle field theory that reproduces
• this S-matrix

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Effective Action II
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Why study string theory ?

• Renormalization properties better
• Quantum gravity
• Unification
• Tool to study other fields of physics and
  mathematics

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Open strings

• State
• Gauge potential
• Fieldstrength

Abelian case
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Chan-Patton factors
labels

• are matrix-valued do not commute
• i.e order of product is important
• In fact

where
Non-abelian case
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D-branes
free
free
Neumann boundary conditions
Dirichlet boundary conditions
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Coinciding D-branes

• Mass open string / length

...
1
2
N
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What about fermions ?

• Supersymmetry
• Symmetry between bosons and fermions
• World-sheet supersymmetry
• Space-time supersymmetry

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Part II
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Massless vector boson

• Action in field theory Yang-Mills

• Effective action in string theory
• Born-Infeld

• Under the conditions
• Abelian
• Constant fieldstrengths

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Yang-Mills in 4d
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BPS solutions

• Bogomolnyi bound
• Background that preserves some supersymmetry
• Born-Infeld

Important properties that we want to generalize
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More than four dimensions

• Generalize

completely antisymmetric tensor

• Breaks symmetry
• Complex case
• Octonionic case

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Complex case

• Introduce complex coordinates

• BPS equations in complex coordinates

holomorphicity
DUY condition
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For Born-Infeld theory

• Generalize the BPS equations ?
• Bogomolnyi bound
• Preserve some supersymmetry
• Yes !
• Complex case

holomorphicity
DUY condition
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Reverse the logic

• Solutions of BPS equations have a natural
  interpretation in D-brane physics
• Link with supersymmetry

• Turn these BPS equations into a tool to construct
  the effective action
• Construct most general action and DUY condition
• Impose constraint

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Abelian case

• Abelian case constant field strengths
• Toy model to study techniques

Born-Infeld unique possibility

• Corrections with 4 derivatives (due to Wyllard)

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Wyllards action
Define

• Actie
• DUY

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Non-abelian case

• No all order result in known
• Work order by order in
• Much more complex
• Order of fieldstrengths important
• Need derivative terms (no slowly varying field
  strength limit)
• Identities between terms in action
• Field redefinitions

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Java program

• So I had to write a computer program

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Flowchart
Most general
Lagrangian
Euler-Lagrange
Subtract
Equations
Resulting
of motions
Lagrangian
Must be 0
DUY must
imply eom
Most general
Apply
DUY
Most general
Comparison
field redefinition
literature
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Optimizations

• Symmetrize in the fieldstrengths using
• Follow abelian calculation more closely

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Complexity

• Internal check
• 1816 equations in 546 unknowns

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Results I

• Order a0 until a2 know for a long time

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Results II

• Order a3 new result

• undetermined coefficient

from string theory

• Confirmed by other groups with various
• methods

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Results III

• Order a4 new result

• Checks internal spectrum check

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Undetermined coefficients

• Study (ir)rationality of and

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Spectrum check

• Strings stretching between intersecting branes

• Calculate spectrum
• String theory
• Effective action
• a4 Sevrin Wijns, Nagaoka

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Conclusions

• Abelian action with derivative corrections
• looks nice and suggestive
• Prove conjecture about more derivatives
• Study also with Weyl invariance method
• Non-abelian result looks complicated and
• rather ugly
• However both actions should possess very basic
  properties like having BPS solutions,
• supersymmetry,...
• Possibility to learn something about the

(ir)rationality of with n odd