対称性をもった南部‐後藤ストリングの古典解

Hideki Ishihara
Department of Physics,
Osaka City University
with S.Hasegawa,T.Koike and Y.Morisawa

12023/2/7

Introduction
Physical significance of extended objects◼ Topological defects

e.g. cosmic strings etc◼ Braneworld universe model◼ ・・・◼ AdS/CFT correspondence

Extended objects (cosmic strings) aredescribed by PDE◼ Nambu-Goto equation, etc.

Cohomogeneity-one （C-1) objects
C-1 universe
universe models with homogeneous 3-spacee.g. Friedmann universe model,
Einstein equations
P.D.E.

Friedmann equation
O.D.E.

C-1 black holeblack holes with spherical symmetrye.g. Schwarzschild black hole4

C-1 String
1-parameter isometry groupof a target space actson the string world sheet
A Killing vector field istangent to the world sheet

Killing vector : generator of an isometry group(infinitesimal version of isometry)We consider cohomogeneity-one strings

Strings with Symmetry

An isometry acts on the string world sheet.

Advantage of C-1 Objects
Tractable and physically interesting
Homogeneous

Cohomogeneity-1

No symmetry

To solve

Simplest(algebraic)

Simple(ODE)

Difficult(PDE)

VarietyPoorRich

RichestPhysics

TrivialNon-trivialGeneral

Example 1:
Stationary Rotating Strings in 4D Minkowski
Target space

Consider a Killing vector

Ogawa, Ishihara,Kozaki,Nakano,Saitoh, PRD78, 023525(2008)

Strings are Rotating

Example 2:
Troidal Spirals in 5D Minkowski

are commutable Killing vectors
We consider C-1 strings with

T. Igata, and H. Ishihara (2010)
T. Igata, H. Ishihara and K.Nishiwaki (2012)

Analytic Solutions

⚫ Cohomogeneity-one strings
with timelike or spacelike symmetry⚫ Classification of Killing vectors(classification of C-1 strings)⚫ Integrability of equations of motion⚫ Cohomogeneity-one strings
with null symmetrycontact structure
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Cohomogeneity-one strings

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World SheetC-1 String
1-parameter isometry groupof a target space actson the world sheet
A Killing vector field istangent to the world sheet

Killing vector : generator of an isometry group(infinitesimal version of isometry)

Nambu-Goto action

We get geodesic action.

Classification ofKilling vectors
17Equivalence of Killing vectors
Equivalence class of isometrys.t.
Conjugacy class
Equivalence of Killing vector

18Isometries in x-y PlaneEquivalence Class

Equivalence classes

Classification of Killing vectors
4-dim. Euclid

4-dim. Minkowski

Ishihara and Kozaki PRD(2005)

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Killing Vector Fields in AdS5C-1 strings in AdS5 are classified in 10 families.
Classification using SO(4,2) ~ SU(2,2)
T.Koike, H.Kozaki, H.Ishihara, Phys.Rev. D77 (2008) 125003

Orbit space

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Geodesics on an orbit space

Classically equivalent

Hamilton equations
Geodesic equations
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Can we integrate a C-1 string?

Can we integrate geodesics
in the orbit space of a Killing vector?
The system with the degree of freedom N
is integrable in Liouville’s sence,
if the number of constants of motion is N.
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が十分な数のLie可換なKilling vector を許せば十分な数のPoisson可換な保存量が存在する．
積分可能
対称性の高いを target space とすれば多くのKilling vector を許す．と可換なのKilling vector ははのKilling vector になる．

を中心とするの isometry の中心化群はのisometry.
その代数の可換な最大部分代数の次元を調べる．
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Symmetry of orbit space
4-dim. Minkowski spacetime

足りない！

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5-dim. Anti-de Sitter spacetime

足りない！

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Results◼

In the Minkowski spacetime, and 5-dimensionalanti-de Sitter spacetime,
All possible orbit spaces with the metricare geodesically integrable thanks to aKilling tensor
in addition to Killing vectors.

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List of Killing tensors
II:

V:

X:

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⚫ Cohomogeneity-one strings
with timelike or spacelike symmetry⚫ Classification of Killing vectors(classification of C-1 strings)⚫ Integrability of equations of motion⚫ Cohomogeneity-one strings
with null symmetrycontact structure31

Spacetime metric admitting a null Killing vector: null Killing vector: null condition

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Spacetime metric admitting a null Killing vector: null Killing vector: null condition33

: geodesic tangent
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Nambu-Goto action

Equation of motion: Nonlinear P.D.E.
Null coordinates on a string world sheet

Equation of motion

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Two cases

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:Reeb vector

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41

42

is a solution.

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An integral curve of the Reeb vector
determines completely.

Equation of motion

is trivial.

If,

on

satisfies
satisfies

44.

Example: Einstein’s static universe

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String in a t = const. slice

Parallel transport

String with null symmetry inthe Einstein static universe istwisted.
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C1 strings with null symmetry
is closely related to thecontact structure.

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