Marta Mazzocco - University of Bristolmaxcu/MartaMazzoccoLMSProspect… · Marta Mazzocco Riemann...

Riemann surfaces Teichmuller space Geodesic lengths Quantisation UK research

Riemann surfaces in Physics

Marta Mazzocco

University of Loughborough

December 16, 2011

Marta Mazzocco Riemann surfaces in Physics


Outline

1 Riemann surfaces



Outline

1 Riemann surfaces

2 Teichmuller space



Outline

1 Riemann surfaces

2 Teichmuller space

3 Geodesic lengths



Outline

1 Riemann surfaces

2 Teichmuller space

3 Geodesic lengths

4 Quantisation



Outline

1 Riemann surfaces

2 Teichmuller space

3 Geodesic lengths

4 Quantisation

5 UK research



What is a Riemann surface?

Definition

A Riemann surface ! is a one-dimensional complex manifold.




Definition


A Riemann surface is a surface with a complex structure.




Definition



Examples

Torus: T




Definition



Examples

Torus: T

T ! C \ "

Di#erent " will give di#erent complex structures



Hyperbolic space

The torus is flat or in other words T ! C \ ".



Hyperbolic space


Not all Riemann surfaces are flat. Most of them have negativecurvature.



Hyperbolic space



Generically ! ! H \$, where H is the hyperbolic space.



Hyperbolic space




What is the hyperbolic space?

Replace the parallel postulate of Euclidean geometry.

Consider H with the metric ds2 = dx2+dy2

y2 .



Hyperbolic space




What is the hyperbolic space?

Replace the parallel postulate of Euclidean geometry.

Consider H with the metric ds2 = dx2+dy2

y2 .The lines are now geodesics:



Hyperbolic space

Given a geodesic ! and a point P there are infinitely many distinctlines through P which do not intersect !.

!

•P



Hyperbolic space

A characteristic property of hyperbolic geometry is that as weapproach the real axis lengths become bigger and bigger.

Examples

The two red arcs have the same length.



Teichmuller space

Given a surface, we can choose many di#erent complex structuresto make it a Riemann surface.



Teichmuller space


Examples

For the torus T, we can choose manydi#erent lattices" = {z = n +m " , m, n " N},for each " " H we have a di#erentcomplex structure on T ! C \ ".



Teichmuller space


Examples

For the torus T, we can choose manydi#erent lattices" = {z = n +m " , m, n " N},for each " " H we have a di#erentcomplex structure on T ! C \ ".

Definition

The Teichmuller space T is the space of all non-equivalentcomplex structures on !.



Coordinates for a pair of trousers

How many di#erent complex structures can we put on a pair oftrousers?





Each pair of trousers is uniquely determined by the three geodesic

lengths of its ”belts”.





Each pair of trousers is uniquely determined by the three geodesic

lengths of its ”belts”.

To see this cut the pair of trousers in two hexagons and take thetop one.




• place the first edge on the imaginary axis starting at i• there is a unique geodesic orthogonal to it at i .• choose its length to be given. Call the end point a.• There is a unique orthogonal geodesic at a.• fix a line of constant distance b from the Imaginary axis.• choose two geodesics meeting orthogonally along the line ofdistance b such that the geodesic orthogonal to them both hasgiven length.

i




• place the first edge on the imaginary axis starting at i• there is a unique geodesic orthogonal to it at i .• choose its length to be given. Call the end point a.• There is a unique orthogonal geodesic at a.• fix a line of constant distance b from the Imaginary axis.• choose two geodesics meeting orthogonally along the line ofdistance b such that the geodesic orthogonal to them both hasgiven length.

ia



Coordinates for a Riemann surface

• Cut the Riemann surface in pair of trousers:

(l2,#2)

(l1,#1) (l3,#3)

(l4,#4)





(l2,#2)

(l1,#1) (l3,#3)

(l4,#4)

• Each pair of trousers is uniquely determined by the threegeodesic lengths of its ”belts” (up to isometry).





(l2,#2)

(l1,#1) (l3,#3)

(l4,#4)

• Each pair of trousers is uniquely determined by the threegeodesic lengths of its ”belts” (up to isometry).• The Riemann surface is uniquely determined by its pairs oftrousers and the twist parameters.





(l2,#2)

(l1,#1) (l3,#3)

(l4,#4)

• Each pair of trousers is uniquely determined by the threegeodesic lengths of its ”belts” (up to isometry).• The Riemann surface is uniquely determined by its pairs oftrousers and the twist parameters.

dim(Tg ,n) = 6g # 6+2n




The twist coordinates (#1, . . . ,#3g!3+n) can be replaced byappropriate geodesic lenghts.

# = 0 # $= 0




The twist coordinates (#1, . . . ,#3g!3+n) can be replaced byappropriate geodesic lenghts.

# = 0 # $= 0

Advantages:

Geodesic lengths have physical meaning: we can measurethem.

They admit Poisson brackets (Weil–Petersson form).

They can be quantised



Poincare uniformsation

! = H/$,

where $ is a Fuchsian group, i.e. a discrete group oftransformations

! : z %az + b

cz + d, s.t. ad # bc $= 0, |Tr(!)| = |a + d | > 2.



Poincare uniformsation

! = H/$,

where $ is a Fuchsian group, i.e. a discrete group oftransformations

! : z %az + b

cz + d, s.t. ad # bc $= 0, |Tr(!)| = |a + d | > 2.

Examples

!1!2



New coordinates: geodesic lengths

Examples

!1!2




Examples

!1!2

Lemma

Hyperbolic elements are in one-to-one correspondence with closed

geodesics.

|Tr(!)| = |a + d | = 2cosh(l!).




Theorem

The geodesic length functions form an algebra with multiplication:

G!G! = G!! + G!!!1 .




Theorem

The geodesic length functions form an algebra with multiplication:

G!G! = G!! + G!!!1 .

G!

G!

=

G!!!1

+

G!!



Poisson structure

Definition

Poisson structure is a skew symmetric binary operation which maps(G! ,G!) to {G! ,G!}, such that

{G!G! ,G!} = G!{G! ,G!}+ G!{G! ,G!},

{G! , {G! ,G!}}+ {G! , {G! ,G!}}+ {G! , {G! ,G!}} = 0.



Poisson structure

Definition

Poisson structure is a skew symmetric binary operation which maps(G! ,G!) to {G! ,G!}, such that

{G!G! ,G!} = G!{G! ,G!}+ G!{G! ,G!},

{G! , {G! ,G!}}+ {G! , {G! ,G!}}+ {G! , {G! ,G!}} = 0.

{G! ,G!} =1

2G!! #

1

2G!!!1 .

!

G!

G!

"

= 12

G!!1!

#12

G!!



Poisson structure

Poisson structures are very important in Physics to quantise.



Poisson structure


A quantisation of a mathematical object is a family of thesame kind of objects depending on some parameter.



Poisson structure



Physicists use the Poisson bracket to deform the ordinarycommutative product of functions, and obtain anoncommutative product suitable for quantum mechanics.



Poisson structure



Physicists use the Poisson bracket to deform the ordinarycommutative product of functions, and obtain anoncommutative product suitable for quantum mechanics.

Examples

A functionG! becomes an operator G !! . While functions commute,

i.e. G!G! = G!G! operators don’t. The commutation rule is givenby:

[G !! ,G

!! ] = !{G! ,G!}

! +O(!).



Quantum version

#

G!

G!

$

= q!12

G!!1!

+ q12

G!!

[G !! ,G

!! ] = q!

12G !

!!1!+ q

12G !

!!



We have a powerful way to quantise Teichmuller spaces.




It relies on the Poisson bracket {G! ,G!} = 12G!! #

12G!!!1 .





12G!!!1 .

By taking the Poisson bracket of two geodesic lengthfunctions, we introduce two new geodesic length functions.





12G!!!1 .


We can eliminate one of the by the multiplication ruleG!G! = G!! + G!!!1 & G!! = G!G! # G!!!1





12G!!!1 .



Open problem 1: How to find a set of closed geodesic which isclosed under Poisson bracket?





12G!!!1 .



Open problem 1: How to find a set of closed geodesic which isclosed under Poisson bracket?Open problem 2: Is it always possible to quantise these sets in agood way to form a quantum group?



UK research on quantum groups and Teichmuller theory

Glasgow: Integrable Systems and Mathematical Physics.

York: Quantum Gravity, Quantum Field Theory, IntegrableModels, and Quantum Groups.

Leeds: Cluster Algebras, Noncommutative Algebra andIntegrable Systems.

Leicester: Moduli Spaces, Noncommutative Geometry,Representation Theory.

Manchester: Supermanifolds, Quantum Groups.

King College London: Quantum groups and Teichmullertheory.



Loughborough research activity


Dr Gavin Brown Algebraic geometry, especially higher dimensional birational geometry and applicationsof computer algebra to algebraic geometry.

Proffessor Jenya Ferapontov

Classical differential geometry (web geometry, projective differential geometry, Liesphere geometry, theory of congruences, conformal structures, geometric aspects ofPDEs); Integrable systems (equations of hydrodynamic type, hyperbolic systems ofconservation laws, multi-dimensional dispersionless integrable systems, Hamiltonianformalism, symmetry methods).

Dr Martin HallnasIntegrable systems and special functions. In particular, in the context of exactly solvableSchrodinger- and analytic difference operators; symmetric functions; and multivariablegeneralisations of hypergeometric functions.

Dr Marta MazzoccoNon–linear differential equations describing monodromy preserving deformations oflinear systems (isomonodromic deformation equations). Quantization of Teichmullerspaces and Frobenius manifold theory.

Dr Vladimir NovikovClassification of integrable nonlinear partial differential equations and differential-difference equations. Integrability tests: symmetry approach, perturbative symmetryapproach in the symbolic representation. Integrable models of mathematical physics.

Professor Sasha Veselov Classical and quantum integrable systems in relation with geometry and representationtheory; solvable Schroedinger equations, special functions and Huygens' principle.

Visiting Fellows

Dr Ian Marshall Poisson structures of integrable systems and Poisson Lie groups.

Geometry and mathematical physics research groupOurresearch:Researchgroups

The theory of integrable systems studies differential equations which are, in a sense, exactly solvable and possess regular behaviour.Such systems play a fundamental role in mathematical physics providing an approximation to various models of applied interest. Datingback to Newton, Euler and Jacobi, the theory of integrable systems plays nowadays a unifying role in mathematics bringing togetheralgebra, geometry and analysis.

The research of the group includes both classical and quantum integrable systems in relation with representation theory and specialfunctions, as well as algebraic, differential and symplectic geometry.

The geometry and mathematical physics research group holds seminars in the mathematical physics seminar series.

Academic staff

Dr Alexey Bolsinov

Integrable tops, bi-Hamiltonian systems and compatible Poisson structures; integrablegeodesic flows on Lie groups and homogeneous spaces, magnetic geodesic flows,symmetries and reduction; obstructions to integrability; symplectic and topologicalinvariants for Lagrangian foliations; singularities of the momentum mapping, theirinvariants and algorithmic classification; projective equivalence in Riemannian geometry.

Dr Gavin Brown Algebraic geometry, especially higher dimensional birational geometry and applicationsof computer algebra to algebraic geometry.

Proffessor Jenya Ferapontov

Classical differential geometry (web geometry, projective differential geometry, Liesphere geometry, theory of congruences, conformal structures, geometric aspects ofPDEs); Integrable systems (equations of hydrodynamic type, hyperbolic systems ofconservation laws, multi-dimensional dispersionless integrable systems, Hamiltonianformalism, symmetry methods).

Dr Martin HallnasIntegrable systems and special functions. In particular, in the context of exactly solvableSchrodinger- and analytic difference operators; symmetric functions; and multivariablegeneralisations of hypergeometric functions.

Dr Marta MazzoccoNon–linear differential equations describing monodromy preserving deformations oflinear systems (isomonodromic deformation equations). Quantization of Teichmullerspaces and Frobenius manifold theory.

Dr Vladimir NovikovClassification of integrable nonlinear partial differential equations and differential-difference equations. Integrability tests: symmetry approach, perturbative symmetryapproach in the symbolic representation. Integrable models of mathematical physics.

Professor Sasha Veselov Classical and quantum integrable systems in relation with geometry and representationtheory; solvable Schroedinger equations, special functions and Huygens' principle.


Loughborough University


!  State%of%the%art%sport%facili/es:%50mt%swimming%pool,%famous%cricket%pitches,%football,%hockey,%rugby,%tennis…..%

!  Brilliant%range%of%arts%ac/vi/es:%contemporary%arts%events,%concerts,%seminars%and%classes.%

!  Best%student%experience%for%5%years%running:%best%employability%academic%life,%support,%ac/vi/es,%loca/on%and%facili/es.%%


Loughborough University


Facul&es)Science)(7)departments)%Engineering)(6)departments)%Social)Sciences)and)Humani&es)(10)departments)%%Number)of)full?&me)students)(approximate)%%%%%%%%Total)14.634)

)Undergraduate )10418))Postgraduate)(taught)programmes) )2912%)Postgraduate)(research))950%)Mathema&cs)(research))30%

Department)of)Mathema&cal)Sciences%Research)Groups:%Dynamical%systems%Geometry%and%mathema/cal%physics%Global%analysis%and%PDE’s%Linear%and%nonlinear%waves%Mathema/cal%modelling%Stochas/c%analysis%


Loughborough PhD programme


!  Degrees:%%•  MPhil%(ini/al%registra/on%for%2%years),%%•  PhD%(normally%takes%3%years)%%

!  Entrance%requirements:%2.1%degree%or%higher%in%Mathema/cs%or%Physics%%!  PhD)funding:%%

•  EPSRC%studentships%for%UK/EU%students%•  Maintenance%grant%£13.290%p.a.%tax%free%plus%tui/on%fee%paid.%

!  Tutorial%Assistance%•  Teaching%experience%•  3X4%hours%per%week%•  £14%per%hour%




)" Choose)a)topic)for)your)research:))))))))h"p://www.lboro.ac.uk/departments/ma/research/index.html88" Fill)out)an)applica&on)form:%)))%h"p://www.lboro.ac.uk/prospectus/pg/essen:al/apply/index.htm8!8




22%February%2012,%from%18:00%to%20:00.%%

#  standard%class%rail%fare%will%be%reimbursed%for%travel%within%GB.%%#  overnight%accommoda/on%provided%for%the%night%of%22%February.%%

#  Please%contact%Prof.%Ferapontov%[email protected]

Marta Mazzocco - University of Bristolmaxcu/MartaMazzoccoLMSProspect… · Marta Mazzocco Riemann...

Documents

Transcript of Marta Mazzocco - University of Bristolmaxcu/MartaMazzoccoLMSProspect… · Marta Mazzocco Riemann...