Chern-Simons appetiser

Jorge Zanelli Centro de Estudios Científicos

CECs-Valdivia z@cecs.cl

27th Indian Summer School of Physics Prague, September 14 - 18, 2015

Gene “Demon” Simmons [Kiss]

Shiing-shen Chern 1911-2004 James H. Simons 1938-

This work, ... grew out of an attempt to derive a purely combinatorial formula for the Pontrjagin number of a 4-manifold...

This process got stuck by the emergence of a boundary term which did not yield to a simple combinatorial analysis. The boundary term seemed interesting in its own right and it and its generalization are the subject of this paper.

... We do not see how this helps to settle the Poincaré conjecture.

Disclaimer (V. I. Arnold)

€

∃ t1 < 0 such that P(t1)∈ D and for t1 ≤ t2 < 0 P(t2)∈ D

Mathematics:

“Peter washed his hands” Physics:

Topological invariants

Euler characteristic

V=4 E=6 F=4

V=8 E=12 F=6

V=6 E=12 F=8

……

  Independent of REGULARITY   Independent of SIZE   Independent of SHAPE   Independent of NUMBER OF ELEMENTS

= V – E + F = 2 χ

The Euler characteristic, , is unchanged under continuous deformations of the surface:

is a topological (homotopic) invariant.

χ

χ

Euler characteristic

……

remains unchanged so long as the topology remains the same. For a torus (donut) =0… χ

χ

where R is the curvature and M is any closed 2-dimensional surface

χ V, E, F can be as large as we wish and remains the same. In the continuum limit, has an integral expression:

χ

€

χ(M ) =1

2π RM∫ dΩ

χ

The Euler characteristic belongs to a family of famous invariant relations:

All of these examples involve topological invariants called the Chern characteristic classes and Chern-Simons forms.

  Sum of exterior angles of a polygon; winding number   Gauss’ law   Residue theorem in complex analysis   Poincaré-Hopff theorem (“one cannot comb a sphere”)

Bohr-Sommerfeld quantization Dirac’s monopole quantization

  Aharonov-Bohm effect   Atiyah-Singer index theorem   Soliton/Instanton topologically conserved charges   Witten index   Etc…

Blackboard

Gauge theories:YM versus CS

Electrodynamics

€

Fµν = ∂µAν − ∂νAµ

where

Field strength (curvature)

Action

€

I[A]= ( −14M∫ FµνFµν − jµAµ )d

4x

Yang-Mills (electro-weak and strong interactions)

€

I[A]= Tr[ 12M∫ FµνFµν − jµAµ ]

where A takes values in a nonabelian Lie algebra, and

€

Fµν = ∂µAν +AµAν − (µ ↔ν)

Action

Field strength (curvature)

What is a Chern-Simons action?

€

I[A]= 14κ ggµαgνβγ abF

aµνFb

αβ

M D∫ dDx

Y-M / EM action [any D]:

€

I[A]= κ A∧dA+ 23 A∧A∧A

M 3∫

Chern-Simons action [3dim]:

€

I[A]= κ A∧ (dA)n +α1A3 ∧ (dA)n−1 + ⋅ ⋅ ⋅+αnA

2n+1

M 2 n+1∫

and in general [(2n+1)-dim.]:

Fixed rational numbers

Comparing Yang-Mills and Chern-Simons actions

€

I[A]= 14g ggµαgνβγ abF

aµνFb

αβ

M 4∫ d 4x

Y-M / EM action (4d):

€

I[A]= κ A∧dA+ 23 A∧A∧A

M 3∫

Chern-Simons action [3dim]:

€

I[A]= κ A∧ (dA)n + c1A3 ∧ (dA)n−1 + ⋅ ⋅ ⋅+ cnA

2n+1

M 2n+1∫

and in general [(2n+1)-dim.]: trace in the Lie alg.

All of these properties are consequences of the last:

(characteristic classes)

€

dC2n−1(A) = P2n (F)

Chern-Simons forms

No dimensionful constants: αk= fixed rational numbers

No adjustable coefficients: αk cannot get renormalized No metric needed; scale invariant Entirely determined by the Lie algebra and the dimension

  Only defined in odd dimensions Unique gauge quasi-invariants: Related to the Chern characteristic classes

€

C2n+1 = A∧(dA)n +α1A3∧(dA)n−1 + ⋅ ⋅ ⋅+αnA

2n+1

€

δC2 p+1 = dΩ2 p Physics

Mathematics

Blackboard

€

δC2n+1(A) = dΩ2n

What makes the CS forms useful in physics is that under gauge transformation, they change in the same way as the electromagnetic potential (abelian transformation):

CS forms transform like abelian connections, e.g., like the vector potential of electrodynamics.

Physicists have been playing with CS forms for 150 years! €

C2n+1(A) = A(∧dA)n + c1A3(∧dA)n−1 + ⋅ ⋅ ⋅+ cnA

2n+1

€

C0+1(A) = AN.B. for n=0, Good old e-m potential!!

CS forms generalize the coupling between a point charge and the electromagnetic field, to the case of charged 2p-branes coupled to (non) abelian gauge fields.

CS forms naturally define Lagrangians for gauge-invariant theories in 2n+1 dimensions

CS forms are by construction metric-independent and invariant under diff. (GCT): Background independent (super-)gravities.

Given a Lie algebra the CS form is uniquely defined in any odd-dimensional manifold.

Blackboard

€

I = AµΓ∫ (z)dzµ , zµ = (z0 = t, zi )

Classical mechanics as Chern-Simons Consider the trajectory of a point in a 2s+1 dimensional space. The projection on the 2s dimensional spatial section can be identified with the position on a phase space.

€

= (A0Γ∫ dt + Aidz

i ), i = 1,..., 2s

Identifying , and , one finds the action of a mechanical system of finite number of degrees of freedom,

€

zi ≡ (q1,...,qs, p1,..., ps); Ai = (p1,.., ps , 0,...,0)

€

A0 = −H (z)

€

I[z]≡ I[q,p]= [pidqi −H (p,q)dt]

Γ∫

z

Γ z0=t

zi

zj

€

Fij ˙ z j = Ei

Classical dynanics The equations read

€

Fij = ∂iAj −∂ jAi , Ei = −∂iA0where

that can be checked to be Hamilton’s equations.

z

Γ z0=t

zi

zj

€

˙ q i =∂H∂pi

, ˙ p i =∂H∂qi .

Also,

= Gauge invariance of CS action

Invariance of mechanical system under canonical

transformations

A good part of classical physics is described by CS forms.

6. Quantization

€

I[A,z]= e AΓ∫

Quantum mechanics Consider

where is a loop. Γ

Γ

Quantum mechanics is the requirement that this integral take integer values, up to a normalization:

€

e ΑΓ∫ = nh

Dirac’s monopole quantization

Aharonov-Bohm effect

Bohr-Sommerfeld quantization

€

AΓ∫ = AµΓ

∫ (z)dzµ , zµ = (z0 = t, zi )

Bohr-Sommerfeld quantization Consider a mechanical system described by a CS lagrangian in a 2s+1 dimensional space.

€

= (A0Γ∫ dt + Aidz

i ), i = 1,..., 2s

If the orbit is periodic, one finds the B-S quantization condition

€

AΓ∫ = [pidq

iΓ∫ −H (p,q)dt]

€

= 2nπ = nh

Γ

€

e AΓ 1∫ − e A

Γ 2∫ = e FM∫

Aharonov-Bohm effect

By Stoke’s theorem,

where .

€

Γ 1 (−Γ 2 ) = ∂M

Hence, for two paths enclosing a quantum of flux,

B

€

Γ 2

€

Γ 1

€

e FM∫ = 2nπ

there is constructive interference (maximum).

Monopole quantization Γ

€

e AΓ∫ = e F

M∫By Stoke’s theorem,

where , and .

€

Γ = ∂M

€

F = dA

For a Dirac monopole, , and therefore

€

F =gr2d2x

€

eg = 2nπ

What is the origin of all this?

7. Summary

Summary

•  CS forms generalize the coupling between a point charge and the electromagnetic field, to the case of charged 2p-branes coupled to (non) abelian gauge fields.

•  CS forms provide a natural, consistent, Lagrangians for gauge-invariant theories in 2n+1 dimensions, without requiring a metric.

•  The simplest CS action can also describe an arbitrary classical mechanical system of finite degrees of freedom.

•  The Aharonov-Bohm effect, the monopole charge quantization and the Bohr-Sommerfeld quantum postulate are the quantization of the simplest CS system.

•  CS forms have evolved from a freak obstruction in algebraic topology to a central structure at the heart of all fundamental systems in physics, from classical mechanics, to electrodynamics, to quantum mechanics and possibly, for quantum gravity.

Bibliography

Shiing-Shen Chern and James Simon: Characteristic Forms and Geometric Invariants, Ann. Math. 99 (1974) 48.

J.Z.: Lecture Notes on Chern-Simons (Super-)Gravities [Second Edition] arXiv: hep-th/0502193v4.

J.Z.: Uses of Chern-Simons Forms, (In 10 Years of the AdS/CFT Conjecture), arXiv: hep-th/0805.1778

J.Z.: Chern-Simons Forms and Gravitation Theory, Proc 7th Aegean Summer School, Paros, 2013 Lect.Notes Phys. 892 (2015) 289-310. DOI: 10.1007/978-3-319-10070-8_11.

J.Z.: Introductory Lectures on Chern-Simons Theories, Cinvestav Summer School, Mexico City, 2011.AIP Conf Proc. 1420 (2012).

to appear in 2015

Thanks!