Introduction to Chern-Simons forms in Physics - I

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1 Introduction to Chern-Simons forms in Physics - I Modeling Graphene-Like Systems Dublin April Jorge Zanelli Centro de Estudios Científicos CECs - Valdivia z@cecs.cl

2 Lecture I: 1. Topological invariants 2. Minimal coupling 3. Chern-Simons forms 4. 1-D Example 5. CS couplings 6. Branes 7. Summary Lecture II: 1. Spacetime geometry 2. Metric and affine structures: e a, ω a b 3. Building blocks 4. Gravity actions: L-L, T, CS 5. CS gravities, BHs, SUGRAs 6. Summary

3 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.

4 1. Topological invariants 1

5 Euler characteristic V=4 E=6 F=4 V=8 E=12 F=6 V=6 E=12 F=8 χ = V E + F = 2 Independent of REGULARITY Independent of SIZE Independent of SHAPE Independent of NUMBER OF ELEMENTS The Euler characteristic, χ, is unchanged under continuous deformations of the surface: χ is a topological (homotopic) invariant.

6 Euler characteristic χ χ remains unchanged so long as the topology remains the same. V, E, F can be as large as we wish... In the continuum limit, 2-dimensional M: χ χ(m ) = 1 2π has an integral expression for any closed R dω = 2 2g M χ =0

7 The Euler characteristic belongs to a family of famous invariants: Sum of exterior angles of a polygon Residue theorem in complex analysis Winding number of a map Poincaré-Hopff theorem ( one cannot comb a sphere ) Atiyah-Singer index theorem Witten index Dirac s monopole quantization Aharonov-Bohm effect Gauss law Bohr-Sommerfeld quantization Soliton/Instanton topologically conserved charges etc All of these examples involve topological invariants called the Chern characteristic classes and Chern-Simons forms.

8 2. Minimal coupling 2

9 Electromagnetic coupling M I[A,z] = j µ A µ d 4 x This coupling is invariant under gauge transformations A µ (x) A µ (x)+ µ Ω(x) provided the current is conserved, µ j µ =0. Gauge invariance Current conservation Lesson 1. Consistent (gauge invariant) minimal coupling between a gauge potential and a charged source is possible only if the source satisfies a conservation law.

10 Minimal coupling for a point charge M I[A,z] = j µ A µ d 4 x = *j A * j = eδ(γ )dξ 1 dξ 2 dξ 3 M Transverse directions Γ e dz ξ 2 ξ 1 ξ 3 z I[A,z] = e, (A A µ (z)dz µ ) A Γ Is this gauge-invariant? (A is not ) + A(x) A(x)+dΩ(x) I[A] I[A] +Ω - PBCs: Ω(+ )=Ω( ) guarantee gauge invariance

11 In other words, δi = e δa = e dω = 0 Γ Γ can vanish under physically reasonable boundary conditions. Lesson 2. A gauge non-invariant integrand can define a gauge-invariant integral, provided the right boundary conditions are satisfied (e.g., periodic b. c.). Generalization to higher-dimensional sources: Chern-Simons forms

12 3. Chern-Simons forms 3

13 Gauge theories All four basic forces of nature share the same mathematical structure Fiber bundle (F ): locally F =M x G Fibers (group G) Connection (Lie-algebra valued 1-form) Interaction Field A µ = A µ a J a Lie alg. generator

14 Electrodynamics Action functional where F = da = 1 2 ( µ A ν ν A µ )dx µ dx ν Field strength (curvature) Yang-Mills (electro-weak and strong interactions) Action I[A] = Tr[ 1 M 2F *F j A] where A takes values in a nonabelian Lie algebra, and F = da + A A = [ µ A ν a + f bc a A µ b A ν c ]J a dx µ dx ν = F a J a

15 What is a Chern-Simons action? Y-M / EM action [any D]: I[A] = 1 4κ M D Chern-Simons action [3dim]: I[A] = κ M 3 and in general [(2n+1)-dim.]: M 2 n+1 gg µα g νβ γ F a ab µνf b αβd D x A da A A A I[A] = κ A (da) n + α 1 A 3 (da) n α n A 2n+1 Fixed rational numbers

16 What is a Chern-Simons action? Y-M / EM action [any D]: I[A] = 1 4κ M D Chern-Simons action [3dim]: I[A] = κ M 3 and in general [(2n+1)-dim.]: M 2 n+1 gg µα g νβ γ F a ab µνf b αβd D x A da A A A Invariant symmetric trace in the Lie algebra I[A] = κ A (da) n + α 1 A 3 (da) n α n A 2n+1

17 Chern-Simons forms C 2n+1 = A (da) n + α 1 A 3 (da) n α n A 2n+1 C 2n+1 No dimensionful constants: α k = fixed rational numbers No adjustable coefficients: α k 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 Their exterior derivatives are invariant polynomials (characteristic classes) dc 2n 1 (A) = P 2n (F) δc 2 p+1 = dω 2 p Physics Mathematics

18 Characteristic classes: Given A = connection, for a certain Lie algebra. Then, under gauge transformations, A A =g -1 Ag +g -1 dg, F F =g -1 Fg the polynomial and closed: P 2n (F) = F n P 2n (F') = P 2n (F) dp 2n (F) = 0. is invariant [Chern-Weil Theorem] These invariants have an additional remarkable property. For a suitable normalization ( # ), # P 2n (F) M 2 n just like the Euler characteristic. = Z

19 What makes the CS forms useful in physics is that under gauge transformations, they change like an abelian connection: dc 2n 1 (A) = P 2n (F) δc 2n+1 (A) = dω 2n The e-m potential is the simplest example of a CS form C 2n+1 = A (da) n + α 1 A 3 (da) n α n A 2n+1 for n = 0 C 0+1 (A) = A = A CS forms provide a generalization of the coupling (between point charges and the e-m field) to higher dimensional objects branesand nonabelian gauge fields. Physicists have been playing with CS forms for 200 years!

20 4. Another 1-D Example 4

21 Classical mechanics (Hamilton, 1833) z 0 =t 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. I = A (z)dz µ, z µ = (z 0 = t, z i ) Γ µ = (A dt + A dz i ), i = 1,...,2s Γ 0 i z i z Γ z j z i (q 1,...,q s, p 1,..., p s ); A i = ( p 1,.., p s, 0,...,0) Identifying, and A 0 = H(z), one finds the action of a mechanical system of finite number of degrees of freedom, I[q, p] = [ p i dq i H( p,q)dt] Γ

22 Dynamical equations z 0 =t Γ The equations read where F ij z j = E i, F ij = i A j j A i, E i = i A 0. Which z z j can be checked to be Hamilton s equations, z i q i = H, p p i = H i q. i Also, Gauge invariance of CS action = Invariance of mechanical system under canonical transformations A good part of classical physics is described by CS forms.

23 5. Chern-Simons couplings 5

24 Generalization to higher dimensions Natural Ansatz: Replace 1-forms by p-forms A µ 1-form A µνλ p-form I[A,z] = M j µ1 µ 2...µ p A µ 1µ 2...µ p then I[A,z] = Γ p A (D-p)-form that projects on the p-dimensional history This works fine for an abelian field: A A =A+dΩ but not for nonabelian one: A A =g -1 Ag + g -1 dg

25 There is no obvious analog of A A' = g 1 (A + )g, g G µ µ µ µ for a p-form: A µ1 µ 2...µ p A' µ1 µ 2...µ p = g 1 (A µ1 µ 2...µ p +? µ1 µ 2...µ p )g This p-form does not define a natural connection (covariant derivative) Nonabelian curvature:? F = da + A A There is no clear relation between current conservation and (nonabelian) gauge transformation. There is no consistent Hamiltonian evolution

26 Chern-Simons forms A consistent (gauge invariant) coupling for any group G and for higher dimensional Γs is provided by a Chern- Simons form C 2n-1 (A): dc 2n 1 (A) = P 2n (F) Then, under a gauge transformation, a closed form: 0 = δdc = dδc C 2n 1 (A) changes by δc = dω Locally exact

27 The Chern-Simons coupling The minimal coupling between a particle and electromagnetism is generalized by the formula I[A] = M *j 2 p+1 C 2 p+1 (A) C 2 p+1 (A) = C 2 p+1 (A) (where we have defined ). This describes the coupling between an extended object, a 2pdimensional (mem)brane and a nonabelian connection. As in electrodynamics, this coupling is invariant under gauge transformations provided *j 2p+1 is conserved, D*j 2p+1 =0.

28 2p-brane in D dimensions: Γ 2p+1 Brane history Σ D-(2p+1) Transverse space Source: *j 2 p+1 = qδ(σ)dω Σ [S a 1a 2 a m J a1 J am ] Source with support at the center of Generators This is a D-(2p+1)-form that couples to a 2p+1-form

29 The Chern-Simons coupling Lemma: I[A] = *j C (A), 2 p+1 2 p+1 Proof: This can be explicitly verified. // M is invariant under gauge transformations, provided j 2p+1 is conserved, d*j 2p+1 + [A, *j 2p+1 ]=D*j 2p+1 = 0. N.B.: If the current is produced by particles or fields that are dynamically governed by an action principle, invariant under the same gauge symmetry, then Noether theorem guarantees that such conserved current can always be explicitly written out. //

30 6. Examples of branes

31 Branes are defects An m-brane is a localized (m+1)-dimensional manifold, embedded in spacetime. They disrupt the homogeneity of spacetime (impurities) They are obstructions that change the topology of spacetime(topological defects) Γ Examples: point particle (m=0), domain wall (m=2), 2p-brane (m=2p)

32 Branes are naked singularities If not wrapped by a horizon, branes are naked singularities (NS). Near naked singularities the energy grows infinitely, the laws of physics break down Green slime, lost socks and TV sets could emerge from them (J. Earman) They better not exist (Cosmic Censorship) But sometimes they do exist and nothing terrible happens...

33 0- and 2- brane sources in EM (4D) 0-brane [point charge]: *j = eδ(γ)dω D 1 j µ = e z µ δ (3) ( x z )dτ 2-brane [interface between two regions in 3-space]: *j = θδ(σ)dn j µνλ = θε µνλ δ(x )d 3 x(σ) θ in 0 θ out =0 n This brane is the 2D surface Σ of a solid volume in 3D, whose history is a 2+1 worldsheet. Σ R

34 Calling the brane charge θ, *j C 3 (A) = θ Σ R C 3 (A) V R 1 = θ 4 ε µνλρ F µν F λρ d 4 x The full Maxwell + brane action is I[A] = 1 2 F *F θ 2 F M 4 V R F θ-vacuum which describes a topological insulator: E = θδ(σ) n ˆ B B t E = θδ(σ) n ˆ E

35 Brane sources in gravity 0-brane in GR = conical defect ~ point particle In 2+1 dimensions, *j a = 1 2 mδ(γ)dω 2 ε abc J bc, J bc = Isometry generator that produced the conical defect. ~ black hole with M<0, M < J, M < Q Ok in 2+1 dimensions; serious problem for D>3 Ok in D>3 for codimension 2 branes Ok for higher codimensions and commuting generators.

36 7. Quantization

37 Quantum mechanics Consider I[A,z] = e where Γ Γ A is a loop. Γ Quantum mechanics is the requirement that this integral take integer values (maximum constructive interference): e Α = nh Γ Aharonov-Bohm effect Dirac s monopole quantization Bohr-Sommerfeld quantization

38 Bohr-Sommerfeld quantization Consider a mechanical system described by a CS lagrangian in a 2s+1 dimensional space. Γ = A (z)dz µ, z µ = (z 0 = t, z i ) µ A Γ Γ = (A dt + A dz i ), i = 1,...,2s Γ 0 i If the orbit is periodic, one finds the B-S quantization condition A = [ p dq i Γ Γ i H( p,q)dt] = 2nπ = nh

39 Aharonov-Bohm effect B e By Stoke s theorem, A e A = e F Γ 1 Γ 2 M where. Γ 1 ( Γ 2 ) = M Γ 1 Γ 2 Hence, for two paths enclosing a quantum of flux, e M F = 2nπ there is constructive interference (maximum).

40 Monopole quantization By Stoke s theorem, Γ e A = e F Γ M where, and. Γ = M F = da For a Dirac monopole, F = g, and therefore r 2 d2 x eg = 2nπ

41 8. Summary

42 Summary CS forms generalize the coupling between a point charge and the electromagnetic field, providing a natural gauge-invariant way to couple gauge fields (abelian or not) to extended sources: 2p-branes. The simplest CS action can also describe an arbitrary classical mechanical system of finite degrees of freedom. Path integral quantization yields the Bohr-Sommerfeld quantum postulate, Dirac s monopole quantization and the Aharonov-Bohm effect. Mathematically, CS forms are related to topological invariants, the Chern characteristic classes.

43 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/ v4. J.Z.: Uses of Chern-Simons Forms, (In 10 Years of the AdS/CFT Conjecture), arxiv: hep-th/ J.Z.: Chern-Simons Forms in Gravitation Theories, Class. & Quant. Grav. 29 (2012) J.Z.: Gravitation Theory and Chern-Simons Forms, Villa de Leyva School, 2011.

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

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