New Gauge Symmetriesfrom String Theory

Pei-Ming HoPhysics Department

National Taiwan University

Sep. 23, 2011@CQSE

Gauge vs Global

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Aμ → A'μ = UAμU−1 +U∂μU

−1

Dμ = ∂μ + Aμ →D'μ = UDμU−1

For covariant quantities

Old definitions in some textbooks:U = independent of spacetime globalU = function of spacetime gauge (local)

Gauge vs Global (2)

• Translation symmetry is usually considered as a global symmetry.

• Different interpretation of the transformation:Translation by a specific length L can be “gauged” (defined as a gauge symmetry) equivalence:

(compactification on a circle).• Gauge potential is useful but not necessary.

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x(t) ≈ x(t) + L

Gauge vs Global (3)

• 2nd example:

base space = R+ with Neumann B.C. at x = 0.

• 3rd example:

space an interval of length L/2 w. Neumann B.C.

Space/(subgroups of isometry) orbifolds

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x ≈ − x, φ(x) = φ(−x)

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φ(x) →φ(x + nL), φ(x) →φ(−x)

Gauge vs Global (4)

New (better) definition:• Gauge Symmetry– Transformation does not change physical state.– A physical state has multiple descriptions.– Transformation changes descriptions.

• Global Symmetry– Transformation changes physical states.

This is a more fundamental distinction than spacetime-dependence.

Non-Abelian Gauge Symmetry

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δA(1) = dΛ(0) +[A(1), Λ(0)]

F (2) = dA(1) + A(1)A(1)

⇒ δF (2) = [F (2), Λ(0)]

Modification/Generalization?

Non-Commutative Gauge Theory

• D-brane world-volume theory in B-field background. [Chu-Ho 99, Seiberg-Witten 99]

• Commutation relation determined by B-field.• U(1) gauge theory is non-Abelian.€

[x i, x j ] = iΘij

Lie 3-Algebra Gauge Symmetry

• BLG model for multiple M2-branes [07].• needed for manifest compatibility with SUSY.

(ABJM model does not have manifest full SUSY.)€

[T a , T b, T c ] = f abcdT d

A = Aiab (T a ⊗T b )dx i

(Diφ)a = ∂iφa + Aibcφd f bcda

Generalization of Lie 3-algebra• All Lie n-algebra gauge symmetries are special

cases of ordinary (Lie algebra) non-Abelian gauge symmetries.

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[T a1 , T a2 ,L , T an ] = f a1a2L an an+1Tan+1

A = Aa1a2L an−1(T a1 ⊗T a2 ⊗L ⊗T an−1 )

φ = φaTa

[A, φ] = Aa1a2L an−1φan

f a1a2L an an+1Tan+1

More Gauge Theories• Abelian Higher-form gauge theories

• Self-dual gauge theories• Non-Abelian higher-form gauge theories• NA SD HF GT w. Lie 3 (on NC space?)

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A(n ) = 1n! Ai1i2L in

dx i1 dx i2 L dx in

F (n +1) = dA(n )

δA(n ) = dΛ(n −1)

δF (n +1) = 0

Dirac Monopole• Jacobi identity (associativity)

is violated in the presence of magnetic monopoles.• Distribution of magnetic monopoles

gauge bundle does not exist 2-form gauge potential (3-form field strength) (Abelian bundle Abelian gerbe)

Q: How to generalize to non-Abelian gauge theory?

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[[Di, D j ], Dk ] +[[D j , Dk ], Di] +[[Dk, Di], D j ] = 0

Self-Dual Gauge Fields• Examples:

type IIB supergravitytype IIB superstringM5-brane theorytwistor theory

• D=4k+2 (4k) for Minkowski (Euclidean) spacetime.

Self-Dual Gauge Fields• When D = 2(n+1), the self-duality condition is

• a.k.a. chiral gauge bosons.• How to produce 1st order diff. eq. from action?

Trick: introduce additional gauge symmetry

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F = *F

F = 1n! Fi1i2L in

dx i1 ∧dx i2 ∧L ∧dx in

*F = 1n!

1(D −n )!ε i1i2L iD F i1i2L in dx in+1 ∧dx in+2 ∧L ∧dx iD

Chiral Boson in 2D[Floreanini-Jackiw ’87]

• Self-duality

• Lagrangian

• Gauge symmetry

• Euler-Lagrange eq.

• Gauge transformation

• 1-1 map btw space of sol’s and chiral config’s

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˙ φ − ′ φ = 0

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L = 12 ′ φ ( ˙ φ − ′ φ )

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φold (x, t) →φnew (x,t) = φold (x,t) + f (t)

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∂x (∂t −∂x )φ = 0

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⇒ (∂t −∂x )φ = g(t)

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φold (x,t) →φnew (x,t) = φold (x, t) + f (t), ′ f (t) = g(t)

• Due to gauge symmetry, f is not an observable.

• k=f’ is an observable the Lagrangian is

• Formal nonlocality is unavoidable, although physics is local.

• Equivalent to a Weyl spinor in 2D in terms of the density of solitons.

• Lorentz symmetry is hidden.

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L = 12 κ (t,x) dy ε (x − y) ˙ κ (t, y)∫ − κ (t, x)( )

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[φ(t,x), φ(t,y)] = iε (x − y)

Comments• No vector potential needed for gauge symmetry

Vector potential is useful for definingcov. quantities from deriv. of cov. quantities.

But it is not absolutely necessary.• The formulation can be generalized to

self-dual higher-form gauge theories.€

Dμ = ∂μ + Aμ

Example: M5-brane theory (D=5+1)

[Howe-Sezgin 97, Pasti-Sorokin-Tonin 97, Aganagic-Part- Popescu-Schwarz 97, …]

[Chen-Ho 10]

[Ho-Imamura-Matsuo 08]

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bμν (μ,ν = 0,1,2,L 5) → 6/2 = 3 independent pol's"5 +1" formulation

bi5 (i = 0,1,L ,4) → 5 comp's omittedbij → 10 comp's dual to ai → 5 comp's

"4 + 2" formulationbij (i = 0,1,2,3) → 6 comp's omitted

bia , bab (a = 4,5) → 4 * 2 +1 = 9 comp's

"3 + 3" formulationbij (i = 0,1,2) → 3 comp's omitted

bia , bab (a = 3,4,5) → 3* 3 + 3 =12 comp's

Questions• What is the geometry of Abelian higher-form

gauge symmetry? (bundles gerbes?)

• How to define non-Abelian higher-form gauge symmetry?

• Can we generalize the notion of covariant derivative and field strength?

• Do we still need the covariant derivative?

Non-Abelian Self-Dual2-Form Gauge Potential

• M5-brane in the (3-form) C-field background. [Ho-Matsuo 08, Ho-Imamura-Matsuo-Shiba 08]

• Non-Abelian gauge symmetry for 2-form gauge potential

• Nambu-Poisson algebra (ex. of Lie 3-algebra)(Volume-Preserving Diffeomorphism)

• Part of the gauge potential VPD• 1 gauge potential for 2 gauge symmetries! [Ho-Yeh 11]

Non-Local Non-AbelianSelf-Dual 2-Form Gauge Field

• Need non-locality to circumvent no-go thms for multiple M5-branes. [Ho-Huang-Matsuo 11]

• In “5+1” formulation, decompose all fields into zero modes vs. non-zero modes in the 5-th dimension.

• Zero modes 1-form potential A in 5D • Non-zero modes 2-form potential B in 5D

Comments• Self-duality: Instantons, Penrose’s twistor

theory applied to Maxwell and GR.• In 4D, 2-form ≈ 0-form, 3-form ≈ (-1)-form,

4-form ≈ trivial through Hodge duality of the field strengths.

• Expect more new gauge symmetries from string theory to be discovered.

• “Symmetry dictates interaction” -- C. N. Yang.