Constructing Self-Dual Strings Christian Smann School of - PowerPoint PPT Presentation

Constructing Self-Dual Strings Christian Sämann School of Mathematical and Computer Sciences Heriot Watt University, Edinburgh EMPG seminar 19.1.2010 Based on: CS, arXiv:1007.3301, CMP ... C. Papageorgakis and CS, arXiv:1101.????

Motivation Find an algorithm for the construction of self-dual string solutions Effective description of M2-branes proposed in 2007. This created lots of interest: BLG-model: >440 citations, ABJM-model: >555 citations Inspired by an idea by Basu-Harvey: Propose a lift of the Nahm eqn. describing D1-D3-system: Basu-Harvey eqn. describes M2-M5-brane system Nahm transform: go from Nahm eqn. to Bogomolny monopole eqn. switch perspective from D1-brane to D3-brane Is there a lift for this Nahm transform? go from BH eqn. to self-dual string eqn. switch perspective from M2-brane to M5-brane Such a transform would open up interesting possibilities: eff. description of M5-branes, new integrable structures, . . . Christian Sämann Constructing Self-Dual Strings

Outline We will discuss the construction of monopoles and lift each ingredient to M-theory. Basu-Harvey lift of the Nahm equation and 3-Lie algebras Monopoles and self-dual strings Principal U (1) -bundles, abelian gerbes and loop space ADHMN construction and its lift Examples of self-dual string solutions Non-abelian tensor multiplet on loop space Christian Sämann Constructing Self-Dual Strings

D1-D3-Branes and the Nahm Equation D1-branes ending on D3-branes can be described by the Nahm equation. dim 0 1 2 3 . . . 6 D1-branes ending on D3-branes: D 1 × × A Monopole appears. D 3 × × × × X i ∈ U ( N ) : transverse fluctuations Nahm equation: ( s = x 6 ) d d sX i + ε ijk [ X j , X k ] = 0 Solution: X i = r ( s ) G i with r ( s ) = 1 G i = ε ijk [ G j , G k ] s , Christian Sämann Constructing Self-Dual Strings

D1-D3-Branes and the Nahm Equation The D1-branes end on the D3-branes by forming a fuzzy funnel. dim 0 1 2 3 . . . 6 D 1 × × Solution: X i = r ( s ) G i D 3 × × × × r ( s ) = 1 G i = ε ijk [ G j , G k ] s , The D1-branes form a fuzzy funnel: G i form irrep of SU (2) : coordinates on fuzzy sphere S 2 F D1-worldvolume polarizes: 2 d → 4 d Christian Sämann Constructing Self-Dual Strings

Lifting D1-D3-Branes to M2-M5-Branes The lift to M-theory is performed by a T-duality and an M-theory lift IIB 0 1 2 3 4 5 6 D 1 × × D 3 × × × × T-dualize along x 5 : IIA 0 1 2 3 4 5 6 D 2 × × × D 4 × × × × × Interpret x 4 as M-theory direction: 0 1 2 3 4 5 6 M M 2 × × × M 5 × × × × × × Christian Sämann Constructing Self-Dual Strings

The Basu-Harvey lift of the Nahm Equation M2-branes ending on M5-branes yield a Nahm equation with a cubic term. 0 1 2 3 4 5 6 M A Self-Dual String appears. M 2 × × × M 5 × × × × × × Substitute SO (3) -inv. Nahm eqn. d d sX i + ε ijk [ X j , X k ] = 0 by the SO (4) -invariant equation d d sX µ + ε µνρσ [ X ν , X ρ , X σ ] = 0 Solution: X µ = r ( s ) G µ with r ( s ) = 1 √ s , G µ = ε µνρσ [ G ν , G ρ , G σ ] Basu, Harvey, hep-th/0412310 Christian Sämann Constructing Self-Dual Strings

The Basu-Harvey lift of the Nahm Equation M2-branes ending on M5-branes yield a Nahm equation with a cubic term. Solution: X µ = r ( s ) G µ 0 1 2 3 4 5 6 M M 2 × × × r ( s ) = 1 M 5 × × × × × × √ s , G µ = ε µνρσ [ G ν , G ρ , G σ ] The M2-branes form a fuzzy funnel: G µ form a rep of SO (4) : coordinates on fuzzy sphere S 3 F M2-worldvolume polarizes: 3 d → 6 d What is this triple bracket? Christian Sämann Constructing Self-Dual Strings

What is the algebra behind the triple bracket? In analogy with Lie algebras, we can introduce 3-Lie algebras. d d sX µ + [ A s , X µ ] + ε µνρσ [ X ν , X ρ , X σ ] = 0 , X µ ∈ A Trivial: A is a vector space, [ · , · , · ] trilinear+antisymmetric. ⊲ Gauge transformations from inner derivations: The triple bracket forms a map δ : A ∧ A → Der( A ) =: g A via δ A ∧ B ( C ) := [ A, B, C ] Demand a “3-Jacobi identity,” the fundamental identity: δ A ∧ B ( δ C ∧ D ( E )) := [ A, B, [ C, D, E ]] = [[ A, B, C ] , D, E ] + [ C, [ A, B, D ] , E ] + [ C, D, [ A, B, E ]] The inner derivations form indeed a Lie algebra: [ δ A ∧ B , δ C ∧ D ]( E ) := δ A ∧ B ( δ C ∧ D ( E )) − δ C ∧ D ( δ A ∧ B ( E )) Bracket closes due to fundamental identity. Christian Sämann Constructing Self-Dual Strings

Monopoles and Self-Dual Strings Lifting monopoles to M-theory yields self-dual strings. M 0 1 2 3 4 5 6 0 1 2 3 4 5 6 M 2 × × × D1 × × M 5 × × × × × × D3 × × × × BPS configuration! BPS configuration! Switch perspective: M2 → M5: Switch perspective: D1 → D3: Self-dual string eqn.: Bogomolny monopole eqn.: H µνρ = ε µνρσ ∂ σ Φ ⇒ ∂ 2 Φ = 0 F ij = ε ijk ∇ k Φ ⇒ ∇ 2 Φ = 0 Only single M5 known: Single D3: Dirac monopole Φ = 1 r 2 ⇒ r ( s ) = 1 Φ = 1 r ⇒ r ( s ) = 1 √ s s ⇒ matching profile! ⇒ matching profile! Christian Sämann Constructing Self-Dual Strings

Dirac Monopoles and Principal U (1) -bundles Dirac monopoles are described by principal U (1) -bundles over S 2 . Manifold M with cover ( U i ) i . Principal U (1) -bundle over M : F ∈ Ω 2 ( M, u (1)) , A ( i ) ∈ Ω 1 ( U i , u (1)) with F = d A ( i ) g ij ∈ Ω 0 ( U i ∩ U j , U (1)) with A ( i ) − A ( j ) = d log g ij Consider monopole in ❘ 3 , but describe it on S 2 around monopole: S 2 with patches U + , U − , U + ∩ U − ∼ S 1 : g + − = e − i nφ , n ∈ ❩ � 2 π i � i � S 1 A + − A − = 1 c 1 = S 2 F = n d φ = n 2 π 2 π 2 π 0 Monopole charge: n Christian Sämann Constructing Self-Dual Strings

Self-Dual Strings and Abelian Gerbes Self-dual strings are described by abelian gerbes. Manifold M with cover ( U i ) i . Abelian (local) gerbe over M : H ∈ Ω 3 ( M, u (1)) , B ( i ) ∈ Ω 2 ( U i , u (1)) with H = d B ( i ) A ( ij ) ∈ Ω 1 ( U i ∩ U j , u (1)) with B ( i ) − B ( j ) = d A ij h ijk ∈ Ω 0 ( U i ∩ U j ∩ U k , u (1)) with A ( ij ) − A ( ik ) + A ( jk ) = d h ijk Note: Local gerbe: principal U (1) -bundles on intersections U i ∩ U j . Consider S 3 , patches U + , U − , U + ∩ U − ∼ S 2 : bundle over S 2 Reflected in: H 2 ( S 2 , ❩ ) ∼ = H 3 ( S 3 , ❩ ) ∼ = ❩ i � i � S 3 H = S 2 B + − B − = . . . = n 2 π 2 π Charge of self-dual string: n Christian Sämann Constructing Self-Dual Strings

Abelian Gerbes and loop space By going to loop space, one can reduce differential forms by one degree. Consider the following double fibration: L M × S 1 � ❅ � ev S 1 � ✠ ❅ ❘ M L M Identify T L M = L TM , then: x ∈ L M ⇒ ˙ x ( τ ) ∈ L TM Transgression � T : Ω k +1 ( M ) → Ω k ( L M ) , S 1 ! ◦ ev ∗ T = � ( T ω ) x ( v 1 ( τ ) , . . . , v k ( τ )) := S 1 d τ ω ( v 1 ( τ ) , . . . , v k ( τ ) , ˙ x ( τ )) An abelian local gerbe over M is a principal U (1) -bundle over L M . Note: Most of the time, we will work on L M × S 1 . Christian Sämann Constructing Self-Dual Strings

The ADHMN construction There is a map between monopole solutions and solutions to the Nahm equations. Nahm transform: Instantons on T 4 �→ instantons on ( T 4 ) ∗ Roughly here: � 3 rad. 0 � 3 rad. ∞ : D3 WV T 4 : and ( T 4 ) ∗ : 1 rad. ∞ : D1 WV 1 rad. 0 Introduce (twisted) “Dirac operators”: d s + σ i ⊗ (i X i + x i ✶ k ) , d s + σ i ⊗ (i X i + x i ✶ k ) / s,x = − ✶ d / s,x := ✶ d ¯ ∇ ∇ Properties: / s,x > 0 , [∆ s,x , σ i ] = 0 ⇔ X i satisfy Nahm eqn. ∆ s,x := ¯ ∇ / s,x ∇ Normalized zero modes: ¯ I d s ¯ � ∇ / s,x ψ s,x,α = 0 , ✶ = ψ s,x ψ s,x yield: � ∂ � d s ¯ d s ¯ A µ := ψ s,x ∂x µ ψ s,x and Φ := − i ψ s,x s ψ s,x I I This is a solution to the Bogomolny monopole equations! Christian Sämann Constructing Self-Dual Strings

Examples: Dirac monopoles One can easily construct Dirac monopole solutions using the ADHMN construction. Charge 1: Nahm eqn: ∂ s X i = 0 , so put X i = 0 . Zero mode: √ � x 1 − i x 2 R + x 3 � ψ + = e − sR x 1 − i x 2 R − x 3 Monopole solution: 1 − x 3 1 − x 3 Φ + = − i i � � � � � � 2 R , A + x 2 , − x 1 i = , 0 2( x 1 + x 2 ) 2 R R Charge 2: Nahm eqn. nontrivial. Choose: T i = σ i X i = − 1 sT i 2i = − ¯ T i with Resulting solution: Φ + = − i R , A + i = . . . Christian Sämann Constructing Self-Dual Strings

Lift of the “Dirac operator” There is a natural lift of the Dirac operator to M-theory. Type IIB (twisted): IIB 0 1 2 3 4 5 6 D 1 × × s,x = − ✶ d d s + σ i (i X i + x i ✶ k ) / IIB ∇ D 3 × × × × Type IIA (twisted): IIA 0 1 2 3 4 5 6 D 2 × × × d d s + γ 4 γ i ( X i − i x i ) / IIA ∇ s,x = − γ 5 ✶ k D 4 × × × × × M-theory (untwisted): 0 1 2 3 4 5 6 M M 2 × × × d / M d s + 1 2 γ µν D ( X µ , X ν ) ∇ s = − γ 5 M 5 × × × × × × M-theory (twisted): d d s + γ µν � � / M 1 2 D ( ρ ) ( X µ , X ν ) − i x µ ( τ ) ˙ x ν ( τ ) ∇ s,x ( τ ) = − γ 5 Christian Sämann Constructing Self-Dual Strings