Global Seiberg-Witten quantization for U(n)-bundles on tori

Transcription

1 Global Seiberg-Witten quantization for U(n)-bundles on tori Andreas Deser 1,2 Based on arxiv: with Paolo Aschieri 2,3 1 Faculty of Mathematics and Physics, Charles University, Prague 2 Istituto Nazionale di Fisica Nucleare, Sezione di Torino 3 Dipartimento di Scienze e Innovazione Tecnologica, Universitá Piemonte Orientale, Alessandria February 2019 Quantum Spacetime 19, Bratislava

2 Context and motivation Well-studied area in mathematics physics: Yang-Mills on noncommutative tori (Connes, Rieffel, 1987). Relation to M-theory (Connes, Douglas, Schwarz, 1998). Study of modules over noncommutative tori, corresponcence Heisenberg-modules to quantized U(n)-bundles, SYM over Morita equivalent torus algebras have the same BPS spectrum (Ho, Konechny, Schwarz, 1998). Open string sector: Gauge theory on the worldvolume of D-branes with backgound Kalb-Ramond field is noncommutative. Relation between commutative and noncommutative theory: Local definition of Seiberg-Witten (SW) map (Seiberg, Witten, 1999). Kontsevich formality theorem used to quantize line bundles over arbitrary Poisson manifolds with SW maps (Jurčo, Schupp, Wess, 2002). In this talk: Define SW map for U(n)-bundles over tori and study compatibility with Morita equivalence/ T-duality.

3 The Seiberg-Witten (SW) map M k-dim. manifold, θ Γ( 2 T M) Poisson structure. E M n-dim. vector bundle, associated to a principal G-bundle with local connection form A Ω 1 (M, g). E M C (M) C (M)-module of sections in E, U M, E U C (R k ) n. : Moyal-Weyl star product on R k θ if θ constant, θ U seen as θ 2 T R k. Definition (Seiberg, Witten, 1999) The SW map locally relates a commutative gauge theory (E U, A U,, θ U ) to a noncommutative gauge theory (Ê C (R k θ) n, Â, ) such that commutative gauge variations are mapped into noncommutative ones: ˆΦ(A + δ εa, Φ + δ εφ) = ˆΦ(A, Φ) + ˆδˆε ˆΦ(A, Φ). (1) where Φ, ˆΦ stand for the connection or sections. e.g. for the connection A itself: Â(A + δ εa) = Â(A) + ˆδˆε Â(A), where δ ɛa µ = µε ia µε + iεa µ ˆδˆε  µ = µˆε iâ µ ˆε + i ˆε  µ. (2)

4 The Seiberg-Witten (SW) map Expanding the SW condition for θ µν θ µν + δθ µν gives the SW differential equations for connection Â(A), gauge parameter ˆε(ε, A) and sections ˆφ(φ, A), e.g. in the fundamental rep.: δθ µν  κ θ = π µν 2 δθµν(  µ ( ν  κ + ˆF νκ) + ( ν  κ + ˆF ) νκ) µ. (3) δθ µν ˆε θ = π ) µν 2 δθµν( µˆε Âν + Âν µˆε. (4) δθ µν ˆφ = π ) 2 δθµν(  µ ν ˆφ +  µ D ν ˆφ. (5) θ µν ˆF µν := µ  ν ν  µ iâ µ  ν + iâ ν  µ. Similar differential equations for fields in adjoint rep. Explicit recursive solutions: Formal power series in θ. Question: Seiberg Witten maps globally for non-trivial bundles? Line bundles (Jurčo, Schupp, Wess, 2002) Now: U(n)-bundles over tori

5 Noncommutative U(n)-bundles via twisted boundary conditions T R 2 σ 1,σ 2/(2πZ)2, 2π-periodic functions C (T ) C (R 2 ). Module of sections of U(n)-bundle over T : Take trivial C (R 2 ) n together with U(n)-matrix valued functions Ω 1(σ 2 ), Ω 2(σ 1 ) satisfying a cocycle condition and determining twisted boundary conditions. This gives C (T )-module E n,m, carrying a connection A with topological charge m. NC torus: C (T θ ) =: T θ C (R 2 θ), e iσ1 e iσ2 = e 2πiθ e iσ2 e iσ1. Module of sections on U(n)-bundle over T θ : Subset of C (R 2 θ) n, s.t. φ θ (σ 1 + 2π, σ 2 ) = Ω 1(σ 2 ) φ θ (σ 1, σ 2 ), (6) φ θ (σ 1, σ 2 + 2π) = Ω 2(σ 1 ) φ θ (σ 1, σ 2 ), (7) Ω 1 (σ 2 + 2π) Ω 2(σ 1 ) = Ω 2(σ 1 + 2π) Ω 1(σ 2 ). (8) Connection, e.g. D 1 = σ 1, D 2 = σ 2 i 2π As bimodule, we write En,m θ End(E M n,m θ ) T. ( θ) mσ 1 n mθ 1n n.

6 The induced SW map: From R 2 to the torus SW map: Local for trivial bundles/bimodules: (E = C (R 2 ) n M End(E) C (R 2 ), A µ, θ) SW map (Ê = C (R 2 θ) n End(Ê) MC (R 2 ),  µ ) (9) θ Idea to induce a SW map for bundles on the torus: For (E End(E) MC (T), A µ), see E as a linear subspace of E End(E) MC (R 2 ) and as bimodule with respect to End(E) End(E) and C (T ) C (R 2 ). Apply the SW map to get (Ê, µ) with Ê as a linear subspace of Ê M End(Ê) C (R 2). θ Prove: The noncommutative twisted boundary conditions are satisfied for this subspace, similarly for the endomorphism algebra of this subspace. Definition (Ê, µ) is called the SW quantization of (E, Aµ), where Ê is the subset of all elements in Ê = C (R 2 θ) n that satisfy the noncommutative twisted boundary conditions with the SW quantized ˆΩ 1(σ 2 ), ˆΩ 2(σ 1 ).

7 Result: Induced SW map for U(n)-bundles on the torus Theorem (Aschieri, Deser, 2018) SW induced The induced SW map on torus bundles (E n,m, A µ) (Ê n,m, µ) satisfies Let φ E satisfy the commutative twisted boundary conditions, then its SW quantization ˆφ satisfies the noncommutative twisted boundary conditions, i.e. ˆφ En,m. θ Let Ψ End(E) satisfy the commutative twisted boundary conditions (adjoint), then its SW quantization ˆΨ satisfies the noncommutative twisted boundary conditions, i.e. ˆΨ End(En,m). θ Consequently, we have the commutative diagram (E End(E) MC (R 2 ), Aµ, θ) SW map i (E n,m End(E n,m) MC (T), A µ, θ) (Ê = E θ End(E M θ ) C (R 2 ), µ) θ i θ SW induced (Ê n,m = En,m θ End(E M n,m θ ) T, µ) ( θ)

8 Sketch: SW map vs T-duality/Morita equivalence The iduced SW map enables to study the relation of T-duality (modules over Morita equivalent tori) and SW quantization. Definition: Gauge Morita equivalence Two torus algebras (A θ, à θ) are gauge Morita equivalent if there exists a Morita equivalence bimodule P Aθ M equipped with a constant à θ curvature bimodule connection. P relates right A-modules with A-connections to right Â-modules with Â-connections via the tensor product over A. Specifying to En,m θ M T( θ) and a (gauge) Morita equivalence bimodule P M T ( θ) T ( θ) we have a duality transformation E θ n,m M T( θ) E θ n,m T( θ) P E θñ, m M T( θ). (10) Result (Connes, Douglas, Morariu, Schwarz, Zumino), that (super-) Yang-Mills theories for the modules E θ n,m and E θñ, m have the same BPS spectrum.

9 Sketch: SW map vs T-duality/Morita equivalence For En,m, θ one can show that the SW map does not change (n, m), so (En,m, θ A θ µ) in general is mapped to (En,m, θ A θ µ ). Natural to investigate SW maps before and after applying the duality transformation T( θ) P. There is a compatibility: Compatibility of SW and duality (Aschieri, Deser, 2018) For En,m θ M T( θ) which is mapped via SW to En,m θ M T( θ ) and gauge Morita equivalence bimodules P and P M T ( θ ) T ( θ ) we have the commutative diagram T ( θ) MT ( θ) (E θ n,m, A θ µ) T( θ) P (E θñ, m, A θ µ ) (11) SW θ θ (E θ n,m, A θ µ ) SW θ θ T( θ ) P (E θ ñ, m, A θ µ )

10 Further remarks, outlook Known fact: SW maps are ambiguous. We wrote the ambiguities in a transparent way and used them to show that explicit solutions for sections in E θ n,m known in the literature (Ho, 1998) can be obtained with SW quantization of E n,m. The Theorem on the induced SW map remains true if one includes these ambiguities. Induced SW maps for bundles over higher dimensional tori (used in studies of T-duality in string theory), and more general for bundles over quotients of R k. Physics meaning of compatibility of SW with duality transformations? Applications to closed string theory? Heisenberg nilfolds etc.

11 Appendix on SW map: Ambiguities Well known fact: SW differential equations are ambiguous (Asakawa, Kishimoto, 1999). We approach this by adding extra terms ˆD µνκ(â), Ê µν(ˆε, Â) and Cµν( ˆφ, Â): µν Âκ δθ θ = π µν 2 δθµν(  µ ( ν  κ + ˆF νκ) + ( ν  κ + ˆF ) νκ) µ + Dµνκ(Â). δθ µν ˆε θ = π ) µν 2 δθµν( µˆε Âν + Âν µˆε + Eµν(ˆε, Â). δθ µν ˆφ θ = π µν 2 δθµν(  µ ν ˆφ +  µ D ν ˆφ + Cµν( ˆφ, ) Â). (12) SW condition is satisfied if ˆD µνκ(â + ˆδˆε Â) ˆD µνκ(â) i[ˆε, ˆD µνκ(â)] = D κ Ê µν(ˆε, Â). Ĉ µν(â + ˆδˆε Â, ˆφ + ˆδˆε ˆφ) Ĉ µν(â, ˆφ) i ˆε Ĉ µν(â, ˆφ) = iê µν(ˆε, Â) ˆφ. Similar for fields in the adjoint representation. In case ʵν = 0, any ˆD µνκ and Ĉµν covariant under gauge transformations are solutions to these conditions.

12 Appendix: Ho s quantum U(n)-bundle via SW There s a known solution to the noncommutative twisted boundary conditions (Ho,1998). φ θ k(σ 1, σ 2 ) = s Z m ( m E ( σ2 + k + ns) + j, iσ1) φ n 2π j ( σ2 + k + ns + n j), (13) 2π m j=1 where φ j (x) are Schwartz functions on R Z m and E(A, B) is a normal ordered version of the exponential function E(A, B) := We have the following Observation 1 1 [A,B] The function φ θ = (φ θ k) k=1...n satisfies the differential equation k=0 1 k! Ak B k. θ φθ π Â 2 1φ θ = 3π ˆF φ θ + iπ D 1D 2φ θ. (14) Hence, taking Ĉ 12 = Ĉ 21 = 3 ˆF φ θ id 1D 2φ θ shows, that φ θ satisfies the SW equation for fundamental sections.