Categorical techniques for NC geometry and gravity

Transcription

1 Categorical techniques for NC geometry and gravity Towards homotopical algebraic quantum field theory lexander Schenkel lexander Schenkel School of Mathematical Sciences, University of Nottingham School of Mathematical Sciences, University of Nottingham GPT Seminar, Cardi University, 20 October 2016 Mathematical models lexander for Schenkel NC in physics Towards homotopical andalgebraic quantum field theory spacetime, Cardi University 1 / 7 Banach Center, Warsaw, November Based joint works with G. E. Barnes and R. J. Szabo [ ; ; ] and with P. schieri [ ]. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

2 Outline 1. Background and motivation 2. Doing algebra in braided monoidal categories 3. Braided derivations and differential calculi 4. Connections and their lifts to tensor products 5. Towards NC vielbein gravity lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

3 Background and motivation lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

4 Recap: Connections on modules Let be NC algebra and (Ω, d) differential calculus over, i.e. Ω = n 0 Ω n with Ω 0 = and d satisfies graded Leibniz rule d(ω ω ) = (dω) ω + ( 1) ω ω (dω ) connection on a right -module V is a linear map : V V Ω 1 satisfying the right Leibniz rule (v a) = (v) a + v da The set of connections Con(V ) is affine space over Hom (V, V Ω 1 ). lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

5 Connections on bimodules Let now V be an -bimodule. What are connections on V? Usual approach: Bimodule connections [Mourad,Dubois-Violette,Masson,... ] right module connection : V V Ω 1 together with an -bimodule homomorphism σ : Ω 1 V V Ω 1 such that (a v) = a (v) + σ(da v) (σ-twisted left Leibniz rule) Bimodule connections lift to tensor products: Given (, σ) on V and (, σ ) on V, then := (id V σ ) ( id V ) + id V : V V V V Ω 1 σ := (id V σ ) (σ id V ) : Ω 1 V V V V Ω 1 defines bimodule connection (, σ) on V V. Bimodule connections form an affine space over Hom (V, V Ω 1 ). such spaces are in general very small! lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

6 Hom (V, W ) is very small For simplicity, consider free -bimodules V = n and W = m. Right -module homomorphisms are -valued matrices Hom (V, W ) = m n L : (v i ) n i=1 ( n ) m L ki v i k=1 Such L s are -bimodule homomorphisms iff all entries are central, i.e. Hom (V, W ) = Z() m n Example: Let = C[x a, p b ] / (x a p b p b x a i δb a ) be 2k-dim. Moyal-Weyl algebra with standard differential calculus Ω 1 = 2k ω a dx a + η b dp b. The center is Z() = C, hence:! Bimodule connections on V = n are affine space over the finite-dimensional vector space Hom (V, V Ω 1 ) 2k n2 = C That s too rigid, in particular for applications to NC field and gravity theories. i=1 lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

7 How can we solve or avoid this issue? For generic NC algebras, the concept of bimodule connections is what is needed for liftings to tensor products [Bresser,Müller-Hoissen,Dimakis,Sitarz]. I will show that for special NC algebras, one can loosen the concept of bimodule connections and still obtains liftings to tensor products. Let me give a first hint what I mean by special by an example: Consider again the Moyal-Weyl algebra = C[x, p] / (xp px i) The product µ : is clearly noncommutative [a, b] = ab ba = (µ µ flip)(a b) 0 when we use flip :, a b b a. However, using the nontrivial braiding τ :, a b e i( p x x p) b a one finds that [a, b] τ = (µ µ τ)(a b) = 0 braided commutative! lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

8 Doing algebra in braided monoidal categories lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

9 Recap: Monoidal categories and monoid objects monoidal category is the following data: a category C, a functor : C C C (tensor product), an object I C (unit object), natural isomorphisms (associator and left/right unitor) α : (c 1 c 2) c 3 = c1 (c 2 c 3), λ : I c = c, ρ : c I = c satisfying the pentagon and triangle identities. Internal to monoidal categories, one can talk about monoids: monoid (or algebra) in C is an object C together with C-morphisms µ : (product) and η : I (unit) satisfying ( ) α ( ) id µ µ id µ µ I η id λ µ id η ρ I Ex: Monoids in Vec K are associative and unital K-algebras. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

10 Braided monoidal categories braided monoidal category is a monoidal cat C with nat. iso (braiding) τ : c 1 c 2 = c2 c 1 satisfying the hexagon identities. (If τ 2 = id, symmetric monoidal category.) This allows us to talk about braided commutative monoids/algebras: monoid (, µ, η) in C is called braided commutative if the product is compatible with the braiding, i.e. µ τ µ Rem: I like to interpret braided commutative monoids (, µ, η) as algebras where the commutation relations are dictated by τ. cf. Giovanni Landi s talk! lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

11 Bimodule objects Let C monoidal category and (, µ, η) monoid in C. n -bimodule in C is an object V C together with C-morphisms l : V V (left action) and r : V V (right action) satisfying the obvious compatibilities. If C has coequalizers, we can equip the category C of -bimodules in C with a monoidal structure where I = and is given by (V ) W r id (id l) α V W V W If C is braided monoidal category and braided commutative monoid, we call V C symmetric iff V τ V and V τ V r V l l V r Prop: The braiding on C descends to a braiding on the monoidal category C sym. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

12 Examples from quasi-triangular Hopf algebras Let (H, R) be a quasi-triangular Hopf algebra. left H-module is a vector space V with H-action : H V V. Tensor products of left H-modules defines monoidal category ( H M,, K). NB: ssociators and unitors are trivial, hence they will be suppressed. Using R-matrix R = R (1) R (2) H H, we obtain braiding τ : V W W V, v w R (2) w R (1) v Braided commutative monoids (, µ, η) in H M are H-module algebras satisfying the commutation relations a b = (R (2) b) (R (1) a). Ex: The Moyal-Weyl algebra, NC torus, Connes-Landi sphere, etc., are braided commutative for Hopf algebra H = UR 2k with R = exp(i Θ lm t l t m ). (More fancy example, see blackboard!) ssociated to each braided commutative monoid (, µ, η) in H M is a braided monoidal category ( H M sym,,, τ ) of symmetric -bimodules. NB: For simplicity, I will focus for the rest of this talk on these examples! lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

13 Braided derivations and differential calculi lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

14 Ordinary vs. braided derivations: first look n ordinary derivation on a braided commutative monoid in H M is an H M -morphism X : satisfying the Leibniz rule X(a b) = X(a) b + a X(b) braided derivation on is a linear map X : (not necessarily H-equivariant!) satisfying the braided Leibniz rule X(a b) = X(a) b + (R (2) a) (R (1) X)(b) where the H-action on linear maps is via adjoint action h X := (h (1) ) X (S(h (2) ) ) Prop: There is a linear isomorphism { ordinary derivations on } = { H-invariant braided derivations on } There are many more braided derivations than ordinary ones! Hence, braided derivations are more flexible for doing geometry on. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

15 Categorical interpretation via internal homs H M has internal homs ζ : HomH M (Z V, W ) = HomH M (Z, hom(v, W )) Explicitly: hom(v, W ) H M is Hom K (V, W ) with adjoint H-action. From abstract non-sense, one obtains H M -morphisms: Evaluation: ev : hom(v, W ) V W Composition: : hom(w, Z) hom(v, W ) hom(v, Z) Let us adjoin µ : to ζ(µ) : end() and define the bracket [, ] := τ : end() end() end(). Braided derivations on are those X end() satisfying [X, ζ(µ)(a)] = ζ(µ) ( ev(x a) ) Formally, der() H M is characterized by the equalizer in H M der() end() ζ([,ζ(µ)( )]) ζ(ζ(µ) ev) hom(, end()) lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

16 Construction of Kähler-style differentials Similarly, one defines braided derivations der(, V ) valued in V H M sym One can show that der(, V ) H M sym (a X)(b) := a X(b), (X a)(b) := X ( R (2) b ) (R (1) a) The functor der(, ) : H M sym where hom (V, W ) H M sym via H M sym der(, ) = hom (Ω 1, ) is internal hom in H M sym ; is representable via (Right -linear maps with adjoint H-action and suitable -bimodule structure.) Ω 1 = Γ/Γ 2 H M sym where Γ := ker ( µ : ) is H-module algebra. (The differential on Ω 1 is the typical one d : Ω 1, a a 1 1 a.) Construct Ω as semifree braided graded-commutative DG over Ω 1. Conclusion: ny braided commutative monoid in H M admits a canonical differential calculus obtained from braided derivations. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

17 Connections and their lifts to tensor products lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

18 Connections on H M sym via internal homs Wanted: Carving out space of connections on V from hom(v, V Ω 1 ). First observe that the right Leibniz rule (v a) = (v) a + v da is equivalent to (with R 1 = R (1) R (2) inverse R-matrix) (a v) (R (2) a) (R (1) )(v) = R (1) v R(2) (da) Both sides are obtained from H M -morphisms to internal homs: lhs := [, ζ(l)( )] : hom(v, V Ω 1 ) hom(v, V Ω 1 ) rhs := ζ ( (id d) τ 1) : hom(v, V Ω 1 ) is the adjoint of the H M -morphism (id d) τ 1 : V V Ω 1 Define the object of connections con(v ) H M by equalizer con(v ) hom(v, V Ω 1 ) K lhs pr 1 rhs pr 2 hom (, hom(v, V Ω 1 ) ) NB: Elements of con(v ) are pairs (, c) hom(v, V Ω 1 ) K satisfying the continuous right Leibniz rule (v a) = (v) a + c v da lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

19 Construction of tensor product connections Question: Given V, V H M sym, (, c) con(v ) and (, c ) con(v ). Can we construct a connection on V V? That s indeed possible! To formalize our construction, we use: Tensor product: : hom(v, W ) hom(x, Y ) hom(v X, W Y ) Fiber product: con(v ) K con(v ) con(v ) con(v ) pr K K pr K Main Theorem [Barnes,S,Szabo: ] Let V, V H M sym. There exists an H M -morphisms (called sum of connections) : con(v ) K con(v ) con(v V ), ( (, c), (, c) ) ( ( ) ) τ , c Moreover, the sum of connections is associative. lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

20 Comparison to bimodule connections How is our notion of connections different from bimodule connections? Braided connections (v a) = (v) a + v da (a v) = (R (2) a) (R (1) )(v) + R (1) v R(2) (da) ( (v v ) = τ 23 (v) v ) +(R (2) v) (R (1) )(v ) Bimodule connections (v a) = (v) a + v da (a v) = a (v) + σ(da v) ( (v v ) = σ 23 (v) v ) +v (v ) lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

21 Further aspects of connections [Barnes,S,Szabo: ] To any connection (, 1) con(v ) one can assign its curvature curv( ) hom (V, V Ω 2 ) The curvature behaves additively under sums of connections curv ( ) = τ 23 ( curv( ) 1 ) + 1 curv( ) Interpreting internal homs hom (V, W ) H M sym as homomorphism bundles, we also would like to induce connections on them: Thm: Let V, W H M sym. There exists an H M -morphism ad : con(w ) K con(v ) con(hom (V, W )) Cor: Denote by V := hom (V, ) the dual module. Then there exists an H M -morphism ( ) : con(v ) con(v ) lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

22 Towards NC vielbein gravity lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18

23 Deformation of vielbein gravity Let M be 4d spin mnf with trivial Dirac spinor bundle S = M C 4 M Let H = UVec(M) F and R = F 21 F 1 for some cocycle twist F Twist deformation quantization constructs -product on C (M) and -bimodule structure on Γ (S), such that = (C (M), µ F, η F ) is braided commutative monoid in H M ; V = (Γ (S), l F, r F ) H M sym. (Note that V = 4 is free.) NC vielbein gravity coupled to Dirac fields requires the following fields: Dirac and co-dirac field: ψ V and ψ V = hom (V, ); Spin connection: (, 1) con(v ) such that = d 1 2 ωab [γ aγ b ]; Vielbein: E end (V ) such that E = E a γ a. Using our constructions, we can define Lagrangian for this NC field theory L = tr (i curv( ) E E γ 5 ( (ψ) ψ ψ (ψ) ) ) E E E γ 5 NB: The noncommutativity is in the composition and evaluation of internal homs! lexander Schenkel Categorical techniques for NC geometry and gravity Warsaw / 18