A HA is a unital associative algebra U over a field k with coproduct - PDF document

An Application of Quantum Groups: A q -Deformed Standard Model or And Now for Something Completely Different... Paul Watts Department of Mathematical Physics National University of Ireland, Maynooth Based on “Toward a q -Deformed Standard Model” J. Geom. Phys . 24 61 (1997) 61 arXiv:hep-th/9603143 Workshop on Quantum Information and Condensed Matter Physics 9 September 2011 NUI MAYNOOTH Ollscoil na hÉireann Má Nuad 1

WHY DREDGE UP THIS OLD STUFF NOW? • QGs and HAs have continued to turn up in several areas of physics, not least of which is condensed matter physics... • The Standard Model is currently being pushed to the limit by the LHC in CERN, so the importance of beyond-the-SM physics can only increase in the next few years... 2

OUTLINE • Review of Hopf algebras (HAs) and quantum groups (QGs): definitions and notation • Recasting familiar “classical” ideas in the language of HAs and QGs: Lie algebras and gauge theories • Construction of a toy S U q (2) gauge theory as a deformed version of the Standard Model (SM) • Agreement and disagreement with undeformed SM 3

WHY DEFORM WHAT AIN’T BROKE? (YET) • Practicality : deformation parameters may give al- ternate ways of – for example – introducing a cut- off in renormalisation or a lattice size. • New physics : special relativity and quantum me- chanics are deformed versions of Newtonian me- chanics (with deformation parameters c and � ); who’s to say there aren’t more deformed theories out there? • Fun : why not? At the very least, it’ll be good exer- cise in seeing how QGs and HAs might play a role in other theories. 4

HOPF ALGEBRAS [E. Abe, Hopf Algebras (Cambridge University Press, 1977)] A HA is a unital associative algebra U over a field k with coproduct (or comultiplication) ∆ : U → U ⊗ U , counit ǫ : U → k and antipode S : U → U satisfying ( ∆ ⊗ id) ∆ ( x ) = (id ⊗ ∆ ) ∆ ( x ) ∆ ( xy ) = ∆ ( x ) ∆ ( y ) ( ǫ ⊗ id) ∆ ( x ) (id ⊗ ǫ ) ∆ ( x ) = x = ǫ ( xy ) = ǫ ( x ) ǫ ( y ) · ( S ⊗ id) ∆ ( x ) = · (id ⊗ S ) ∆ ( x ) = 1 ǫ ( x ) *-HA: includes involution θ : U → U θ 2 ( x ) = x θ ( xy ) θ ( y ) θ ( x ) = θ (1) = 1 ∆ ( θ ( x )) = ( θ ⊗ θ )( ∆ ( x )) ǫ ( x ) ∗ ǫ ( θ ( x )) = S − 1 ( x ) θ ( S ( θ ( x ))) = ( ∗ is the conjugation in k ) 5

SWEEDLER NOTATION [M. E. Sweedler, Hopf Algebras (Benjamin Press, 1969)] ∆ ( x ) is generally a sum of elements in U ⊗ U , but sum is suppressed and we write � x i (1) ⊗ x i (2) = x (1) ⊗ x (2) ∆ ( x ) = i So � � ( ∆ ⊗ id) ∆ ( x ) ⊗ x (2) = ∆ x (1) � � � � x (1) (1) ⊗ x (1) (2) ⊗ x (2) = and � � (id ⊗ ∆ ) ∆ ( x ) x (1) ⊗ ∆ = x (2) � � � � x (1) ⊗ (1) ⊗ = x (2) x (2) (2) Coassociativity ( ∆ ⊗ id) ∆ ( x ) = (id ⊗ ∆ ) ∆ ( x ) gives both as x (1) ⊗ x (2) ⊗ x (3) (like ( ab ) c = a ( bc ) = abc ). Similarly, � � · ( S ⊗ id) ∆ ( x ) = ǫ ( x )1 → S x (1) x (2) = ǫ ( x )1 6

QUASITRIANGULAR HOPF ALGEBRAS A QHA is a HA U together with an invertible element, the universal R-matrix, R = r α ⊗ r α ∈ U ⊗ U satisfying ( ∆ ⊗ id)( R ) R 13 R 23 = (id ⊗ ∆ )( R ) = R 12 R 23 R ∆ ( x ) R − 1 ( σ ◦ ∆ )( x ) = where σ ( x ⊗ y ) = y ⊗ x , and r α ⊗ r α ⊗ 1 , R 12 = r α ⊗ 1 ⊗ r α , R 13 = 1 ⊗ r α ⊗ r α . R 23 = R satisfies the Yang-Baxter equation (YBE) R 12 R 13 R 23 = R 23 R 13 R 12 We can construct the special element u ∈ U via · ( S ⊗ id) ( R 21 ) = S ( r α ) r α u = which has the following properties: r α S 2 ( r α ) u − 1 = S 2 ( x ) uxu − 1 = [ uS ( u )] x = x [ uS ( u )] 7

EXAMPLE: A CLASSICAL LIE ALGEBRA If g is a “classical” Lie algebra with generators { T A } , then the universal enveloping algebra U ( g ) is a quasi- triangular Hopf algebra with ∆ ( T A ) = T A ⊗ 1 + 1 ⊗ T A ǫ ( T A ) = 0 − T A S ( T A ) = R 1 ⊗ 1 = If the hermitian adjoint is defined on g , then U ( g ) is a *-Hopf algebra with T † θ ( T A ) = A 8

DUAL PAIRING OF HOPF ALGEBRAS Two HAs U and A over the same field k are dually paired if there is a nondegenerate inner product � , � : U ⊗ A → k such that � xy , a � � x ⊗ y , ∆ ( a ) � = � 1 , a � ǫ ( a ) = � ∆ ( x ) , a ⊗ b � = � x , ab � ǫ ( x ) = � x , 1 � � S ( x ) , a � = � x , S ( a ) � � x , θ ( S ( a )) � ∗ � θ ( x ) , a � = x , y ∈ U , a , b ∈ A 9

REPRESENTATIONS OF HOPF ALGEBRAS A faithful linear representation ρ : U → M ( N , k ) of a HA can be used to dually pair U with another HA A , generated by the N 2 elements � � A i j , via ρ i j ( x ) � x , A i j � = so ∆ ( A i j ) = A ik ⊗ A k j ⇒ ρ ( xy ) = ρ ( x ) ρ ( y ) ǫ ( A i j ) = δ i j ⇒ ρ (1) = I S ( A i j ) = ( A − 1 ) i j � � � � ρ = ǫ ( x ) I ⇒ S x (1) x (2) The multiplication in A is determined by the comultipli- cation in U , but little can be said of that without more info. Which leads us to... 10

QUANTUM GROUPS [V. G. Drinfel’d, Proc. Int. Cong. Math., Berkeley (Berkeley, 1986) 798 S. L. Woronowicz, Commun. Math. Phys. 111 (1987) 613] A quantum group (QG) is a HA A generated by the elements A i j is dually paired with a quasitriangular HA U by means of a representation ρ . The N 2 × N 2 numerical R-matrix is the universal R- matrix in this representation: R ijk ℓ � R , A ik ⊗ A j ℓ � = The dual pairing between U and A gives the commu- tation relations between the generators of A as R ijmn A mk A n ℓ A jn A im R mnk ℓ = or RA 1 A 2 = A 2 A 1 R The numerical version of the YBE is R 12 R 13 R 23 R 23 R 13 R 12 = 11

QUANTUM LIE ALGEBRAS [D. Bernard, Prog. Theor. Phys. Suppl. 102 (1990) 49] A (left) action of U on itself, the adjoint action, is de- fined as � � x ⊲ y = x (1) yS x (2) It satisfies ( xy ) ⊲ z = x ⊲ ( y ⊲ z ) , x ⊲ ( yz ) = ( x (1) ⊲ y )( x (2) ⊲ z ) x ⊲ 1 = ǫ ( x )1 , 1 ⊲ x = x When U is the UEA of a “classical” Lie algebra, then T A ⊲ T B = T A · T B · 1 + 1 · T B · S ( T A ) T A T B − T B T A = [ T A , T B ] = so ⊲ generalises the commutator. 12

The projectors P 0 ( x ) = x − ǫ ( x )1 P 1 ( x ) = ǫ ( x )1 , decompose U into k 1 ⊕ U 0 . U is a quantum Lie alge- bra (QLA) if (a) U 0 is finitely generated by n elements { T 1 , T 2 , . . . , T n } (b) U 0 ⊲ U 0 ⊆ U 0 If U is a quasitriangular HA whose universal R-matrix depends on a parameter λ such that R → 1 ⊗ 1 as λ → 0 and there is a dually paired QG A , then U is a QLA generated by the elements of the matrix 1 X i j � 1 ⊗ 1 − R 21 R , A i j ⊗ id � = λ [P . Schupp, PW, B. Zumino, Lett. Math. Phys. 25 (1992) 139] The deformation parameter q is usually defined via λ = q − q − 1 , with q → 1 giving the “classical limit”. 13

THE KILLING METRIC There is also an invariant trace for such QLAs, defined by tr � ρ ( u ) ρ ( x ) � tr ρ ( x ) = such that tr ρ ( y ⊲ x ) ǫ ( y )tr ρ ( x ) = which vanishes if y ∈ U 0 . This means that the Killing form η ( ρ ) ( x , y ) = tr ρ ( xy ) is invariant under the adjoint action of U 0 : η ( ρ ) � � ǫ ( z ) η ( ρ ) ( x , y ) = 0 z (1) ⊲ x , z (2) ⊲ y = and we may define a U 0 -invariant Killing metric η ( ρ ) tr ρ ( T A T B ) = AB [PW, arXiv:q-alg/9505027] 14

DEFORMED GAUGE THEORIES Mathematically, gauge theories are described in terms of fibre bundles... • Fibre F : where the matter fields live. • Connection Γ : how we move between fibres; the gauge fields. • Structure group A : the group of transformations on the fields. • Base space M : the manifold on which the fields live. We wish to generalise the structure group to a HA, and so the others must be generalised as well. 15

THE FIBRE AND STRUCTURE GROUP Take F to be a unital associative *-algebra (with invo- lution ¯ ) and A a *-Hopf algebra which acts on F via a linear homomorphism L : F → A ⊗ F as ψ (1) ′ ⊗ ψ (2) L ( ψ ) = satisfying ψ (1) ′ ⊗ L ψ (1) ′ � � ψ (2) � � ⊗ ψ (2) = ∆ ψ (1) ′ � � ψ (2) ǫ = ψ ψ (1) ′ � � � � ⊗ ψ (2) L ψ = θ 1 ⊗ 1 L (1) = 16

THE EXTERIOR DERIVATIVES AND CONNECTION Suppose d and δ are exterior derivatives on F and A respectively. The coaction of A on differential forms on F is given recursively by � ψ (1) ′ � � δψ (1) ′ ⊗ ψ (2) + ( − 1) � ψ (1) ′ ⊗ d ψ (2) � � � � L ( d ψ ) = A connection is a linear map taking p -forms on A to ( p + 1) -forms on F satisfying Γ (1) = 0 Γ ( δα ) − d Γ ( α ) = � � � � �� � � � α (1) � α (3) � α (2) � + 1 � + � � � � � � � � � � L ( Γ ( α )) = ( − 1) α (1) S α (3) ⊗ Γ α (2) � � � − δα (1) S α (2) ⊗ 1 17

The FIELD STRENGTH AND COVARIANT DERIVATIVE The field strength is given by � � � α (1) � � � � � � F ( α ) = d Γ ( α ) + ( − 1) � Γ α (1) ∧ Γ α (2) Thus, � � � � � α (2) � α (3) � � � � � � � � L ( F ( α )) = ( − 1) � α (1) S α (3) ⊗ F α (2) . � The covariant derivative D of a p -form ψ on F is ψ (1) ′ � � ∧ ψ (2) , D ψ d ψ + Γ = Thus, ψ (1) ′ � D 2 ψ � ∧ ψ (2) = F and � ψ (1) ′ � � � ψ (1) ′ ⊗ D ψ (2) � � � � L ( D ψ ) = ( − 1) 18