Pos(CNCFG2010)005 † ∗ -Lie Algebras and Their Enveloping Algebras

PoS(CNCFG2010)005 http://pos.sissa.it/ † ∗ -Lie algebras and their enveloping algebras. We illustrate these ax- n [email protected] , [email protected] Speaker. Report numbers: HWM–11–1 , EMPG–11–03 ioms by describing extensions of Berezin–Toeplitzquantum quantization spaces to of produce relevance various to examples themensions. of dynamics We briefly of describe preliminary M-branes, steps such towards makingmanifolds as the rigorous fuzzy notion by of spheres extending quantized in the 2-plectic groupoid diverse approach di- to quantization of symplectic manifolds. We propose generalized quantization axiomsgeometric for interpretation Nambu–Poisson of manifolds, which allow for a ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Corfu Summer Institute on Elementary Particlesand and Gravity Physics – Workshop on NoncommutativeSeptember Field 8–12, Theory 2010 Corfu, Greece Christian Sämann and Richard J. Szabo Branes, Quantization and Fuzzy Spheres Heriot–Watt University, Edinburgh, U.K. E-mail: PoS(CNCFG2010)005 2 2 4 5 5 6 6 7 8 9 9 12 12 14 11 11 12 10 10 10 Christian Sämann and Richard J. Szabo 2 4 S -Lie algebras n : The fuzzy sphere 3 2 4 3 R S S S -Lie algebras Quantization of 2-plectic manifolds Transgression to loop space Quantization of hyperboloids Quantization of Fuzzy scalar field theory on Generalized Nahm equations n Geometrical meaning of Quantum geometry of branes Axioms of quantization Axioms of generalized quantization Quantization of Quantization of Quantization of A large variety of quantum field theories arise in the low energy limits of string theory. Many 6.1 6.2 5.1 5.2 4.4 1.1 1.2 1.3 1.4 2.1 2.2 4.1 4.2 4.3 . Gerbes and quantization of loop spaces . Quantization of non-compact manifolds . Quantization . Berezin–Toeplitz quantization . Quantization of spheres . Brane dynamics and fuzzy spheres properties and phenomena of stringsame theory time, embedding are field reflected theories in ining these string to theory low unexpected provides energy insights interesting descriptions. and newof points conjectures. At of string It the view, theory, lead- also collectively called provides “branes”, atheory, in means such terms of as of realizing monopoles, conventional solitonic topological vortices, objects nonperturbative solitons and regime of of instantons. field string theory. The dynamics of these objects describe the 1. Brane dynamics and fuzzy spheres 1.1 Generalized Nahm equations 6 5 2 3 4 Contents 1 Branes, Quantization and Fuzzy Spheres PoS(CNCFG2010)005 ] 3 , ] 2 5 , [ 1 [ 4, representing , 3 , 2 , D1-branes. From the 1 N . = ] i l 1 supersymmetric Yang– Nahm equations , τ i ] , = k , X k τ τ φ , , , j l j N Basu–Harvey equations ∂ τ 0 τ [ [ , i jkl = i jk ε i jkl 0 , Christian Sämann and Richard J. Szabo ε ε 0 i jk = = . This describes the polarization of the H = i ] = s 1 i l which can be identified with the worldsheet i i jk ∂ τ ] = is the three-form field strength on the world- X τ k H φ φ , k H X D . Points in the worldsheet of the D1-brane are , X j ) , 3 2 j = X ( [ ]. X [ 4 with su F i jk and with ∗ with radius ε i jkl 2 ε + S 0 and i i + s i τ X φ ]. These equations have a “fuzzy funnel” solution given by i d τ i = d s s 2 1 ∂ X s d d a 2 1 matrix fields describing the transverse fluctuations inside the D3- = X √ 2 4 supersymmetric Yang–Mills theory in four dimensions, obtained i N ) = describes a generalization of the notion of Lie algebra to an entity ∂ s ] 01 = ( × i H − ) = N X , of the Lie algebra s N ( i − i , τ X − 3 are [ , 2 . These monopoles can be interpreted as D1-branes ending on the D3-branes. , φ 1 = i denotes the zero-mode of the Higgs field , ) s s are coordinates along the membranes and ( i a X X While there is an effective description of D-branes, a complete such theory is lacking for M- For example, the low energy effective field theory on a collection of coincident D3-branes in which describes the M2-brane as ative solitonic of the configuration M2-branes, inside one the postulates M5-brane. the existence From of the four perspec- scalar fields volume of the M5-brane. For a soliton solution we take using T-duality as the dimensional reduction of ten-dimensional brane, and spatial coordinate of the D1-branes [ the transverse fluctuations inside the M5-brane, and satisfying the Since these objects are BPS configurations, one can trivially form a stack of with Higgs field perspective of the D1-branes, the effective dynamics is described by the in terms of generators thus blown up into fuzzy two-spheres D1-branes which carry magneticRamond–Ramond charge two-form gauge due potential [ to the anomalous coupling of the D3-branes to a where branes. The situation justM5-branes described are has conjecturally a described natural by lifta Nahm-type flat to equations. M5-brane M-theory The are wherein analog given by M2-branes of BPS ending equations on on called a 3-Lie algebra. Similarly to the D1–D3 system, these equations have a solution Type IIB string theory is Branes, Quantization and Fuzzy Spheres where where the 3-bracket Mills theory. A class ofmonopoles BPS which solutions obey to the the Bogomol’nyi supersymmetric equations Yang–Mills equations are magnetic PoS(CNCFG2010)005 ] 8 ” of 3 S -ary map n  n b . Note that fixing is addressed in [ ) into a single space ) ,..., A 2 corresponds to the 1 1 ) ( + + A i = n g b A n , ( , ] . i b ], noncommutative Gröbner A of End U A , ( 9 1 A ∈ − obeying g n b 1 a via ]. Gauge transformations are built + . 6 n A 1 τ + Christian Sämann and Richard J. Szabo ∈ ,..., n , i A i 1 ] τ . a a [ 1 b ,..., ) , , + 1 n 1 1 1 i τ −→ realized as a quotient of the free associative n − − i i + n ) (or “strong homotopy Lie algebra”). In this ... b n a 1 n 1 -Lie algebras. In [ i ∧ -Lie bracket. The case ( + equipped with a totally antisymmetric n ) ε A n 1 4 so A : A ,..., ,..., ( defined for -Lie algebras appearing in brane models and strong . Its associated Lie algebra is ] 1 − 1 n = b -Jacobi identity” [ ) n C ] = a − U  with basis n n ( A i 1 -algebra 1 ( 1 + τ n ∞ n = = [ ∑ + i A : L n ,..., g Der C ) -structures may play an important role in the quantization of = − ,..., [ b -Lie algebra relevant to the quantization of spheres is denoted ∞ 1  ( i n L ] 1 τ −→ n − [ n b ) . a 1 1 − + n n ( ∧···∧ ,..., A ∧ 1 . These 1 -Lie algebra over a . We now explain this concept in more detail. ]. One can combine the pair of algebras ) A b n δ 7 [ : R , ( 1 δ • is an (ungraded) is a complex vector space − n Ω a L ⊗ -Lie bracket reduces it to an L ,..., n fundamental identity 1 , which can be regarded as an “ a n-Lie algebras  A . s ∈ 2 n-Lie algebra 1 j √ -algebra b -Lie algebras , ∞ . It is defined as the vector space i An One of the goals in the following will be to provide a geometrical meaning and proper defini- The basic example of a metric There is a close connection between the n 1 L a . The fundamental identity and the Jacobi identity are then part of a chain of homotopy Jacobi + n -Lie algebras, as we describe later on. tensor algebra on a basis of 1.2 tion of these noncommutative spheres. We willare work from built the on fact that generalized Nahm equations The problem of constructing associative universal enveloping algebras satisfying the L identities. Hence n A The fundamental identity ensures that they form a Lie subalgebra setting, the generalized Nahm equationsthe are precisely the homotopy Maurer–Cartan equation for This is the unique simple one slot of an usual notion of a Lie algebra. homotopy Lie algebras [ from the point of view ofbases quantization are of used duals to of give a monomial basis for radius from inner derivations Branes, Quantization and Fuzzy Spheres We interpret this solution again as a “fuzzy funnel”, but this time with “fuzzy three-spheres for PoS(CNCFG2010)005 -Lie . For n n S Nambu– to be the A 2. A . The Nambu– 1 = + n 1 supersymmetric n x , regarded as an em- = 2 n f S generates isometries of ,..., } N 1 ) 1 x 1 + n should also help in pinning f + 3 n S -sphere ( ) n ,..., so M -Lie algebras by taking 1 ( 1 f n + ∞ = { n Christian Sämann and Richard J. Szabo i C x 1 + 1 + } n + n 1 A i n g + i −→ n provides a geometric realization of the ... f n 1 n i ∧ S ε -bracket ) -Lie algebra structure n n 5 M ,..., = ( 2 that appears in the M2–M5 system. It is expected f ∞ with Cartesian coordinates } 3 { n C 1 S i 1 is an : x f + n or of the canonical Nambu–Poisson structure on M R 1 = −} + ,..., n 1 } i A 1 x + { ,..., ] it is shown how to interpret the Nahm equations for the D1–D3 n f -spheres as the appropriate quantization (to be discussed below) of -Lie algebras 12 {− n n ]. In this article we explore the question of what is the appropriate ,..., 3 11 f , , 2 -Lie algebra f 10 n 1 ; this is the most studied and best understood fuzzy space. The question of f 2 { S on a smooth manifold . ) . In particular, the associated Lie algebra M 1 ( + ∞ n A C ∈ coincident D1-branes obtained as the two-dimensional reduction of i The basic example of a Nambu–Poisson manifold is the unit We have seen that the effective geometry of the D1–D3 system in string theory is that of a Another context in which theories on quantized spaces arise in low-energy effective descrip- Nambu–Poisson manifolds yield geometric realizations of 2 both these quantizations are well-known and equivalent. f N . Thus we may regard fuzzy = n Poisson bracket is defined as the extension of the for bedded submanifold of Euclidean space by linearity and the generalized Leibniz rule. Thus n 1.4 Quantum geometry of branes fuzzy two-sphere a low-energy description of stacksover the of past M2-branes few has years, beenmultiple and has the M2-branes culminated focus [ in of the much Bagger–Lambert–Gustavsson (BLG) work theory in of M-theory down the gauge structure in an effective description of stacks of multipletions M5-branes. is through modifications toground generalized supergravity Nahm fields. equations In when [ branessystem are as subjected boundary to conditions for back- open strings;on they are BPS equations in the effective field theory that the quantization of the three-spherethe will current give effective interesting descriptions hints of on multiplebranes how M2-branes. ending to on Moreover, appropriately by stacks amend considering of stacks M5-branes, of the M2- properly defined fuzzy notion of noncommutative or deformedbranes geometry that and appears M5-branes, in in thetheory. effective the description context The of answers of M2- to howand these the could questions D-brane significantly might improve realizations the provideproper of current deep definition monopoles understanding of insight lift of the into to M-theory. fuzzy the M- three-sphere nature In of particular, M-branes we seek a satisfying, in addition to the fundamental identity, the generalized Leibniz rule S either the dual of the Branes, Quantization and Fuzzy Spheres 1.3 Geometrical meaning of algebra of functions. They generalize the usual notion of Poisson manifold for algebra Poisson structure PoS(CNCFG2010)005 ], ∧ i ∂ 19 . At i j M θ and the ]. These -field on 2 1 3 8 B S = θ a function of the ] k ∂ is Hermitian; 18 [ † ∧ ˆ f j ∂ ; . In the following we will = -field on the M5-brane is ∧ ; and H 3 i ) C ˆ f ∂ R M ( ˆ i jk f ∞ ]. Θ C , and observables are operators on 1 3! 17 ∈ , 7−→ H Christian Sämann and Richard J. Szabo g , = , 16 f f 1 b , is real then . , and observables are functions on Θ 1 b ∗ ]), which in the present context arises via 15 i jk f M for i j , H 13 Θ θ , } i i 14 ∈ = g ) [ , L 1 and noncommutativity bivector b is a symplectic manifold, then we should further f f 2 ψ 6 Nambu–Heisenberg algebra H \ { R ] = ( M ] = ¯ h k map to automorphisms of the associative operator j i X in ten dimensions. In particular, a constant X , − End M , j i ) X X N , [ ( i ] = to vectors −→ X U ˆ g [ , ) M ˆ -linear, and if f [ M 1 is mapped to the identity operator id C ⊂ ( -field. The effective noncommutative geometry on the D3-brane ∞ is L B = ˆ C f f ] it is suggested that the associated noncommutative geometry should 7−→ 12 1 and noncommutativity 3-vector b f -field. In [ C Dirac’s “wish list” was that of a full quantization, which is a map satisfying: In an analogous way, the Basu–Harvey equations may be regarded as boundary conditions The problem of quantization is highly non-trivial and far from being fully understood. Quan- . Thus quantization provides a map 1. The assignment 2. The identity function 3. Correspondence principle: a function of the two-form j the quantum level, states are rays in a complex Hilbert space focus on the geometries mentioned above,geometry and of discuss the how to Nambu–Heisenberg make algebra sense following of for both a fuzzy large part the treatment of [ which contains the IKKT matrix model as a strong coupling limit. 2. Quantization 2.1 Axioms of quantization tization is essentially a processclassical that level, turns states “classical” are objects points into on their a “quantum” Poisson analogs. manifold At the H ∂ in this case is the well-known Moyal space (see e.g. [ three-form arise from quantization of the constant Nambu–Poisson 3-bracket on that we should subject todemand a that list Poisson of diffeomorphisms axioms depending of on the specificmap applications. Lagrangian submanifolds We could also on the boundary string fields, with central element quantization of the constant Poisson bracket on algebra of quantum observables. If in addition of open membranes. In this case the modification due to a constant Branes, Quantization and Fuzzy Spheres Yang–Mills theory with gauge group accounted for by postulating the relations of the with 3-central element and other quantized Nambu–Poisson manifolds have recently beentions derived in as a supersymmetric 3-Lie solu- algebra reduced model derived from dimensional reduction of the BLG model [ the D3-brane induces a shift in thealgebra Nahm relation equations, which is accounted for by writing a Heisenberg PoS(CNCFG2010)005 n R ∗ T as the al- ) to obtain a ] deformation of “quantiz- i is Hermitian; . Finally, one H x [ ) ( † ˆ f C M ( ∞ = , satisfying: C ˆ f ) . and End ( M G ; and ( 0 Σ ∞ H
H C = -norm is taken with respect K 2 } ( 2 ) L L n . A natural choice can often be ˆ Σ f

. ) is real then , ( } ) ∗ , that corresponds on local charts n σ f f H H ; this is the earmark of ( geometric quantization H ¯ h Christian Sämann and Richard J. Szabo ( = ; this involves a choice of line bundle ,..., ,..., f ) 1 1 End f ˆ f ( , we may also wish to add: σ − { polarization −→ { H
(i.e. a restriction of the deformation parameter ] ¯ h Σ / n i 7 1 ˆ : f G − − Q ¯ h , and leads into = act irreducibly on prequantization 1 -linear, and if U i ,..., ∗ − x 1 C -Lie bracket on End ˆ M f T σ n is the symbol map and the [ is ) Σ = f σ : ( i ¯ h ] n Q 1 is mapped to the identity operator id

ˆ −→ is a Poisson manifold, then these axioms are always satisfied for f 0 ]. Denote the truncated Nambu–Poisson bracket to polynomials of ) = = M → Σ 20 lim ¯ h ˆ ). Secondly, one quantizes only a subset ( f carries a representation of the isometry group f ,..., , and introduce × Q 1 ] for a list of references). Here we will extend Berezin quantization to ˆ K . If f : C 8 [ H M 7−→ σ −} the Nambu–Poisson algebra on the algebra of polynomials f ; this amounts to choosing a M as a cotangent bundle ,..., -Lie algebra [ M , where {− n requires the choice of an ) 0 . M truncating by ( ∞ K ] we proposed a generalization of the quantization axioms to Nambu–Poisson manifolds. C 8 , which can lead to a quantization of ≤ . ∈ 2 M i Axiom 3 There are various loopholes to the obstructions to full quantizations. We shall use Berezin A Nambu–Poisson quantization is a map In [ . The assignment . Correspondence principle: . The identity function S f 0 0 0 4. The quantized coordinate functions ˆ 1 3 2 to a discrete subset of 4a. The Hilbert space ¯ to a chosen measure on Berezin quantization. obtained by corresponding degree This problem is notoriously difficult,metric quantization and (see many [ previous effortsNambu–Poisson manifolds, were keeping devoted the to data of extending a geo- complex Hilbert space quantization which is a hybridgeometry. of It utilizes geometric three and weakenings deformationducibility of assumption, quantization, the which and axioms amounts constructs for to quantization. fuzzy a Firstly, one drops the irre- Branes, Quantization and Fuzzy Spheres The problem with this “wish list”the is Groenewold-van Howe that theorem full there quantizations is in no general full quantization do for not the exist; symplectic in manifolds particular, by for When dealing with homogeneous spaces gebra of quantum observables. The extensionaxioms, requires that the we formulation now of describe. generalized quantization to representing invokes the correspondence principle onlyquantization to linear order in 2.2 Axioms of generalized quantization able functions” on h over and PoS(CNCFG2010)005 and ) = L n ( 1 c CP , the states consists of n of ]. L n . z CP 21 H f [ | = n -dimensional har- i z = ) z ,..., : | 1 ih 1 z M CP z . It obeys the conver- z h | k , , n + 0
n n z ! H i ( n CP ω 0 . For | L k 0 M † α ˆ Z H with first Chern class a , we take as Hilbert space the ) = M ··· ω i ∈ , 1 z HS ) = | † α corresponding to the Fubini–Study ˆ f M

a ( Christian Sämann and Richard J. Szabo
( over ω } C Q i L ]. g , 0 , | ) f 22 ) k = L [ {
span , Z ˆ ) f , † α k ˆ M a Q = ( M -particle Hilbert space of “lowest Landau level α H 0 ( ¯ − z ( k 0 2 8 H  = i H ) ! n α 1 g k ) = ( and 0. Then the correspondence principle always holds ∈ k α Q L ] = z , ) ω H in the homogeneous coordinates i [ −→ f ··· z | i ( k 1 ¯ h z i α | Q z z ˆ |  f ( z | k C h z i h

∞ span = . Then the finite-dimensional Hilbert space → ) lim k
k = ˆ f ( there is a corresponding coherent state k O . -bracket is in general a deformation of the totally antisymmetric operator σ M is quantizable only if this constraint is met. n H n i ˆ , we take the natural Kähler two-form f ) = ∈ n : ω z , ··· ) = L CP this quantization map defines the fuzzy projective space 1 z in the classical limit i M ˆ ( f ( n are the standard creation and annihilation operators of an f n ∞ = i † α ˆ ··· CP a 1 , i M ε α −→ = a K For Generally, an overcomplete basis of the Hilbert space is given by the Rawnsley coherent states. We now review the basic aspects of Berezin quantization that we shall need, particularly of M ; this is the choice of holomorphic polarization. The existence of this holomorphic line bundle ] ω metric, and set homogeneous polynomials of degree and the space can be presented as with respect to the Hilbert–Schmidt norm on End of global holomorphic sections of a “quantum[ line bundle” where ˆ imposes the quantization condition monic oscillator. This spacestates” coincides in with the generalization the of the quantum Hall effect (Landau problem) on are the usual Perelomov coherent states. Theas quantization a is now bridge set between up by classicaland using and quantization the coherent maps quantum states observables in two ways through the Berezin symbol Associated to any Branes, Quantization and Fuzzy Spheres with For gence property automatically. This product 3. Berezin–Toeplitz quantization complex projective space. Generally, on a Kähler manifold vector space PoS(CNCFG2010)005 . n = of g , the CP i 2 αβ ? S λ f ∼ = ] 1 24 . The Poisson k CP ≤ l ]. For 26 with . , m , one can factor the algebra 1 , l n − 25 k Y ρ ˆ a CP . ··· on the Hilbert space [ 1 .

3 ρ

ˆ M 0 with the Gell-Mann matrices Christian Sämann and Richard J. Szabo a 0 R in powers of curvatures approximates ··· β = . k ˆ a i ˆ ⊂ αβ ∞ x | + i i k ) = 2 z σ 0 , given by the zero locus x µ z | 2 R S i jk | n ( z ih ˆ ε p h p i jk 1 −→ 0 ¯ | h ∈

ε −

CP k 1 2 i k n log − L n k β . This construction generalizes to any projective − ⊂ ω = † ρ 9 z ∂ = ˆ H ∞ a 2 CP + ∂ } M | i j ¯ αβ h n z k 1 ∈ ··· | x ] = i ∈ σ j , ω 1 z −→ i µ α ˆ † ρ x x | ¯  ˆ z = k a , i { =  ˆ † α x g [ is the celebrated fuzzy sphere [ ˆ = i a = = z 1 | i ]. Pauli spin matrices z i M αβ x M h ) is spanned by spherical harmonics σ 27 CP 2 H ! ( in the homogeneous coordinates of Σ 1 k ) ] . The corresponding (non-formal) coherent state star-product z SU ) ( = 23 1 p i ˆ x + Lie algebra : The fuzzy sphere n 2 ) ( S 2 by the corresponding ideal and quantize ( n SU su CP by replacing the n is computed explicitly in [ CP maps to the ) ˆ The Rawnsley coherent states also have geometrical applications. The Bergman metric for The quantization map is described by the Jordan–Schwinger transformation which sends the The Berezin quantization of g 2 ˆ S f ( σ The expansion of this metric in the classical limit To leading orders one has [ Einstein metrics for projective Kähler manifolds, e.g. for Calabi–Yau manifolds embedded in A straightforward calculation gives its isometry group of a holomorphic function of functions on space of quantizable functions bracket under the correspondence principle. local coordinates to the operators in this case, so thatspace the classical limit is on For holomorphically embedded submanifolds Kähler manifolds is given by Branes, Quantization and Fuzzy Spheres 4. Quantization of spheres 4.1 Quantization of PoS(CNCFG2010)005 . ) ]. k ⊂ 4 S 1 28 ( ∞ . This k CP C 0. This H × ∩ 1 = Σ 5 to CP x k in H 3 S via the map ∈ 0 is not holomorphic, 3 ˆ f = , given by CP ) , 5 ˆ f 3 . 1 x ⊂ ( . S − l k ˆ 4 x σ 1 ρ S ˆ a i jkl −→ = ε ··· 4 i ) 1 S x ρ k i ˆ ( a Christian Sämann and Richard J. Szabo x ¯ , h β i by a holomorphic ideal. Thus we must k ˆ a | ) H 0 k id ]. ih ] = H ] 0 (  5 5 | 31 ˆ x 1 k 4 with , , ]. In our case, we choose to keep these modes. − k k is proportional to the identity operator id ˆ † ρ + x 30 29 ˆ 10 i , a , 1 j ˆ x allows for consistent solutions to the Basu–Harvey . However, the constraint ˆ i  x 3 β 29 , ··· to isometrically embed x 3 i z ]. [ 1 S ˆ ) x = † ρ α [ 4 5 CP ˆ ¯ 28 z a consists of the noncommutative polynomials of degree 3 S i − R ˆ x † α ) ⊂ ( 2 i ˆ i k αβ a ˆ 1 S x γ −→ = Cl : H 2 4 3 i αβ ( | ] CP 1 S γ z k | ! ˆ x × CP 1 k , 1 j onto a maximal set of irreducible representations of the isometry = ˆ x k , = i yields the noncommutative spheres of Guralnik and Ramgoolam [ i CP : ˆ x x H 4 [ i ⊂ S ˆ x 3 3 4 S S S ] or are dynamically suppressed [ ]. on which the Casimir operator ˆ . ) 28 k 33 5 , ( −} 32 SO was dealt with appropriately; either the radial modes are projected out after operator multipli- In addition to their relevance in the low-energy descriptions of brane dynamics, quantum field The corresponding 3-Lie algebra is defined by [ To quantize the three-sphere, we consider the surjection Our quantization of The restricted coherent state projection of the Berezin symbol 3 ,..., In previous works, the additionalCP radial fuzziness in the normalcation directions [ to This embedding is not holomorphic. Anconsidering the alternative sphere quantization fibration in the same spirit is obtained by and so we cannot factorizeproject the the operator algebra Hilbert End space group yields a nonassociative operator algebra [ theories on Berezin-quantized spacesLet provide us an consider interesting the alternative particular to example lattice of regularization. the noncommutative four-sphere. The Hilbert space and the endomorphism algebra End and the 4-Lie bracket isthis identical case to the the totally correspondence{− antisymmetric principle operator is product satisfied at with affine level. the In truncated Nambu–Poisson4.3 4-bracket Quantization of induces an embedding 4.4 Fuzzy scalar field theory on equations [ For this, we use the Clifford algebra Then the radial fuzziness of our quantum Branes, Quantization and Fuzzy Spheres 4.2 Quantization of gives the quantization map This map yields the quadratic Casimir eigenvalue PoS(CNCFG2010)005 ) ∆ k , is k H ( H in the ) id 5 2 k ( R SO ]. In this in- ] with radius = 35 36 i ˆ x , i isometry group of x 35 ) 4 , ( ) ˆ f . of quantizable functions. SU ( ) . The Laplace operator ) tr k 4 4 S ) S . H 4 ( ( k S ∞ ( k H C . The corresponding path integral ) End id ∩ 3 vol 1 in this case, there is no quantization Σ Christian Sämann and Richard J. Szabo k ∞ = by fuzzy spheres [ are given by
6i . k which was defined in [ M CP ) ¯ 3 h . Define 3 M ( 1 b k ¯ 3 h − k . ) = ∞ ( 6 R ˆ R x ¯ = ¯ h hk f R C i σ O ( = 3 i jk , ◦ r ρ − in A ε k 1 ˆ ∆ x k 2 3 ) k j 2i 4 isometry group of ◦ 3! ˆ ω x CP S + ) i ] = − ( 11 Q 3 ˆ 1 3 x 5 0 ∞ ˆ ( x is infinite-dimensional. CP , H C = r i jk : and Z 2 ε ˆ ] = SO x ∈ j ) , = B = H ˆ 2 f x 1 ∆ ˆ , S x k k i [ ( 1 ˆ x ] = k R [ H 3 ˆ x corresponds to the quadratic Casimir operator of vol H , ) 1 2 ˆ 4 x ∞ = = , S k M ( 1 ˆ f with Hilbert space x ∞ as a “discrete foliation” of [ obeying = 2 4 C i ¯ 3 h S S ] x µ R d carries both an irreducible representation of the 34 H 3 4 S R )) Z k and the Hilbert space ( O C , 3 ∈ of this fuzzy sphere, defined by the quadratic Casimir eigenvalue ˆ ¯ h k is the image of the function CP R ( ) 0 f ( H ρ . and the spinor representation of the = The geometry of the quantized Nambu–Heisenberg algebra Take the fuzzy sphere ¯ 3 h 3 R . Note that, as expected from the non-compactness of k k spinor representation. It carriesvia over the to “Berezin a push” linear [ operator on the endomorphism algebra End and coordinate generators ˆ can be realized in terms of the noncommutative space CP on the algebra of functions This describes the space R condition on is defined by taking theThus integration fuzzy domain quantum to field theories bequantum can the projective be spaces. space used to dynamically distinguish quantum spheres from 5. Quantization of non-compact manifolds 5.1 Quantization of stance there is no truncated 3-bracket ensuringstructure the correspondence is principle. only Hence realized a 3-Lie atof algebra affine level. One possible model for this space is as follows in terms The radius given by where The Hilbert space and algebra of quantized functions on Branes, Quantization and Fuzzy Spheres H An action functional for scalar fields may now be constructed through the integral PoS(CNCFG2010)005 . . ) i ) is ( of ω Z q i A and , I , p ∈ 3 π i M A S ) 2 i ( between 2 − U of global ( ) H 1 ) , which can = ( q L , , p is quantizable U . For example, ∇ M ) L H F ( ω 0 −→ , -Lie algebra j H n M which gives rise to a U , again endowed with ( ) = ∩ ϖ i ϖ , U : -algebra of linear operators . . M H ∗ ) ] for further details. ( 1 j k ji of curvature 8 C − ( U U g ∇ ∩ ∩ , which is a characteristic class of j = i ) ) ) U U q to the L i j , ( , Christian Sämann and Richard J. Szabo ( 1. See [ ∩ . The associated 1 g 1 i i ¯ 3 h M c − U U − R , the local one-form gauge potentials q p ∇ × 1 on ( + F 2 , p ¯ 1 h = SO . A quantization is completed by specifying a R on L  = ) ) µν i j 12 q ( n η , g ν p ˇ Cech cohomology. Given an open cover ) . In this case we must allow for an indefinite metric in x ( i 1 on ) ( µ q A x , SO = d d log p with a unitary connection ( , where ) = = = ki R L ( ) ˇ q g Cech one-cocycle, i.e. they obey the cocycle condition j ∇ , ) ( p F jk A H ( , and the transition functions g − i ) ) i U i j ( ( A of signature g is then constructed by restricting the space of sections of on ) , and which is an element of the integer cohomology group ∇ M µν H . Such a manifold comes with a closed three-form η I ( ∈ -Lie algebra over i n −→ ) i L U , we construct the curvature two-form ( M , and the quantized M5-brane geometry with metric q . AdS + Let us attempt to generalize this picture to a 2-plectic manifold The Hilbert space We will now sketch some ways in which one may arrive at a complete quantization of Our construction can also be extended to the quantization of the hyperboloids p H R on They are all related through prescription of how to map a certain subset of functions on the manifold defining the connection neighbouring charts which define a These relationships are obtained directly from repeated applications of Poincaré’s lemma. we saw that inholomorphic Berezin sections quantization of it the is quantum line given bundle by the vector space an open cover Nambu–Poisson bracket, analogously to the way in which a symplectic structure gives rise to a This two-form is a representativethe of line the bundle first ChernThis class class can be constructed explicitly in if it admits a Hermitian line bundle the unique simple other Nambu–Poisson manifolds. As we discussed, a symplectic manifold be realized as the non-compact homogeneous space 6. Gerbes and quantization of loop spaces 6.1 Quantization of 2-plectic manifolds Branes, Quantization and Fuzzy Spheres 5.2 Quantization of hyperboloids or alternatively as the quadric in the corresponding Clifford algebra. Hence theand quantization non-unitary map representations. involves Important non-Hermitian examples operators infuzzy quantum gravity and string theory involve PoS(CNCFG2010)005 2 ∩ = S j ) U ˇ Cech i j on to the ]. On ( ∩ i ∇ ω 37 U TM . l together with U . −→ -algebra corre- ∩ ∗ M k k E C is a representative U U : ∩ , H ∩ −→ j ρ j k U E U U , ∩ ∩ ∩ l would therefore construct i j ; the connection one-form , i representing the first ) 2 U U j j U U , where , and hence in this case one S ˇ Cech cohomology. From the ∩ ϖ U U κ ∩ , H k ˇ i on triple intersections Cech two-cocycle such that , d ∩ ∩ π i U M ) i i i U 2 −→ ( ∧ ik ∩ U U U ( 3 on j κ on L S = ; the natural notion of “polarization” U : Christian Sämann and Richard J. Szabo = ∩ ϖ π i on H −→ . ) 3 is a vector bundle on
on ) U S ) ) i jk jk µ ( Z ( M jkl , this may be attained by pullback from the h , ( L ) ) of the canonical symplectic two-form ◦ h 1 M ) ) ) ⊗ ) , i ( i jk i j ( ) ( ( and an “anchor” morphism ⊗ ik 3 1 over ( ω ) i j g ∇ B 13 ) ( ( H i j − d F ∇ ( ∗ ( L E L π , : , a 0-connection, i.e. a family of connections ∈ = = = i × id ) 1 on d log M ] 2 ) U
= ( j are equivalent to the specification of transition functions i jk S ) H ◦ ( ( ϖ ∞ = = ) ) [ jk h B κ ( C ) ) . However, in general all of these objects need to be defined i jl i jk ∼ = ( ) ( ∇ jk − 1 jkl h ( h ) κ − ( i 2 over intersections of patches Lie algebroid ⊗ , ( A g 3 U ) 1 B ) , with volume form d = S ji + 3 through the Hopf fibration ∩ ( ∗ ( ikl on patches ) + S
( 1 L 2 ) on triple intersections which define a ik ) 1 g i ( kl S ( U ( ) ) is a characteristic class for a bundle gerbe with connection and curving A L ⊗ 1 1 B = i jl ( ) − ( id − ) i j g ) U ( i j with ) ⊗ ( i j ∇ ( ) . L 2 i jk ) , we obtain by repeated application of Poincaré’s lemma also a 1-connection, from “global holomorphic sections” of the gerbe, and the A ( i jk U −→ i j ◦ g ( ( H ) k h ∪ A H U 1 d i jk ( ◦ U ∩ h ) j = ikl U ( = ) i j h ∩ ( 3 i ∇ ] and references therein. A S F U : on line bundles 38 A naive approach to the quantization of a 2-plectic manifold To find a proper quantization of 2-plectic structures, one can alternatively regard a gerbe as a Dixmier–Douady class ) ) = obeying the coherence condition ji on this bundle defined via pullback d . In the setting of abelian local gerbes this also has a definition in i jk ∗ ( k ( They are all related through sponding to quantized functions fromnow linear appears operators on to be provided by the first cojet bundle rather than the cotangent bundle [ Note that the bundle isomorphisms where a Hilbert space g ∇ cohomology, and bundle isomorphisms Just like the first Chernthis class is a characteristic classH for a Hermitian line bundle with connection, i.e. a family of two-forms closed three-form a Lie bracket on the space of sections κ defines a contact structure on M corresponding structures on Branes, Quantization and Fuzzy Spheres Poisson bracket. We thus expect the quantizationof condition a class in the integer cohomology group sheaf of groupoids with locallytation isomorphic of objects. geometric In quantization this in settingsee terms one [ of may integration extend of the Lie reinterpre- algebroids by Hawkins and others; quantizes the contact manifold precisely. U PoS(CNCFG2010)005 . . I M E ∈ i } ) i . This ( −→ g B { E -invariant ) 1 ( , this algebra 3 U S with differential = M to the M . Given the difficulties )) −→ ) 1 4 ( . Geometric quantization E A of this exact sequence, and U ( M is a vector bundle / ) 1 U ( TL , U , / M ( -algebra. For TL ∞ Christian Sämann and Richard J. Szabo TM , ∗ ], and more generally in the setting of C C 40 ρ −→ one builds the standard Courant algebroid TM is a point a Lie algebroid is the same thing −→ i −→ in [ ) ) U M ρ 1 TM 3 M ( −→ : S ( Courant algebroid s U ∞ E 14  C = which is an extension of the tangent bundle ∗ . When -algebra structure with de Rham differential d, which M ]. The appearence of the loop space is very natural in TL ρ M ∞ −→ 41 TM L id −→ -bundle. The price one has to pay is that the base manifold M −→ ) ∗ -algebra arising in the quantization of symplectic manifolds E of this exact sequence given by the 1-connection 1 ) ∗ T = ( L C E . In this manner, one can define polarizations for Lie groupoids, ( U ρ L ad defines a splitting −→ and L has a natural ) -algebra associated to the Courant algebroid TM TM ∞ ϖ : , L s = M ( E , and then glues together on double intersections using the 0-connection curvatures i U ∗ -algebra which is the convolution algebra of one of the Lie groups integrating T ∗ ] it was demonstrated that the Lie algebroid of a symplectic manifold is replaced by on the line bundle d. Although the issue of integrating such Courant algebroids has not been completely C ⊕ ∗ i 39 ∇ ρ TU . This gives a Courant algebroid In [ In this setting one starts from the observation that the quantization of the dual of a Lie algebra An alternative approach to the quantization of Nambu–Poisson structures employs a trick to ) = = i j is replaced by an infinite-dimensional manifold, the loop space of : ( yields a i ∇ E the context of the quantization of Nambu–Poisson manifolds. Firstly, this was originally observed as well as a splitting infinite-dimensional Kähler geometry in [ the Courant algebroid on a 2-plectic manifold; a E F Branes, Quantization and Fuzzy Spheres tangent bundle which implements a Leibniz ruleLie for algebroid the bracket; with the simplest example is the tangent can be generalized in the followingnatural manner. Lie algebroid, Every the quantizable Atiyah symplectic Lie manifold algebroid comes which with is a an extension of the tangent bundle one can construct a Lie algebra homomorphism vector fields on the total spaceand of reconstruct both geometric and Berezin quantization. that generalizes the structure ofGiven a a Lie bundle algebroid gerbe with equiped connection, with on an each inner patch product on the fibres of as a Lie algebra. g and which can be integrated to aevery Lie morphism groupoid, is abstractly a invertible. category with The should smooth structure therefore in be which identified withnection the convolution algebra of the Atiyah Lie groupoid. The con- of loop spaces is discussed in the example d settled yet, it isOne tempting advantage to of try the toconstruction groupoid generalize of construction, Hawkins’ a approach besides Hilbert to its spaceshould quantization generality, and represent to is comes the this the directly proper setting. to fact quantizationwith the of that defining the the it notion enveloping avoids of algebra the globalfruitful. holomorphic sections of a gerbe, this approach might prove very 6.2 Transgression to loop space substitute the gerbe by a principal M The 2-plectic space embeds in a canonical PoS(CNCFG2010)005 , 1 S H ) to a first M ( M 1 + k ]. Secondly, the Ω available on loop 43 gives rise to a line ]. ∈ M α 45 M , L ], yields a noncommuta- T 44 12 , = .
) 1 b . and the pushforward along τ ) ( M ˙ x ev TM σ , was formulated as the dynamics of ) L − M L τ 3 Christian Sämann and Richard J. Szabo ( τ , k R ( ∈ L % v K 1 b δ ) K K τ ) 1 K S ( i jk . τ K H x ( K Θ K k M ,..., i K ˙ x ) 1 K K τ S is the obvious evaluation map of the loop at the has proved to very helpful in generalizations of on ( i jk = 1 3 × 15 Θ α v ev  S i k M x u , = u L j and τ α u x u -form d  4 u . Put differently, a bundle gerbe on ) , . Here ) 1 and works as follows. Consider the correspondence ev u i 1 R S u σ M x M I u (  + u z j ]. In contrast to the point particle endpoints of open strings, L k x L = ( , amounts to the pullback along M 17 ) )
τ ) ]. ( M τ i ( x 47 k  L , v ( -field [ is the integral over the angle parameterizing the loop. The transgression ] canonical quantization of the M5-brane theory was considered with the • transgression B ] to quantize open membranes ending on an M5-brane in a large constant 1 46 Ω S 45 H . Explicitly, it is given by , 45 ∗ ,..., , ) 44 −→ ev τ 44 ( ) ◦ , and 1 is some 3-algebraic structure, e.g. the bracket of a 3-Lie algebra. Its transgression ! 1 v M . In principle, the latter can then be quantized in the usual way. The difficulty here
] S ( 1 , we can thus map the Dixmier–Douady class on a 2-plectic manifold 1 x − M S ) , H + and its free loop space T • α L − , where we used the natural tangent vector ˙ , Ω M T : M = − ( [ ] where an action principle for Nambu mechanics on ∈ T Using Nevertheless, the transgression map can be used to connect the different perspectives on the This trick is called a T x 42 -field background of M-theory proceeded in analogy to the quantization of open strings ending for This coincides with the noncommutative loop space structure derived in [ is that one is working with the infinite-dimensional base space Chern class on the freebundle loop on space quantum geometry of M5-branes. The conventional approach, cf. e.g. [ space to fill one of the slots of the i.e. loops; Nambu mechanics has also beenstrings and recently membranes proposed in as constant a three-form low and energy four-form effective flux description backgrounds of [ map given angle in between tive space with the description where corresponds to noncommutative loop space relations original attempts [ the ADHMN construction of monopoles (D1-branes) totheory the on self-dual an strings M5-brane of [ the six-dimensional on a D-brane in a large natural Poisson structure on loop space, leadingFinally, to a the complicated map nonassociative algebraic to structure. the loop spaces of the boundaries of openspace membranes structure. are In closed [ strings, and this leads to a noncommutative loop Branes, Quantization and Fuzzy Spheres in [ C PoS(CNCFG2010)005 ]. , J. , Phys. Rev. D , Nucl. Phys. 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