Heriot-Watt University Research Gateway

An Introduction to Nonassociative Physics

Citation for published version: Szabo, R 2019, 'An Introduction to Nonassociative Physics', Proceedings of Science, vol. 347, 100. https://doi.org/10.22323/1.347.0100

Digital Object Identifier (DOI): 10.22323/1.347.0100

Link: Link to publication record in Heriot-Watt Research Portal

Document Version: Publisher's PDF, also known as Version of record

Published In: Proceedings of Science

General rights Copyright for the publications made accessible via Heriot-Watt Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights.

Take down policy Heriot-Watt University has made every reasonable effort to ensure that the content in Heriot-Watt Research Portal complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim.

Download date: 26. Sep. 2021 PoS(CORFU2018)100 -flux R -flux back- R https://pos.sissa.it/ ∗ EMPG-19-10 [email protected] Speaker. We give a pedagogical introduction toments the in nonassociative quantum structures mechanics arising with from magnetic recentgeometric monopoles, develop- in fluxes, string and theory in and M-theory M-theoryoverview with with of non- non-geometric the Kaluza-Klein main monopoles. historicaltheory appearences After and of a M-theory, we nonassociativity brief provide inelectric a quantum charges detailed in mechanics, account the string backgrounds of ofreciprocity the various to classical distributions map and of this quantum magnetic system dynamics charge. to of We the apply phase Born space of closed strings propagating in grounds of string theory, andbackgrounds then of describe M-theory. the Applying liftphase Born to space reciprocity the of maps phase M-waves this probing space M-theoryfour a of configuration perspective non-geometric M2-branes to systems Kaluza-Klein in the are monopole unified background.space. by These a covariant 3-algebra structure on the M-theory phase ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2018 "School and(CORFU2018) Workshops on Elementary Particle Physics and31 Gravity" August–28 September 2018 Corfu, Greece Preprint: Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom. Maxwell Institute for Mathematical Sciences, Edinburgh, UnitedThe Kingdom. Higgs Centre for Theoretical Physics, Edinburgh, United Kingdom. E-mail: Richard J. Szabo An Introduction to Nonassociative Physics PoS(CORFU2018)100 2 2 4 5 6 8 2 29 31 31 32 33 27 27 26 25 23 23 10 11 11 12 14 16 20 Richard J. Szabo 1 -flux model R -flux model R Magnetic monopoles in quantum gravity The M-wave phase space The covariant M-theory phase space 3-algebra Quantization of the M2-brane phase space M-theory lift of the Nonassociative gravity How closed strings see nonassociativity The The octonions Jordanian quantum mechanics Nambu mechanics Nonassociativity in string theory Nonassociativity in M-theory Outline Magnetic Poisson brackets Magnetic monopoles Classical motion in fields of magneticQuantization charge of magnetic Poisson brackets Nonassociative quantum mechanics Nonassociative algebra 5.1 5.2 5.3 4.2 4.1 3.3 3.2 3.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 1.1 . M-waves in non-geometric Kaluza-Klein monopole backgrounds . M2-branes in locally non-geometric flux backgrounds . Electric charges in magnetic monopole backgrounds . Closed strings in locally non-geometric flux backgrounds . A brief history of nonassociativity in physics 5 4 2 3 Contents 1 An Introduction to Nonassociative Physics PoS(CORFU2018)100 , 6= C ]] (1.1) z 0. In , y [ , x ] = [ z , y , and x Richard J. Szabo [ ] x , y 6= [ ] y , x [ ,
] 0. A well-known example of a on a vector space is generally a y , ] ] 6= · z , , themselves, the complex numbers · x yx [ R − ] + [[ of (commutative) Jordan algebras, which are commutative and associative algebras, z xy ) , ] C = y : yx , ( ] x 2 y [[ and , x 2 is both noncommutative and nonassociative. The x [ − R = ]] O x z , ) y y [ , 2 x . Both x [ ( O 3 1 = : ] z , y , x [ of real numbers: The real numbers R , and the octonions H , but rather it is antisymmetric and satisfies the Jacobi identity. If we package the quantity ] z , Nonassociative algebras first appeared around the middle of the nineteenth century and have Algebras whose associativity is controlled by identities like the Jacobi or alternative identities Nonassociative algebras actually have a long and diverse history of appearences in physics, For later use, let us begin by recalling the standard mathematical example of the octonions, as ] y is noncommutative but associative, while , x since formed an independent branch ofof mathematics. a By nonassociative far algebra the is simplest a and Lie best known algebra. example A Lie bracket An Introduction to Nonassociative Physics 1. A brief history of nonassociativity in physics 1.1 Nonassociative algebra noncommutative and nonassociative binary operation in the sense that then Lie algebras are characterized by the feature that this 3-bracket vanishes: [[ which vanishes by the Jacobi identity into a ternary operation called the ‘Jacobiator’, our present context this willthe be binary considered operation to be on ais the trivial associative. Lie example group In of which fact, athat in nonassociative integrates has algebra, this the non-vanishing as paper Jacobiators, Lie just we algebra like willcharacterized the via be multiplication by the in looking a a exponential at non-vanishing noncommutative map cases algebra commutator noncommutative can of be and algebras nonassociative with algebra a in multiplication stitutes this an sense important is example the of algebra(and an of ‘alternative hence octonions, algebra’, the where which Jacobi associativity con- ofwhich identity) the generalizes is multiplication the generally Jordan violated, identity but it possesses the alternativity condition the quaternions were of central mathematical interestalgebra in does the not last lead century, to ase.g. a arbitrarily good [1]. relaxing mathematical associativity theory in They unless an havequantum some mechanics, recently other string structures made theory are and their present, M-theory.of way see The these into main recent insights aim the and of to forefront these highlighttogether lectures some of of with is some the the to interesting many recent survey physical important some consequences developments of open the in avenues theory, awaiting further investigation. which is perhaps notreview so some of widely their appreciated. occurences withments a in The personally physics biased purpose with point of of andevelopments view: eye this that We to we only opening the shall discuss section treat main develop- in topics is more that detail to we in briefly will subsequent pursue sections. 1.2 later The on, octonions and to overview the it will play an importantnice introduction). role A in well-known some theorem in of algebraalgebras the states over that the discussions there field throughout are only this four paper normed division (see e.g. [2] for a we review below. H PoS(CORFU2018)100 = A (1.2) (1.3) (1.4) (1.5) (1.6) (1.7) , is only A e ABC by 1 and an η R Richard J. Szabo and represent the (the Lie algebra of 3 f O , 2 f ⊂ , 1 , H f generated by the inclusions i e = H 2 , of the rotation group in seven . i 6 is generated over − f ⊂ e
) , (with only one non-zero compo- , ) 7 C 2 C 5 7 ( C e ] = e e i is of the form , i jk ⊂ 7 f , B 4 ε ] = SO i a , , e e can be written in the form R O D e 7 ( , , C e ⊂ e + A [ O 7 e of k e 2 e e [ ··· G O − whose structure constants are invariant under ABC i jk ABCD + C . ε η H η e 2 ) ) 4 e + + k B f , the symmetry group of the tensor and 2 3 − i j e ) a AB also having only seven non-vanishing components. and δ 3 A i jk δ ( e + ε − ] = ( is generated by 1 and three imaginary quaternion units 1 and 1 − C k e SO − = e − is then equivalent to the statement that its Jacobiators are e 2 H ABCD 1 = 7 j , k a e η B e e B = O i jk i e e i j ε ] = 2 i + , e . i jk δ A e 2 C 0 A ε ( form a completely antisymmetric tensor with only seven non- e e e H a [ 2 − 2 , is generated by 1 and seven imaginary octonion units B 1, while = e ABC O , − ] = ] = ] = O η j j j A f f e √ e [ , , , i i i 1, so that a generic element f = e e [ [ [ − . The multiplication in the algebra explicitly, it is convenient to rewrite 3, which obey = , R 2 A 2 e , ∈ ABC 1 7 η property of the algebra ), analogous to the hierarchy of embeddings a = ) i 3 , ( i is the Levi-Civita symbol in three dimensions. ,..., σ is generated over itself by a central identity element 1, 1 SO 1 a R i jk , − ' ε 0 7, satisfying ) a √ Analogously, the algebra To specify alternative 2 ( = ,..., i i, j and k, familar from quantum mechanics in their representation in terms of Pauli spin matrices 1 e An Introduction to Nonassociative Physics algebra where with algebra of octonions in terms of the commutators imaginary complex unit i where the structure constants vanishing components, generalizing the structurenent) of constants the algebra of quaternions The first set of commutation relationsSU describes a quaternion subalgebra of 1 and i; this followsThis from demonstrates the the important Cayley-Dickson feature construction that, of unlike the real normed division algebras. the full three-dimensional rotation group the 14-dimensional simple exceptional Lie subgroup dimensions. Using these commutators one can derive the non-vanishing Jacobiators with antisymmetric structure constants proportional to its ‘associators’: The PoS(CORFU2018)100 + 2 (1.8) (1.9) A . The (1.11) (1.12) (1.10) . are Hermi- special B Jordan identity Richard J. Szabo and A formally real as a commutative power-
2 B − noncommutative Jordan algebra 2 A -algebra, which can be represented ∗ − ) C 2 A ) Jordan algebra ◦ B , . B , + ( m ) ◦ + A A n ( yx 2 ◦ ( A A B x 1 2 4 in general. However, it satisfies the = = = = property ) m A x ) = C B A . Hermitian operators also have the property that ) ◦ ◦ ◦ ◦ ) m B BA n xy A B ( ( A ◦ + ◦ 2 and A A n AB ( 6= ( on an associative algebra, as above, it is said to be is an example of what may be called a noncommutative Jordan 1 2 C ◦ ◦ O = ) [4, 5], B B ◦ ◦ power-associativity A 0, i.e. the symmetrized product on observables is A ( = B alternative = A 0 implies The basic observation is that the symmetrized product Abstracting these properties leads to the notion of a The starting programme of Jordanian quantum mechanics was then to find examples of non- The algebra of octonions = Hermitian. This product is obviously commutative, 2 special Jordan algebras as algebras of observables for quantum systems. These hopes were dashed by operators on a finite-tian operators, or then infinite-dimensional their separable operator Hilbert productneither is is space. not their a If commutator. Hermitian Thus operatorobservables, neither (unless and they of so commute), these they and binary arethe operations not system. closes an on Jordan the intrinsic set algebras partquantum of system. were of physical introduced the physically to meaningful formalize characteristics the of properties of a finite-dimensional for all non-negative integer powers B associative algebra; in other words,the although usual the multiplication of set operators, of they do observablesproduct. form do a If Jordan not the algebra form multiplication with on an respect to a algebraform the Jordan under of symmetrized algebra a can symmetrized be product brought via a suitable isomorphism to the By abstracting this property wewherein may we relax thus the define commutativity the of notion the multiplication. of a algebra, which we now proceedciativity to in define. physics, in Jordan the algebrasthem early were was days the to of first define quantum appearencehence theory an of an [3]. algebraic nonasso- overall structure Jordan’s new motivation algebraic on for settingany the introducing for quantum set the system foundations of are of observables given quantum in by theory. Hermitian quantum The elements mechanics, observables in of and a An Introduction to Nonassociative Physics 1.3 Jordanian quantum mechanics is but it is nonassociative: which is equivalent to the power-associative property implies that itJordan algebra is is sufficient to demand that the binary operation on a PoS(CORFU2018)100 (1.14) (1.17) (1.16) (1.13) (1.15) on the Nambu- , ·} , 0 · , = } {· Richard J. Szabo 2 h , Hermitian matrices 1 h n , } × 3 h n , 3 Hermitian matrices over given by g 3 , × f R = , g M } } − {{ , 3 k called a ‘3-Lie algebra’ [10]. , h h ) , h k , 1 ∂ f M h ( g } , { j 2 ∞ } ; likewise, a Nambu-Poisson bracket defines ∂ ) 2 H + C f ~ ω h , i } M , 1 ( k ∂ × × × g H , , ∞ 5 L , h ~ f i jk f C , ε = g { {{ { t = L . The hope was to discover new (generalized) integrals ~ f = : + d 2 d . These axioms extend the axioms of a Poisson bracket } f } t H ) = 3 h d d , h } M , g ( k 2 , and , ∞ f h h 1 , C , { is a completely antisymmetric ternary operation } H 1 ∈ f g M h k , { , g 3 , h f which obeys the Leibniz rule, , 2 h M , 1 h , }} − {{ h 3 , , the exceptional 27-dimensional Albert algebra of 3 h on a manifold g , and the Clifford-type (or ‘spin’) Jordan algebras; among these, only the Albert al- , , H 2 f O h , 1 and h { C , , g , R f Nambu’s idea was to reformulate the equations of motion of complicated non-integrable dy- Let us now fast-forward ahead many years to a very different kind of nonassociative structure Nambu’s original example was the canonical 3-bracket on { namical systems as a bi-Hamiltonian dynamics with flow equations a higher version of a Lie algebra on the vector space which defines a Lie algebra on the vector space for functions and the motivating dynamical problem is given by the Euler equations algebra of functions on and a higher version of the Jacobi identity for Lie algebras called the “fundamental identity”, that takes place directlyNambu at [9]; the level they of were a formalised ternary and operation. generalised 20 Such years structures later were by introduced Takhtajan by [10]. A over the octonions gebra, equiped with the symmetrized product,based is on non-special. the An Albert ‘octonionic quantum algebraof mechanics’ was quantum later theory, developed despite by thereOperators [7] being which acting no satisfies on Hilbert the a space vonWigner separable formulation Neumann theorem for Hilbert was axioms generalized nonassociative space quite algebras: necessarily some timecase. associate. later This by implies Zelmanov The that [8] there Jordan-von to is the Neumann- modate no infinite-dimensional infinite-dimensional the non-special observables Jordan of algebra quantum thatfrom would mechanics, playing acco- thus a eliminating viable the role in prospects the of foundations Jordan of algebras quantum1.4 theory. Nambu mechanics Poisson bracket involving a pair of Hamiltonians of motion. Consistency ofalso these the time fundamental identity evolution if equations oneso necessitates requires that both time they derivatives the play to Leibniz act crucial asPoisson rule roles derivations bracket analogous and of in to the ordinary the 3-bracket, Hamiltonian roles mechanics. of the Leibniz rule and Jacobi identity of the An Introduction to Nonassociative Physics by the famous work of Jordan,mally von real Neumann finite-dimensional and Jordan Wigner [6] algebras who are proved the that Jordan the algebras only simple of for- PoS(CORFU2018)100 = i ∂ (1.18) (1.19) ) with and angular L ~ Richard J. Szabo 1.16 . This perspective will be the . A ] ~ ω ) instead of the less natural half- C · , L ~ 1.1 B 1 2 + [ = B ] T

C T , = , with orbital angular momentum A 2 2 [ 3 L H ~ , − i R 6 L C ]  and B 2 , = L to be the “half-Jacobiator” [9] ~ A i t L = d d = [ 1 H NH ] nonassociative quantum mechanics C , B , Noncommutative geometry first made its appearence in string theory A [ of the angular momentum with respect to the 3-bracket ( i L Nambu-Heisenberg bracket about its principal axes. Nambu’s observation was that these equations are equivalent to ~ ω , and the pair of Hamiltonians i String theory has been known for a long time to lead to new notions of geometry which have Attempts to quantize the Nambu-Poisson bracket brings us closer to our earlier discussions In light of these issues, Nambu suggested to use nonassociative algebras to quantize his 3- L ∂ / in the mid-1980s with Witten’s work on bosonic open string field theory [12]. Shortly thereafter, driven many developments and fieldsit of also mathematics. offers As a host a ofciative candidate structures modifications quantum in and theory string deformations of theory of gravity, have themodern a notion developments relatively of long that spacetime history we itself. ofso shall Nonasso- appearences widely predating discuss appreciated. the in more Here subsequentnonassociativity we in sections, shall the highlight which context some of is string of again theory. the perhaps most important not Closed string selected field examples theory. of Jacobiator. These arguments thus demonstrate thatintimately the related problem to of quantizing formulating Nambu mechanicsdriving is theme throughout the rest of this article. 1.5 Nonassociativity in string theory of nonassociativity. If one wishescorrespondence to principle use of quantum a mechanics, correspondence thenbrackets, principle the analogous by natural to extension mapping the of them usual quantizingNambu-Poisson Poisson Bohr to brackets to commutators Jacobiators, of butHilbert operators this space on does always not a associate work and Hilbertcan because so define space, their operators the Jacobiators on would vanish, a be as separable to mentioned before. map Instead, one bracket, which we can then do via the Jacobiator defined in ( for the components ∂ consisting of an antisymmetric combinationthe of half Jacobi of identity. all terms Thisthe which proposal fundamental would unfortunately otherwise identity meets vanish is by with notbetween several preserved the problems, under Leibniz the most rule correspondence notably andfied principle. that the by fundamental There a identity is Poisson as, aconstraints bracket, in tension on contrast the the to allowed fundamental Nambu-Poisson the identity brackets. Jacobigebra Hence imposes of identity it algebraic operators is satis- that difficult in would to addition quantize findquantization the a to Nambu-Poisson of suitable 3-bracket differential Nambu-Poisson 3-Lie of brackets al- functions. to In thismodern this day perspective sense remains on the an this open problem problem; within see our [11] present context, for together a with more further references. An Introduction to Nonassociative Physics describing the motion of a rotatingvelocity rigid body in the bi-Hamiltonian equations PoS(CORFU2018)100 controls the B Richard J. Szabo d -ary operations. n = -fields in [22, 23], B H -algebra for short, and has ∞ L -algebras in [26]. ∞ L 7 experiences a noncommutative deformation of its B Witten’s inception of noncommutative geometry in open Around the middle of the last decade a mathematically rigorous approach phenomenon: In the integrated on-shell open string scattering amplitudes the effects of off-shell An Introduction to Nonassociative Physics nonassociativity originally appeared in string theory throughthat Strominger’s work the [13] algebraic which showed structures ofnot gauge symmetries possess underlying the bosonic usual closed algebraicstring string features field theory, theory of which do conventional are symmetries governedtheory in infinitesimally include quantum by local field Lie spacetime theory diffeomorphisms, algebras.the and and associator The the of gauge-invariant gauge Witten’s action open symmetries is stringassociator of constructed field is the star as non-vanishing product. on Whereas closednonetheless open still string string leads states states to associate, due a the tocarried consistent out theory. anomalies later on The in by complete associativity Zwiebach analysis [15] [14],full of using action which the the of Batalin-Vilkovisky gauge closed formalism, who string symmetries showed field was thaton theory the is the defined off-shell by an state infinitestructure space chain was of satisfying identified higher as homotopy string a field deformations strong products homotopy of Lie Jacobi-type algebra [16,17], identities. or Such a Jacobiators of star products.an As in the case ofnonassociativity (but string not field noncommutativity) theory, vanish nonassociativitynonassociative due in star to this products appropriate case [24], cyclicity is consistently propertiesworldsheet with conformal of the field requirements the theory of of open crossing stringsnecessarily symmetry (which associate); of still see involves the e.g. ordinary [25] quantum fields forThis that a review physical of caveat these and will othersubsequent reappear developments sections. along in The these lines. gauge various symmetries contexts underlyingtheories of these arising as and our decoupling other limits discussions nonassociative of Yang-Mills of opento string Lie nonassociativity theory algebras, also in but do have not been close recently in shown the to conventional way naturally form Topological T-duality. been realised in the mathematicsa community Lie to algebra be the which most genericallySee natural exhibits [18] and an for general infinite a higher hierarchy recent versionstring of survey of theory. nonassociative of the developments, together with both old andD-branes new applications in in curved backgrounds. string field theory wasthrough reinvigorated the in seminal a work much oflimit simpler Seiberg which and and decouples Witten precise the [19]. settingbackground open of over They and a ten showed closed constant that, years Kalb-Ramond string ingeometry, later field modes, and a the the particular low-energy scaling worldvolume effective ofof field Yang-Mills a theory theory, a D-brane on non-local the in gauge D-branequantum the theory field is which theory, a retains such stringy noncommutative as features version anexciting in exact activity the open in framework the string of physics T-duality invariance. communitycommutative This that field has sparked theory’ evolved the into (these flurry the days of e.g. area a [20, of 21] subject research for class dubbed reviews ‘non- of ofdevelopment many the was high accelerating energy subsequently developments physics through extended the journals); towhich turn see the showed of that case the the of millenium. noncommutative star non-constant This brane products 2-form worldvolumes deforming become the nonassociative, multiplication where of fields the on NS–NS D- 3-form flux PoS(CORFU2018)100 B Richard J. Szabo -flux background can be H ; this latter quantum geometry 3 S 8 . In the lift to M-theory, this describes a self-dual string 2 -flux is turned off; rather one only recovers functions up to S H In the last decade substantial progress was made in understanding the 0 can be regarded as a background magnetic field, and ‘topological T-duality’ can be = B 1 dimensions can be embedded into string theory as a collection of open D1-branes end- -flux as the Dixmier-Douady characteristic class of a gerbe, just like a Kalb-Ramond field M-theory is the as yet unknown 11-dimensional quantum theory which unifies all of the con- + H -flux are aligned in a particular way along the fibre directions on which T-duality transforma- wherein open M2-branes ending ontative an quantization M5-brane of the polarize canonical into Nambu-Poisson fuzzy 3-bracket 3-spheres on obtained as a pu- sistent 10-dimensional superstring theories, and ittum has deformations likewise suggested of many geometry. classical Nonassociative and structures quan- a have sense also more appeared naturally independently in and M-theory in indevelopments somewhat we different discuss fashions, in also predating the the following, more whose modern most important highlightsMultiple we summarise M2-branes. here. An Introduction to Nonassociative Physics to T-duality was developed forabelian principal isometries torus [27–29]. bundles, anddescribed also The by more algebra a general of mildly backgrounds functions noncommutativeof with ‘continuous in trace an algebra’ NS–NS whichwith incorporates d the effects defined in a purelyH algebraic way on thesetions functions. are performed, the It algebra was ofalgebras realised functions in can that the map when to T-dual background. noncommutative thethat and legs even However, such nonassociative the algebraic of phrase T-duality the transformations ‘topological arestring T-duality’ not refers theory the to literature. same the as fact the Inof geometric particular, functions maps on one found the cannot in the fibration recover‘Morita when equivalence’, the which conventional is commutative a algebra natural “symmetry” isomorphism (in between the the corresponding physical K-theory sense) groups.in only This viewing was topological insofar the T-duality as original as motivation it agroups provides weakened classify version a the of charges open of string D-branes T-duality,the as in phenomenon these the K-theory should given backgrounds. be more Itto was, general D-brane however, and worldvolumes, suggested but also that also apply togued to the to the background be spacetime closed global itself, string versions andin sector, of such conventional i.e. Riemannian algebras the geometry. were ‘non-geometric’ not These ar- deformations string only ofin backgrounds closed more which string recent theory have years have no reemerged in description equivalence. a somewhat They different will way which constituteshall avoids some review the them of unphysical in our features more primary of detail there. Morita examples in subsequent sections1.6 and Nonassociativity we in M-theory quantum geometry probed by thegrees membranes of freedom of from M-theory, type whichwho IIA lift string proposed the theory. a fundamental This lift string began ofM-theory. de- with the the A D1–D3-brane work BPS of system magnetic Basu inin monopole and string solution 3 Harvey theory of [30] to maximally the supersymmetricing M2–M5-brane Yang-Mills on theory system a in D3-brane whichof polarize the into canonical fuzzy Poisson 2-spheres, bracket obtained on as the well-known quantization is still not fullyattempt understood, to see lift [11] the for nonabelian a gauge related theory discussion which and governs some the partial low-energy attempts. description As of an multi- PoS(CORFU2018)100 ; to or 2 1.4 G ABC that the η 1.2 -field of 10- B Richard J. Szabo ) combine with -field. The connection be- 1.6 B . For a detailed introduction, ) 8 ( SO ⊂ ) 7 ( of the Jacobiator ( Spin 9 ABCD η ) of octonions which specify a nonassociative binary of the M5-brane worldvolume. These expectations were 1.4 The natural lift to M-theory of the situation in which open -field of 11-dimensional supergravity. Whereas the boundaries A somewhat more indirect and subtle appearence of nonasso- is precisely the automorphism group of the algebra of octonions C ) 7 ( loop space SO ⊂ 2 -structures will reappear in some of our later discussions of non-geometry G G of the commutator ( backgrounds. -algebras is reviewed in e.g. [36]. -field [39], in the same way that the noncommutative geometry on a D2-brane -field backgrounds. ∞ ) ABC ) ) C L C 7 C 7 C 7 η ( ( ( n n in i pi p Sp S -structure on a seven-dimensional manifold is defined locally as a 3-form using the struc- S , which are real Ricci-flat manifolds, see e.g. [43, 44]. Recall from Section 2 ) G 7 ( and 2 2 .A 2 G G operation. Likewise, the structure constants see e.g. [45]. These O ture constants exceptional Lie group locally define a self-dualframe 4-form bundle which of gives an a eight-dimensional reduction manifold to of the structure group of the oriented M5-branes in An Introduction to Nonassociative Physics ple D2-branes in string theory, thestructure famous which Bagger-Lambert was theory proposed then to generalizedof this underlie multiple 3-bracket a M2-branes nonassocative take algebrasymmetries values in [31] in which the these and fields theories it of [32], wasview a initially subsequently with theory in gauged more the to background form governthe and of first higher further a explicit gauge appearence references). 3-Lie of algebra nonassociativitythe (see This in relation e.g. M-theory appearence between in [33] of the the for 3-algebras sense 3-algebrasof a discussed constitutes Nambu-Poisson occuring re- in brackets in Section was the explored Bagger-Lambertwhich in explicit model exact [34] and localization through computations the were the quantization 3-algebras developed framework and in of [35]. reduced The relation models, between for these strings ending on adimensional D2-brane, supergravity, is in the the scenariobackground background of in a which of constant open 3-form a M2-branesof constant open end NS–NS strings on are 2-form an pointD2-brane, M5-brane, the particles in boundaries and the of hence M2-branes probecommutative are geometry a closed on strings noncommutative the and worldvolume hence theory are on expected a to probe a non- confirmed explicitly in [37, 38],formation which of worked out loop a space. somewhat Itthat complicated was the noncommutative proposed de- noncommutative almost geometry 10compactly probed years and later, by simply using understood open the in membranes Basu-Harvey termsM5-brane equation, on of worldvolume a the geometry direct itself, loop nonassociative 3-algebra akin spacethat deformation to can of this what the be 3-algebra appears more in shouldto the arise the Bagger-Lambert from theory, constant a and quantizationarises of as a the quantization Nambu-Poisson of the 3-bracket Poisson associated bracket associated to a constant tween these noncommutative loop space andestablished nonassociative worldvolume precisely 3-algebra structures in was [40, 41]this using associative loop transgression space techniques; perspective ontheir we nonassociativity quantization, will later have on. comment appeared further Nambu-Poisson in brackets,mid-1990s, about various and see other e.g. [42] contexts for in a M-theory review. since its inception inG the ciativity in M-theory has11-dimensional emerged supergravity on over Riemannian the manifolds with years holonomy through valued in the the well-known groups compactifications of Spin PoS(CORFU2018)100 - and 2 G . Richard J. Szabo 2 ; they are the topic 1.6 -field background is completely B , and also to our discussion of the role of 10 1.2 -flux; this extends, to the closed string sector, the well- H In the course of this decade it has been suggested that closed A pedagogical but rich example of nonassociative quantum mechanics, . 3 which were encountered in closed string field theory at around the same time. As in . 4 1.5 -holonomy manifolds in compactifications of M-theory from Section ) 7 The noncommutative geometry of D3-branes in a constant The locally non-geometric backgrounds of string theory have lifts to M-theory, where it is Let us now come to an overview of the main topics that we shall discuss in more detail in the ( Spin of Section analogous to that of the motionintroduction. of Likewise, electric the nonassociative charges geometry in of constantbe non-geometric magnetic string understood fields; backgrounds from can see a also e.g. certain “dual” [62]the perspective, for electron-monopole in an system a discussed sense above. explained An below, which electric relates charge them propagating to in a distribution of purported that closed M2-branesspacetime probing geometry also a capture noncommutative locally non-geometric andcompactifications backgrounds, nonassociative of at deformation least M-theory for of [60, four-dimensional 61].our discussion These of nonassociative the structures octonions are from intimately Section related to the setting of string fieldtheories theory, when suitably it interpreted. was Recent argued developmentsillustrate have that gone that these beyond a these instances interpretations nonassociative still to version leadalso of has to quantum the sensible mechanics potential quantum to is be not experimentally only detected. physically This sensible, isLocally but the non-geometric content of fluxes. Section strings should probe ain noncommutative order and to nonassociative capture deformation atheory, global of as perspective spacetime originally on geometry suggested locally non-geometric bygeometric flux backgrounds [51], compactifications with at an of least NS–NS string for those which are obtained via T-duality from following. In these lectures, we willin focus physics on which two intimately have related recently occurences received of intensive nonassociativity investigation: Magnetic monopoles. unrelated to the Jordanian framework, arises inmoving the in quantization the of background the of dynamics fields ofdates of electric back magnetic charges to charge. the This mid-1980s observation [46,47].simpler is framework, This in the work itself commutator was not anomalies inspired recent that at and arisefermions the in which time certain lead gauge in to theories part violations coupled to of tosymmetries, understand, the chiral in see Jacobi a e.g. identity among [48–50]. thequantum These currents generating mechanics models local and also gauge quantum provide field muchSection theory, simpler of examples, the at associativity the anomalies level that of we discussed in An Introduction to Nonassociative Physics in M-theory, though in a somewhatstring different theory. context which relates them directly to non-geometric 1.7 Outline established case of open strings probing aof noncommutative the (but (on-shell) worldvolume associative) geometry deformation on ation D-brane has in by the now presence been ofcontent interpreted a of from Kalb-Ramond Section many field. distinct This points sugges- of view, see e.g. [52–59]. This is the PoS(CORFU2018)100 M are ∗ i (2.1) (2.2) (2.3) T p = and is) but not M 0 M ω Richard J. Szabo defines the magnetic are defined in the usual 1 − B ω M = B whose phase space are coordinates on θ d i x R = M g J is nondegenerate (because , where . ∂ ) , B i i f x I p ω B d , ∂ 0. Its inverse i together, with profound physical consequences, − x ∧ IJ B 0 i 4 = θ 11 p – ω = ( B d 2 = I = B = X B } 0 g ω ω , f { , not necessarily closed, will be refered to generally as a ‘magnetic ) 0 if and only if d . M 5 ( = 2 B Ω -dimensional configuration space d ω ∈ to an almost symplectic form B 0 ω The purpose of this section is to describe a very simple example of nonassociativity which We work on the These lecture notes are pedagogically written and unapologetically geared at a general physics arises in the classical and quantum dynamicswe of begin an by electric descibing charge the in underlying fields geometryis of of also magnetic the tailored charge. kinematical to problem But our in later a discussions general of way non-geometric which string theory. and is the topic of Section audience, without assuming anygrounds detailed or requisite of technical the knowledge interestingrecent of review but of sophisticated non-geometric non-geometric mathematics back- backgrounds of mostin relevant the [64]. to More higher our expository structures current technical involved. exposition detailsrelation can on A the to be quantization found double of non-geometric field backgrounds,quantum theory their mechanics in and these contexts the areinclined found higher in reader structures the interested lecture involved, notes in as [65]. higherthe The well ideas structures more presented as may mathematically in consult the of following. thealong. nonassociative We mathematical will give introduction further [66] pointers to to relevant literature as we move 2. Electric charges in magnetic monopole backgrounds 2.1 Magnetic Poisson brackets An Introduction to Nonassociative Physics magnetic monopoles can be embeddedD6-branes, into giving string another theory perspective as onverse a the to D0-brane the nonassociative probing worldvolume deformation of a of the distributionM-wave D6-branes. the probing of a The spacetime non-geometric lift trans- Kaluza-Klein of monopole this background of D0–D6-braneall M-theory system of [63], to our which M-theory ties three is main an examples from Sections way by the inverse of the canonical symplectic 2-form is equiped with local Darboux coordinates (cotangent) momentum coordinates. The canonical Poisson brackets on field’ in what follows,symplectic for form reasons that will become clear soon. The magnetic field twists the Any given 2-form necessarily closed: d Poisson brackets where the adjective ‘almost’ refers to the fact that PoS(CORFU2018)100 ) 0 in = 2.3 B ~ (2.4) (2.8) (2.5) (2.6) (2.7) B } h , 3 dimen- with g , A ~ = f d { Richard J. Szabo , . This modifies the ) 3 x ( R i j B − H-twisted Poisson brack- . = A B ) as } j p 2.3 , i p at the origin of . { g ) . x ) ( x ~ ( i jk ) , 3 0 corresponds in three dimensions to the clas- H , a ‘magnetic charge’, also for reasons that will ( and A ~ − 0 δ = B . There are three distinct cases to consider, all of g × d × × 3 B = = 12 π R ∇ ~ B = B ~ 4 j } · i = k on H δ = p ∇ ~ B ~ , g B ~ j = B ~ p for a globally defined 1-form · B , } i A j ∇ ~ p d p { is a symplectic 2-form, i.e. when the magnetic field is closed, , i = x B B { ω . In this case the magnetic Poisson brackets govern the motion of an can be written using Poincaré duality in terms of a vector field k , . On the local coordinates of phase space these brackets read as that B ) B 0 d when i jk = M ε ) ) to ( ; in particular, on the local coordinates one finds that only triples of momenta B e Dirac’s semiclassical modification of Maxwell’s theory postulates a magnetic ∞ } this case reads as j M The associative case where d 2.6 C = M ( x B ~ , ∞ ∈ i j i in a static magnetic field it implies the existence of a globally defined magnetic vector potential x C B g e which is sourced by a point magnetic charge { , 3 ∈ f 3 R is that they govern the Hamiltonian dynamics of electrically charged particles in back- h R , = g , on 2.1 f M g 0. In general, we may call the 3-form B ~ The noteworthy feature here is that these brackets only fulfill the Jacobi identity One of the main motivations for the general definition of magnetic Poisson brackets from [67, 68]. = B d for all become apparent very soon, and it controls the associativity of the magnetic Poisson brackets ( and they provide a deformationcal of model the based canonical on these Poissonmodel magnetic brackets [47]. Poisson on brackets phase is space. a generalization The of dynami- the Günaydin-Zumino on the phase space grounds of magnetic charge in classical and quantum mechanics. This is the case of can nonassociate with the nonvanishing Jacobiators which will be described in some detail in the following. Maxwell theory. Section An Introduction to Nonassociative Physics on functions In Poisson geometry oneets would thereby refer to the brackets ( 2.2 Magnetic monopoles sions wherein the 2-form components as and on or in general dimensions Dirac monopoles. field Maxwell equation ( electric charge sical Maxwell theory of electromagnetism, inthe which magnetic there field are no magnetic monopoles. In terms of PoS(CORFU2018)100 , d ), n ~ 1 (2.9) (2.10) Richard J. Szabo given by } 0 ~ , \{ n ~ 3 · n R ~ x ~ × × × = x | − ~ × x ~ | M | g x ~ | = . n ,~ 3 g | x A ~ ~ x ~ | 0 is a smooth function, or in general dimensions 13 g 6= = B ~ · with g B ~ ∇ ~ In this article we are primarily interested in the case of non- n ,~ . g × A ~ M × × × Spin ice pyrochlore lattice with magnetic dipoles. ∇ ~ = g B ~ 0 is smooth; in this instance the Jacobiator of the magnetic Poisson brackets Figure 1: 6= B d = , n ,~ H g A ~ one can write the Dirac monopole field in terms of a locally defined magnetic vector × Magnetic monopoles are hypothetical particles with a single magnetic pole whose existence M in non-vanishing throughout the smooth submanifoldThese supporting are the the magnetic cases charge of distribution. theory, interest but in our we later shall discussions also of describe nonassociativity other in string potential theory physical and and M- even observable consequences of On potential are subjected to an externaldipole magnetic moment. field Local and configurations scatterobserved neutrons, of through monopoles which the have form neutron their andcontext scattering own of the interference this magnetic resulting paper patterns; Dirac together see with strings more [71] can references for be to the further most details important experiments. in the the 3-form Smooth sources of magnetic charge. singular monopole distributions, wherein would explain the quantization of electric charge,in as theoretical we physics, discuss they below. have In not spitefirst yet of observed their been in vast analog observed interest condensed with matter full systems over successstates ten in of years experiment. ago matter where They they [69, were appear 70]. asspin emergent In ice these pyrocholore lattice, experiments, consisting certain ofthe rare an corner earth array atoms elements of such with tetrahedra that the with the magnetic total structure magnetic dipoles of charge arranged a through at each tetrahedron vanishes (Figure which has additional singularities alongwhich the is the infinite celebrated line Dirac inbrackets string the in singularity. this direction case Note of vanishes that a on the fixed Jacobiator unit of vector the magnetic Poisson An Introduction to Nonassociative Physics This describes a singular distributionmagnetic of monopole, magnetic can be charge solved which, by by the removing magnetic field the on location of the PoS(CORFU2018)100 , ), m 0 is / | 2.9 in the (2.13) (2.11) (2.12) (2.14) B = ~ | e m B ~ · ∇ ~ Richard J. Szabo with mass e , by first discussing the 2.2 of a single Dirac monopole ( with the angular momentum of the g B ~ L ~ = with a time-dependent angular frequency B ~ ), these equations of motion are equivalent . B ~ K B ~

B . ~ 2.4 2 H × t × × x ~ , p d ~ I d p ~ X m m e 14 1 m  2 = = = = p ~ I t p ~ . In this case the classical dynamics is integrable and the t H d 3 X d d d (with or without sources) is governed by the Lorentz force R B ~ on B ~ where the orbits are circular trajectories. Thus the motion follows B ~ 3 dimensions, the motion of an electric charge The textbook example of a source-free background with is constant, as these are the cases which are typically amenable to exact = In the background of the field ρ d = B ~ · ∇ ~ . , which is the sum of the orbital angular momentum 2a ~ K Although we are primarily interested in the quantum mechanics of electric charge in monopole law background of a static magnetic field for the Hamiltonian Dirac monopole fields. the classical dynamics isvector also integrable. The integralselectromagnetic field of due motion to are thePoincaré provided electric vector by confines charge the and the motion of the Poincaré the Dirac electronof to monopole. the the monopole, surface The which of conservation precesses a cone, around of with the the apex direction at the location to Hamilton’s equations classical mechanics. In backgrounds, we can gain adifferent lot of types physical of intuition sources about the of qualitative magnetic differences between charge, the discussed in Section analytic solution; then nonassociativity persists throughout all space. 2.3 Classical motion in fields of magnetic charge An Introduction to Nonassociative Physics such models. We shall becharge, particularly for interested which in the cases of uniform distributions of magnetic for the kinematical momentum of the electron.electric We charge. ignore Using the the magnetic Poisson effects brackets ( of magnetic backreaction due to the acceleration of the The particular behaviour ofdepends the drastically trajectories on the determined nature by of the these magnetic dynamical field evolution equations Uniform magnetic fields. given by a constant magnetic field motion is uniform along the direction of theof magnetic a field. two-dimensional The equations harmonic of oscillator, motion reduce with to angular those frequency the Larmor frequency a helical trajectory with uniformFigure velocity along the direction of the magnetic field, as depicted in in the plane perpendicular to PoS(CORFU2018)100 of the (2.16) (2.15) n ~ , the closest ρ Richard J. Szabo of the magnetic n ~ (c) with n ,~ ρ B ~ = B ~ with an additional time-dependent g B ~ . , so that 3 . n ~ R ) x ~ n ~ ρ · in 1 3 x ~ n ~ ( 15 (b) = ρ The situation is drastically different in the case of a rot = , ρ which sources the rotationally symmetric magnetic field n ,~ B ~ the equations of motion are those of a harmonic oscillator ρ ρ . B ~ n ~ 2c and nonassociativity of the magnetic Poisson brackets plays no × M In the case of a uniform distribution of magnetic charge . In this case the electric charge never reaches the magnetic monopole, 2b (a) Classical electron trajectories in the background of: (a) a uniform magnetic field; (b) a Dirac with rot , ρ B ~ = B uniform distribution of magnetic charge Rotationally symmetric magnetic fields. so the motion takes placerole in [72]. This dynamical systemand was generalized monopoles recently in to the [73],Poisson case where brackets of vanishes it multiple only electric was if charges all shownif charges an that have electron identical the never ratio collides Jacobiator of with electric of a to monopole. the magneticAxial corresponding charge, or magnetic magnetic fields. analog to the casealigned of along the a direction uniform of a magnetic fixed field unit vector considered above is when the magnetic field is In [72] it was shownization that of the the Lorentz trajectories force intomotion equation the of can an equation electric be of charge transformed in motion via the of field a a of time a dissipative reparameter- Dirac system, monopole which governs the as depicted in Figure An Introduction to Nonassociative Physics Figure 2: monopole field; and (c) an axial magnetic field. The corresponding Lorentz force equation wasis studied again in [74], completely where integrable. it Again wasfield, the shown and that motion in the is the motion uniform plane along perpendicular the to direction magnetic field, as depicted in Figure with a time-dependent angular frequency,this which case the can trajectories be follow solved an in Euler terms spiral of with Fresnal uniform velocity integrals. along In the direction ~ PoS(CORFU2018)100 0, do v ) on 6= ¯ (2.21) (2.17) (2.18) (2.20) (2.19) h breaks P 2.3 B , together ) in terms of . ) M x ( from functions M Richard J. Szabo ), they formally O f ∞ ( C i j O 2.19 ∈ with unit; the precise ¯ hB 7→ g i ◦ f , f at the quantum level: − d = R ◦ ] j ∈ p , . In this case, the motion is no v O ρ , , i f g p B . Instead, by formally iterating the } O = d , g [ ,
i R v f f , should satisfy the algebraic property + { i
O x v O · ◦ = p O g
O and ◦ O = ] i ¯ h v g + O 16 ◦ g P , f . By the commutation relations ( j O ◦ i ◦ i . As noticed originally by [46], the operators ◦ exp O x δ [ ◦ f ¯ O h = 1 i O ◦ v − 1 1 = , as expected because the background 2-form field − . This is a generalization of Dirac’s quantization framework P O v − d f ◦ i ¯ ] h j O R P O is intentionally left vague for the moment, as this will depend p 0 2 1 ◦ f O → g 0 , lim ¯ h i O x O → lim ¯ h O − which admit some binary composition operation [ g f denotes the usual formal power series expansion of the exponential func- O ◦ O , ◦ 0 f = O , physically understood as Planck’s constant. As usual, the quantization map ◦ ¯ ] h = j x ◦ ] O g ) of local coordinates as the quantum commutators , to symbols i O x ) is the usual pointwise multiplication of phase space functions , 2.4 f O [ M . Quantization can be generally regarded as a one-to-one linear map O f g ( d [ ∞ R Let us examine the general problem of quantizing the magnetic Poisson brackets ( To study the representation of the quantum kinematics, it is useful to introduce, for any We shall now turn to the quantization of the dynamics and ask, in the case of distributions of C magnetic translation operators = ∈ f not represent the translation group where the inverse is taken with respect to the translational symmetries of the dynamical system on with the generalized Bohr correspondence principle where does not generally preserve the classical magneticbrackets Poisson ( brackets. However, it does represent the some parameter where which deforms the algebraic and geometric structures of the classical phase space the where the operation exp tion with powers taken with the composition implement translations on configuration space by constant vectors magnetic charge, if there is a sensiblethe version classical of analysis nonassociative above. quantum mechanics accompanying 2.4 Quantization of magnetic Poisson brackets An Introduction to Nonassociative Physics frictional force accounting for thelonger background confined magnetic in charge anyinterpretation direction, of and this dissipative the nature of dynamics the is dynamics later not on. integrable. We shall discuss another M specification of the operators crucially on the quantization scheme chosen. The map PoS(CORFU2018)100 , ) 1 )) w , and 1 v ( ) = , x (2.26) (2.22) (2.24) (2.23) (2.25) x u U ( , ; w x , ( M v , ( 3 u ∞ 4 Ω C ) with values in based at x Richard J. Szabo d ( ) R w ψ , ) v x ; ( i x ( A 2 , 4 ) + u x with values in , ( given by the usual associative d ) P ψ v ◦ R ◦ 0, the coboundary i x ) − ∂ v = to operators acting on the quantum =  ∂  x . An explicit calculation verifies the ( ¯ B H h B x P M defines a 2-cochain of the translation i ) d ) ψ M , ◦ ) w to w , , −  x w v w v = ; v ( , A x + u w ; ( v ] P , H − 2 x v x -dependence of the 2-cocycle that modifies ( ) = , x x v 3 4 x P 0, so there exists a globally defined 1-form )( Π − Z 4 x ) in the group cohomology of x , or equivalently by Stokes’ theorem, the total )( ( [ Z i x = ¯ d h ) Z w ( ψ , i ¯ i x h R v B w , i p ¯ 17 ( , h − u v  w O ,  in v − Π Ω ( , ) of magnetic translation operators. u w  exp = , exp Ω v v ) = , u 2.22 exp u from the point ) = P defines a 2-cocycle on , while x ) = d P ), one finds that they satisfy the composition and association ) ◦ ( x w and ◦ x ( R w , ) = w , ( v w v v x , , , with the composition operation w v , u 2.19 P d v )( P Π ( Ω R Π ψ ) ◦ v x ) in this case where, since ( w on -dimensional generalization of the classical Maxwell theory, the mag- P of square-integrable functions on configuration space with respect to the is a closed 2-form, d ψ . The magnetic translation operators in this case can be represented by d x ( i P ) 2.22 ) using ( x d H M R ( 2 ∈ ) = 2.21 = In the . The quantum theory can then be described via canonical quantization. The L x ψ A )( = M d , the adjective ‘weak’ refering to the ψ is the straight line in i = x ] H on H . From this point of view, defining a quantization map means providing a representa- x B O , )) ( B v and spanned by the vectors with coboundary the 3-cocycle on 1 ( x − d d x U [ R R , Let us now explicitly confirm these formal calculations in our specific cases of interest. M ( such that ∞ where composition relation in ( is trivial. Thus in this instance and the magnetic translation operators generategroup a weak projective representation of the translation where spanned by the translation vectors is a phase factor determined by the total magnetic charge enclosed by the tetrahedron the Wilson lines is a phase factor determined by the magnetic flux through the triangle based at tion, in a suitable sense, of the algebra ( on wavefunctions flux through the faces of the tetrahedron. In general, Maxwell theory. group netic field relations usual Lebesgue measure d An Introduction to Nonassociative Physics conjugation action ( Hilbert space composition of linear operators.magnetic This translations constructs where the the Schrödinger position representation and kinematical of momentum the operators algebra are of represented as A quantization map in this case takes functions on phase space C PoS(CORFU2018)100 (2.32) (2.27) (2.28) (2.30) (2.31) (2.29) themselves, f parameterized Richard J. Szabo , H Z d on ). The magnetic Weyl Y d ) . Its inverse can be used d ) 2.1 X ) ( by Z X H π O d . This is called the magnetic 2 − . ∆ ( H ) X ) H ( y ∈ X g )( ( ) . The deformation from classical ψ O ψ B given by Y x ? ∆ f (see e.g. [75]). In the case where the − P  O ) ( X . d x ( y ) · ( , ) f ) p Y π w ) w i , ) X d , v 2 X v Z − , ( ( − − Π Y B e ( ( ( ) f ¯ h i B x is the Darboux 2-form ( O 2 · Y ω p on O ); see e.g. [75] for the explicit expression in the ( 0 ∆ − to a noncommutative product of the corresponding ¯ ¯ 18 h h , which are integrable with respect to the Lebesgue f 2 2i i d e ω ) on the Hilbert space g = ) = − 2.2 e f R Y † O † , e = ) , M ) X O f ( X ( w ( , X 0 ∞ v f M ) = ( O ω y Z i C Π O , where ( O e )
∆ ∈ M ψ f Z ) M M given by the star product it recovers the ordinary convolution product between integrable ( p Z d ◦ ∞ , ) B  x ¯ h is constant is given explicitly by the twisted convolution product C 1 ( π B O ∈ are constant, the 2-cocycle simplifies to ( M f ∆ which are defined on wavefunctions Z B . This defines the associative magnetic Moyal-Weyl star product through is unitary, g = ) = , M f f f X O O of )( 0. ) g , to trace-class operators , and we obtain the standard projective representation of translations in the back- 7→ p = B x , f ? ¯ h x M , which when f g ( on = ( O ) it follows that this operator satisfies f is the twisted symplectic 2-form ( X X O B 0, while for any 2-form ω 2.22 = = g The magnetic translation operators also provide the general form of the quantization map This shows that the magnetic translation operators also provide a bridge between the canoni- B B ? f where measure d Weyl correspondence, and it isby determined points by a family of operators general case. Thisfor generalizes the standard Moyal-Weyl star product on canonical phase space from generic phase space functions O ground of a constant magnetic field. phase space functions functions when cal formulation of quantum mechanicsquantum and mechanics, the which is somewhat historically less the knownsee origin phase e.g. of [76]. space deformation In quantization formulation the and of phase star space products, quantization scheme, the operators are the functions components of the 2-form independently of From ( for suitable class functions correspondence and it is invertible as a consequence ofto completeness pull of back the the Hilbert product space of two operators with the composition operation to quantum observables is then characterized by the noncommutative and associative magnetic An Introduction to Nonassociative Physics the usual cocycle condition by the action of Then the magnetic Weyl correspondence defines operators PoS(CORFU2018)100 f Σ C O × (2.33) 7→ M . , while f ) can be . Recall d = } R M 0 L ~ 2.25 = 1, and which in the state \{ M f Richard J. Szabo 3 = R O X d = ) × X ( M , and the magnetic trans- Σ H M ∈ R ψ . . X for 0 is not straightforward, because the d M ) ψ 6= , we observe that the quantization of i X ) × B A )( d M D Σ = . Then the quantum Hilbert space of states B ? = ( A . This is just a reformulation of the canonical H d f L to functions on phase space, which determines ψ ( i 0, so that the quantum mechanics is also confined of square-integrable (global) sections of this line ~ = p 19 M ) H which is normalized, O B Z L , as . All of canonical quantum mechanics can be rephrased = f M M A ). Recall that in this case the magnetic Poisson brackets Σ ( O i 2 D f L 2.9 that the electric charge in this background follows a classical O 0 on h = > 2.3 H Σ . They heuristically give the quantization of the constant energy surfaces M ) define parallel transport in Quantization in the case when 2.26 can be thought of as a connection on a (necessarily trivial) line bundle A ) taking quantum operators on 2.30 . plays the role of astate density function matrix originates determining through the the correlations inverse of of the the quantum magnetic system. Weyl correspondence The are given by the magnetic Wigner functions representingdistributions the on density matrices that are quasi-probability in phase space through the star product. Expectation values of operators observables are real functions. The noncommutative composition of operators becomes thefunctions. noncommutative star product of Traces of operators become integrals of functions over in ( Wavefunctions are given by arbitrary complex-valued functions on phase space There is a state function , with field strength the magnetic field × To understand how to quantize the dynamics on • • • • M M quantization problem as geometric quantization, which is of course superfluous on are associative away from locationalso of from the the monopole analysis at of the Section origin, i.e. on bundle, the Schrödinger representation of thewritten kinematical in momentum terms operators of from covariant ( derivatives trajectory that never reaches the monopole. Inof the the quantum electric theory charge this should means vanish that the at wavefunction the origin can be thought of as the space Dirac monopoles. to Maxwell theory discussed above canform be potential recast in a moreon geometric way: The globally defined 1- lation operators ( An Introduction to Nonassociative Physics Moyal-Weyl star product of functions oncontains phase the space, necessary which information substitutes about the the pointwiseproduct corresponding product deforms quantum and system. the magnetic In Poisson particular, the bracketsto star in the commutator such of a quantum way operators asin to the contain language all of information phase space related for quantum later mechanics, use with some we caveats, briefly through summarise a as dictionary follows: which operator/state formulation of canonical quantum mechanics cannotPoisson handle brackets: nonassociative magnetic Operators acting on athe separable case Hilbert space of always the associate. Dirac The monopole exception field is ( PoS(CORFU2018)100 1 × = = × M w (2.36) (2.34) (2.35) , M v defines , u . For any n ,~ Ω g in this case A f O Richard J. Szabo , ; we refer to the ¯ h 7→ Z ), formulated alge- d f ) in three dimensions. Y d 2.34 ) on the phase space 2.16 Z ) , standard canonical quan- X ) − ( M X O ( ( ∆ 3 g ) Ω deformation quantization Y ∈ . Line bundles on a 2-sphere can be − ) H X L ( , , f , which becomes increasingly complicated × . , which is hence the field strength of a line k ) ¯ h ¯ h x Z M , Z / ( ) in turn implies that the 3-cocycle Y , and the quantum Hilbert space is again the 2 ( i jk B × ∈ eg L , where the locally defined 1-form that is obtained by integrating the field strength H ω n ,~ M 3 1 ¯ 2.34 h 20 g 2i = Z . The 2-form corresponding to the Dirac monopole ¯ eg A h − 2 given as a formal power series in ∈ → , associate, consistently with their representability on d 2 S e ) = H L ) n L gives 2 x = of electric charge, formulated geometrically in this way = ( M 2 g M i j n Z S ( B B ∞ [55]. In [81] it was shown that this star product can be written M C Z over ) as d 2.1 , Kontsevich formality provides a noncommutative and nonasso- g ) ) are constant, we can choose the magnetic field to have components B ¯ h 1 , which induces an associative phase space star product constructed M 2.10 H π ( × ( 3 M Ω ) = ∈ 1, which implies the Dirac quantization condition ( X Dirac quantization B . Turning this argument around, the associativity of magnetic translations is d )( = from Section g w B = H , H v θ , ? u H f Ω ( , the Kontsevich formula can only be written as a formal power series expansion and if and only if , which is topologically a 2-sphere H } ) is trivial, and hence that the corresponding magnetic translation operators, defined by par- L . Integrating the 2-form 0 2 ~ S For generic smooth distributions of magnetic charge When the components of However, this has the virtue of generalizing to non-trivial configuration spaces such as 2.22 \{ 3 in ( in the form of aan twisted asymptotic convolution series) product and which reads as gives a convergent expression (as opposed to equivalent to originally by [77]. The quantization condition ( the Hilbert space This is the celebrated allel transport in the non-trivial line bundle tization breaks down. However, asinstances first can still observed be by quantized [55], by appealing thesmooth to 3-form magnetic the formalism Poisson of brackets inciative these star product on functionsoriginal in treatment of [55]generic and to the reviewcomputed order [65] by order for in details the deformation of parameter the Kontsevich expansion. For In this case the Kontsevich seriesthe can nonassociative be star summed product explicitly in toPoisson terms give bivector of a closed the formal bidifferential expression operator for determined by the magnetic explicitly by [79, 80]. 2.5 Nonassociative quantum mechanics as the order increases. An Introduction to Nonassociative Physics R field can be expressed froma ( non-trivial connection on a line bundle braically in this way originally bycan [46] be (see constructed also via [78]). aof The suitable the quantization magnetic configuration map Weyl space correspondence which is the generalization of the rotationally symmetric magnetic field ( over space of square-integrable (local) sections non-trivial with a non-zero Chern number bundle PoS(CORFU2018)100 ) is (2.38) (2.39) (2.37) 2.36 . This star H Richard J. Szabo themselves, and ) [85]. This formalism M using the nonassocia- ( ) H ∞ , w , C 2.4 v u , u ∈ ( P f H ¯ h H i 6 . The nonassociative magnetic ? e H
v ? = P , w , H v w , ? u ), and subsequently proven to be a gen- + v w Ω P . P 2.36 ) v · x p w are the functions ( , i ¯ h v w framework which gives novel predictions, such f , , u v e 21 and O Ω Π = . an alternative algebra: This was shown in [83] for a : d , because the algebra of functions on phase space with = = v . Note that this is formally identical to the star prod- R v
, with the difference here that the star product ( ) ) u 1.3 not P w ) with B , P v , M P x H ( ( ) is H ? quantitative ∞ 2.22 H ? w ¯ h C i v 6 P 2.36 ∈ − P e g , is the nonassociative star product H f ? ◦ ) = w x ( w P , v Π , with the nonassociativity controlled by the constant magnetic charge ) which is written for constant 2.32 One undesirable feature of this deformation quantization approach, aside from the usual tech- Although there is no standard canonical quantization framework available in this case, one can In this framework, the quantum operators where nical and conceptual issues arising in conventional phase space quantum mechanics [76], is that for the composition operation translation operators in this case are given by the phase space functions was subsequently used to developin certain [86, 87]. testable In features particular, of thebeen nonassociative possibility considered quantum of extensively, see mechanics producing e.g. magnetic [88] monopoles forthe a in recent present particle discussion treatment colliders and of has further the references. physicsto In of this develop smooth way measurable distributions predictions of of magneticLarge nonassociativity, monopoles such Hadron may Collider as be (LHC) in of which use the aimsat MoEDAL LHC to experiment energies pursue at [89, the 90]. the quest for magnetic monopoles and dyons now adapt the formalism of phase space quantization outlined in Section define a 3-cocycle of the translation group the nonassociative star product ( class of nonassociative star productseral which feature includes of ( nonassociative deformation quantization by [84].nonassociative algebra At of leading semi-classical functions order, is the alternative,Jordan and algebra this whose may quantum be moments can used be to developed study perturbatively in a noncommutative nonassociative product was subsequently computed by a variety of other means, see e.g. [72, 82]. They satisfy the algebraic relations ( tive star product. In [81]mechanics it obtained in is this shown way that isquantum the physically theory, phase such sensible: space as It reality, formulation passes positivity of all and(real nonassociative completeness preliminary functions). quantum of tests eigenstates for of It a physical sensible isas observables modified a uncertainty completely relations. Seefor [71] the for physics a of discussion electron ofapplication propagation some later in of on fields to the of explain interesting magneticof some predictions monopoles; string of we theory. the This shall unusual nonassociative give physics quantum aquantum in theory mechanics concrete locally is discussed in non-geometric in no backgrounds Section conflict with the studies of Jordanian uct ( An Introduction to Nonassociative Physics for suitable class functions PoS(CORFU2018)100 into (2.42) (2.40) (2.41) M 3 with the = d to an extended Richard J. Szabo M -invariant Hamilto- ) d , d ( O , ) for the physical degrees of by eliminating the auxiliary ×
) ) B x d ( , ω 2.12 d i jk ( , H O j i − ) and ( δ ) x . ( , = ) i j J 2.11 } x j B p (  ˜ k p i j ∂ IJ , i B , taking phase space functions to operators on a η x k ) 1 2 I 0 1 1 0 ˜ { 22 x p X  2 1 ( = = 1 m O } = } j j ∆ and the symplectic brackets ) + p = η p x ) , i , ( i i ˜ b H p ˜ ˜ i j p x , i { { B p , = = = i . From this point of view deformation quantization is not a true ˜ x } } } ¯ under the map which embeds the original phase space h j j j , i , since a proper Hamiltonian formulation of dissipative systems also ˜ p p p x B , , , ( i i i ω 2.3 x p p Symplectic realization is a well-known method from Poisson geometry { { { and polarisation of the extended symplectic algebra which is consistent with both the . Some mathematical aspects of this symplectic realization are reviewed in [66]. ) ) i i ˜ ˜ p p no . This mimics the situation in double field theory [91], with one crucial difference. , , ) i ) then reproduce the Lorentz force law ( i i ˜ -invariant metric. The corresponding Hamilton equations of motion for p x p ) ( , i d , x 2.40 = ( ( d I ( p O 0: There is On the extended phase space one can then introduce the = A consistent Hamiltonian reductionsuitable of first the or auxiliary secondH degrees class of constraints, freedom, is possible viaLorentz force if the law and and imposition the only nonassociative of magnetic ifrelated Poisson to brackets there the [74]. dissipative is This nature feature no of is thethat seemingly magnetic classical we dynamics charge, discussed in in a Section rotationally symmetric magnetic field is the brackets ( freedom requires the introduction of auxiliary variables, representing a reservoir, in order to conserve the “Hilbert space”. We conclude thisspace section formulation by of briefly nonassociative summarising quantum three mechanics. such attempts atSymplectic a realization. Hilbert that enables one togeometric set quantization of up phase the spaces.to problem In the case of [74] of it magnetic quantizing Poisson was generic brackets. demonstratedphase The space how Poisson idea with to is brackets coordinates to extend “double” in the the technique phase the space realm of These brackets define an associativeback algebra to and the are twisted inverse to 2-form a symplectic 2-formcoordinates which pulls where quantization in the physical sense,It where would Planck’s therefore constant still can beavailable, be desirable in set to order to have to some an pursue fixedseek alternative better finite further form the value. testable of foundations aspects canonical of of quantumlog this the mechanics of nonassociative model. quantum the What theory, magnetic is and Weyl correspondence missing to in the theory is a nonassociative ana- An Introduction to Nonassociative Physics magnetic charges beyond the constanttotic case, series the in star Planck’s product constant can only be developed as an asymp- the extended phase space, or more concretely which restricts to nian PoS(CORFU2018)100 ; this map- M , with 2-form Richard J. Szabo M was worked out in algebra, where B whose field strength is ) are interpreted as con- M is sent to the field strength on 2.22 M under the transgression map, ) M , higher structures 1 gerbes S ( ∞ 0, to C into the configuration space 1 = appearing in ( S ) B x ( w , v , 23 u Ω 0. In this case the usual quantum Hilbert space of sections 6= B d In higher geometry it is well-known that the nonassociativity fea- = H of embeddings of a circle ) M , on the 2-Hilbert space; this utilizes a higher notion of parallel transport in a 1 v S satisfying the Bianchi identity d can be traded for more conventional noncommutative features of a line bundle ( Nonassociativity implies the appearence of P A ∞ M d C = B -flux model R We now turn to our attention to somewhat more conjectural applications to non-geometric ping is called transgression and it hasfor a closed natural string interpretation degrees of of trading freedom. particleof The degrees a field of strength line freedom of bundle a on gerbeand the on on closed the string latter configuration oneapproach space can was proceed pursued to in apply [40, the 41].2-Hilbert standard However, space this techniques of approach of sections simply of geometric hides a quantization; gerbe thedimensional this in higher configuration a structure Hilbert space of space of the of sectionsphysical closed of interpretations strings, a line unwieldy. again bundle over Once making an more,to explicit infinite- although quantization constructions the is and formally problem their solved, offurther many a technical exploration. Hilbert and space conceptual approach The difficultiesnonassociativity remain discussion structures awaiting in of for terms this of closed paragraph strings, to motivates which considerations we now of turn. 3. these Closed higher strings in locally non-geometric flux backgrounds 3.1 The on the loop space of a line bundle shoulda be ‘categorified’ replaced Hilbert by space an calledometric analog a quantization of 2-Hilbert of sections space the of which magnetic ain is gerbe, Poisson this a setting that brackets certain by has for monoidal [75], the category. anytrolling where structure magnetic suitable Ge- the of field higher 3-cocycles versions oftranslation functors a (weak) projective representation, implemented by magnetic tures of a gerbe on gerbe. While again this approach formallytive captures quantum a mechanics, Hilbert the space physical formulation meaningare of at and the present development nonassocia- unclear. of A the concise higher review structures of involved thisTransgression approach to can loop be space. found in [66]. An Introduction to Nonassociative Physics total energy. However, in theof present this case analogy is the not totalsolves entirely energy the clear. is problem already Thus of conserved, while ameaning the so Hilbert of formalism the the space of auxiliary meaning formulation degrees symplectic of of realisation freedom,contribution the formally which of nonassociative cannot V. Kupriyanov quantum be to eliminated, mechanics, these remains the proceedings unclear. for See further the details2-Hilbert spaces. of this method. in addition to non-vanishingternary binary Jacobiators (or commutators associators). oftheory a of These monoidal multiplication structures categories, one are whereinof natural one encounters vector works spaces, from non-zero where with the associativity algebra of pointassociativity objects tensor holds of not products only holds view in on up the of the to usualof the nose, a category geometric but natural in quantization, a isomorphism this category called corresponds where field the strength to associator. the passage From from thecontrolled by line perspective a bundles closed on 3-form PoS(CORFU2018)100 on ) -flux (3.5) (3.4) (3.3) (3.1) (3.2) R M ( , where , and so 2 ) ∗ B Ω ) and takes ω M . Applying 3 spacetime , and it con- ( ∈ f 3 . For this, we = B Richard J. Szabo 2.42 , Ω d 2.1 0 ∈ . In this case the d = β R d β (and vice-versa). Start- } j = = ) R-flux model i p ∗ -flux R , p i 3 s ( H M p ` { , i jk -flux’ R R . of order 4. This map preserves the and ) -flux’. A final T-duality then results in a k p T Q ( M -algebra which was developed in [55,93]. ←→ -th coordinate direction which sends string ∞ as a para-Hermitian manifold [92] where it i jk i ] ] j k L which sends the R i jk jk i j M 3 s δ ∗ ` B β Q i jk i i -field and a metric flux (torsion) [ [ T = T − , called the ‘ to momentum modes B ˆ ∂ ∂ ∗ j 24 β = = Z T = } = M j ←→ β , a T-duality takes this to the Heisenberg nilmanifold ⊕ p M } ) i jk i jk , k jk Z i i H H R x x f , , on phase space , but it does not preserve the twisted 2-form ⊕ j { 3 on the dual momentum space ) x Z T , x M i i ( ) T ∗ x ) of local coordinates, depending on a 2-form − { ←→ , ) = , on M p ( 2.4 Z ) ( 0 2 , p i jk 3 ω ( Ω H T 7→ i j -flux. This originates from a type II background containing a 3-torus ( ∈ ) 1 β R -transformation which preserves the Lorentzian metric ( p ) -flux (proportional to the volume form) through a chain of successive T- H β − , d , which interchanges the roles of local position and momentum coordinates x H , ∈ ( , to the twisted Poisson brackets = d ) i ( β M } w O j ( x , i x { denotes a T-duality transformation along the with constant i -flux -flux. The triple T-duality transformation Born reciprocity 3 T R R T The dynamical system captured by these dual brackets is called the = , which is a geometric background with no is the string length. This now gives a nonassociative configuration space with the nonvanishing 3 s ˜ to the ing from the geometric background another T-duality then results in awell-defined with T-fold the [95], transition a functions controlled non-geometric by backgroundbackground a which ‘ which cannot is be not described even globally locallyby in an conventional geometric terms, characterised T winding numbers where M dimensions with constant dualities which gives rise toas geometric [94] and non-geometric fluxes that are depicted schematically depending on a 2-form twisting is by a 3-form` on momentum space coordinate Jacobiators These brackets have a natural formulation as a 2-term jecturally describes the phase space ofbackgrounds closed strings of propagating string in theory, ‘locally which non-geometric’ havefor no the formulation case as of a the conventional spacetime magnetic [53, Poisson 55]. brackets, As the prototypical example arises in configuration space invoke through the mapping canonical symplectic 2-form is not a naturalinterpretation symmetry by of regarding the the phase magnetic space Poisson brackets. Instead, itthe has magnetic a Poisson brackets natural ( geometric corresponds to an An Introduction to Nonassociative Physics string theory. At the algebraica level, certain‘duality’ the transformation pertinent to phase the space magnetic brackets Poisson are brackets from obtained Section by applying PoS(CORFU2018)100 ) (3.8) (3.7) (3.6) 2.36 -flux, R to fields M on for an arbitrary Richard J. Szabo i f Σ X -field to a bivector i with nonvanishing i d -flux to the B x 3 ) H 3 dimensions via the T X −h ( i =
x , with the replacements of d Σ everywhere; in the case of 0 = = R β i p 2.5 ? x

) R ∆ ? x ] ( k with 3 x f B ∆ R , j ? x
) ∆ x , -flux model through , i ( R x 2 -flux model raises some curiosities concern- f i jk ∆ R [ R R 1 2 -transformation applied to the magnetic Poisson ? 3 s ) ) ` in the d 2 x 2 25 , ( ¯ M h 1 d f Z ( = O = , where a particular bracketing of the nonassociative star i jk Σ ) ) =

and the magnetic field V x R M ) ? ( , D3-branes map to D0-branes and the i ] )( triproducts . This corroborates the expectations that particles cannot be x k ∞ 3 f i jk x Σ C − ∆ T , i , M ∈ j p 2 3 x f f . The phase space formulation of nonassociative quantum mechanics ∆ , R , = ( 2 i M ? f I x 1 can then be used to quantitatively explore the physical implications of , f ∆ X 1 -flux background [97]. This can be argued in [ f ( one can express this triproduct as a twist of the usual Fourier convolution R 2 1 2.5 ˜

M = . This quantity computes the quantum volume of the tetrahedron in configuration i jk Σ V -flux, we denote the corresponding nonassociative star product obtained from ( R denoting ‘modified derivatives’ with respect to the coordinates dual to the string winding i ˆ ∂ The nonassociative phase space structure of the In the quantum theory, a suitable substitute for canonical quantization of locally non-geometric on Fourier space with -flux. Under a triple T-duality i ˜ f product has been chosen. By Fourier transforming the configuration space fields for any three functions used as probes of the a constant independently of the chosen state Freed-Witten anomaly, which forbids D3-branesH from wrapping a 3-torus and hence D0-branes cannot propagate in locally non-geometric string backgrounds. 3.2 How closed strings see nonassociativity ing basic string physics, insofar thattheory at the one level does of not the two-dimensional expectconformal worldsheet any field conformal such field theory). structures The issue to was arisefrom clarified nonassociative (these by phase would [98] space which otherwise star demonstrated products violate explicitlysociativity to axioms how certain of to of ternary configuration pass products space which fields capture inthis, the a one nonas- defines way configuration that space is consistent with conformal invariance. For closed strings is provided by the deformation quantization of Section brackets above. phase space coordinates with these substitutions by discussed in Section determined by the Jacobiators of thestate position uncertainty function operators space spanned by the coordinateobviously uncertainties vanish in in the the associative given case,non-zero directions. in quantum the of While present volume this case [81] it quantity can would be explicitly computed to give a the local non-geometry ofexpectation the values of configuration the space oriented volume [81]. uncertainty operators As an application, let us compute the modes. This is completely analogous to the An Introduction to Nonassociative Physics is best understood in double field theoryβ [96], where it maps the 2-form NS–NS PoS(CORFU2018)100 ) . 3 . 3.9 R (3.9) d (3.14) (3.10) (3.11) (3.13) ) 3 k π ) on d 2 ( d 1.16 ) 2 k Richard J. Szabo π , and indeed the d 2 R ( d ) 1 k π d 2 ( , . x ) · x 3 ) d ( 3 . k σ ) f +  x 2 ) ( k 3 3 M -flux background. Computing the + f k ) 1 , R k 2 ) 2 ( ( : (3.12) x i k σ ( ,
f e 2 . ) 1 f ) ¯ z k 3 ) M , ( k i jk , z x ) 2 R ( ( 1 R k ( , x 3 s 1 3 s 1 ` f · σ ` 2 k 4 f ( ¯ h k 2 i | R M i ¯ h σ Z 3 s | ` − ) -flux model: 4 2 26 = ¯ h = 1 i R  M x − − ] : exp d ( k e 3 x exp ) S ) , x j 3 ∈ ∑ ) was independently postulated by Takhtajan [10] over 20 ) = = k x σ )( ¯ z ( , ) of the 3 R i , 3 f 1 3 ˜ z 3.9 effect, invisible to the on-shell closed strings. See [65] for a f x

[ ( 3 3.2 ) k k M = 2 V V k 2 2 M f ( k ] 2 3 ˜ V f f M 1 ) , k 1 off-shell 1 2 f V f k

( ( , 1 1 ˜ M f f Z [ ˜ M ) ): x Z )( ˜ 3.9 M 3 f Z ˜ M M Z 2 f = M 3 dimensions, the triproduct ( 1 f ( = Given that the closed string sector contains gravity, it is natural to ask if there is a consistent These triproducts were rediscovered by [54] through calculations of the scattering of momen- d nonassociative theory of gravity which governs the low-energynon-geometric dynamics of backgrounds. closed strings in This locally endeavourof has nonassociative been (Riemannian) pursued differential at geometry length through by a developing generalization a of theory the usual twist Thus nonassociativity is an detailed discussion of various otherstring caveats nonassociativity. associated with these and other derivations of3.3 closed Nonassociative gravity on configuration space is inheritedstar from product the and nonassociative star the product triproductthus on violate forbids phase the a space. strong conventional constraint Both spacetime of the theory interpretation, double amplitudes but is field on-shell retrieved theory as associativity [57, a 99], of result which conformal of field the integral identity [98] One can introduce a natural 3-bracket via antisymmetrization of the triproducts to get where the sum runs throughdefined all by the permutations classical of Jacobiator degree ( 3. This naturally quantizes the 3-bracket In The precise meaning of this formula was thus subsequently clarified by [98]: The triproduct ( years earlier as a candidate quantization of the canonical Nambu-Poisson 3-bracket ( tum states of closed string tachyon vertex operators An Introduction to Nonassociative Physics product of three fields as [98] in a linearisation of conformal field3-point theory functions on of flat such space states in thenall the reproduces orders the expression for triproduct to theexact linear multiplication expression order of ( in vertex operators was conjectured to be given by the PoS(CORFU2018)100 , (4.1)  (3.16) (3.15)
n lk Γ b ∂ ) km Richard J. Szabo g , l . This expression i j x k Γ i j d a Γ ∂ 
n m − i j ik lifts to M-theory on the ··· Γ Γ b c km + M ∂ ∂ g ) ) a φ k i ∂ lk lm l Γ i j g Γ b φ ∂ i lm ( ∂ ∂ g ( ( − 6 1 a kl n l j ∂ g − a Γ b − ∂ ∂ i jk n j ik ) Γ H ∂ 0 b km  ∂ − with Christoffel symbols g M i jk M ) l
/ ik H k l j m M Γ T-duality. One then wishes to find an equivalent 3 i j Γ a / 27 Γ ) ∂ φ c on lm Z − ∂  g ; − ) e i j ( d a g 1 , lm km ∂ is the string dilaton field, and we have written only the 1 S d g g 12 ( is a supersymmetric noncommutative Yang-Mills theory b a φ ∂ ∂ − mn d ( l SO g ik T c kl Γ ∂ g Ric -field. In that case the low-energy effective theory of D-branes a  B ∂ + ( + g -flux, which is also the order at which the conformal field theory k √ ∂ -flux model. One can construct a Ricci tensor as well as a unique R  R -flux model M Z R abc , this results in a non-trivial real deformation of the spacetime Ricci tensor R G 3 s π 1 3.2 ` 4 2 16 ¯ h -dimensional torus = + d of a circle bundle S i j 0 is the Ricci tensor of a metric M i j Ric is the gravitational constant, = -invariant (off-shell) nonassociative version of the closed string effective action G ) ◦ i j d The final goal of nonassociative gravity is to mimic what happens in the well-known case of the Type IIA string theory on a background containing the spacetime The main result has been the development of a metric formulation of nonassociative gravity , d Ric ( holds to linear order in the O open string sector with constant calculations of [54] are reliable.theory, as The the fact Ricci that tensorthe on this projection phase quantity to space is spacetime. is real complex-valued is but a its imaginary remarkable parts feature vanish of under the wrapped on a which is invariant under open string where Ric where bosonic part of the fullfrom type such II an supergravity action effective theory, for webackgrounds brevity. can in Although move their much on uplifts remains and to to seek M-theory. be the seen generalizations of these string4. theory M2-branes in locally non-geometric flux backgrounds 4.1 M-theory lift of the total space given by [104] on the phase spacemetric-compatible of torsion-free the connection, playingdinary the Riemannian role geometry of [104]. thedescribed Levi-Civita Using in connection a Section in projection to or- configuration space analogous to that An Introduction to Nonassociative Physics deformation techniques to the case wherewhose the coboundary twist is is provided a by 3-cocycle98, a controlling 100–104]. quasi-Hopf nonassociativity Progress algebra in 2-cochain, in a this similar directionthe has way technical been constructions as required. slow previously due [81, to the extremely complicated nature of PoS(CORFU2018)100 . 1 1 of S S ) 4 -flux (4.5) (4.6) (4.2) (4.4) (4.3) × x R , i M x that there = 0 . ) = ( M ) 3.1 µ 4 Richard J. Szabo ( x , SO = ( i x p ~ , i x . The lifts of the three- s 1234 2 , g 4 λ R 3 s = . The explicit choice of non- ` ) and whose remaining four in- λ 4 symmetry to λ } ) ( i ) -flux. Since D0-brane charges in 4 p = ( 5 exceptional field theory [106, 107]. R , SO ( 4 λ ) x SO } 5 . i SL { ( , x ] with , which sends membrane wrapping modes , -flux, but otherwise cuts out a one-dim- i jk 4 SL , R ε x M { ραβ 0 R µνρ of 11-dimensional supergravity to a trivector 3, as previously, and on the M-theory circle Ω µναβ U = = , ν and ε [ 2 µ µ , R i jk p µνρ ˆ 28 1 ∂ R C = and -flux = = (and vice-versa). The lift of the non-geometric string k i R x ) jk νραβ , i , i jk T p µ x i µναβ −−−→ -field ( , , k R νραβ 4 , C k p by R µ jk λ ε p 4 i R f + M i jk i jk , 4 4 lifts to a U-duality x j R λ ε i jk 3 s a 5-vector but rather a mixed symmetry tensor whose first index with quasi-Poisson brackets given by ` δ − T 0 = = = not M λ λ λ } } } -flux backgrounds to locally non-geometric backgrounds of M-theory was j j j R x p p takes the 3-form , , , i of the circle fibres geometrizes the string coupling i i x x p to momentum modes 0 vanishes along the M-theory direction. We can write this constraint on the { { λ { ) µνρ -flux background: = Z U R , 0 4 p M denotes a ‘modified derivative’ with respect to the coordinates conjugate to M2-brane ( 2 to the -flux to M-theory can be described explicitly in µν H ˆ R 3 ∂ which is a potential for the M-theory ˜ ∈ T ; we collect these coordinates into the four-dimensional coordinate vector The quasi-Poisson brackets of the M2-brane phase space in the locally non-geometric 4 ) x . In this case the T-duality = i j µνρ 0 w ensional subspace of thedimensional phase phase space. space In this case the membrane is purported to have a seven- which is trivially satisfied in the absence of M-theory type IIA string theorymomentum lift to momentum modeslocally along non-geometric M-theory the background M-theory more circle, covariantly in this the form implies that the are no D0-branes permitted on the string background An Introduction to Nonassociative Physics where the radius dimensional constant For simplicity we consider the lift of this chain to M-theory on the trivial circle bundle which we shall often make in what follows, then breaks the background in this case were conjecturally described in [60]. Recall from Section considered for compactifications to four dimensionsfocus in here [60] on and the to former lift. higherM dimensions One in starts [105]; from a we double T-duality taking the Heisenberg nilmanifold We denote the local coordinates on by vanishing components where wrapping modes; this is transforms as a vectordices under transform four-dimensional in rotations the in totally antisymmetric representation of ( M theory The U-duality Ω PoS(CORFU2018)100 O -flux (4.7) (4.8) (4.9) (4.10) R , defined k , p 3 k x 4 R ), we equip . i jk i jk ,  on 4 ε 1.4 i Richard J. Szabo e 3 R × × × 3 s λ -flux background λ . Given a vector ` R 2 − O = , from ( λ , λ 7 k } e = 3 x ABC 3 x 4 λ , / η j } i jk R 4 , p 4 x 3 s , , i ` R j p 3 3 s x ) is invariant under the subgroup { ` , λ i correspond to a quaternion subal- 2 p i , whose properties can be abstracted q λ 4.9 { p , − , i i jk f ε C = 3 P and -algebra which is discussed in [93]. Note / λ ) of the M2-brane phase space was carried B ∞ } which stems from the automorphism group R K . 4 L ) by a suitable change of basis. and 3 s 4.6 x cross product ` A , e j
ABC 4.9 λ j x A η , x i 0 these brackets reduce to those of the magnetic q 29 K k
x i -structure on a seven-dimensional real vector space j =  { 2 δ = = p A 1 2 -structure G with the ) k 2 ~ K − i λ 1 becoming a central element of the algebra). Since the i 7 P ~ . This extends the usual cross product δ G O x 7 ) = R η k = .A − j A R 7 × 4 × × k e δ x and R ( p in ~ K j through the redefinition of the imaginary unit octonions given ( i λ Λ ) δ A − P 1.2 4 4 ) = i x x . Given the octonionic structure constants p k 4 = ( 1234 , , O i j 4 4 i jk P ~ ε , x R corresponds to the radius of the M-theory circle, this limit is the weak 4 2 , 3 s i R ` λ . λ x 3 s and 0 − λ ` 0 which reduces M-theory to perturbative type IIA string theory in the ) A = = = = ) = ( K → λ λ λ λ A s } } } } X g k k k k = ( x x x p of the rotation group of , , , , j j j ~ j K = ( ) x x p p 7 , , , X ~ ( i i , i i x p , define the octonion element p p { 7 { SO { { In fact, these brackets originate from the nonassociative alternative algebra of octonions The cross product has a natural representation on the octonion algebra The quantization of the quasi-Poisson brackets ( R ⊂ ∈ 2 ~ These brackets also admit athat natural the description brackets as involving an solely thegebra. momentum coordinates which we discussed in Section analogously in terms of the quaternionic structureinto constants a general set of axiomsthe independent real of normed dimension division [45,61]; algebras, similarly which to onlyexist the exist on pattern in vector followed dimensions by 1, spaces 2, of 4product real and of 8, dimensions two cross 0, vectors products 1, is only defined 3 to and be 7 0). (where The in cross the product first ( two instances the cross The crucial observation of [60] is that at of the algebra of octonions the seven-dimensional real vector space for vectors K Poisson brackets for the six-dimensional phase space of closed strings in the (with the M-theory circle coordinate out in [61]. It is based on the notion of a G by An Introduction to Nonassociative Physics which have the Jacobiators contraction parameter coupling limit background. 4.2 Quantization of the M2-brane phase space is a cross product that can be brought to the form ( PoS(CORFU2018)100 . ) O ( (4.14) (4.17) (4.13) (4.11) (4.12) (4.16) (4.15) between ) X ~ ( O Richard J. Szabo , ∆ 7 ) P ~ π 1 one finds that the d 2 ( = ). This star product has 7 0 λ ) = ~ K π 4.8 p d

2 ) ( i ) to explicitly compute p ), for , X ~ , we can unambiguously define , 1 1.4
µ − , O ) , x , Λ P ~ · is somewhat complicated, see [61] ( 3.10 ~ K , ∈ ) ) β
~ K ) x P O ~ 3 ( X ) one can now compute explicitly the f O P η ~ | Λ . , , )( ~ λ B | ~ K ~ K is defined in (  g ~ | K ? P O ~ 4.12 ~ K µναβ ( Λ R ) | ( Λ ε η ? 2 O η sin f , f ~ ~ for all R B B ~ K λ ¯ 3 s h exp + ) i ` ? O | octonionic Weyl correspondence  2 e 1 = ~ K f 30 ¯ h ) = ( 2 1 | ) OO
( ( P P X ~ ~ ~ = = ( O cos ˜ O 1 )( g P ~ M g ) . Again the explicit form is much more cumbersome and ] η ) = = = ) λ 0 α × × × x ~ K ~
? x ( ( O exp ~ K ˜ M , ~ K
f f ) 1. (
ν 3 O ( 0 O f x ∞ ˜ ~ K 0 , = M C λ O OO µ 4 → Z ( x lim x M ∈ λ [ 0 exp ˜ 2 3 7 transformation matrix f f M exp , defined via a twisted Fourier convolution product [61] Z × λ ) naturally appears as the closure condition of these elements under the 2 ) f 0 M , 4.9 1 1 ) = f f M ), one may use the unit octonion relations ( ( X ~ ∞ )( C g 4.10 λ ∈ ? g f , f ( To understand the spacetime significance of this nonassociative phase space star product, let -flux model [61]: where the nondegenerate 7 functions on the seven-dimensional M2-brane phaseoctonionic space products in and the octonions, usual and way one byof derives pulling a functions noncommutative back and the nonassociative star product 3-brackets of local configuration space coordinates are given by octonionic Baker-Campbell-Hausdorff formula for its explicit form. With it one can construct an the non-trivial feature that in the contractionR limit it suitably reduces to the star product of the string complicated compared to the string3-brackets theory triproducts, via see antisymmetrization [61] of for these the technical triproducts details. as Defining in ( An Introduction to Nonassociative Physics Then the cross product ( octonionic commutator: where the expression for the composition element Using the alternativity property powers of any octonion, and viaFor formal the power elements series ( expansion an octonionic exponential exp by a calculation completely analogousquaternion to relations the of familiar the one Pauli in spin quantum matrices. mechanics From which ( uses the for any three functions after setting the central element us introduce the M2-brane triproductssimilarly via to before: projection to the four-dimensional configuration space PoS(CORFU2018)100 0, on in a 4 = (5.3) (5.1) (5.2) (5.4) p e ρ ) for . , together with 5.1 0 k Richard J. Szabo p we considered the M . ) to i jk 2 i x on , 5.1 λ ε 2 i ) i x λ − x ) defines, after restriction ρ i j -flux backgrounds, and in = − e , δ R ρ µ λ 2 4.17 } p ) for an electric charge 4 p − ~ ( p 2 , = i 2.4 λ 7→ x ρ ) { − } i i . p 1 p , 0 , µ p 4 x = p − . ( { 0 1 } and = -invariant Nambu-Poisson 3-bracket on the 3- 4 ) 0 ) to get = p we then applied Born reciprocity to map these with the rotationally symmetric magnetic field 4 } , j 2 4 i ( . We would now like to understand what physics ) describe the kinematics of a seven-dimensional 3 3 p 4.6 p 4 p generate a quaternion subalgebra with commuting 31 k R , p { i i SO and p 5.1 + x x 2 = { i jk p ~ 0 2 } j λ ε k λ p x , , − i k 4 i jk p x and p { ρ ε i jk i j e δ , the brackets ( , to the brackets ( − − λ ε 1, the canonical ρ k λ x e = = = = is an important missing ingredient in the understanding of the dynamics ) reveals that the brackets describe a noncommutative but associative | ρ ρ ρ i jk commutes with the other momenta, we can solve this equation for x is easily computed to be a central element of the algebra ( } } } ~ | j j j of magnetic charge on 5.1 4 1.6 2 4 with x p p λ ε p p , , , ρ i R i i = x x 3 s p + { ` 0 { { 2 0 in ( } p j ~ x 2 = , i λ ρ x 0, these reduce to the magnetic Poisson brackets ( { we lifted these brackets to the phase space of M2-branes. In this final section we ask [11]. Thus the triproducts in this case provide a definition of a quantum 3-sphere, which = 3 4 S λ ). For generic values of For this, we apply again an order 4 transformation Setting Let us recap where we have gotten to in the present exposition. In Section which appears as the nonassociative gauge symmetry underlying the theory of a pair of M2- 4 2.16 Since the momentum the upper hemisphere of the ellipsoid and reduce the remaining brackets of ( When phase space with an “extra” momentum mode The quantity so without loss of generality weof may the restrict ellipsoid the four-dimensional momentum space to the surface sphere as discussed in Section deformation of spacetime whose coordinates momenta to vectors of unit length magnetic Poisson brackets describing thein phase magnetic space monopole kinematics backgrounds. of electricallybrackets In charged Section onto particles those describingSection the phase space ofwhat is closed the strings fate of in the M2-brane phase space kinematics whenthe we substitution reapply of Born reciprocity. uniform distribution ( this system represents. of M5-branes in M-theory. Theseproduct on triproducts the are seven-dimensional M2-brane naturally phase inherited space from in the the locally nonassociative non-geometric star background. 5. M-waves in non-geometric Kaluza-Klein monopole backgrounds 5.1 Magnetic monopoles in quantum gravity An Introduction to Nonassociative Physics which we recognise as a quantization ofA the defining relations (up to rescaling)branes of in the the 3-Lie Bagger-Lambert algebra theory [33]. The classical version of ( PoS(CORFU2018)100 0, → s (5.8) (5.5) (5.7) (5.6) → g acting 3 / and the i 11 1 x P R ` is given by ∼ ) P x ~ and the string ` ( Richard J. Szabo s U ∼ g ) of the algebraic λ are related through 5.5 ) x ~ ( U ). The precise gravitational , 2 5.5 .
, x 2 ~ / d 3 ρ ·  A ~ = P 11 + ` U R 4 2 x  ∇ ), with the conjugate coordinates , which lifts a single D6-brane. In that case ~ d ) x = ~ ) and the harmonic function 1 5.3 ( s ) . − of the three-dimensional spacetime through 3 g ( P U ¯ P h 2.10 δ ` ` 32 + and g x = ~ π d . The electric probes of this background are M-waves and the harmonic function 4 · λ ρ ) and x ~ x ~ d ( ) = U A ~ x ~ U ∇ ~ ( ρ + = 3 P 2 7 11 ` s A ~ R , which lift the D0-branes. d , through the identifications [109] 1 × × × S = 11 = ∇ ~ R 2 s is given locally by ( ∈ ` 2 11 n 4 s ,~ x g d finite. This is compatible with the identification ( A ~ as the Planck length, since then the limit corresponds to s λ = ` A ~ 0 with 0, they are the natural invariant Poisson brackets on the six-dimensional phase space is identified with the Planck length → ) is just the standard Taub-NUT metric. 6= λ is the metric of the transverse space to the M-theory compactification to four dimensions. 0 these are just the canonical phase space Poisson brackets of the configuration space , are related to the parameters of M-theory, the 11-dimensional Planck length 2 7 λ 11 s s 5.6 ` R = The standard Kaluza-Klein monopole solution corresponds to the embedding of a single Dirac To understand how the purported locally non-geometric background fits into this solution, we Combining these observations together, we may infer that the uncontracted octonion algebra, A magnetic monopole can be embedded into type IIA string theory as a D6-brane, with the λ . For 3 as required. the vector potential monopole, with singular distribution the Green’s function of themetric ( three-dimensional Laplacian. Then the four-dimensional part of the The proper reduction which0 takes and M-theory tocontraction type parameter IIA string theory is the limit where along the M-theory circle for a given distribution of magnetic charge first observe that the parameters of type IIA string theory, the string coupling after Born reciprocity, is in some sense relatedtime to of the three-dimensional dynamics of quantum magnetic gravity monopoles in with the the space- identification ( electric probes provided by D0-branesKaluza-Klein monopole, inside which the is a D6-brane. background of The 11-dimensional supergravity D6-brane given by lifts the to metric M-theory as a as covariant derivatives. Theof quantization a of Ponzano-Regge spin these foamparameter brackets model has of appeared three-dimensional before quantum in gravity the [108], provided context the system which realises this situation wasM-theory, recognised and by we [63] shall in now turn the to setting some of of locally the non-geometric 5.2 details. The M-wave phase space where d The three-dimensional vector potential length radius of the M-theory circle An Introduction to Nonassociative Physics For R whose momentum coordinates live on the ellipsoid ( PoS(CORFU2018)100 , - ) = 7 ˆ A ( (5.9) ξ ) by a (5.10) (5.11) , which Spin 5.9 ρ into O Richard J. Szabo of the M-theory 0 ˆ . This corroborates ), see [63] for a for- M 2.5 -structure on an eight- 5.6 ) ) and the structure con- 7 ( ) are symmetric under the 1.4 ). This splitting breaks the 5.9 Spin 4.8 ABCD , η
or the metric ( which realizes the constant magnetic 0 = , ξ A 3 ~ ˆ given by non-geometric Kaluza-Klein monopole D 2 λ R ξ [45, 61]. A O ˆ − D ABCD -flux or Kaluza-Klein monopole charge, in ˆ , C 8 φ ˆ R B is defined in ( ξ ˆ R ~ A φ Λ Λ = ). The 3-brackets ( 33 = φ -gerbe over and }
1.6 ˆ 1 C µ S ξ p , , ) onto 3-brackets for the eight-dimensional phase space ˆ B µ ξ x of the octonionic commutators ( , . ) as governing the phase space dynamics of the M-waves; 5.9 ˆ 1 A ABC S ξ η = 5.1 { ABC ∈ ˆ A = η 4 X x has components ABC on the real vector space 0 of the eight-dimensional rotation group and define a ternary operation φ φ ) 7 transformation matrix 8 ( × SO , to the total space of an ⊂ and defining the 3-brackets 3 ) ) 7 R A ( . In [63] it was then subsequently shown how this 3-algebra provides a unified of the octonionic Jacobiators ( e , 1.6 1 Spin which is defined by collecting all eight generators of the octonion algebra triple cross product ABCD η ) = ( Let us now map the 3-brackets ( In [61] it was proposed to view the full eight-dimensional phase space In the present case we are interested in a uniform distribution of magnetic charge ξ ~ , -bundle over 0 1 ξ structure ( given by the structure constants suitable change of basis. coordinates through the redefinition of the generators of where the invertible 7 terms of a natural 3-algebra structure, akinin to Section those expected to arise indescription of M-theory the that M2-brane we and discussed M-wave phase spaces. The starting point is the notion of a called a 5.3 The covariant M-theory phase space 3-algebra compactification to four dimensions, with or without where the self-dual 4-form stants subgroup dimensional real vector space is a triple cross product that can be brought to the form ( that the phase space is nowwell-defined lacking localised a position position coordinate stems from the fact that the wave has no An Introduction to Nonassociative Physics can be understood as ainstance smearing there of is Dirac no monopoles local expression throughout formal three-dimensional non-local the space. expression. vector potential This In smeared this solution is called a the interpretation of the brackets ( which lifts a uniform distributionsponds of to D6-branes the in passage from type the IIAS four-dimensional string Taub-NUT space, theory. realised Geometrically, as it the corre- total space of an charge [63], similarly to our discusson of geometric quantization in Section PoS(CORFU2018)100 -flux (up to (5.12) (5.14) (5.15) (5.16) (5.18) (5.19) (5.21) (5.13) (5.17) R 4 gives the A 4 -symmetry x ) 7 ( , ) = Richard J. Szabo i define a Nambu- µ p p , Spin , k µ not ) of the string x 1234 x 4 , ( 4 3.2 i jk G R , 4 , 3 s R ` 4 2 2 3 s p decomposes according to the with its 3-algebra to a seven- ` . λ 0 where quantum gravitational ) 2 2 0 k i jk 7 ˆ − λ p ( → M λ ε = P = i jk ` 2 ε φ , φ Spin ). This breaks the } 2 k ∼ } = 4 2 p 4 λ x gives the M2-brane phase space of Sec- φ φ λ x , 4 , i , 4 } } 0 (5.20) 5.21 j ) does not hold and its failure is controlled k = x k p i jk G , , x p p = φ , 4 , 4 1 (5.22) 1 1 , , i 4 } g j p
R j x 1.14 , ˆ
8 ⊕ ) = p { i jk j { 3 s f p , , µ ` X 7 7 7 4 x , i { i 2 p 2 k 34 p R i p , = λ δ { = 3 s , µ : 2 ` 4 1 are just the brackets of the 3-Lie algebra x 2 ,..., − G − p ( ˆ G , 1 i λ

4 { } = and ). On the other hand, the choice x k and G 8 8 8 X . It naturally defines a 2-bracket ) are non-zero. These 3-brackets do p g 0 x k j − ˆ j , λ i δ ), which reduce to the Jacobiators ( 4 f jk 4.6 G δ i M p and and results in the 3-brackets { 5.19 ε 4 k 2 ) 4 λ 2 p 4.17 4 x ) for 2 i jk 1234 ( λ and , , 4 4 − -flux or Kaluza-Klein monopole charge, in which case only the 4 4 4 k R R R i jk i jk + x , , ) and ( SO x 5.12 4 4 4 3 s 3 s k x ` ` , which describes the reduction of × R R i i j i jk j ) 2 2 2 2 i ) 0 x ε ε 3 s 3 s 5.18 ˆ 2 δ 4 ` λ ` λ 2 3 3 2 ( 2 2 2 2 M λ λ λ λ λ − − − ), ( ( SO ∞ ======C φ φ φ φ φ φ φ φ 5.16 1 is central. These 3-brackets even describe the free eight-dimensional M-theory } } } } } } } } ∈ j j k k k 4 4 4 ), ( x x x x x x x x g = , , , , , , , , , i i j j j i j j 4 f x p p x x x x p p , , , , 5.15 , , , , i i 4 i i i 4 4 x p p p p p p p subgroup, under which the spinor representation of { { { { { { { { -symmetry to 2 ) with the quasi-Poisson brackets ( 7 G For example, the constraint function Consider now any constraint ( 4 appropriate to the reduction to a seven-dimensional subspace. model when phase space in the3-brackets absence ( of by a higher 5-bracket [61,63]. Theyeffects all are vanish turned in off. the limit to its branching rule tion on the eight-dimensional phase space The first set of 3-brackets in ( Poisson bracket, because the fundamental identity ( on functions dimensional submanifold with quasi-Poisson brackets ( An Introduction to Nonassociative Physics Spin rescaling) that we encountered in ( PoS(CORFU2018)100 . ) , 8 ( 61 , SO , which ) 160 ⊂ µ ) x 7 , 524–555 ( − , 59 -Lie algebras,” Richard J. Szabo µ n p Spin ( , 253–294 (1986). 7→ 268 ) µ p ) (under the identification , µ 5.1 x ( , 2405–2412 (1973). , 89–104 (1983). 7 , 93–112 (1987). 24 (Dover Publications, New York, 1995). 294 , 145–205 (2002) [Erratum: Bull. Am. Math. 39 35 are likewise related by rotations valued in 0 ˆ M and are related by Born reciprocity , 29–64 (1934). 0 -transformation. Other linear seven-dimensional slices of the eight- , 69–85 (1978). ˆ ) 35 7 M 61 ( Spin , 122303 (2010) [arXiv:1001.3275 [hep-th]]. 51 An Introduction to Nonassociative Algebras , 213 (2005)] [arXiv:math-ra/0105155]. ). The two phase spaces are just different seven-dimensional slices of the same eight- 42 ρ e 295–316 (1994) [arXiv:hep-th/9301111]. J. Math. Phys. formalism,” Ann. Math. Commun. Math. Phys. (1946). 469–484 (1955). Göttingen, 569–575 (1932). Soc. We would like to thank the organisors of the Workshop on “Dualities and Generalized Ge- = R [9] Y. Nambu, “Generalized Hamiltonian dynamics,” Phys. Rev. D [8] E.I. Zelmanov, “On prime Jordan algebras II,” Sibirsk Mat. J. [5] R.D. Schafer, “Structure and representation of nonassociative algebras,” Bull. Amer. Math. Soc. [6] P. Jordan, J. von Neumann and E. Wigner, “On an algebraic generalization of[7] the quantumM. mechanical Günaydin, C. Piron and H. Ruegg, “Moufang plane and octonionic quantum mechanics,” [4] A.A. Albert, “On Jordan algebras of linear transformations,” Trans. Amer. Math. Soc. [1] R.D. Schafer, [2] J.C. Baez, “The octonions,” Bull. Amer. Math. Soc. [3] P. Jordan, “Über Eine Klasse Nichtassociativer Hyperkomplexer Algebren,” Nachr. Ges. Wiss. 3 s [11] J. DeBellis, C. Saemann and R.J. Szabo, “Quantized Nambu-Poisson manifolds and ` dimensional phase space [10] L. Takhtajan, “On foundation of the generalized Nambu mechanics,” Commun. Math. Phys. [12] E. Witten, “Noncommutative geometry and string field[13] theory,” Nucl.A. Phys. Strominger, B “Closed string field theory,” Nucl. Phys. B An Introduction to Nonassociative Physics M-wave phase space of this section with the quasi-Poisson brackets ( ometries” for the invitationand to comments deliver which these have lectures, helpedlarly shape and grateful the all to structure the David and Berman, participants contentKupriyanov, Bernd for of Dieter Henschenmacher, this Lüst, their Jeffrey article. Nick questions Heninger, Mavromatos We Chrisdence are and Hull, on particu- Vladislav Philip the Morrison contents forQSPACE of discussions from this the and European correspon- paper. Cooperationidated in Grant Science This ST/P000363/1 and work from Technology (COST) the was and UK by supported Science the and in Consol- Technology Facilities part Council (STFC). byReferences the Action MP1405 is realised here as a dimensional M-theory phase space Acknowledgments PoS(CORFU2018)100 , 0403 , 32 , 977–1029 , 032 (1999) Richard J. Szabo 73 9909 , 207–299 (2003) , R199–R242 (2006) 23 378 , 045020 (2007) 75 -fluxes via noncommutative H , eds. R. Penner and S.-T. Yau (International , 643–706 (2008) [arXiv:hep-th/0607020]. 277 , 33–66 (2002) [arXiv:hep-th/0101219]. 225 36 , 45–51 (1987). , 097 (2018) [arXiv:1803.00732 [hep-th]]. 185 , 705–721 (2004) [arXiv:hep-th/0401168]. 1805 253 , 41–69 (2006) [arXiv:hep-th/0412092]. structures: Then and now,” arXiv:1809.02526 [math.QA]. 264 ∞ A Perspectives in Mathematical Physics -algebras,” JHEP and ∞ ∞ L L , 136–150 (2005) [arXiv:hep-th/0412310]. , 1–100 (2013) [arXiv:1203.3546 [hep-th]]. 713 , 33–152 (1993) [arXiv:hep-th/9206084]. 527 , 014 (2001) [arXiv:hep-th/0106159]. , 065008 (2008) [arXiv:0711.0955 [hep-th]]. 77 390 0109 [arXiv:hep-th/9908142]. (2001) [arXiv:hep-th/0106048]. [arXiv:hep-th/0109162]. in curved backgrounds,” Commun. Math. Phys. 003 (2004) [arXiv:hep-th/0312043]. [arXiv:hep-th/0606233]. theories from Press, Boston), 265–288 (1994) [arXiv:hep-th/9304061]. 1087–1104 (1993) [arXiv:hep-th/9209099]. moduli space,” in: [arXiv:hep-th/0611108]. D open string field theory,” Phys. Lett. B B Phys. B Rept. Commun. Math. Phys. noncommutative manifolds,” Commun. Math. Phys. topology,” Commun. Math. Phys. [23] M. Herbst, A. Kling and M. Kreuzer, “Star products from open[24] strings in curvedM. backgrounds,” Herbst, JHEP A. Kling and M. Kreuzer, “Cyclicity of nonassociative products on D-branes,” JHEP [20] M.R. Douglas and N.A. Nekrasov, “Noncommutative field theory,” Rev. Mod. Phys. [21] R.J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. [22] L. Cornalba and R. Schiappa, “Nonassociative star product deformations for D-brane worldvolumes [25] R.J. Szabo, “Symmetry, gravity and noncommutativity,” Class. Quant. Grav. [26] R. Blumenhagen, I. Brunner, V.G. Kupriyanov and D. Lüst, “Bootstrapping noncommutative gauge [18] J. Stasheff, “ [19] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP [27] V. Mathai and J.M. Rosenberg, “T-duality for torus bundles with [17] J. Stasheff, “Closed string field theory, strong homotopy Lie algebras and the operad actions of [32] J. Bagger and N. Lambert, “Gauge symmetry and supersymmetry of multiple M2-branes,” Phys. Rev. An Introduction to Nonassociative Physics [14] G.T. Horowitz and A. Strominger, “Translations as inner derivations and associativity anomalies[15] in B. Zwiebach, “Closed string field theory: Quantum action and the BV[16] master equation,”T. Nucl. Lada Phys. and J. Stasheff, “Introduction to sh Lie algebras for physicists,” Int. J. Theor. Phys. [30] A. Basu and J.A. Harvey, “The M2–M5-brane system and a generalized Nahm’s[31] equation,” Nucl. J. Bagger and N. Lambert, “Modeling multiple M2’s,” Phys. Rev. D [33] J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, “Multiple membranes in M-theory,” Phys. [28] P. Bouwknegt, K. Hannabuss and V. Mathai, “Nonassociative tori and applications to T-duality,” [29] J. Brodzki, V. Mathai, J.M. Rosenberg and R.J. Szabo, “D-branes, RR-fields and duality on PoS(CORFU2018)100 , , 015401 19 -field 44 C , 046 (2011) Richard J. Szabo 2011 Old and New , 225–228 (1986). , 3–69 (2002) 167 625 , 06A104 (2016) , 159–162 (1985). 2016 (Quaderni, Pisa: Scuola Normale 54 , 1590 (1988)]. -field background and , 497–552 (2009) [arXiv:0708.2648 C 60 13 Noncommutative Geometry and , 014 (2000) [arXiv:hep-th/0005123]. 37 -manifolds 1,” Nucl. Phys. B ) 7 ( 0007 , 097 (2009) [arXiv:0901.1847 [hep-th]]. Spin , 770–778 (1985). 0904 60 -manifolds and four-dimensional physics,” Class. Quant. Grav. , 075 (2011) [arXiv:1012.2236 [hep-th]]. 2 , 173–197 (2000) [arXiv:hep-th/0005026]. G 590 1104 , 113B01 (2013) [arXiv:1307.1663 [hep-th]]. , 680–683 (1988) [Erratum: Phys. Rev. Lett. 60 2013 , 1330005 (2013) [arXiv:1211.0395 [hep-th]]. 25 , eds. Y. Maeda, H. Moriyoshi, M. Kotani and S. Watamura (World Scientific Publishing), models,” PTEP 171–210 (2017) [arXiv:1609.09815 [hep-th]]. 5-brane,” Nucl. Phys. B Physics 4 from multiple membranes,” JHEP [hep-th]]. reduced model,” JHEP Phys. Rev. Lett. and D-branes in bivariant K-theory,” Adv. Theor. Math. Phys. possibility of correct quantization of a theory with anomalies,” Phys. Lett. B noncommutative boundary strings,” JHEP [arXiv:1203.5921 [hep-th]]. Math. Phys. [arXiv:1603.09534 [hep-th]]. 5619–5653 (2002). [arXiv:hep-th/0109025]. (2017), 1–85 [arXiv:1005.2820 [math.RA]]. Problems in Fundamental Physics: Meeting inSuperiore), Honour 43–53 of (1986). G.C. Wick anomalies,” Theor. Math. Phys. (2011) [arXiv:1010.1263 [hep-th]]. [35] J. DeBellis and R.J. Szabo, “Reduced Chern-Simons quiver theories and cohomological 3-algebra [36] C. Saemann, “Lectures on higher structures in M-theory,” in: [37] E. Bergshoeff, D.S. Berman, J.P. van der Schaar and P. Sundell, “A noncommutative[38] M-theory S. Kawamoto and N. Sasakura, “Open membranes in a constant [52] R. Blumenhagen and E. Plauschinn, “Nonassociative gravity in string theory?,” J. Phys. A An Introduction to Nonassociative Physics [34] J. DeBellis, C. Saemann and R.J. Szabo, “Quantized Nambu-Poisson manifolds in a 3-Lie algebra [39] C.-S. Chu and D.J. Smith, “Towards the quantum geometry of the M5-brane in a constant [50] G.W. Semenoff, “Nonassociative electric fields in chiral gauge theory: An explicit construction,” [51] J. Brodzki, V. Mathai, J.M. Rosenberg and R.J. Szabo, “Noncommutative correspondences, duality [40] C. Saemann and R.J. Szabo, “Groupoid quantization of loop spaces,” PoS CORFU [49] L.D. Faddeev and S.L. Shatashvili, “Realization of the Schwinger term in the Gauss law and the [41] C. Saemann and R.J. Szabo, “Groupoids, loop spaces and quantization of 2-plectic manifolds,” Rev. [43] B.S. Acharya, “M-theory, [44] S. Gukov and J. Sparks, “M-theory on [45] D.A. Salamon and T. Walpuski, “Notes on the octonions,” Proc. 23rd Gokova Geom.-Topol.[46] Conf. R. Jackiw, “3-cocycle in mathematics and physics,” Phys. Rev. Lett. [48] L.D. Faddeev and S.L. Shatashvili, “Algebraic and Hamiltonian methods in the theory of nonabelian [42] P.-M. Ho and Y. Matsuo, “The Nambu bracket and M-theory,” PTEP [47] M. Günaydin and B. Zumino, “Magnetic charge and nonassociative algebras,” in: PoS(CORFU2018)100 , , -flux , 040 , R , 151 326 19 1302 451N7174 2017 , 084 (2010) Richard J. Szabo , 027 (2016) 1012 1611 , 341–344 (2002) 43 , 385401 (2011) [arXiv:1106.0316 44 38 , 141 (2014) [arXiv:1312.0719 [hep-th]]. , 144 (2017) [arXiv:1705.09639 [hep-th]]. 1404 1710 -structures and quantization of non-geometric M-theory 2 G , 012 (2012) [arXiv:1207.0926 [hep-th]]. , 099 (2017) [arXiv:1701.02574 [hep-th]]. , 091 (2015) [arXiv:1405.2283 [hep-th]]. 1209 , 171 (2014) [arXiv:1309.3172 [hep-th]]. 1702 1506 1401 , “Dirac strings and magnetic monopoles in spin ice Dy2Ti2O7,” Science , 012041 (2018) [arXiv:1709.10080 [hep-th]]. et al. -flux backgrounds,” JHEP 965 R ˇ cík and T. Strobl, “WZW–Poisson manifolds,” J. Geom. Phys. , 145–154 (2001) [arXiv:math-sg/0107133]. , 066003 (2017) [arXiv:1707.00312 [hep-th]]. 411–414 (2009) [arXiv:1011.1174 [cond-mat.mtrl-sci]]. Conf. Ser. string vacua,” JHEP 1837–1862 (2004) [arXiv:physics/0401142]. M-theory (2018) [arXiv:1803.08861 [hep-th]]. [arXiv:math-sg/0104189]. 144 42–45 (2008) [arXiv:0710.5515 [cond-mat.str-el]]. (2013) [arXiv:1207.6412 [hep-th]]. geometry in double field theory,” JHEP 96 backgrounds, the algebra of octonions, and[arXiv:1607.06474 missing [hep-th]]. momentum modes,” JHEP backgrounds,” JHEP backgrounds,” JHEP [hep-th]]. flux backgrounds,” JHEP [arXiv:1010.1361 [hep-th]]. asymmetric strings and nonassociative geometry,” J. Phys. A [71] R.J. Szabo, “Magnetic monopoles and nonassociative deformations of quantum theory,” J. Phys. [72] I. Bakas and D. Lüst, “3-cocycles, nonassociative star products and the magnetic paradigm of [63] D. Lüst, E. Malek and R.J. Szabo, “Non-geometric Kaluza-Klein monopoles and magnetic duals of [66] R.J. Szabo, “Quantization of magnetic Poisson[67] structures,” arXiv:1903.02845 [hep-th]. C. Klim [68] P. Ševera and A. Weinstein, “Poisson geometry with a 3-form background,” Prog.[69] Theor. Phys. Suppl. C. Castelnovo, R. Moessner and S.L. Sondhi, “Magnetic monopoles in spin[70] ice,” Nature D.J.P. Morris [64] E. Plauschinn, “Non-geometric backgrounds in string[65] theory,” arXiv:1811.11203 [hep-th]. R.J. Szabo, “Higher quantum geometry and non-geometric string theory,” PoS CORFU [62] R.J. Szabo, “Magnetic backgrounds and noncommutative field theory,” Int. J. Mod. Phys. A [57] R. Blumenhagen, M. Fuchs, F. Haßler, D. Lüst and R. Sun, “Nonassociative deformations of [59] L. Freidel, R.G. Leigh and D. Minic, “Noncommutativity of closed string[60] zero modes,” Phys.M. Rev. Günaydin, D D. Lüst and E. Malek, “Nonassociativity in non-geometric string and M-theory [61] V.G. Kupriyanov and R.J. Szabo, “ [58] C.D.A. Blair, “Noncommutativity and nonassociativity of the doubled string in non-geometric [55] D. Mylonas, P. Schupp and R.J. Szabo, “Membrane sigma-models and quantization of non-geometric [56] A. Chatzistavrakidis and L. Jonke, “Matrix theory origins of non-geometric fluxes,” JHEP [54] R. Blumenhagen, A. Deser, D. Lüst, E. Plauschinn and F. Rennecke, “Non-geometric fluxes, An Introduction to Nonassociative Physics [53] D. Lüst, “T-duality and closed string noncommutative (doubled) geometry,” JHEP PoS(CORFU2018)100 , 107 , 095205 51 , 966 (2018) , 103 (2015) 78 Richard J. Szabo , 201602 (2018) 1509 121 , 137–144 (2004) 19S1 , 093 (2015) [arXiv:1411.3710 1503 , 099901 (2016)] [arXiv:1510.07559 , 122301 (2014) [arXiv:1312.1621 [hep-th]]. implementation and perturbativity via 117 55 RAPH 39 G A Concise Treatise on Quantum Mechanics in Phase AD of 13 TeV proton-proton collisions at the LHC,” Phys. Lett. B 1 − , 3614–3618 (2010) [arXiv:0912.2197 [math-ph]]. , 045005 (2018) [arXiv:1803.00405 [hep-th]]. 98 374 [MoEDAL Collaboration], “Search for magnetic monopoles with the MoEDAL et al. , 028 (2017) [arXiv:1610.08359 [math-ph]]. , 2293–2301 (2018) [arXiv:1802.05808 [math-ph]]. , 220402 (2015) [Erratum: Phys. Rev. Lett. (World Scientific Publishing, 2014). 1704 108 115 , 510–516 (2018) [arXiv:1712.09849 [hep-ex]]. [quant-ph]]. charges and monopoles in nonassociative quantum[arXiv:1810.06540 mechanics,” [hep-th]]. Phys. Rev. Lett. heavy-ion collisions,” arXiv:1902.04388 [hep-th]. 782 photon fusion and Drell-Yan processes: M [arXiv:1808.08942 [hep-ph]]. forward trapping detector in 2.11 fb [hep-th]]. Lett. velocity-dependent coupling and magnetic moment as novel features,” Eur. Phys. J. C Phys. Uncertainty relations and semiclassical evolution,” JHEP 365–380 (1976). [arXiv:hep-th/0212058]. a monopole,” Phys. Lett. A (2018) [arXiv:1708.05030 [math-ph]]. nonassociative quantum mechanics,” J. Math. Phys. [arXiv:1506.02329 [hep-th]]. JHEP [math-ph]. distributions,” Phys. Rev. D translations,” arXiv:1804.08953 [hep-th]. Space [87] M. Bojowald, S. Brahma, U. Büyükçam, J. Guglielmon and M. van Kuppeveld, “Small magnetic [88] O. Gould, D.L.-J. Ho and A. Rajantie, “Towards Schwinger production of[89] magnetic monopoles in B.S. Acharya [90] S. Baines, N.E. Mavromatos, V.A. Mitsou, J.L. Pinfold and A. Santra, “Monopole production via [86] M. Bojowald, S. Brahma and U. Büyükçam, “Testing nonassociative quantum mechanics,” Phys. Rev. [82] V.G. Kupriyanov and D.V. Vassilevich, “Nonassociative Weyl star products,” JHEP [84] D.V. Vassilevich and F.M.C. Oliveira, “Nearly associative deformation quantization,” Lett. Math. [85] M. Bojowald, S. Brahma, U. Büyükçam and T. Strobl, “States in nonassociative quantum mechanics: [78] R. Jackiw, “Dirac’s magnetic monopoles (again),” Int. J. Mod. Phys. A [79] J.F. Cariñena, J.M. Gracia-Bondía, F. Lizzi, G. Marmo and P. Vitale, “Star product in the presence of [81] D. Mylonas, P. Schupp and R.J. Szabo, “Non-geometric fluxes, quasi-Hopf twist deformations and [83] M. Bojowald, S. Brahma, U. Büyükçam and T. Strobl, “Monopole star products are non-alternative,” An Introduction to Nonassociative Physics [73] J.M. Heninger and P.J. Morrison, “Hamiltonian nature of monopole dynamics,” arXiv:1808.08689 [74] V.G. Kupriyanov and R.J. Szabo, “Symplectic realization of electric charge in fields of monopole [80] M.A. Soloviev, “Dirac’s magnetic monopole and the Kontsevich star product,” J. Phys. A [75] S. Bunk, L. Müller and R.J. Szabo, “Geometry and 2-Hilbert space[76] for nonassociativeT.L. magnetic Curtright, D.B. Fairlie and C.K. Zachos, [77] T.T. Wu and C.N. Yang, “Dirac monopole without strings: Monopole harmonics,” Nucl. Phys. B PoS(CORFU2018)100 , 1503 , 019 , 085 (2005) -flux string R , S773–S794 1607 Richard J. Szabo 0510 24 , 85–126 (1995) , 111–152 (2015) 89 , 065 (2005) 443 -algebras and their field ∞ 0510 L , 099 (2009) [arXiv:0904.4664 , 234–255 (2016) [arXiv:1507.02792 exceptional field theory,” JHEP ) 106 5 0909 ( SL 40 , 9282905 (2018) [arXiv:1709.10004 [math-ph]]. , 012004 (2015) [arXiv:1504.03915 [hep-th]]. , 036 (2018) [arXiv:1710.11467 [hep-th]]. 634 2018 , 1150–1186 (2012) [arXiv:1204.1979 [hep-th]]. 60 1802 ´ quantum gravity and effective noncommutative quantum field c and R.J. Szabo, “Nonassociative differential geometry and gravity D ´ Ciri ´ c , 015 (2018) [arXiv:1802.07003 [hep-th]]. , 221301 (2006) [arXiv:hep-th/0512113]. 96 , 050 (2018) [arXiv:1710.05919 [hep-th]]. 1807 , 081 (2016) [arXiv:1601.07353 [hep-th]]. , 1800093 (2019) [arXiv:1810.03953 [hep-th]]. 1801 67 2015 [math.QA]]. PoS CORFU categories I: Bimodules and their internal homomorphisms,” J. Geom. Phys. [hep-th]]. Fortschr. Phys. theory realizations,” Adv. Math. Phys. [arXiv:hep-th/0508133]. [arXiv:hep-th/0406102]. double field theory,” Fortsch. Phys. (2007) [arXiv:0708.3984 [hep-th]]. compactifications,” J. Phys. Conf. Ser. [arXiv:1409.6331 [math.QA]]. categories II: Connections and curvature,” J. Geom. Phys. 144 (2015) [arXiv:1412.0635 [hep-th]]. threebrane sigma-models,” arXiv:1901.07775 [hep-th]. sigma-models,” JHEP with non-geometric fluxes,” JHEP theory,” Phys. Rev. Lett. M-theory,” JHEP (2016) [arXiv:1604.03253 [hep-th]]. [arXiv:hep-th/9503124]. [102] G.E. Barnes, A. Schenkel and R.J. Szabo, “Working with nonassociative geometry and field theory,” [100] G.E. Barnes, A. Schenkel and R.J. Szabo, “Nonassociative geometry in quasi-Hopf representation [103] R. Blumenhagen and M. Fuchs, “Towards a theory of nonassociative gravity,” JHEP [92] V.E. Marotta and R.J. Szabo, “Para-Hermitian geometry, dualities and generalized flux backgrounds,” [93] O. Hohm, V.G. Kupriyanov, D. Lüst and M. Traube, “Constructions of [95] C.M. Hull, “A geometry for non-geometric string backgrounds,” JHEP [96] D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, “Non-geometric fluxes in supergravity and [98] P. Aschieri and R.J. Szabo, “Triproducts, nonassociative star products and geometry of [101] G.E. Barnes, A. Schenkel and R.J. Szabo, “Nonassociative geometry in quasi-Hopf representation [107] A. Chatzistavrakidis, L. Jonke, D. Lüst and R.J. Szabo, “Fluxes in[108] exceptional field theoryL. and Freidel and E.R. Livine, “3 An Introduction to Nonassociative Physics [91] C.M. Hull and B. Zwiebach, “Double field theory,” JHEP [94] J. Shelton, W. Taylor and B. Wecht, “Non-geometric flux compactifications,” JHEP [97] B. Wecht, “Lectures on non-geometric flux compactifications,” Class. Quant. Grav. [99] A. Chatzistavrakidis, L. Jonke, F.S. Khoo and R.J. Szabo, “Double field theory and membrane [106] C.D.A. Blair and E. Malek, “Geometry and fluxes of [105] D. Lüst, E. Malek and M. Syväri, “Locally non-geometric fluxes and missing momenta in [109] E. Witten, “String theory dynamics in various dimensions,” Nucl. Phys. B [104] P. Aschieri, M. Dimitrijevi