PoS(CORFU2016)131 http://pos.sissa.it/ gauge N -dimensional space-time can be reformulated d gauge theory on a ) N ( SU . 2-dimensions with the extra two dimensions describing a fuzzy closed sur- N + / d ∗† [email protected] Speaker. Talk presented at the Ioannis BAKAS memorial Conference, Ringberg, November 2016. We show that a classical as a theory on face. The coordinates of this surface satisfyproportional a to Heisenberg 1 algebra with the deformation parameter † ∗ Copyright owned by the author(s) under the terms of the Creative Commons c theories Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2016 "School and31 Workshops August on - Elementary 23 Particle September, Physics 2016 andCorfu, Gravity" Greece École Normale Supérieure, 75005 Paris, France E-mail: John Iliopoulos Non-commutative geometry from large PoS(CORFU2016)131 ], 2 John Iliopoulos ] ], but there was no fol- 3 5 ] who gave it as a problem to his 4 1 ] but there are some open problems remaining. 1 He talked instead to Oppenheimer[ ]. In fact, as history evolved, Pauli was probably right. The motivation 1 6 In a letter to Bohr he commented : “....it seems to be a failure for reasons of physics.”[ There may be several motivations to study a quantum theory in a space with non-commutative I met Yannis for the first time in 1983. I was visiting Imperial College for a seminar and These notes give a summary of a discussion we had together in the first days of August 2016, I am reporting here on some on-going work done in collaboration with Prof. Manolis Floratos. 1 suggested that non-commutativity among space coordinatesgularities. could He eliminate tried the to short convince distance Peierls sin- not and Pauli think to much work of on the this idea. problem,PhD but, student apparently, H.S. Pauli Snyder. did Snyder publishedlow a up thorough for study many of years. it Snyder inas himself 1947[ left an non-commutative accelerator geometry engineer[ and had abased successful on career short distance singularitiesthe development did of not the renormalisation prove programme fruitful inproblem for the of framework ultraviolet elementary of divergences took quantum particle a field physics. completely theories,theories different the turn. finite, With While the a renormalisation space programme cut-off applies makesIt all to is very a few most and remarkable verybut specific fact rather field that lack theories. they of are sensitivity precisely to the unknown ones physics chosen at by very Nature. short It distances is that not turned finiteness out to be the geometry. The first proposal goes back to W. Heisenberg who, in a letter to Peierls in 1930[ the late Tom Kibble, thengraduate Chairman students of in the Theoretical Department,I Physics. recommended to had Yannis me the was a occasion among fewas to them. bright in confirm Greek Conferences During this and the summer opinion.in schools following our and years We Yannis field. met soon established often Incommitment himself in recent as and École years a responsibility. I Normale, leading discovered scientist atscientists He another CERN working facet was as in of deeply well Greece his concernedauthorities under personality: with the of his the current his profound harsh crisis sense nativeExcellency”, conditions and of town, and he of asked Kalamata, decided me young to to to jointo establish help. this two a effort. young He special The high convinced first prize, energy suchguests the Jim prizes “the physicists, Cronin were one Messinian and awarded David experimentalist in Prize Gross. October andYannis. for I 2015 In one promised his to theorist, quiet continue and by this simple effort, our manner but distinguished it he won’t was be able easy to without inspire,just convince two and days before mobilise. he entered thediscussion. Hospital for the He last was time.we It optimistic was promised probably and Yannis’ to last confident scientific meet thatwhen again his I in heard health a the problems few newscolleague would weeks of answering be his to that death. continue soon this our was I over impossible:in sent discussion. and the just a ArXiv. very message So, He recently was to I Yannis right. École hadprivilege felt Bakas Normale to published will a know and two remain him, great I papers in will remember shock the always scientific a remember literature, Yannis. young but we, who have had2. the Introduction The main results have already been published,[ Non-com. geometry from large N gauge theories 1. Forward PoS(CORFU2016)131 (3.2) (3.3) (3.1) (2.1) ] showed 8 ] solved the 7 John Iliopoulos gauge theories ) N ( SU (Lie algebra case, the one ρ x ) ] and co-workers and aims at x ρ ( µν 11 f is constant (canonical, or Heisen- iF ][ ν 9 y ∂ x θ = x ] = eH µν cp | ν ) θ x i y − , ( µ g = x ) [ c µν x )] = y ( θ x i f ( ] who showed the existence of a map between f ν ∂ ; y , 10 2 ] = µ µν ν x θ [ x x µ , , (sometimes called quantum space case). ∂ x + µ σ i 2 x x y e [ ⇒ ρ x eH cp = µ ν g µν ρσ = δ ∗ R c 1 f x = − q ν x = µ ∂ ν x µ ] has become quite fashionable. A new element was added a few years x 9 ]. The more recent advances have been presented in several CORFU meetings. 12 and matrix models. N We also define the derivative: Space non-commutativity means that we have a commutation relation of the form: In the simplest case, the one we will consider in this talk, However, almost at the same time, a new motivation for studying theories in a non-commutative and a * product which is formally given by: constructing a gauge theorycommutative with geometry. spontaneously The result broken which symmetry relatesbetween using the different symmetry the branes breaking techniques has parameter been of toarticles, first the non- both distance obtained from in the this mathematicsgiven as approach. in well reference[ There as the have physics been point many of review view, and anat incomplete large list is 3. Some elementary formulae berg case). Other choices have also been considered: Here I want to mention a different but related motivation which comes from problem of the motion of anthe electron in energy an levels, external constant the magnetic so-calledoperator field and, “Landau of besides levels”, the computing he electron do showedclassical not that case commute. the where components the A of electron simpledicular the follows to way velocity a the to spiral field visualise trajectory is thisare: a whose result circle. projection is on to In a think Landau’s of plane quantum the perpen- mechanical solution the centre’s coordinates which shows that thecommutative two structure coordinates on do space not itself.that, commute. at Following The Heisenberg’s least suggestion, magnetic the R. fieldSince lowest Peierls[ the has Landau presence induced of level, a non-vanishingmodern non- can supergravity magnetic-type be and external string obtained fields models, by the iscommutative study a using geometry[ of common this field feature theories space formulated inago non-commutativity. on many with spaces with the non- work of N. Seiberg and E. Witten[ analysed by Snyder), or space appeared, although only recently it was fully appreciated. In 1930 L.D. Landau[ Non-com. geometry from large N gauge theories important criterion. The geometry of physicalpresence space is may not still relevant produce for an the ultraviolet calculation cut-off, of but physical its processes among elementary particles. gauge theories formulated in spaces with commutingindependent and line non-commuting coordinates. of A approach somehow has been initiated by A. Connes[ PoS(CORFU2016)131 ) 1 N x 2 ( + and i g , the NN (3.4) (3.5) (3.6) (3.7) ] and 1 φ d ∞ 1 σ ,..., → 1 N with ) = 2 ) i John Iliopoulos 2 σ , there exists a , a ) σ t 1 , . x ) 1 ( 2 σ x i , we can write ( σ σ structure constants, φ x , µ a ( x ) N A ( N µν and ( µν 1 F ) = } σ } ) F x ) SU ) 2 ( 2 2 σ µ σ → σ 2 , , A the , ) 1 1 )) 1 x σ σ x ( 2 ]. Written explicitly we have: σ ∞ , , , , 1 x  x µν x ( σ ( 2 ( → ( F ν belonging to the adjoint representation )) µν φ x W A N ( ) , , , F x , which, in the Landau paradigm, is pro- and σ ) ) ( σ 2 2 2 θ d ) is local[ ( µ σ 2 σ σ π . The proof of this statement is essentially φ A , , d 2 σ N as the Fourier components of a field in 2 ( 1 : 1 1 0 , σ ) Z 1 σ σ σ σ σ x , , d d σ ( 3 x x , i x π ( ( x 4 -dimensional space 2 φ and µ µ ( -dimensional space. At large 0 d d → µ d Z ]. We can also show it explicitly for the sphere[ 1 A A 2 Z σ  ) is established for the classical theories. To go to the A ) x ( → { → { → 3.7 i → SDiff ) φ → )] )] x ) x x ( x ( (
µ ( ) ν 4 i . As a result, the perturbation expansion cannot be defined. This x A W A 2 φ φ ( ) we have not imposed ’t Hooft’s rescaling condition in which , , 1 ) σ ) x ∞ µν = x ∑ i ( 3.7 ( F ] is to prove that at the limit µ µ ) and A x A [ 13 1 ( [ , i.e. we can consider σ µν π Yang-Mills theory in a F 2 ) r N ≤ +2 dimensions ( T gauge invariance has become invariance under area preserving diffeomorphisms d σ x ) ] 4 SU -component field defined in a N d ≤ group, the resulting expression at large ( 14 N ) 0 Z ][ N ) is an element of the gauge group. We can show that the Yang-Mills action becomes: 1 SU ( x , W be an σ SU ( A final remark: The equivalence ( Given an All computations can be viewed as expansions in The main remark which underlines our approach is the following: Let The crucial remark is that, for a Yang-Mills field The ∞ φ appropriately chosen coordinates on a compact 2-dimensional surface, such that, at 2 reformulation in appropriately rescaled, go to those ofthe [ torus[ is not surprising. In proving ( is kept fixed and we recover, already at lowest order, the infinite number of graphs. It is possible, quantum theory we should first finda a second suitable problem: gauge and The thisrespect can to quadratic be the part done. variables of But then the we new are 6-dimensional facing action has no derivatives with Non-com. geometry from large N gauge theories portional to the inverse ofcomputational the tools external can magnetic be field.nature developed of which It space would is and not take this clear fullhave seems a whether advantage to quantum any of be field more theory the a in efficient severe non-commutative around a limitation ordinary space of space. with the non-commutative This power geometry may which ofbe be is limited the not the to approach. an reason the expansion We classical why do theory most and not formulations it of is gauge not theories easy seem→ to extend to to them the BRST symmetry. However, the interaction term will be non local in σ matrix commutators become the usual Poisson brackets with respect to → dimensions with the extra dimension being a circle. In this case we have: of an where of the 2-dimensional surface spanned by algebraic. A direct way[ PoS(CORFU2016)131 i T and (4.2) (4.1) θ , will have 1 generators N / John Iliopoulos − 2 00 . They satisfy the N i ) m , S N 00 ( l 2 1. This in turn shows S 1 1 linearly independent N 0 00 = m m + , − , 2 3 0 l 00 l x l 1 ; ) m N + , = ( l 2 2 f . The corresponding spherical 2 3 x i . θ goes to infinity of the generators T are represented in the standard way by l 2 i + ) S x + N 2 1 2 ] = cos 0 ( x 2 2 ... there are 2 ) m 1 , T N i 0 l = SU ( l x S 3 + l i x 2 , ) 1 . A different approach would be to expand ) ... , we can parametrise the three operators m T 1 m ( N N i , ( l N whose generators we call α and ]. In practice this means that we must choose a suitable = S [ l 3 2 , 2 15 2 θ , ,..., 1 , the limits as T 1 4 ∑ = . In this section I want to point out that we can 2 = k k i sin ⇒ 3 ∞ rx , rescaled by a factor proportional to 1 and T i φ l k S → i T subgroup[ = S subgroup ) = structure constants in a somehow unusual notation. It is sin ) N 2 ) i jk ... 2 φ . We can use them as a basis to build the ) k ( 2 1 ε , k = i x ( N S θ S N 2i 2 k ( SU ( as follows: l x i i jk SU m ) , , 2 ε ... l SU m i z ] = 1 ( θ i with the price of introducing non-commutative geometry for the j Y generators α T , an ) , limit. For any value of l N 3 ] = ) sin i , 2 and j 2 , ]. I will sketch the proof for a “fuzzy” sphere and a “fuzzy” torus. T ( ,..., N 1 S [ N 1 1 . 1 φ ( ∑ = , z l = are the i k k i SU goes to infinity: S ) [ ) with the closed surface being a sphere SU cos N ( ,..., N which commute and are constrained by l f = i = 3.7 ) x − implies 1 m N , i x ( l = S Lie algebra such that the generators of the chosen S 2 N ) m N and non-commutative geometry , is a symmetric and traceless tensor. For fixed ( l = i l ) i i N ) ... ]. SU in the fundamental representation: m T 1 ... ( m i 1 ) ( i 16 α → N α i ( matrices forming an irreducible representation. S To be precise, it must be the principal A similar parametrisation has been used by T. Holstein and H. Primakoff in terms of creation and annihilation In other words: under the norm So much for the large For the classical sphere a convenient choice of coordinates is given by the usual angles The previous results were valid at We choose, inside 2 3 N SU are three objects × . We can write i of where the constants well-defined limits as now clear that the three N operators[ basis in the φ T 4.1 A fuzzy sphere extend them to any valueextra 2-dimensional of space[ Non-com. geometry from large N gauge theories although we have no explicit proof, that wescheme, can but absorb these it divergences is in a not clevertheory renormalisation clear we whether started from any is new renormalisable insight for can any finite be obtained this way. The 4-dimensional harmonics are given by: where that the classical Yang-Mills theory becomes thephisms theory of invariant equation under ( area preserving diffeomor- in terms of two operators, around a non-trivial solution which,the hopefully, Yang-Mills theory. captures Such part a of “master the field” non-perturbative has not dynamics yet of been found. 4. Finite tensors commutation relations: PoS(CORFU2016)131 . 2 ) z an 2 ( ) (4.3) (4.4) (4.5) (4.7) (4.8) (4.9) (4.6) N and ( SU algebra 1 ) z SU 2 ( ) 2 John Iliopoulos SU z , 1 z , x ( µν             satisfy the 0 1 0 0 F i i N 2 ∗ T odd (a similar construction ) ...... Moyal Moyal 2 imply the Heisenberg algebra } N } z ] = ) i ) , 2 1 2 k T 2 z 2 1 z i z T 1 z z . We define the two matrices: 2 , z , i , , − 1 ...... 1 N 1 0 0 0 0 0 0 1 0 0 0 1 1 i jk x e e z / z i z [ ( ε , 1 2 1 2 , π             gh x 4 x ) ) N 2i ( µν ] with respect to the operators ( e 2 2 2 2 ω ν z z = F and 17 = W A = − − ] = 2 , , h j z ) ) ) 1 1 ω T d 2 2 2 hg ( ( ; , z z z 1 2 i , , , z 1 1 z 2 2 z z T 1 1 i i 1 d [ . Let us take first z z z x e dimensional space, there exists a reformulation in 5 ) − = , , ,             4 e 1 x x x 1 ; 3 1 − d = ( ( ( ( T ⇔ − d = µ µ 2 generators in terms of the three generators of = 0 0 0 0 0 0 U Z N µν T 2 ) i A A N T i ω × F i h N N 2 ) algebra for the operators + ( → 1 1 − ...... ) = → { → { ( → T 1 2 th root of unity: 2 SU ) ] = N ( T U )] )]
0 2 x ] to prove the following algebraic statement: g = N ) ω x x z ( 1 ( ( x = , SU + ( ν 1 ω µν T − z W A [ F , T µν ...... , 1 0 0 0 0 0 0 0 0 ) ) F x be the x ) (             ( x satisfy the Heisenberg algebra, the operators µ ω µ ( ) 2 A 2 A [ = z [ z µν , F Yang-Mills theory in a g 1 z ) and , r x N 1 T ( ( z x . From that point we can go on and show a formal equivalence: µ 4 2 SU z A d even) and let Z → N and ) x 1 ( z subgroup and express all the µ ) A Given an such that The case of the fuzzy torus is even simpler. For the sphere we had isolated inside 2 ( +2 dimensions in which Then it is straightforward algebra[ in other words, if and the opposite is alsoamong true, the d and the action becomes: Non-com. geometry from large N gauge theories with an appropriately defined *-product. 4.2 A fuzzy torus SU They satisfy quantum group commutation relations: For the torus we isolateapplies a to quantum where the brackets are the symmetrised Moyal brackets[ PoS(CORFU2016)131 N are / , in 2 g ] can z ) (4.14) (4.15) (4.13) (4.11) (4.10) 2 . Could T and and N ( ) is indeed 1 h z , by: John Iliopoulos g 4.11 SDi f f and generators: h and [ ) ) N matrices, but, obviously, N to be an integer. It could ( ( 2 N N m SU . Note that the later imply the z × SU − , ZZ (4.12) = 1 N N w m | ]. In this case, we can define two ∈ n − ) algebra by the finite dimensional with the commutation relations of 2 + ] that the algebra ( S 19 w and the group elements ) 2 n m ( i z , 18 N S = g z gh 1 ( ) 2 n z  ω 2 m ( , z n and SU 1 f ) π = 1 † m ) 2 × 2i z j z ]. integers mod S w − 2 hg ∂ 2 m it becomes the algebra of the area preserving e i n z 20 n i ; π ∂ N ∞ π = 2 i j ⇔ 2 2 6 g  and m → ξε + h i 1 i 1 1 1 ( . We can show[ n z N N 2 i z m 2 1 e g n 2isin n 2 1 exp i / = ] = ( 2 m with 2 h m ] = 2 z 1 xp − ) = n , n z m e appear. The *-product can be written as 1 g S 2 ( z , [ ω g m = N m 1 of the two circles, and expanding all functions on the torus on the ∗ / 2 S = and n n [ 2 ) , 1 2 z z 1 powers of these matrices to express the n ( n = m . The Yang-Mills action can be written again in the form of equations , h and at the limit f h 1 N N ). If we define the corresponding group elements ) m m and / S 2 N 1 4.4 × ( z = n , the first being the position operator on the discrete configuration space and p SU ξ ). They form a discrete subgroup of the Heisenberg group and they have been and . The generators of the Heisenberg algebra ) and ˆ and ) 4.10 2 ) N q ) and, as before for the sphere, this equivalence is exact at any order in the 1 n ( 2 , z 1 , 4.7 1 n SU z ) and ( = ( At the formal level, all the previous expressions do not require = ( 4.8 n z As we found for the sphere, the Moyal bracket can be defined by symmetrising in Let me finish with some randomly• chosen remarks. ) and ( 4.5 be made explicit by choosing afor pair example, of the variables angles forming local symplecticbasis coordinates of on the the eigenfunctions torus, of the laplacian: Here we are interested in thethe fuzzy Heisenberg torus, algebra so ( we endow infinite dimensional operators, but weones can ( represent the ( for the set of group elements algebra of with diffeomorphisms of a 2-dimensional torus. This connection between new operators ˆ the second its finite Fourier transform.they They do can not be satisfy represented anymore by the Heisenberg algebra[ used to construct quantum mechanics on a discrete phase space[ equivalent to that of we can prove again the equivalence: expansion. 5. Conclusions and remarks be any, even complex, number. Sothis this offer writing a new offers insight? a way to define the theory for any Non-com. geometry from large N gauge theories We can use the integer mod with which case only odd powers of 1 PoS(CORFU2016)131 , 49 , 566 B632 John Iliopoulos , Vol. III, p. 380. , Vol. II, p. 15. (Ed. , Vol. II, p. 414. (Ed. Karl , 553 (2010); A.H. 58 ] was more ambitious. It gave a 11 , 1190 (1952) ][ 9 88 , 098 (2014) 1412 , 237 (1988); E.G. Floratos, J. Iliopoulos and G. 7 B201 Wolfgang Pauli, Scientific Correspondence Wolfgang Pauli, Scientific Correspondence , 032 (1999) 09 Wolfgang Pauli, Scientific Correspondence , 285 (1989); E.G. Floratos and J. Iliopoulos, Phys. Lett. , 38 (1947). Apparently Snyder had completed this work as early as 1943. , 629 (1930) B217 71 64 , 763 (1933) 80 , 977 (2001); J. Wess, “Gauge theories beyond gauge theories”, Fortsch. Phys. 73 This brings me to the last point which applies to all the formulations I have seen so far. The non-commutative geometry we found here appears to be only formal, it translates the The approach initiated by A. Connes and co-workers[ Chamseddine, A. Connes and V. Mukhanov, JHEP Proceedings, SUSY02; J. Madore, “An IntroductionPhysical to applications”, non-Commutative Cambridge, Differential UK, Geometry University and Press,Rev. 2000; Mod. M. Phys. R. Douglas and N. A. Nekrasov, school, 1992 (2006) Karl von Meyenn, Springer-Verlag (1985)) von Meyenn, Springer-Verlag (1985)) (Ed. Karl von Meyenn, Springer-Verlag (1993)) Tiktopoulos, Phys. Lett. • • • [8] R. Peierls, Z. Phys. [7] L.D. Landau, Z. Phys. [9] For a review, see A. Connes, “Non-Commutative Geometry and Physics”, Les Houches summer [2] Letter of Heisenberg to Peierls (1930), [3] Letter of Pauli to Bohr (1947), [4] Letter of Pauli to Oppenheimer (1946), [5] H.S. Snyder, Phys. Rev. [6] E.D. Courant, M.S. Livingston and H.S. Snyder, Phys. Rev. [1] E.G. Floratos and J. Iliopoulos, Phys. Lett. [10] N. Seiberg and E. Witten, JHEP [11] See, for example, A.H. Chamseddine and A. Connes, Fortsch. Phys. [12] For some reviews see: J. Wess, “Non abelian gauge theories on non-commutative spaces”, We do not have ageometry correct without quantum referring field to the theory limiting defined commutingamount directly space. to All in expansions formulations, in a one the space way non-comutativenon-perturbative or with parameter. aspects another, non-commutative of This gauge way theories, we which cannotquestions are expect I to precisely would capture the have liked the ones to we discuss want. with It Yannis. is the kindReferences of new qualitative insight by associating thebetween scale different of branes, spontaneous a symmetry picture breakingfactory which to quantitative has the predictions emerged were distance also obtained. in WeStandard know string Model now theories. form that an the However, irreducible no seventeen set parametersas satis- from of we the the stay renormalisation with group the usual pointbe perturbation of stable. expansion view in and, commuting as space, long no relation among them can fact that the gauge fieldsequivalent in way to a write Yang-Mills the theory theory. are matrix valued. 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G.[20] Athanasiu, E.E. G. G. Floratos Floratos and G. K. Leontaris, Phys.Lett