On the Generalized Geometry Origin of Noncommutative Gauge Theory

Home , Bruno Zumino, Matrix decomposition

JHEP07(2013)126 y Springer July 4, 2013 July 22, 2013 : : March 25, 2013 , : braic Geometry, d geometry point Accepted Published ve and semiclassically Received 10.1007/JHEP07(2013)126 doi: ralized geometry of D-branes. s, Charles University, c,d Published for SISSA by [email protected] , and Jan Vysoký c Dedicated to Bruno Zumino on the occasion of his 90th birthda Peter Schupp a,b D-branes, Non-Commutative Geometry, Differential and Alge We discuss noncommutative gauge theory from the generalize [email protected] Czech Technical University in Prague, Faculty of Nuclear Sciences andPrague Physical 115 Engineering, 19, Czech Republic E-mail: Mathematical Institute, Faculty of MathematicsPrague 186 and 75, Physic Czech Republic CERN, Theory Division, CH-1211 Geneva 23, Switzerland Jacobs University Bremen, 28759 Bremen, Germany [email protected] b c d a Open Access Sigma Models Keywords: of view.noncommutative We DBI argue actions is that naturally encoded the in the equivalence gene between the commutati On the generalized geometrynoncommutative origin gauge of theory Abstract: Branislav Jurˇco, JHEP07(2013)126 ] 3 6 7 ]. ], 1 3 8 5 7 10 11 12 12 , 6 , a more ]. In [ θ 11 sted) Poisson ]. Field equations 4 ], it has been shown 17 ac structures. This was and a bivector B naturally involves structures Lie algebroid morphisms [ to be intimately related to a ]. In [ d (complex) geometry [ ar if one lifts the topological ations ] and also generalized to Dirac es. Here we mention only few urn, at least in some instances, tion of Poisson manifolds (and 16 8 a has been shown to close under ated to the present paper. Topo- , geometry. For instance, the topo- -field [ H ], or coisotropic submanifolds [ 10 , 9 – 1 – ], it was observed that in the first order (non- 15 ]. In [ 14 ] recently appeared to be a powerful mathematical tool for 2 , ] and [ 1 13 ] and are at the heart of deformation quantization [ 3 ], where the graph defined by the corresponding (possibly twi 7 2.1 Fiberwise metric,2.2 generalized metric Factorizations of2.3 generalized metric, open-closed Dorfman rel bracket, Dirac structures, D-branes general form of world-sheet currentsa appears. more Their general algebr bracket, the so called Roytenberg bracket [ further developed in [ topological) Poisson sigma model characterized by a 2-form that the structure constants of the Roytenberg bracket appe Poisson sigma models can besigma twisted models by [ a 3-form it has been observed thatof the current generalized algebra of geometry, sigma such models as the Dorfman bracket and Dir lot of interesting differential,logical in Poisson particular sigma generalized modelsLie are algebroids) of [ interest forof the (topological) integra Poisson sigmaand models as can such can be further interpreted be as generalized in terms of generalize the description of variousinstances aspects of of its relevance string that andlogical are field and more non-topological theori or less Poisson directly sigma rel models are known Generalized geometry [ 1 Introduction 4 Non-topological Poisson-sigma model and5 Polyakov action Seiberg-Witten map 6 Noncommutative gauge theory and DBI action 3 Gauge field as an orthogonal transformation of the generalized metric Contents 1 Introduction 2 Generalized geometry structure is replaced by aD-branes more can general be Dirac related structure. to Dirac In structures t [ JHEP07(2013)126 . ], M 10 ∗ . As T F ⊕ ns of an d metric to a three- and TM θ θ and a possibly ty of D-branes ′ ], where, among F 21 d geometry origin of ]. We will discuss this berg-Witten map. In on D-brane volume cor- quivalence of commuta- , the closed background 25 ndpoints, in the presence θ F ivalence between the com- , natural generalized metric mmutative gauge theories ized geometry. We empha- etric defined by the closed and a bivector ual sections, let us note that y is relevant for discussions tring endpoints in the open ty in string theory we refer, m, in our case or le for a direct generalization 24 Dirac structures. d Witten [ oded in two different ways of d naturally interpreted within ertain orthogonal transforma- e gauge field and closely related issues) and ctions to the usual DBI action, B n the noncommutative version ransformations of er we follow the proposal of [ Poisson sigma models play a role. and Φ. It comes as a relation between two G ]. A thorough discussion of noncommutativity – 2 – 20 ] and [ ], which fits naturally to their role in both the integration 19 3 , . Finally, we recall the definition of the Dorfman bracket, 18 θ . The natural question is whether the so obtained generalize ). The positive answer is given by two different factorizatio ′ B θ and ]. Let us also note that the (semiclassical) noncommutativi g 23 and the open string backgrounds , 22 B , -field was recognized in [ g Before going into a more detailed description of the individ The paper is organizedIn as the follows. second section, we review basic definitions of general The purpose of the present paper is to unravel the generalize In the third section, we observe that adding the gauge field Noncommutativity of open strings, more precisely of their e B modified bivector can again be rewritten in the open string variables (with som expressing a generalized metric on a D-brane. mutative and semiclassically noncommutative DBIargue actions that ( the necessity of suchgeneralized an geometry. equivalence can In be the seen discussion,Roughly an non-topological speaking, wetive intend and to semiclassically convince noncommutative the DBI actions reader is that enc the e noncommutative gauge theory. We will mainly focus on the equ on the D-brane, thebackgrounds pullback of the generalized space-time m where D-branes correspond to leaves of foliations defined by responds to an action of an orthogonal transformation on the in detail in a forthcoming paper. can be seen astopological the Poisson sigma (semiclassical) model [ noncommutativityas of well the as s deformation quantization of Poisson structures. other things, alsowas the discussed equivalence via ofparticular, a commutative it field and was nonco argued redefinitionof that known the the Dirac-Born-Infeld under higher (DBI) action derivative the canthe terms be name effective i viewed D-brane as Sei corre action.e.g., For to reviews [ on noncommutativi in string theory followed in the famous article of Seiberg an part of first order Poisson sigma characterized by a 2-form of a orthogonal transformations defined by a bivector and a 2-for This allows us to recover the formulas relating, via a bivect almost everything in thisto paper Nambu-Poisson is structures presented and in M-theory a membranes, form cf. suitab [ generalized metrics, which are connectedtion by induced the action by of the a bivector c dimensional WZW term.of It non-geometric this backgrounds. respect, generalized geometr fields Dirac structures and their relation to D-branes. In the latt size the behavior of a generalized metric under orthogonal t JHEP07(2013)126 . ′ ]. is × θ ) 26 n (2.1) (2.2) f the , E 2 and θ ) : Γ( . · } , , defined as ) ), where · i E 2 n, n , e on 1 e ivalence is a direct h ·i is a symmetric non- , me relating the non- eneral framework, we = ve and semiclassically h· tween open and closed R anifestly invariant un- i -branes, i.e. those which 2 he semiclassical Seiberg- fined by graphs of Pois- D er, what we believe is new → be a smooth manifold and ical Poisson-sigma model, y correspond to the same sidering D-branes which are p , Oe ions of Seiberg and Witten, )-bilinear map ( ] should replace the Poisson , 1 E M D-brane which is a symplectic 7 ) tions combined with a Seiberg- ized geometry, see, e.g., [ vity. e metric M ξ wo (Poisson) bivectors r discussion of DBI actions, also Witten map, once one recalls its × ( ( Oe its main ingredients as developed h . This interpretation is the most p ∞ W ′ i E θ C )) ( : E ) + p ) η and is a · Γ( ( , ) the set of vector bundle automorphisms · θ V ∈ E i 2 ,( = , the latter one closely related to the nocom- n, n ′ ( , e i M – 3 – 1 F ) on η O · e ∈ , ∀ · + ( p | and )) F ξ, W ). This fiberwise metric has signature ( E ( + E V Γ( h Aut ∈ Γ( ) η ∈ , the Seiberg-Witten map is naturally interpreted in terms o + O θ { W ( . Hence, we denote by . A fiberwise metric ( , ) = ) ), such that for each M ξ M ∗ M + ( n, n T ( ∞ V O ⊕ C → TM We believe that analogous results hold also for more general In the final section, we discuss the equivalence of commutati In the fifth section, we briefly recall the interpretation of t In the fourth section, we use the above mentioned relation be ) = E leaf of the Poisson structure, describing the noncommutati noncommutative DBI actionconsequence of of a the (gauge D-brane. fieldWitten dependent) map. open-closed We The rela show discussionand here that interesting is is this not the completely clear equ new.in generalized geometry previous Howev origin sections. of Everything works very naturally for a for every ( relevant one for our discussionsymplectic in leaves the of final section. When con corresponding Dirac structure. commutativity parameters (Poisson bivectors) Witten map as a local diffeomorphism on the D-brane world volu mutative gauge field strength. Thishints equality, towards crucial the for appearance ou ofinterpretation the as semiclassical the Seiberg- local coordinate change between the t variables (including gaugeits fields) Hamiltonian to andder show the the that corresponding open-closedgeneralized non-topolog Polyakov metric. field action redefinitions are as m they geometricall focus on a particularE choice of vector bundle. Namely, let 2 Generalized geometry 2.1 Fiberwise metric,In generalized this metric section we recallAlthough some most basic of facts the regarding general involved objects can be defined in a more g sigma models. a dimension of are related toson more tensors. general Dirac For structures such than D-branes, the Dirac ones sigma de models of [ Γ( degenerate bilinear form. There exists a canonical fiberwis which includes the field strengths a consequence, we find a generalization of open-closed relat preserving this fiberwise metric. That is JHEP07(2013)126 = 0 (2.3) (2.4) (2.7) (2.6) (2.9) (2.8) (2.5) (2.12) (2.11) (2.10) 12 A = 1. For T 22 2 . A τ 1 + − 11 A 22 21 A e uniquely decom- A T 12 − A = B = 0, . In the block matrix form 1 21 . − , and A ) ! , N . ξ , T 11 ) . ( 12 , ξ V ) ! ) A T ! V . A ξ , ( i − ξ 1 ( ) V ξ + 2 V θ ,V − 11 B · N , ξ ! ( · , e 11 A be any invertible smooth vector bundle + ( ) + T . B A 1 ! − θ ξ θ ! is given as ξ − ) + 0 e as 1 T 21 − ( ) be a bivector. The induced vector bundle θ = = = + V N e B + ) transformations, which we will use in the θ τ TM A N ( 1 0 e h 1 0 1 e B θ θ M N B V O ξ 0 ( V by the same letter, i.e., we define , N N i V →

)-linear map of sections, such that n, n i X – 4 – := 11 ( − B 2

. In this paper we will always denote the induced − M = τ ) = = M A − ) = O Λ ) ∗ ( ξ ), i.e., ) := e ξ = M 2 TM ! ! T ξ = ∞ ∈ + : ) := ) = + , e ! ξ ξ ξ V θ C V + n, n 1 to ( N V V ( ξ e N V V ( θ (

(

( V θ O ( θ B

B e B e e N e TM in N ) be a O O denotes the map transpose to E 12 22 Γ( A A M ∗ 11 21 ) be a 2-form on = 1, we find → T A A ). Finally, let ) ). Similarly, let M 22 → ( E E E A 2 ). Correspondingly, we have Γ( ). Correspondingly, the map Ω Γ( T 11 M : Γ( ∗ A ∈ ∈ M ∈ M T ( τ ( ) ) 1 + : ), we put ξ B ) transformation with the invertible upper-left block can b ξ X Ω E T 12 + + ∈ − ∈ A Γ( n, n V V ( V ξ N T 21 ∈ O A Let now 2 , e 1 e for all and More explicitly, for posed as a product of the form morphism is again denoted by the same letter, that is Any In the block matrix form In the block matrix form for all vector bundle morphism from where morphism over identity. We define the map for all ( for all ( sequel. Let There are three important examples of JHEP07(2013)126 , , − is − V M V τ , we ) · = ·i (2.13) (2.14) (2.15) (2.20) (2.16) (2.18) (2.17) (2.19) , , · ⊥ and + h· , we can V e ( is positive + B generalized V ·i ). Moreover, = 0, and the 1 eigenvalues , − h· and , M ) n, n , ) can be written ∗ . ξ g ( , } a skew-symmetric . ( T } 1 O ! M ! − B ∩ 2.19 ∗ M ∈ η g 1 W ∗ + T τ is positive definite, set a ( T V B ∈ and 1 0 ) + . ·i ∈ − ξ ! ξ , is a Riemannian metric on ( A ! ξ 1 | 1 h· 1 g | − ξ − ξ V − ! . ξ 1 Bg E = 0 and as covariant 2-tensor field on 0 − ) + , ! g ) + ξ i B Bg A ( ) 1 ξ 1 ) + 1 0 g ( 1 2 B g − − TM 1 e 1 − V ′− ( . − ( − ∩ , on which A − Bg g A B , τ ! . Using the orthogonality of we get that n , corresponding to +1 and ). Finally, we can observe that V + 1 1 − − B Bg V e − − − B M + 1 ⊕ h V g V g B g V − 1 0 1 } ≡ { 1 + } ≡ { can be viewed as graphs of invertible smooth is positive definite, we get that =

− V −

– 5 – i τ can then be written in the block matrix form ) Bg T g − and 2 ) · TM = V of rank = τ V − is a symmetric part of ! TM − , ) , e + · · ∈ )( ) E ξ g g V ∈ ! = 1, it is orthogonal and thus V , E 1 , we have · B 1 e 2 V and

+ 1

1 ( V − τ of | V − τ − | + = h = ) ) V V τ Bg V ! ) V ( , where ′ ( B Bg η ξ − V 1 B A A B g g + − , respectively. We can view

1 . From this equivalent formulation, i.e. using + + + 1 to get a generalized metric on − τ g B M Bg g V V − ∗ . This will give ) = ( { { = − + ξ, W − T τ ). In the block matrix form, ξ and vice versa, and using the knowledge of the signature of = g g A + E = = + ·i ⊥ → ). Also, because −

, V V Γ( − + ) and negative definite on Γ( E | V ( h· V V ( = can be an arbitrary 2-form on τ + ∈ τ Γ( ′ defines a positive definite fiberwise metric, we refer to it as a is a symmetric map, that is, V TM ) B . From now on, we will always assume that this is the case. Sinc A : . This means that ξ ∈ τ τ ) ′ − E 2 + V . From the positive definiteness of ., . , e = 1, we get two eigenbundles A V 1 A, A 2 e τ From positive definiteness on The important observation is that the block matrix in formul := +1 and , respectively. Using the fact that ( τ , whereas V | with respect to of definite on Γ( from for all get the direct sum decomposition Conversely, for any subbundle symmetric, same for τ vector bundle morphisms: as a product of simpler matrices. Namely, and write uniquely metric on If such ( where The corresponding fiberwise metric ( uniquely reconstruct for all ( part of we see that M JHEP07(2013)126 , G B . (2.22) (2.21) (2.23) (2.26) (2.27) (2.25) (2.24) τ ) · , · ( -flux. The is globally O H is changed by B = ′ H τ and a 2-form τ ) · on the generalized . g , ] is trivial. · θ . ! H τ − θ e )) = 1 and − rphism 2 sume that 2 e ′ dles: 0 1 1 ( , τ . . } n metric ,O ! ! ) map ) θ, M ! 1 ∗ 1 have to be closed, and this will , e Φ) T ( Φ Φ 0 1 n, n B 1 0 Φ 1 1 ( 1 0 O ∈ 1 0 − , we get the matrix equations − ohomology class [ − − − O . Clearly ) on the space of generalized metrics. ξ G is invertible, we can proceed the other G G . , described by a Riemannian metric = ( | Φ θ . θ Φ τO i ! ) − H ! n, n − 1 ) ! H − does not ξ + 1 e ( e 2 ) of − 1 ( − 1 V − e 0 1 O G 0 − H − θ ( , which has the form B 0 O G ( τ − g G e ) θ G G 2.23 = e − 0 g ,O = 0 B = G – 6 – ′ 0 1 G )) τ = B 1 − + G + 1 e ! g G G ( V − ! τ ! ) and ( O B ( + ( Bg and Φ. Since 1 0 1 τ = Φ define 1 Φ 0 1 ξ h 1 Φ 0 1 − 1 2.22 { G

. The action of the − g We thus consider only the models with trivial = = = i = M 1 ! 2 Bg G + 1 G H , e V ) θ 1 0 1 -ﬂux will be discussed elsewhere. e

( H ′ globally. τ = ) and given h G dB n, n ( = O is again a generalized metric. We may use the notation ( H ′ ∈ τ ) gives us a new generalized metric · O , · H be a 2-vector field on θ There exists a natural action of the group Note the important fact that the 2-form More precisely, we assume that the corresponding integral c . Namely, we have 1 θ − By the previous discussion, there exists a unique Riemannia metric and a 2-form Φ. Hence such that Let Comparing the two expressions ( Let us start with a (different) generalized metric 2.2 Factorizations of generalized metric, open-closed relations From that, we can easily find the relation between +1 eigenbun Since e remain true throughoutdefined, the i.e. whole paper. Nevertheless, we as Hence ( For each which can be uniquely solved for case of the non-trivial way around as well. We also know how the corresponding endomo JHEP07(2013)126 , . . is g ). G M θ 2.1 (2.29) (2.28) (2.30) (2.31) (2.32) . Define ic ( ]. ). If nd only if M 27 as its graph, , there exists 2.28 θ E of θ G , , Φ and can be equipped with e geometric interpreta- , D G is a closed 2-form on ) )] M η ). dξ B nt bracket is the antisym- , . In particular, for given ∗ ( } θ . If we exchange the tangent T + W y fman bracket and Dirac struc- G i M . 2.29 ⊕ has the same properties as ∗ } W . . − 1-eigenbundles. The open-closed ( 1 } T 1 xt of duality rotations see [ ) − B ± − η ∈ , which is maximally isotropic with M . As before, define a vector bundle TM TM ( , e ∗ G ξ ) E V M ∈ T ξ | = + Φ) L of ) ∈ V + ξ G E | ξ ( L ] + V 1 | ) relating closed and open string backgrounds ( − + ( ) B 2 V ). Define a subbundle θ ξ ( e ), one has ( V,W – 7 – , ξ B θ · = M ( + Φ) = [ be any rank-2 covariant tensor field on ( 1 θ + + = [ 2 , respectively, − G D ξ ] ) B Ω V D { η ] M { B + ( η ∗ of the generalized metric ∈ ) = = + as ξ ξ = 1 T + and Φ, and conversely, for given ( + θ B { − θ B g B G ( G G = G G is a Dirac structure, if and only if ξ, W and by ξ, W H B + + ) are the symmetric and antisymmetric parts of ( V ]. Our vector bundle G + V TM [ TM 28 have identical graphs as well as V [ 2.25 B . , e ). The corresponding pairing is the canonical fiberwise metr G → B is a Dirac structure with respect to the Dorfman bracket, if a 26 E θ , M Γ( 2 G and ∗ and and involutive under the Dorfman bracket ( ) and ( ∈ 1 T ) and its graph g · ) is the Seiberg-Witten formula ) − ·i : ξ , V, G 2.24 θ h· be a rank-2 contravariant tensor field on ( + 2.28 B θ , we can find a unique V ]. Consider the inverse θ 10 ) = Let A Dirac structure is a (smooth) subbundle For Φ = 0 the open-closed relations can be given a slightly mor Furthermore, for any closed For an earlier appearance of this type of formulae in the conte V 2 and is a Poisson bivector. Similarly, let ( Obviously, respect to for all ( in the presence of a noncommutative structure represented b B a structure of ametrization Courant of the algebroid. Dorfman bracket: The corresponding Coura that is and cotangent bundles and morphism tion [ Formulae ( we get using the above formula that Poisson, ( It is known that 2.3 Dorfman bracket, DiracHere we structures, briefly D-branes recall sometures, relevant facts see, concerning the e.g., Dor [ relations, for Φ = 0, is a simple consequence of that. θ a unique pair B Again, one can show that JHEP07(2013)126 . ∈ is F θ H (3.2) (3.1) + (2.33) B . supporting = ′ ! M , where 1 B , where zed metric θ e be a 2-form (at H F , but by gauged 1 0 by the action of − F H ′ , , leaves of foliations G ! ! as ∼ ! ′ 1 by presympletic leaves, θ θ G will be given as a graph , − F − 1 0 ). The generalized metric i M L − η 1 0 1 . 0 1 1 is an automorphism of the + ! 2.23 er, . Hence, in this case we will will be the foliation generated 1 B θ . Further, let ! ! ! e ′ 1 B T M Φ . As shown before, we can always 1 0 ξ, W − 0 , see ( B F Φ 1 − 1 0 1 0 + N − − G − e ) is no longer true for V 0 N h ! , such that we get 1 !

! on the generalized metric = ′ 2.32 M − 1 0 1 ) action. In particular, there exists a vector i θ 0 and 2-form − ) − G H 0 − g η will make sense only when considered on the D-brane. g on e G ! 0 n, n G 0 g ′ – 8 – F + ( . This can be achieved under some assumptions, 1 0 ′ θ G − ) transform O 0 F W ) has the following block matrix form: ! N ! ( ). ′ F ) of the Dirac structure under the anchor map. We ! B n, n T B ( 0 will be closed and deﬁned only on a submanifold of L + , e ( provides a singular foliation of N 1 0 1 , such that 1 Φ 0 1 O ) 2.21 , which is described by the same F ). In the other words, ρ ′ 1 Φ 0 1 ξ L

E H g, B + TM ! ! = Γ( ! 1 ′ V ). The gauge transformation deﬁnes new 2-form F ( → 3 ′ by the 1 ) transformation ∈ H 1 0 θ B 1 0 1 θ 1 0 e ) G h

η n, n TM ( and Φ, see ( and the corresponding foliation of = : = + ! O ′ θ ′ for some gauge ﬁeld G ) holds. ′ N F G G W ) corresponds the generalized metric F ( 1 0 1 , 2.33 )

ξ g, B = ] for arguments in favor of the identification “D-branes is related to + ′ = Φ+ ′ 10 ′ V G G ), but ( by action of M on the generalized metric One may ask, if there is a bivector Generally, a Dirac structure ( G Later, when discussing DBI action, corresponding to the pair ( ′ 3 X θ ′ 2 − a D-brane. In which case, all expression involving described by fields bundle morphism however, only up to a certain additional get where that is, 2-form Φ Indeed, examine the block matrix decomposition: G e defined by Dirac structures”. In the case we will consider lat and Λ corresponding Courant algebroid. Note that ( by Hamiltonian vector fields, i.e., by symplectic leaves of identify the symplectic leaves and D-branes. 3 Gauge field asLet an us orthogonal start with transformation a of given Riemannian the metric generali of a Poisson tensor To the pair ( this point an arbitrary one for all ( refer to [ which is generated by its image JHEP07(2013)126 . is 2.3 θ (3.4) (3.5) (3.6) (3.8) (3.9) (3.7) (3.3) . (3.10) (3.12) (3.11) ! 1 . . ′ F 1 0 )) ′ − Nθ we have F )) ′ N ′ ) we see that the θ ! )) F ′ θ (Φ + F 2.28 − 1 ices, can be uniquely and ar block matrix — of . − 0 1 1 1 θ (Φ + o ( : G 1 − ′ ) ) ′ (Φ + − tion, we assume that F 1 of the DBI action and non- ! e F N G ′ − ve expression. Since we want ) ) 1 , ′ ′ F, F 1 G ′ 1 − F ) F − ng the DBI action. 1 0 ′ − e F ) ) (Φ + F , ′ has to be again a Dirac structure of )). What we find are the following F θ θ − F θ (Φ + ′ ! − θ G + Φ + e − (Φ + . . G TM N ′ ′ ) 1 (1 + G ′ θ F ( G − O θ using the tools described in subsection det( ′ ( F T G G ′ · = (1 + F . For the graphs of T G = θ ( 2 N 1 − 1 0 = = ) θ N T e M − 1 Γ(Aut( θ ) – 9 – F N + ( (Φ + − − N = and G ) ′ ∈ G ′ − T F θF θ ) is invertible. For this, use the decomposition of θ F e − e F θF −

) = N = θF. e θF N ) as F θ 1 = 1 (1 + − − − + e ) )) = det( F ! G 2.11 F F 1 ) and B ) ), we ﬁnd that ′ = = 1 + = (1 + ( 1 + + ′ ′ F M F 1 0 − ( 3.7 θ is a bivector on N : F − g B B X ′ ) ( ), we get the equalities 2 N θ 1 + F ) can equivalently be written as Λ (Φ + − g ! 3.2 , we may proceed by inserting 1 = g T ( + ′ ) ∈ θ 3.8 and N F ′ F B ′ − ( , θ = F + 0 1 1 ) ) according to ( − , 1 ′ ) and ( is closed. B g − θ M ( ) and ( ( g is a Poisson bivector. Similarly, one can see that ! n, n ) 3.1 F 2 ′ ( − 3.7 Ω F θ g O is an automorphism of Dorfman bracket, Φ 1 + ∈ 1 0 ∈ ′ − F B det( e F ( F

− Furthermore, following the same type of arguments leading t Finally, let us examine the objects e θ . Hence, − expression for Comparing ( Taking the determinant of ( This equality will play the central role when later discussi and with decomposed into acourse, product only of if a wee assume diagonal that and (1 + an upper triangul Now it is enough to note that the product of the last three matr E We will concentrate oncommutative the case gauge important theory. for the Therefore, discussion in the rest of this sec equations ( Poisson and to modify Φ to Φ+ It suffices to consider the three rightmost matrices in the abo Since JHEP07(2013)126 i ) ). η n 1 r in (4.5) (4.2) (4.3) (4.4) (4.1) , σ 0 σ , . . . , y 1 y , equipped ). We also ˜ η M is therefore a η, ′ with signature owing notation: F h ). Integrating out := ( e coordinates. For erves the (maximal) and 3.5 T V. ). Finally, we define a 1 E cal coordinates ( − according to their index X . ˜ 1 G M ! T ′ ) and ( 0 V. V, θ X, ∂ a model from the generalized T T ′ σV 0 3.8 2 θ ∂ 0 and introduce auxiliary fields 1 d − is closed. − ), (

Z N ! F := ( ′ we will use the following notation: Ψ + Ψ Ψ + Ψ 2 1 3.7 + F F T ¯ ˜ G G 1 T − M − V T T − N by ( = Ψ Ψ ! B ′ ′ → 1 2 2 1 -dimensional target manifold − ¯ V coupling to the boundary (and extending to F := F n σ σ 1 2 2 ′ still makes perfect sense as a Dirac structure. − F g − ¯ A d d ) can equivalently be written as G ¯ g F – 10 – :Σ ¯ θ doesn’t need to be closed. The last remark: In G ¯ + Φ − ′ Z Z T G 4.2 − B X F F -dimensional column vector Ψ ′ is an isotropic subbundle of and is Poisson nor that e

n σV , Φ and ¯ ′ F ). We also choose some local coordinates ( θ 2 ] := ] := ¯ G G F d N = − G dA Φ ˜ Z ¯ G e e Φ + η, X η, X T = 1 2

N η, η, [ [ − F we obtain the Polyakov action expressed equivalently eithe and a 2-form Φ. We can assume Σ with a non-empty boundary = S S θ η ), the action ( ¯ G . Hence ] := ¯ Φ := F 3.8 , X -dimensional column vector and ˜ are Cartesian coordinates for a Lorentzian metric [ G n ), ( S η µ ) is not invertible, GN being related to σ 3.7 T will still define an almost Dirac structure. , 2-vector 4 F θF N F G . Let us also note, that G assume an abelian gauge field ). In this section it will be convenient to introduce the foll θ := M and e X ¯ M ( matrix G i B y , which transform under change of local coordinates on , n . Lower case Latin characters will always correspond to thes j 2 g e η Let us consider a 2-dimensional world-sheet Σ with a set of lo = +) on Σ. Furthermore, we consider an M Here, we neither need to assume that × , 4 i Σ. On n − the components of the smooth map on and ∂ Σ, the field strength being with a metric open or closed variables We put 2 ( Similarly, 4 Non-topological Poisson-sigma modelIn and this Polyakov section action wegeometry review point of the view non-topological developed Poisson-sigm in the previous sections. X structure. We combine them in a 2 case that (1 + We assume that where Using relations ( the auxiliary fields introduce another 2 2-form on isotropy property of This is no more an automorphism of Dorfman bracket but it pres with Our (non-topological) Poisson-sigma model action is JHEP07(2013)126 i - ) – η 34 4.2 ned. onal = 0) (4.6) (5.1) holds . The F θ ˜ G ). G = tF tF θ e leads to the expressed as ). Hence, we ¯ G ′ t ]. = ) can be found, 3.2 G 37 t is applied to the θ 3.2 G s, are used. As can 1 η − ) and ( ) with 1 1 1 ) appeared (with r this, note that rela- ). The auxiliary fields 3.1 4.5 on from the commutative oordinates given by a map variables. Let us note that, 4.5 given by ( X, η l methods akin to the ones of elds, the ˜ ′ 1 atrix ∂ tical. To write down the result G , ) with respect to Υ ′ := ( 5.1 ]-flux. The interested reader may find some G 1 T , t T H . ) with − ]. Here we follow the approach of [ θ t ) ) Υ 1 A 33 4.6 , simply observe that ( F θ t t t , tF θ ) and the action ( θ dσ θ 32 − 4.2 Z −L 2 1 (1 + = – 11 – θ = t θ t t This map can be derived using a generalization of = ], see [ θ ] = t ∂ and the related non-commutative gerbe in [ t ∂ 31 θ 5 . H θ X, η [ ) or in the Hamiltonian corresponding to the action ( H can explicitly be seen either in the Hamiltonian correspond ) = ′ ′ is a bona-fide Dirac structure even for non-invertible (1 + 4.5 Partial differentiation of ( θ ]. The Hamiltonian ( G θ ( ∗ 6 G ) and their transposes are obtained from the nonzero off-diag 29 ρ 1]. Of course, we have to presume that the formula is well-defi tF , e -dimensional column vector Υ ]. Polyakov actions like the first one in ( 3.10 [0 n are indeed Poisson for all 17 t ∈ θ t . ) can alternatively be expressed as the equality of matrices ˜ G 3.8 , such that = 0, in [ , this can be rewritten as = M the matrix given by the two equivalent decompositions ( F ), ( ¯ dA G ′ → 3.7 = ) can be found in [ G = 0, the relation between the action ( ] in the context of Poisson-Lie T-duality. The generalized metric F M 3.1 F As said before, here we assume only topologically trivial [ Let us note again that 30 5 6 ], where it was shown that the Seiberg-Witten field redefiniti : ing to the Polyakov action ( equality where straightforwardly be checked, thesewe Hamiltonians introduce are a iden new 2 for the respective graphs. To see that these relations in the form ( become the canonical momenta and the Hamiltonian is in ( For ρ Moser’s lemma: Consider the family of Poisson bivectors to the non-commutative setting has its origin in a change of c again for have the same Hamiltonian usingfor either the closed or the open after the equations of motions for one half of the auxiliary fi blocks after the similarity transformation with the block m differential equation relevant discussion concerning nontrivial in [ parameterized by For an approach to theZumino’s non-abelian famous case, decent using equations cohomologica 36 [ 5 Seiberg-Witten map tions ( Actually, all this can be seen rather straightforwardly. Fo JHEP07(2013)126 . D . (5.3) (5.2) } . We ] that ′ . Also, θ M 36 ∗ ij (i.e., the ∗ A explicitly θ ρ ), and the ]. T ) transfor- . To avoid θ ρ A A ∈ 38 ( := ρ t . Actually, all n, n θ t ], to define the ij ( , η ) ) O in order to carry 10 η θ ( T structure − N y the . Obviously, 1 that we can safely restrict φ ) + . This differential equation ertheless, according to our ralized geometry. Namely, n map is the map of graphs θ eeded for our discussion of η he matrix ( ( = spective D-branes. In such a ) we see that so is T = ρ w for quite some time [ suggestion of [ − nguish between the tensor itself oncommutative DBI actions fol- 0 3.4 θ N . ′ . ) . 1 θ , are Morita equivalent [ ij ′ ) still makes sense and we can argue − Nθ θ ) θ . Thus { ′ j θ det l ) and the (semiclassical) Seiberg-Witten 5.3 θ ) and has understood the (semiclassical) = 2 J ∗ A i J } and k ρ 3.7 only changes coordinates on the symplectic ) = 3.7 ( t J ), if one considers D-branes which are sym- θ M θ θ A ( ∗ ), ( ) = ρ , hence we shall use the notation ), ( ∗ t ′ T – 12 – A ) = θ φ 2.11 ( ∈ kl 2.25 ′ ∗ A 2.25 θ ρ = det ( ( is invertible. From ( ), ( is a diffeomorphism of the D-brane world-volume 2 the formula ( ∗ A η, η ), ( − θ ′ ρ det A θ ), with initial condition J ρ 2.24 ) + 2.24 η ,A · ( , such that is infinitesimally generated by the vector field ( t θ t φ θ A ρ have in fact the same symplectic foliations for all t θ } 7→ { ) := A , including in particular M ( t ∗ t θ θ T and hence also . We have i k ∈ θ ’s are the same, we see that t ∂ρ ∂x θ . More explicitly, entering the decomposition ( θ η, η ). We can thus restrict ourselves to the non-degenerate case = G θ N k i ) + O 7→ J η ( θ ′ ′ In the next section, we will discuss the case when the Poisson Finally, on the level of Dirac structures, the Seiberg Witte θ G { : ∗ For degenerate immediately have that denote corresponding Dirac structure) will be used, following the leaves (of we have seen that thehave relations their ( root in generalizedthe geometry. equivalence of Actually, the it commutative is andlows kno (semiclassically) once n one hasSeiberg-Witten established map ( as a (local) D-brane diffeomorphism. Nev plectic leaves. 6 Noncommutative gauge theoryIn and DBI the action previousnoncommutativity sections of we D-branes have as described a all consequence ingredients of n their gene our discussion without thecase, loss the of generality (Seiberg-Witten) to map any of the re ρ D-branes as its symplectic leaves. The above argument shows as follows: Since the map We thus get that can be integrated to a flow possible confusion, we will forand a its moment components notationally in disti coordinates. Therefore we introduce t Let us assume for a moment that depends on the choice of gauge potential out the computation of the Jacobian. Hence, the Seiberg-Witten map can be seen as the map induced b with a vector field mation kernels of all Poisson structures The Poisson structures JHEP07(2013)126 . (6.4) (6.2) (6.1) (6.3) , one nc DBI S and an S , ) =: ) to rewrite ) ′ ′ ˆ F F . ction, we have Seiberg-Witten 2.24 ) ′ ˆ F Φ + ), again using the ic matrix + + Φ + Φ). ]. utative and noncom- 3.7 ˆ etric and an antisym- G G 39 G, ( ( cture having constant rank. 2 + Φ + 2 / / 1 1 semiclassically noncommu- G ism. ( n between the D-brane and 2 As argued before, under this ound fields (open and closed det he equivalence of the commu- / ffective closed and open string 1 es, see [ ) det 8 y is new. Moreover, the discus- or the sake of completeness and . and corresponding field strength . ˆ θ θ θ 2 θF , and the relation ( the D-brane is a submanifold, such that / A ) det 2 1 ] and will be elaborated in detail in . , i.e, a submanifold of target space- 2 / closed relations, here we start from a given 4 / d 1 / | 1 25 θF 1 ) (1 + A , 2 1 B det / ) and the relation ( 24 1 − g G s + Φ) + (1 + 1 ˆ 2 G 6.1 ]. These are related as g G det / AS det x 1 det s 21 d − 1 d G – 13 – det( s det( det S | x g s of dimension 2 Z d 1 / s d G = 1 g | D s S ) = Z = | G F ) = s -theory branes [ = F G ) = + | M F + A and determine uniquely the open variables ( B , respectively [ s θ + + B + G S g B | + ( , g 2 Namely, assume that our D-brane is of a particular kind, i.e. + ( / A and 1 2 g 7 / ( . s ), pick a 1 2 θ g det / 1 s g, B det 1 to it defines a regular Poisson structure, i.e. a Poisson stru g s θ det 1 x g d s ), and performing the change of coordinates according to the 1 d g . While describing the Seiberg-Witten map in the previous se x 5.3 Z d D equipped with a line bundle with a connection d := Z M Before we turn to the discussion of the DBI action and its comm Assume that we have a D-brane It is straight-forward to modify everything to the case where Recall, in accordance with our above discussion of the open- c 8 7 DBI . Also, consider the restrictions (pullbacks) of the backgr S Integrating over the D-brane world-volume the restriction of A most intriguing relation is obtained from ( We can use the formula for the determinant of a sum of a symmetr map, we finally obtain atative relation DBI between the actions commutative and above mentioned formula formetric the matrix: determinant of a sum of a symm this as assumption, the Seiberg-Witten map is a D-brane diffeomorph mutative description, we discuss thecoupling relation constants between the e which comes as symplectic leaf of the Poisson structure a forthcoming paper. Heretative we and will include (semiclassically) the noncommutative DBI derivationthe of actions reader’s t f convenience. For related work based on dualiti closed background ( antisymmetric matrix time sion generalizes to the case of best knowledge, the direct relation to generalized geometr recalling ( seen that it isthe quite Poisson natural tensor to assume that there is a relatio ones) to F JHEP07(2013)126 ˜ λ is ′ ˆ (6.5) F and . This ′ θ ommons G nition, (2003) 281 . This makes ¯ G 54 ) is the effective it is defined as = θ 6.4 ˇ ˜ CR P201/12/G028. G AD (PPP) and ASCR ts on an earlier version of Gravity”. h Technical University in t GA tative and semiclassically efully acknowledge support t the necessity of a relation ccasion of his 90th birthday. ve the details to the reader. . nder the gauge transformation g to the Poisson tensor d reproduction in any medium, =0 t | ). Expressed directly in terms of ) λ on the l.h.s. in ( ( 4.5 n ) t c DBI ∂ . Hence, an alternative — but completely s S Quart. J. Math. Oxford Ser. ˜ + 1)! g G , 4 ) + n / ( 1 , A ( } t – 14 – ˜ λ θ det , ( ′ ij ) decompositions of the generalized metric ˆ F X { 3.2 = = ) is to start from the matrix equality ) can equivalently be expressed using either the “commu- ′ ˜ λ ij is the integral of 6.4 ˆ 4.6 F δ ]. c DBI S SPIRE IN ][ This article is distributed under the terms of the Creative C Generalized Calabi-Yau manifolds , the action ), and similarly for the other objects. As a result of this defi ˜ ij G ) or “noncommutative” ( θ ( transforms semiclassically noncommutatively, i.e., 3.1 A ∗ ). ρ dλ math/0209099 6.4 [ Finally, the Hamiltonian ( Let us note: The commutative DBI action := = [1] N. Hitchin, ij ˆ equivalent — way of obtainingnoncommutative DBI the actions relation ( between the commu up to the inverse of the closed coupling constant is maybe the most directlike ( hint from generalized geometry abou the relation to the Polyakov action more transparent. We lea tative” ( D-brane action obtained fromthe the matrix Polyakov action ( We would like to dedicate thisWe article would to Bruno like Zumino to on the thankof o Satoshi the Watamura manuscript. for important The commen research of B.J. was supported by gran Acknowledgments The hat “ˆ” has the following meaning: On matrix elements of Here, the curly bracket is the Poisson bracket correspondin & MEYS (Mobility) forfrom supporting the our DFG collaboration within and the grat Research Training GroupOpen 1620 Access. “Models The research of J.V. wasPrague, supported grant by No. Grant SGS13/217/OHK4/3T/14. 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