Kurantovi algebroidi u bozonskoj teoriji struna

[I. Ivanišević]

IX, 128 str.

Physics - String theory / Fizika -: Teorija struna

Generalized geometry is a new mathematical paradigm in which vectors and 1-forms are united and investigated as single objects - generalized vectors. In this dissertation, we explore symmetries of bosonic string theory and their relations with T-duality in the formalism of generalized geometry. The generator of both diffeomorphisms and local gauge transformations is constructed and expressed as an O(D, D) invariant inner product of two generalized vectors. In the same way that the Poisson bracket algebra of generators of diffeomorphism gives rise to the Lie bracket, the algebra of the extended generators gives rise to the Courant bracket. Taking into account the T-duality relation between two string symmetries, we interpret the Courant bracket as the T-dual extension of the Lie bracket [1]. We then develop a simple procedure for twisting the Courant bracket with any O(D, D) transfor- mation, allowing us to obtain Courant brackets deformed with different fluxes. The crux of this method consists of expressing the generator in the basis of non-canonical currents, which are connected with canonical variables via the O(D, D) transformation. We show that the Poisson bracket algebra of gen- erators in the basis of currents closes on the appropriate twisted Courant bracket. We prove that there is a natural way to define a Courant algebroid using these twisted Courant brackets. We provide many examples of O(D, D) transformations and their corresponding twisted Courant brackets, including the B-twisted Courant bracket and the θ-twisted Courant bracket. The B-twisted Courant bracket is characterized by H flux appearing in the algebra of currents, while the θ-twisted Courant bracket is characterized by the so-called non-geometric Q and R fluxes. It has been shown that these brackets are mutually T-dual [2]. In addition, we construct the generator that produces the Courant bracket twisted simultaneously by B and θ in its Poisson bracket algebra. This generator is expressed in terms of currents that contain all string fluxes in their Poisson bracket relations. Moreover, we show that the Courant bracket twisted simultaneously by B and θ is invariant under the T-duality [3]. We also demonstrate that all fluxes can exist on the Dirac structures associated with the Courant algebroid for this bracket, without any restrictions imposed on fluxes. In the end, results are generalized to a double theory, in which variables depend on both initial and T-dual coordinates. The algebra of generators that include both initial and T-dual diffeomorphisms ii closes on the double field extension of the Courant bracket called C-bracket. Following the same pro- cedure as in the single theory, we obtained the B-twisted and θ-twisted C-brackets [4]. We demonstrate that by projecting the twisted C-brackets to the initial and T-dual phase spaces, the mutually T-dual twisted Courant brackets are obtained.

Generalisana geometrija podrazumeva novu matematičku paradigmu u kojoj se vektori i 1-forme ob- jedinjuju i razmatraju kao jedinstveni objekti - generalisani vektori. U ovoj disertaciji istražujemo simetrije bozonske teorije struna i njihove veze sa T-dualnošću korišćenjem formalizma generalisane geometrije. Konstruisan je jedinstven generator difeomorfizama i lokalnih gradijentnih transformacija i predstavljen kao O(D, D) invarijantan skalarni proizvod između dva generalisana vektora. Na isti način kao što u algebri Poasonovih zagrada generatora difeomorfizama nastaje Lijeva zagradi, alge- bra proširenog generatora daje Kurantovu zagradu. Uzimajući u obzir T-dualne veze između ove dve simetrije, Kurantova zagrada je interpretirana kao ekstenzija Lijeve zagrade invarijantna na T-dualnost [1]. Zatim razvijamo jednostavnu proceduru za pronalaženje Kurantovih zagrada zavrnutih proizvoljnim O(D, D) transformacijama, što nam omogućava da dobijemo Kurantove zagrade deformisane ra- zličitim fluksevima. Osnova metode je predstavljanje generatora u bazisu nekanonskih struja, koje su O(D, D) transformacijom povezane sa kanonskim promenljivama. Pokazano je da se algebra Poa- sonovih zagrada između generatora izraženih preko struja zatvara na odgovarajućoj zavrnutoj Kuran- tovoj zagradi. Dokazano je i da takva zavrnuta Kurantova zagrada definiše na prirodan način Kuran- tov algebroid. Dali smo više primera O(D, D) transformacija i odredili njima odgovarajuće zavrnute Kurantove zagrade, uključujući i B-zavrnutu i θ-zavrnutu Kurantovu zagradu. Kurantovu zagradu za- vrnutu poljem B karakteriše pojavljivanje H fluksa u algebri struja, dok Kurantovu zagradu zavrnutu poljem θ karakteriše pojavljivanje takozvanih negeometrijskih Q i R flukseva. Pokazano je da su ove dve zagrade međusobno T-dualne [2]. Dodatno, konstruisan je i generator koji daje Kurantovu zagradu istovremeno zavrnutu poljima B i θ. Ovaj generator izražen je preko pomoćnih struja u čijim algebarskim relacijama izraženim preko Poasonovih zagrada se dobijaju svi fluksevi teorije struna. Dodatno, pokazali smo da je na ovakav način zavrnuta Kurantova zagrada i invarijantna na T-dualnost [3]. Takođe smo pokazali da svi fluksevi mogu postojati na Dirakovim strukturama Kurantovog algebroida definisanog ovom zagradom, bez ikakvih ograničenja na tim fluksevima. Na kraju, uopštili smo rezultate na duplu teoriju, u kojoj sve promenljive zavise i od početnih i iv od T-dualnih koordinata. Algebra generatora koji obuhvata difeomorfizme i T-dualne difeomorfizme zatvara se na C-zagradi, što je generalizacija Kurantove zagrade na dupli fazni prostor. Koristeći se istom procedurom kao i u nedupliranoj teoriji, dobili smo C-zagrade zavrnute poljima B i θ [4]. Projektujući ove zagrade na međusobno T-dualne fazne prostore, dobili smo međusobno T-dualne zavrnute Kurantove zagrade

Bosonic string, T-duality, Generalized geometry

Bozonska struna, T-dualnost, Generalisana geometrija

539.120.5:539.129.3(043.3)

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