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İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü

# d=2n boyuta genelleştirilmiş Gürsey modelde monopol çözümleri

## A Monopol-like solution of Gürsey model generalized to d=2n dimensions

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##### Özet:

ÖZET İki bölümden oluşan bu çalışmada, 2n boyutta saf fermiyonik bir modelde klasik monopol çözümlerinin bulunup bulunmadığı araştırılmıştır. Dirac tarafından teorik esaslarının kurulmasından beri magnetik monopoller yaklaşık 60 yıldır önemli bir çalışma konusu olmuştur. Magnetik monopollerin teoriye dahil edilmesi halinde Maxwell teorisindeki hareket denklemleri simetrik bir yapı gösterir. Ayrıca magnetik monopollerin varlığı elektrik yükünün niçin kuantumlu olduğunu da açıklamaktadır. Bununla beraber şimdiye kadar deneysel olarak gözlenmemişlerdir. Magnetik monopollerin hesaba katılması bir sicimi gerektirir. Bu da monopol teorisinin özelliğini bozduğundan Dirac sicimin elde edilmesi araştırmaların ana hedefi olmuştur. Ulu-Yang'ın buldukları monopol çözümleri sicim içermemektedir. İlk bölümde UluYang'ın Saf SU (2) Yang-Mills hareket denklemini çözmek için yaptıkları varsayımın gerçekten bir monopolün vektör potansiyeli olduğu ispatlanmıştır. İkinci bölümde Gürsey modelinin standart metodla yüksek boyutlara genelleştirilebileceği gösterilmiştir. Bunu göstermek için bileşik vektör alanının Ulu-Yang tipi monopol olması halinde 2n boyutta konform değişmez saf spinör teoride klasik çözümler araştırılmış ve bunların bağlanma sabiti "g" nin özel bir değeri için varolduğu görülmüştür. -iv-

##### Summary:

A MONOPOL-LIKE SOLUTION OF EÜR5EY MODEL GENERALIZED TO d=2n DIMENSIONS SUMMARY In this thesis, we present a static solution to the classical field equations of a purely spinorial model with SD(d) symmetry in d=2n (n ^ 2) dimensions. The model contains composite vector fields which have solutions of the Uu-Yang ntrcpnle type. Magnetic monopoles have always recieved great deal of interest in the last nearly sixty years ever since Dirac laid down the theoretical foundations for them. In the non-abelian pure gauge theories, which are classically conformally invariant, whole sets of monopole, solutions were found. the first solution is the Uu-Yang monopole. Later finite enery single monopole solutions were found. Finite energy multi- monopole solutions were found more than ten years later. Nahm applied the ADHM method to the construction of multi-monopoles. When alternative models for pure gauge theories in even higher dimensions were suggested by several authors, it was a logical step to search for such solutions in these theories. Dibekçi-Hortoçsu and Kalaycı showed that indeed monopole solutions exits in six dimensions. The most interesting set of these sdutions read J Zr*- These solutions give non-vanishing field tensor, Fuv. Another development is the study of purely fermionic models where vector particles appear as composites. These models are generalizations of the conformally invariant Thirring model in higher dimensions, with the hope of making Heisenberg's dream cometrue by describing the world only by spinors. Giirsey proposed such a model which was conformally invariant in four dimensions with non-integer powers of the field. Kortel found cassical solutions to this model which were later classified as instanton and -v-tnerons. Gürsey model can also be written as a vector- vector coupling. To reqularize such models one has to introduce auxiliary fields, which turn out to be composite vector fields, in the classical sense. Arık-Hortaçsu and Kalaycı showed that in such models one can construct a classical solution -for the spinor field which gives a magnetic monopole when the prescribed composite is calculated. In this thesis, we investigate if one can find classical solutions in a conformal invariant pure spinor theory in 2n ( n^ 2) dimensions such that the expression for the composite vector field is of the Idu-Yang monopole type. Our connection with the alternative g ajge models in higher dimension in this respect is just in spirit. The composite vector field has dimension (-1) in d dimension, which is possible only in alternative pure gauge models. We find that for a single value of the coupling constant, a classical solution exists, which gives us a composite vector field of the Idu-Yang monopole type. This is very suggestive. A similar situation occurs in nonlinear sigma models in two dimensions. The self coupled o(N)xD(M) symmetric sigma models, also named projector models, have an infinite number of non-local conservation laws only if the coupling constant in front of the quartic term is equal to unity. The similarity between these two results may hint to integrability in our case with quantum connotations. Perhaps a better known example is the LJess- Zumino-lilitten model. Another example occurs in Thirring model, where one has strong conformal symmetry at the quan tum level at particular values of the coupling constants and spins. We start with the L'agrangian m where Y* represents a fermion field in d dimensions with Jim ^= d-1 r /** are Dirac matrices and <T~, are generators of the symmetry group 50(d). Saçlıoğlu stresses the importance of 5U {k) for alternative models in six dimensions. Our derivation hinges on finding matrices which simultaneously serve as the chiral generators of the internal symmetry group. In eight and ten dimensions, we need groups with 2B snci 45 -VI-generators respectively. We could not find such groups in the 5U(N) series to generalize our calculations made in six dimensions to arbitrary 2n dimensions. That ^ is the reason why we switch to the spinor representations of the SO (2n) groups for d=2n. Since ue markin Euclidean space we translate our equation to its Euclidean counterpart &=? where ¦A, From now on we rename l to be I and drops the hats on '. LJe use the following hermitean representation for the matrices r _ where yu =1...d-1 P are the 2. X 2. Dirac matrices and They satisfy d/z-1 d/a. - <! I £, £ J = ^O^v.p iV=l d The SD(d) generators &]. can be constructed from f~" s where i, j = l...d-2. With these conventions the equation of motion which is meaningful in the classical level becomes where A named composite vector potential is given by -vii-at, ^a<rot^ lwrv°-al°v)(yr\vj}^ 6-Z_ 2. ot We require that the composite field A Wu-Yang type monopole. ' is given by a A0=so i, b = o,.... d- 1 Amjd _,.- *-i y im 4r* L. mn om o,d-1 ro,d-1 Ad"1=siJîa- uhere i^j m n= l...d-2. We investigate if the equation of motion has a classical solution under these constraints, Dur ansatz for T* is t = d-1. 2. ol-t itl (M^i -+-xd-iaf/«'-*») i- II u where ı =l...d-2 and a^b^a and b are functions of r The vector potential given above is equivaletnt to -Vlll-where a, b^s o... d-1, Then we find that d+2> d-1 it -2. d-z (2d -4) *- îi-(U*."^*n) rrijd-1 d+2. d-n rr^d-1 d-2. (2d -4-) *? d+2- d-2. C2d-4-) 2" *±a £ km The monopole solution constraint requires that d-Z. 56-2. 2L *- After some trivial algebraic manipulations, we get two sets of equations for the coefficients. One, can solve these equations and get d-1. 2A-4r a«K,i- Uf^./- _ d-S _, 2jd-^j- s) 3) where K^ and K" are integration constants -IX-The constraint given -for "ab" forces us to select 6-2. KilC, = (2.d-4) N1'\2. 5d-3- and g=- 2.d-4 which shows that the monopole solution exits only.for a particular value of "g". Furthermore, we can express the composite field as a Wu-Yang monopole foT this solution, For d=6, one can find a» K-i h i+^ t= Ki(- -î+z3 and g=- Note that this model is only globally gauge invariant. To make it locally gauge invariant, we haveto include a pure gauge term g"3 ğ1. Then if when If* - >5-j'lf,jg - ^g,g, this mo del is locally gŞuge invariant. One may fix the gauge to make this extra term vanish. One may take our solution as a solution only in this gauge. -x-