Universita degli Studi di Torino

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1 Universita degli Studi di Torino Scuola di Scienze della Natura Dipartimento di Fisica Corso di Studi in Fisica Teorica Tesi di Laurea Magistrale Standard Model-like Scenarios in String Theory: Non Abelian D-brane Rotations and the Classical Bosonic String Towards a Fully Stringy Computation of Yukawa Couplings on Non Factorized Tori and Non Abelian Twist Correlators Relatore: Prof. Igor Pesando Controrelatore: Prof. Carlo Maccaferri Candidato: Riccardo Finotello Anno Accademico 2016{2017

2 Standard Model-like Scenarios in String Theory: Non Abelian D-brane Rotations and the Classical Bosonic String Riccardo Finotello Prof. Igor Pesando October 20, 2017 Abstract In the framework of open Superstring Theory, we consider stacks of intersecting D-branes as a description of Standard Model-like scenarios. Yukawa couplings and Higgs mechanism arise in the presence of twisted strings, stretched between the branes. Even though chiral fermions present at the intersections are one of the most remarkable features of these models, we focus on the classical bosonic string, responsible of the instanton contribution to the couplings. We consider the geometrical setting of three D6-branes in the 10- dimensional Minkowski spacetime M 1,9 seen as M 1,9 = M 1,3 R 6 where R 6 is not factorized as R 6 = R 2 R 2 R 2 as in the literature. We focus on non abelian twist fields with arbitrary monodromies. Following recent works, we start from SU 2 monodromies and we generalise to SO 4 SU 2 SU 2 rotations of the branes. From the mathematical point of view, the relevant configuration can be effectively described by three euclidean D2-branes in R 4. We use conformal invariance to write the solution in terms of products of two hypergeometric functions with the prescribed monodromies. Using the equation of motion of the string we show the consequences on the string classical action and the implications for the Yukawa couplings. 2

3 Riccardo Finotello CONTENTS Contents Introduction and Conclusions 5 I Towards String Phenomenology 8 1 Extra Dimensions and Compactification Calabi-Yau manifolds Compact manifold Complex manifolds Kähler manifolds Holonomy group Calabi-Yau manifolds D-branes and open strings Kaluza-Klein compactification T-duality and open strings D-branes String Phenomenology Light spectrum and symmetry enhancement Building the Standard Model gauge group with branes Chiral fermions and D-branes at angles Physics in 4 dimensions and family replication Yukawa couplings The Higgs mechanism Towards non abelian rotations II Non Abelian D-brane Rotations 35 3 D-branes at Angles and the Geometrical Set-up D-branes geometry and parameters D-branes embedding and string boundary conditions Doubling trick in real coordinates and string action Complex Isomorphism SO4 group isomorphism Doubling trick and complex structure

4 Riccardo Finotello CONTENTS 5 Building the Classical Solution From rotations to hypergeometric function From monodromies to parameters of hypergeometrics Monodromy in the origin Monodromy at infinity Constraints on the rotations parameters Global conditions and stress-energy tensor Hypergeometric functions: independent solutions Independent hypergeometric functions Consequences on a previous constraint The finiteness of the action Behaviour in ω z = Behaviour in ω z = Behaviour in ω z = Solving the inequalities Some particular cases The abelian case The non rotating sector and the untwisted string The Implications for the Yukawa Coupling The classical action The abelian case The general case Conclusions 88 Acknowledgements 89 Appendix 90 A On the Determinant of the SO4 Matrix 90 B On the Dependent Monodromy 92 C On Hypergeometric Contiguous Functions 94 References 96 4

5 Riccardo Finotello Introduction and Conclusions Introduction and Conclusions Since the beginning the first aim of String Theory has been the description of reality in a unified framework of gauge and gravitational interactions. Even though it may sound an oxymoron [1], the construction of semi-realistic and phenomenological models in the string formulation is the key to our understanding of the theory, which must be able to reproduce known results of quantum physics as we know it. For what concerns particle physics, String Theory should properly describe the Standard Model SM, which is arguably one the of the greatest theoretical developments in modern physics: strings should therefore recreate the known gauge group SU 3 C SU 2 L SU 1 Y at currently accessible energies, as well as the experimental evidence which endorses the astonishing robustness of the model. Furthermore, our present understanding of the theory has led to the suggestion that the SM could be embedded in representations of larger gauge groups at very high energy. String Theory should therefore provide the correct unified framework, if any, in which to account for every aspect of one of the most proven and successful theories. The first phenomenological models in String Theory usually were derived from the compactification of the heterotic string 1 on orbifolds, which however led to several inconsistencies in the results, such as the presence of supersymmetry at low energies which has not been proved yet by experimental evidence and instability of the proton [3] following from baryon and lepton number violations. More recent developments have suggested the use of D-branes in the open string sector: the so called Intersecting Brane Worlds. Not only do these models account for the correct features of particle in the SM e.g.: chirality, family replication, anomaly cancellations [4, 5] but also for more complex phenomena such as proton stability [6]. In this work we deal with intersecting D6-branes at angles and type IIA strings. In these models chiral fermions and bosons can arise at the intersections of the branes, thus leading to a computable formulation of Yukawa couplings and Flavour Changing Neutral Currents [7], at least in the simplest cases. In the literature the issues related to this construction have been vastly studied from several perspectives such as conformal field theories on orbifolds [8, 9] and compactification on factorized tori e.g.: 1 A string with different algebras acting on left and right moving fields, which present different symmetries [2]. 5

6 Riccardo Finotello Introduction and Conclusions T 6 = T 2 T 2 T 2 [10, 11, 12]. In these cases the rotations performed by the branes at angles are abelian, planar rotations. In this Master s thesis we try to generalise to non abelian SO 4 rotations in a non factorized space R 6 = R 4 R 2. We look at the bosonic classical string in order to write the classical action of the string and show the mechanism connected to the mass hierarchy in the Yukawa couplings. The first part of this Master s thesis is dedicated to string phenomenology. In the first chapter we describe the operational instruments we use to consistently build a semi-phenomenological model of particle physics: the compactification of the extra dimensions leads to the introduction of Calabi-Yau CY manifolds and we show how D-branes are mathematical and physical entities naturally inserted in a theory of open strings. In the second chapter we use D-branes in order to describe a viable SM-like model of intersecting D6-branes at angles in a CY manifold over a 4-dimensional space. We show how gauge interactions are connected to stacks of coincident D-branes and how chiral fermions and various bosons live at the intersections. We then deal with the problem of reproducing 4-dimensional physics through the compactification of the extra dimensions. We also stress the issues arising from the factorized case before moving to its generalisation in the second part of the thesis. Finally we define the Yukawa couplings and we briefly sketch the Higgs mechanism inside a model of intersecting branes. The second part of the thesis deals with non abelian D-brane rotations in R 4. In the initial chapter we build the geometrical set-up with D-branes rotations in the Grassmannian [13, 14, 15] SO 4 SO 2 SO 2 and show the issues related to the boundary conditions emerging from the configurations of the branes: we introduce the doubling trick in real coordinates and argue that finding a solution directly is not convenient. In the following chapter we show that the isomorphism [16] SO 4 SU 2 SU 2 Z 2 can be used in order to write the solution in terms of a basis of hypergeometric functions and SU 2 SU 2 rotations acting on matrix-valued coordinates: the problem configures as a successive generalisation of the SU 2 SO 4 rotations shown in [17]. We then look at how to link the parameters of the hypergeometric functions to the boundary conditions and we work out the constraints on the possible geometry of the system and the conditions to 6

7 Riccardo Finotello Introduction and Conclusions retrieve the Nambu-Goto action. Since there are multiple functions leading to the same monodromies, we consider a virtually infinite sum of all of them as a possible bosonic field. In order to write the solution, we find the independent hypergeometric functions, labeled by integers, with the correct boundary conditions and impose several physical requests on it as further constraints. We then try to recover previous simpler cases [10, 17] from this general construction. In the last chapter we show the implication for the Yukawa coupling using the string classical action and its equation of motion e.o.m.. In conclusion, in this Master s thesis, we show how more general, non abelian D-brane rotations can be described in terms of simple hypergeometric functions and how this general case compares with the known abelian formulation. Using properties of contiguous functions and requiring the action to be finite, we show how to reduce the number of terms to a finite subset and link the coefficients of the sum to physical requests on the strings, thus fixing completely the solution. We finally prove that, differently from the known abelian case, the fields are no longer strictly holomorphic but involve both coordinates z and z on the complex plane. We eventually write the string action, stressing the differences between the holomorphic case and the generalisation we presented. In this case the Yukawa couplings are suppressed with respect to the abelian case: the difference comes entirely from the non abelian geometrical set-up and the fact that the strings move on a non planar minimal surface with respect to the simpler case where the minimal surface is strictly flat. 7

8 Riccardo Finotello Part I Towards String Phenomenology 1 Extra Dimensions and Compactification In this work we are interested in Yukawa couplings in the intersecting D- branes at angles scenario leading to the SM of particle physics. It is therefore obvious to consider the framework of superstrings since fermions play a protagonist role in this model. In this case, strings propagate and interact in a 10-dimensional Minkowski spacetime. However we need to recover the results of a 4-dimensional supergravity in the low energy limit starting from the string formulation. Since we need the theory to be valid in 4 dimensions and lead to physical results, we consider the 10-dimensional Minkowski spacetime M 1,9 as a product of two different subspaces: M 1,9 = M 1,3 X 6, where M 1,3 is the usual 4-dimensional, physical spacetime and X 6 is a 6- dimensional manifold. In order to find a realistic outcome in the usual approach, X 6 must satisfy very strict requests such as [18]: it must be a compact manifold with a compactification radius R much smaller than currently accessible lengths, it should preserve a N = 1 supersymmetry in 4 dimensions in order to be computable, it must lead to a realistic spectrum of chiral fermions and gauge interactions. These constraints lead to the request for X 6 to be a CY manifold, whose existence has been first conjectured by Eugenio Calabi [19] and then proved by Shing-Tung Yau [20], hence the name. The compactification of the extra dimensions on these manifolds plays a central role in order to reproduce the correct results at field theory level. The idea behind this mathematical construction is a Kaluza-Klein reduction from a 10-dimensional space to the usual 4 dimensions applied to String Theory. Differently from the usual field theory calculations, String Theory presents some peculiarities which are the keys to reproduce the physics of SM-like scenarios through a unified theory of gravity and gauge interactions. 8

9 Riccardo Finotello 1.1 Calabi-Yau manifolds 1.1 Calabi-Yau manifolds CY manifolds are a particular set of complex manifolds. Specifically, X m is a compact CY manifold of complex dimension m or CY m-fold and real dimension d = 2m if: X m is a complex, Kähler manifold, X m is Ricci-flat or equivalently has SU m holonomy group. We now summarise briefly these definitions since they are significant to String Theory Compact manifold A manifold X is compact depending on the choice of its topology T : every collection of covering open subsets U α T must have a finite subcover. That is, let {U α } α A be a cover of X, i.e.: U α = X, α A such that U α T. Then there must be a subcover of X {Ũβ } β B {U α } α A such that B is finite. If A is already a finite set, then the condition is automatically satisfied Complex manifolds Starting from a real differentiable manifold X of even dimension d, we can extend its definition in order to give the topological space a complex structure. Consider the situation in Figure where φ α, U α and φ β, U β are two different charts φ αβ : U αβ X V αβ R d, such that {U λ, φ λ } λ A, α, β A, is an atlas for X. We can then express any point p on X in local coordinates as a d-dimensional point in R d : p αβ U αβ X x 1 αβ, x 2 αβ,..., x d αβ R d. At this point we can introduce a set of complex coordinates: zαβ a = 1 x a αβ + ix d 2 +a, 2 9 αβ

10 Riccardo Finotello 1.1 Calabi-Yau manifolds X U α U α U β U β R d 1 φ α R d 1 φ β V α V β R ψ αβ = φ β φ 1 α R Figure 1.1.1: Let X be a differentiable manifold, we can equip the topological space with a complex structure requiring the transition functions ψ αβ to be holomorphic. where a = 1, 2,..., m and m = d, together with their complex conjugate 2 z a = z a, in order to express every map φ α or φ β as a function φ C α or φ C β, such that: φ C αβ : Uαβ C X Vαβ C C m, where the apex C is a simple remainder that every mathematical object must be defined through an m-tuple of complex coordinates zαβ a and za αβ. In order to simplify the notation, we assume to deal with complex coordinates and the additional index can be understood. The last step in the definition is to equip the topological space X with a complex structure: we demand the transition functions ψ αβ shown in Figure to be holomorphic functions. Therefore ψ αβ : φ α U α U β C m φ β U α U β C m between charts U α, φ α and U β, φ β satisfies the Cauchy-Riemann equations. These are actually encoded by an endomorphism [21] on the tangent space of X in a point p X J : T p X T p X such that Jv p a = J a b vb, v p T p X, J 2 = id and [v p, w p ] + J [Jv p, w p ] + [v p, Jw p ] [Jv p, Jw p ] = 0 v p, w p T p X. 10

11 Riccardo Finotello 1.1 Calabi-Yau manifolds J is known as complex structure and the Cauchy-Riemann equations can be expressed in terms of it: for example consider m = 1 and let f : X C a smooth function, then J f p [v p ] = i f p [v p ] are the Cauchy-Riemann equations f p [v p ] is the pushforward of v p T p X in p X through the function f. As a matter of fact, suppose m = 1 d = 2: let z = x + iy, f x, y = u x, y + iv x, y and 0 1 J =. 1 0 Then becomes J b a f p b = i f p a x u x, y = y v x, y, y u x, y = x v x, y, that is the usual Cauchy-Riemann equations in C. This proves that f x, y is a holomorphic function Kähler manifolds Kähler manifolds are complex manifolds. Let X be a complex manifold of dimension m with complex structure J and Riemannian metric g, then g is Hermitian if g v p, w p = g Jv p, Jw p g ij = J k i J l j g kl. for v p, w p T p X. This condition becomes clear if we write the real-valued metric g = g ij dx i dx j, i, j = 1, 2,..., d in terms of complex coordinates z and z as follows: g C = g C abdz a dz b + g C a b dza d z b + g C ābd z a dz b + g C ā b d za d z b, where a, b = 1, 2,..., m m = d/2 and g C ab = g a,b ig m+a,j ig a,m+b g m+a,m+b, g C a b = g a,b ig m+a,j + ig a,m+b + g m+a,m+b, g C āb = g a,b + ig m+a,j ig a,m+b g m+a,m+b, g C ā b = g a,b + ig m+a,j + ig a,m+b + g m+a,m+b. 11

12 Riccardo Finotello 1.1 Calabi-Yau manifolds Then g C is Hermitian if g C ab = g C ā b = 0. For the sake of simplicity, we omit the complex field index from now on. In this case, we define the Hermitian 2-form ω as ω v p, w p = g Jv p, w. The condition for X to be a Kähler manifold is dω = + ω = 0, where in complex coordinates the operator d factorizes in a holomorphic operator acting on the holomorphic part of the basis and an anti-holomorphic operator acting on the anti-holomorphic part. Under such condition ω is the Kähler form. It can also be expressed locally in terms of a smooth function K on X, known as the Kähler potential, since is a closed form i.e.: dω = Holonomy group ω = i K The holonomy group is related to the idea of the parallel transportation of a vector field over a manifold. Let X be a manifold, at least differentiable, of real dimension d, with a metric g and let v p T p X be a vector located at p X. We can define a connection over the tangent bundle of X from its metric. We can then choose a smooth curve γ : [0, 1] X and move v along the curve such that γ v p = 0. In general we have: v p 1 = P γ v p 0. If γ is a closed loop around p X, then P γ : T p X T p X, i.e.: P γ End T p X. The holonomy group of X with metric g, around p is thereupon the group: Hol p g = {P γ γ is a closed loop around p}. 12

13 Riccardo Finotello 1.1 Calabi-Yau manifolds If X is orientable, Hol p g is in general a subgroup of SO d [22]. Moreover, if X is simply-connected, the holonomy group does not depend on the base point: we assume this to be our case. As we have seen, a Kähler manifold must satisfy the strict condition on the ω 2-form and its metric. As a consequence if X is a Kähler manifold, its connection has no mixed index Christoffel symbols, i.e.: only symbols with the same kind of holomorphic or anti-holomorphic indices are different from zero: the action on the complex coordinates vector v p = v i + vī T p X z i z i does not mix holomorphic and anti-holomorphic coordinates. The holonomy matrices can be written in terms of the action on the two set of coordinates, which leads to: Hol g U m SO d, where, as usual, m = d/ Calabi-Yau manifolds CY manifolds of complex dimension m are Kähler manifolds with holonomy group Hol g = SU m and where a holomorphic m, 0-form Ω called the holomorphic volume form can be defined. Let ω be the usual Kähler form, then the condition for X to be a CY manifold is: ω m m! mm 1 = 1 2 i 2 m Ω Ω, where Ω is the anti-holomorphic analogous of Ω, obtained by complex conjugation. The choice of the holonomy group has a reflection on the connection and its Christoffel symbols as well. As a matter of fact, a Kähler manifold of complex dimension m with holonomy group SU m, is also Ricci-flat, i.e. [18, 21, 22]: Ric a b = Γ a c c = 0. z b As shown in detail in [18], the existance of these set of complex manifolds is crucial to Stirng Theory as it allows the presence of unbroken supersymmetries and realistic physics to be built. Tori of complex dimension m are usually the simplest example of CY m-folds and are extensively used in String Theory: toroidal compactification is the starting point for many interesting 4-dimensional models. 13

14 Riccardo Finotello 1.2 D-branes and open strings 1.2 D-branes and open strings Dirichlet branes, better known as D-branes, are another crucial mathematical and physical entity in String Theory. They are extended objects where the open string endpoints are restrained: they are extremely important in the consistent formulation of string phenomenology and, thanks to their configurations, it is possible to recreate realistic results of particle physics. We are interested in their definition and their properties as a starting point for the construction of a SM-like scenario of Yukawa couplings Kaluza-Klein compactification One way to introduce D-branes in String Theory is to study the small radius compactification of closed strings and then look at the consequences of such action on open strings [23], thus showing that D-branes are naturally included in the theory. Consider the two usual worldsheet coordinates < σ 0 < + and 0 σ 1 π, then the action of the bosonic string on the worldsheet Σ is: S = 1 4πα Σ d 2 σ ab Xµ X ν h h σ a σ η µν, b where a, b = 0, 1 and µ, ν = 0, 1,..., D 1 in D spacetime dimensions and h = det h ab. Furthermore η = diag 1, 1, 1,... and h = diag 1, 1. The variation with respect to X µ should return the e.o.m. of the string and possibly its boundary conditions: δs = 1 2πα Σ d 2 σ a a X µ δx µ + 1 2πα Σ dσ 0 [ 1 X µ δx µ ] σ1 =π σ 1 =0, where we contracted the indices by means of the worldsheet and target space metrics. Setting δs δx µ = 0 we find the usual e.o.m. a a X µ = 0 and the Neumann boundary condition Even though it seems that 1 X µ σ 0, σ 1 σ1 =π σ 1 =0 = δx µ σ1 =π σ 1 =0 = 0X µ σ1 =π σ 1 =0 =

15 Riccardo Finotello 1.2 D-branes and open strings satisfies the condition as well, following [23] we can show that this Dirichlet boundary condition arise automatically from the theory and identify hypersurfaces where the string is constrained: the D-branes, that is. Consider the Wick rotation σ 0 σ 0 E = iσ0 and z = e σ0 E +iσ1, z = z = e σ0 E iσ1, then the open string with Neumann boundary conditions has a general solution: α X µ z, z = x µ iα p µ α µ m ln z z + i z m + z m, 2 m m Z\{0} where x µ and p µ = 2α α µ 0 are the position and the momentum of the centre of mass of the string. Moreover the canonical commutation relations hold: [x µ, p ν ] = iη µν, [α µ m, α ν n] = iη µν δ m, n. On the other hand, for the closed string the boundary condition translates into: X µ σ 0, σ 1 + π = X µ σ 0, σ 1 and shows a difference in its left-moving modes of oscillation identified by α m: µ α X µ L z = xµ + i α µ 0 ln z + α m µ 2 m z m m Z\{0} and right-moving modes of oscillation identified by α m: µ α X µ R z = xµ + i α µ 0 ln z + α m µ 2 The complete mode expansion is of the form: m Z\{0} X µ z, z = X µ L z + Xµ R z. m z m In the usual case where X µ z, z is free to move in the entire 10-dimensional Minkowski space 2 M 1,9, we can compute the momentum of the centre of mass of the string: p µ = 1 2α αµ 0 + α µ 0, 2 We consider directly the superstring formulation, where consistency conditions impose the dimension of the spacetime D to be equal to

16 Riccardo Finotello 1.2 D-branes and open strings and show that: Suppose now to consider: α µ 0 = α µ 0. M 1,9 = M 1,8 S 1 in such a way that under σ 1 σ 1 +2π, the coordinate relative to S 1 changes according to X 9 ze 2πi, ze 2πi = X 9 z, z + 2πmR, where m Z and R is the radius of S 1 where the string can wind an integer number of times. For such coordinate we impose this last condition to show: α α 0 9 α = 2n, n Z, 2 R α 0 9 α = mr, m Z, α where the first equation comes from a quantization condition on the compact momentum and the second from the compactification. This procedure is the simplest and most naive example of Kaluza-Klein compactification, where n are Kaluza-Klein modes and m are the winding numbers T-duality and open strings To simplify the notation we choose α = 2 and compute the mass spectrum M 2 = p µ p µ = α N 1 = = α Ñ 1, where N is the number operator for all left-moving modes and Ñ for all rightmoving modes. As a consequence of the compactification, we can show the insurgence of T-duality: the mass spectrum in a theory with small compactification radius R 0 is identical to a theory with large radius R under the exchange of winding numbers and Kaluza-Klein modes, that is the theory is symmetric under R 2 R, m n. At oscillator level, this implies the maps: α 9 0 α 9 0, α 9 0 α

17 Riccardo Finotello 1.2 D-branes and open strings Figure 1.2.1: From the mathematical point of view, D-branes are particular hypersurfaces where the open string enpoints can end. Were we to consider a field theory, depending on the size of the compactification radius R 0 or R, some states would become infinitely massive, while others would go to a continuum. The remaining states would be independent of the compact coordinate and the effective dimension of the theory would be reduced. The presence of T-duality is a peculiarity of String Theory and shows that strings behave differently: closed strings still propagate in 10 dimensions since the compact coordinates do not disappear. We can now turn the attention to open strings and suppose to describe the same Kaluza-Klein procedure. While closed strings can wind around the periodic dimension, open strings do not have any winding quantum number. On the other hand, the compactification acts on them in the same way. Consider the R 0 limit and this time choose α = 1/2, then: α 9 0 = p 9 = 2n R. The states with n 0 become infinitely massive but no continuum appears. The compactified dimension vanishes: the Kaluza-Klein compactification reduces the number of spacetime dimensions and from a D = 10 theory, we are left with D = 9 [23]. Open strings are therefore different from closed strings, which live in D = 10 spacetime dimensions even after the compactification in this case one of them lies in a compact manifold: their endpoints are restricted to a D 1-dimensional space. The X 9 coordinate is instead bounded to a 1-dimensional hypersurface and satisfies the Dirichlet condition 1.2.2: 0 X 9 σ 0, σ 1 σ 1 =0,π = 0, instead of the Neumann condition, from which we started. This is the consequence of the action of T-duality on the compact dimension. Clearly one could generalise the procedure to more dimensions, bounding the open 17

18 Riccardo Finotello 1.2 D-branes and open strings string endpoints to a D n-dimensional hypersurface, where n is the number of dimensions where T-duality has been applied: these are known as D-branes i.e.: Dirichlet branes and are naturally part of any string theory D-branes In conclusion, D-branes can be introduced as mathematical entities with the property that open strings end on them Figure The usual notation for such hypersurfaces of dimension p + 1 including the time-like dimension, where the string is usually free to move is Dp-brane, where p is the number of space-like dimension spanned by the motion of the string. Moreover we have shown that D-branes are intrinsic to String Theory and arise naturally from mathematical considerations. On the other hand, D-branes show also several physical properties. Even though in this work we are not interested in them, they are significant to the development of the theory and to superstring models of particle physics which reproduce a SM-like scenario. As a matter of fact, Dp-branes are topological solutions of the low-energy action of some kind of string theory we especially focus on type II string theories, but it can be generalised to type I [24, 25, 26] and carry the charge of a p + 2-form field strength [27]. Moreover D-branes are BPS states which break half of the supersymmetries of the theory [27] when introduced in spacetime. 18

19 Riccardo Finotello 2 String Phenomenology We now turn our attention to the issues related to string phenomenology. We would like to show how the objects presented earlier CY manifolds and D-branes can be used to build a model of particle physics: we are interested in reproducing correctly the spectrum of a SM-like scenario using properties of open strings and D-branes. The search for such models is key to our understanding of the theory: strings are an effort to get to a unified description of gauge and gravitational interactions and as such must lead to known results in order to be considered valid. Even though part of the community is sceptical about it, String Theory tries to suggest solid bases for the construction of particle physics models. Unfortunately the theory predicts the existence of many possible vacua the string landscape, but only one of them corresponds to our universe [28]. The purpose of a good string phenomenology is therefore to show and to motivate that one of these vacua actually has the properties of the SM of particle physics [29]. Among the properties any phenomenological model should present, the existence of chiral fermions is arguably one of the most crucial: String Theory must be able to give reason of the presence of different charges between left handed and right handed leptons and quarks. We show how intersecting D- branes models account for this asymmetry and reproduce the correct chiral spectrum. Then we focus on Yukawa couplings and their peculiarities: we introduce the boson-fermion couplings, isolating the issues we address in the second part of this work, and briefly describe the dynamics of the Higgs mechanism in these models. 2.1 Light spectrum and symmetry enhancement As already stated before, string phenomenology should give reason of the gauge interactions and possibly gravitational interactions of the SM, described by the gauge group: SU 3 C SU 2 L U 1 Y, representing colour charge, left weak isospin charge and hypercharge, respectively. Moreover it should show the correct spectrum 3 of chiral fermions with the correct replication of generations. With respect to 2.1.1, and for each family of particles, we should find the spectrum of Table and a 3 Remember the Gell-Mann Nishijima formula for the hypercharge Y = 2 Q I 3, where Q is the electric charge and I 3 is the third component of the isospin vector. 19

20 Riccardo Finotello 2.1 Light spectrum and symmetry enhancement Particle Quantum numbers Q L 3, 2 1/3 ū R 3, 1 4/3 d R 3, 1 2/3 l L 1, 2 1 e + R 1, 1 2 ν R if any 1, 1 0 Table 2.1.1: Spectrum of SM particles one generation with related quantum numbers scalar particle H with quantum numbers 1, 2 1, the Higgs boson. Interactions between these particles and the Higgs boson are very specific and must respect gauge and accidental symmetries in the string formulation. On the other hand, String Theory is a high energy theory propagating in 10-dimensional space and presenting a high degree of supersymmetry. This leaves arbitrariness in the choice of the model which reproduces particle physics in 4 dimensions. In fact, there are several ways to compactify the theory on a CY manifold in order to get to the content of the SM or similar models. For example, one of the first attempt in string phenomenology was the compactification of heterotic string theories on CY 3-folds. We are instead interested in models involving D-branes and type II and possibly type I string theories compactified on tori. As seen in Section 1.2, D-branes introduce preferred directions of motion of the strings, breaking the SO 1, d 1 Lorentz symmetry. Since we are interested in the 4-dimensional effective theory we should consider the case of Dp-branes filling entirely Minkowski spacetime M 1,3 in 4 dimensions and then extending in the compact CY 3-fold X 3 : it is natural to consider 3 p 9 in order to preserve the 4-dimensional Lorentz symmetry. Moreover we are interested only in those modes which result in the 4- dimensional supergravity after the compactification, that is we look at the massless states of the theory GSO projection takes care of the tachyon modes [2, 30]. For type II string theories and a single Dp-brane, the light spectrum of the theory is made of: one U 1 gauge boson transforming under the little group SO p 2, 9 p real scalars and a set of p + 1-dimensional fermions. Scalars and fermions can be regarded as Goldstone bosons and Goldstinos of the translational symmetries broken by the introduction of the brane [31]. 20

21 Riccardo Finotello 2.1 Light spectrum and symmetry enhancement D 1 D 2 a c.1 b c.2 Figure 2.1.1: The light spectrum of the theory is made of gauge bosons and fermionic partners coming from the oscillation modes of string living on the D-branes, such as a and b, and modes generated by the stretched strings between branes D 1 and D 2, as c, when the branes are coincident. For the bosonic sector, in lightcone gauge, this can be seen from the action of the creation operators { α µ n, α i n} n > 0, µ = 1, 2,..., p, i = 1, 2,..., 8 p of the broken SO 1, p SO 9 p symmetry on the vacuum 0 defined by α µ n 0 = α i n 0 = 0, n 0: Vector A µ α µ n 0, µ = 1, 2,..., p, Scalars a i α i n 0, i = 1, 2,..., 8 p. These states are generated by strings in different configurations Figure 2.1.1: strings with both endpoints on the same brane D 1 or D 2 : in this case A 1,1 µ, A 2,2 µ, a 1,1 i and a 2,2 i are massless vectors and scalars the indices p, q show the position of the two endpoints on brane D p and D q ; strings stretched between the two branes: states A µ 1,2, A 2,1 µ, a 1,2 i are massive, with mass M such that: a 2,1 i and M 2 c α c α, α = 0, 1,..., 9, where, as already done for open strings, we chose α = 1/2 and c is vector of the distance between the branes. What seems to be a U 1 U 1 symmetry with massless gauge bosons A 1,1 µ and A 2,2 µ changes deeply when c 0: there are more massless 21

22 Riccardo Finotello 2.2 Building the Standard Model gauge group with branes N separate D-branes N coincident D-branes U 1 U 1... U 1 }{{} N times U N Figure 2.1.2: Symmetry enhancement is a peculiar trait of String Theory, contrary the usual field theories. Instead of a simple replication U 1 [U 1] N, the symmetry changes to the larger U N. vectors, including those arising from the stretched string. Since the charge with respect to these vectors remains untouched, the symmetry of the 4 massless vectors is enhanced to U 2 SU 2 U 1. The procedure can be generalised to a stack of N coincident identical D-branes: the N 2 massless gauge bosons and their fermion superpartners transform under the adjoint representation of U N Figure The massless spectrum on the world volume of N coincident Dp-branes is therefore made of [32]: N 2 U N gauge bosons, N 2 9 p scalars, N 2 sets of p + 1-dimensional fermions. 2.2 Building the Standard Model gauge group with branes As we have seen in the last section, the fields on the world-volume of the D- brane fill a U N multiplet with respect to the unbroken supersymmetries. Using this property we examine the SM of particle physics at string level [33] and analyse the branes configuration which lead to the known SU 3 C SU 2 L U 1 Y We start from the gauge bosons of the theory. In detail, we begin from the carrier of the strong force, the gluons, i.e.: a 4-dimensional SU 3 Yang-Mills theory. This is describerd by the low-energy limit of 3 coincident D-branes, which lead to a world-volume gauge theory: U 3 SU 3 U

23 Riccardo Finotello 2.2 Building the Standard Model gauge group with branes The same reasoning can be applied to the electroweak sector, i.e.: the 4-dimensional SU 2 Yang-Mills theory. This time the description at low energy is suported by 2 coincident D-branes, which lead to a gauge group: U 2 SU 2 U Clearly the two stacks of D-branes must be separate 4 in order to lead to U 3 U 2 SU 3 SU 2 U 1 U 1. Even though it resembles the usual SM gauge group, the two U 1 factors or their combination cannot be identified with the usual hypercharge in 2.2.1: we must remember that the construction of a phenomenological model in String Theory must reproduce correctly the matter content as well. However this configuration does not take into account the existence of quarks and leptons. Two stacks of 3 and 2 coincident D-branes cannot effectively explain the correct quantum numbers of the SM. Consider now the SM fermions. These particles are characterised by their helicity which classifies them into left handed particles and right handed particles according to their value 1/2 and 1/2 in adimensional units, respectively. According to their nature quarks or leptons, fermions interact with different gauge bosons in a different way. In general these particles transform under the bi-fundamental representation N, M of the gauge group SU N SU M, that is they carry a charge with respect to the gauge bosons involved in the interactions. In String Theory this is realised by open strings with endpoints on different stacks of D-branes: string states are characterised by their charge q 1, q 2,..., q N with respect to the Maxwell fields that live on the N D- branes [33], i.e.: by the charge carried by the endpoint of the string on the N-th D-brane. The charge of the decoupled U 1 is proportional to q 1 + q q N, while the other charges are linearly independent combinations of q i, i = 1, 2,..., N. The corresponding antiparticles are simply oppositely oriented strings. Consider for example the left handed quarks of the SM which transform under the representation 3, 2 4 If the stacks were to coincide we would find a U 5 gauge theory. 23

24 Riccardo Finotello 2.2 Building the Standard Model gauge group with branes G rb r g b u d u g L W ud d r L Figure 2.2.1: This is a pictorial representation of SU 3 SU 2 gauge bosons and left handed quarks. Event though the presence of two stacks of parallel branes can explain the presence of gluons G ij, i, j {r, g, b} and weak bosons W ab, a, b {u, d} as well as quark states u L and d L stretched between the stacks, the model lacks a way to account for the chirality of the SM fermions. of SU 3 C SU 2 L. Their string states are formally realised by a string with one endpoint on the stack of 3 D-branes in and the other endpoint on the stack of 2 D-branes in 2.2.3, such as in Figure The quarks on the stack of 3 branes would be then identified by the pair q 1 q 2, q 2 q 3, where q 1, q 2, q 3 are the charges of the string endpoint with respect to the Maxwell field on each D-brane, forming the representation 3 of SU 3 [33]: 3 : 1, 0, 1, 1, 0, 1. The idea is the similar for the other endpoint of the string, ending on the second stack of D-branes and forming the representation 2 of SU 2. The two stacks of D-branes would seem to be sufficient to explain the quantum numbers of quarks, but if we consider the leptons in the 1, 2 representation, we immediately realise the necessity of an additional D-brane, since leptons are not subject to colour charge and cannot have one of the endpoints on the same stack of D-branes as quarks. Moreover stacks of parallel D-branes do not allow the treatment of chirality in particle physics. In fact, we need a more refined model in order to take into account all the aspects of the SM. 24

25 Riccardo Finotello 2.3 Chiral fermions and D-branes at angles 2.3 Chiral fermions and D-branes at angles In the last example we implicitly dealt with one of the most remarking features of the SM of particle physics: chirality. That is, in the SM of particle physics the left and right chiral projections of 4-dimensional fermions do not present the same quantum numbers. Moreover, as seen in 2.1.2, a string stretched between parallel D-branes is massive, which is not allowed for chiral particles: an explicit mass term would mix left and right components. There must be a symmetry preserving mechanism, such as the Higgs mechanism, which encorporates the mass terms for fermions. Constructions with parallel D-branes are therefore not enough in order to explain most of the features of the SM. In String Theory there are multiple ways to build a phenomenological model with chiral fermions. For example [31]: D-branes located at singular points in the internal compact space, such as D3-branes lying at singularities of orbifolds e.g.: C 3 /Z N [34]; Sets of intersecting stacks of D-branes with fermions stretched between the stacks and localised at their intersection, in order to minimise their tension. We are interested in the latter case and in particular to stacks of D6- branes intersecting over a 4-dimensional subspace of their volumes in a CY 3-fold e.g.: a complex 3-torus, in the simplest case. The strings that stretch between two stacks of D6-branes are localised at the intersection and lead to the creation of chiral fermions: suppose to start with both stacks of branes coincident and parallel, then start to increase the value of the angle between the two stacks; as the intersection angle grows some states that are massless at zero angle become massive. Under the right conditions only left handed particles and right handed antiparticles remain massless, thus leading to a chiral massless spectrum [33]. Consider for example two intersecting D6-branes in M 1,9 and suppose their intersection is in the X 6, X 7 plane with an angle πθ 0 θ < 1. Following [35], we find that the mode expansion for X µ σ 0, σ 1, µ = 0, 1,..., 9 is: X µ σ 0, σ 1 = i 2 pµ σ 0 i α µ m 2 m e imσ0 cos nσ 1, µ = 0, 1,..., 5 m Z\{0} X 6 σ 0, σ 1 = i α 6 m+θ 2 m+θ e im+θσ0 cos m + θ σ 1 + α6 m θ m θ e im θσ0 cos m θ σ 1 m Z X 7 σ 0, σ 1 = i α 6 m+θ 2 m+θ e im+θσ0 sin m + θ σ 1 + α6 m θ m θ e im θσ0 sin m θ σ 1. m Z X ν σ 0, σ 1 = i α ν m 2 m e imσ0 sin nσ 1, ν = 8, 9 m Z\{0} 25

26 Riccardo Finotello 2.3 Chiral fermions and D-branes at angles Fermions, for example, in the NS sector become: ψ µ + σ 0, σ 1 = i 2 ψ+ 6 σ 0, σ 1 = i 2 m Z ψ 7 + σ 0, σ 1 = 1 2 m Z ψ+ ν σ 0, σ 1 = i 2 m Z b µ m+ 1 2 e im+ 1 2 σ0 +σ 1, µ = 0, 1,..., 5 b 6 m+θ+ 1 2 e im+θ+ 1 2 σ0 +σ 1 + b 6 m θ+ 1 2 b 7 m+θ+ 1 2 e im+θ+ 1 2 σ0 +σ 1 + b 7 m Z b ν m+ 1 2 e im+ 1 2 σ0 +σ 1, ν = 8, 9 m θ+ 1 2 e im θ+ 1 2 σ0 +σ 1 e im θ+ 1 2 σ0 +σ 1 while for the R sector we should exchange m r Z + 1/2. From this we can read the mass operator α = 1/2 as usual: 2MNS 2 = N θ + 1 θ 1, 2 where N θ is the total number operator, which depends on the rotation parameter θ, for bosonic and fermionic modes: N θ = N bos θ + N fer θ. Clearly we can generalise to a larger number of rotation angles between the branes. Consider for example the case of the 10-dimensional Minkowski space M 1,9 seen as the product of a 4-dimensional M 1,3 and R 6 factorized as: R 6 = R 2 R 2 R 2, where two intersecting D6-branes are embedded in order to completely fill M 1,3 and then as a line in each plane: the situation is therefore described by the 3 rotation angles θ 1, θ 2, θ 3, which identify the relative rotation of the second brane with respect to the first in each R 2. This implies the change for the mass operator: 3 2MNS 2 = N NS θ + 1 θ i 1, 2 where also N θ includes the sum over the three angles, for the NS sector, and 2M 2 R = N R θ in the R sector. We can now check that there is only one massless fermionic state in the Ramond sector [36], which in terms of 4-dimensional physics corresponds to a Weyl fermion of definite chirality: the construction is therefore suitable to describe chiral 4-dimensional particle physics. The appearance 26 i=1

27 Riccardo Finotello 2.3 Chiral fermions and D-branes at angles Y = 1 2 Y = 1 Y = 0 Y = 1 3 ul dl ū R dr baryonic Y = 0 ν e L el e + R ν R leptonic Y = 1 ν R leptonic left right right Figure 2.3.1: Using intersecting branes it is possible to build a model with content similar to the SM of particle physics. of chirality is therefore linked to the preferred orientation introduced by the branes, that is by the relative rotation between the branes [32]. It should be noticed that apart from the chiral fermions, at the intersections there are also several light scalar, whose masses depend directly on the value of the angles they can be massive, massless or tachyonic. The case of a massless scalar shows a mass degeneracy with the fermionic spectrum, yielding a supersymmetric spectrum. The tachyonic state is instead linked to an instability of the system which presents an explicit non supersymmetric spectrum. Using intersecting branes is therefore possible to introduce chiral fermions in the theory. Combining accordingly different stacks of branes we can then reproduce the correct quantum numbers of the theory. For example, left handed quarks, which transform under the representation 3, 2 of SU 3 C SU 2 L, have one endpoint on a stack of three baryonic branes such as and the other on a stack of two left branes as in Right handed positrons have instead one endpoint on a leptonic brane to account for the absence of colour charge and one endpoint on a right brane. Different stacks of different branes also account for the hypercharge factor which comes from different combinations of the decoupled U 1 factors in U 3 U 2. The situation is schematically described in Figure It would appear that not only does the model reproduce the SM spectrum at least one gener- 27

28 Riccardo Finotello 2.4 Physics in 4 dimensions and family replication ation for each fermion: in the next section the presence of three generations of fermions is investigated, but there are also several additional particles. For the sake of simplicity and for the purpose of this work we do not deal with them, but it is possible to build particular models where this peculiarity is canceled by orientifolding the type II string theory considered [37, 38]: the symmetry introduced by the O6-planes produces cancellations which ensure the model to present the SM spectrum only. Moreover it can be shown that different D-branes at angles preserve different degrees of supersymmetries [39, 40] and particles arising from their intersections form multiplets with respect to the supersymmetry of the configuration. 2.4 Physics in 4 dimensions and family replication As discussed, the intersecting branes model leads to the definition of the chiral matter content. Being localised at the intersection of the branes, the fermions arising from strings stretched between the stacks are naturally 4- dimensional they intersect over their common M 1,3 space. However gauge interactions on a stack of Dp-branes are p + 1-dimensional and gravitational interactions remain untouched and still propagate in 10-dimensional spacetime. Unless we hide the extra dimensions in a compact space, the theory is not consistent. For example we could consider the compactification on a CY 3-fold X 3 such as a 6-dimensional real torus T 6. Furthermore, for the sake of simplicity, we could consider T 6 factorized as: T 6 = T 2 T 2 T 2, and the D6-branes embedded as lines in each one of the T 2. Each 2-torus is identified on a plane X i, X i+1 R 2 or z, z on C by the identifications: X i X i + 1, X i+1 X i Each line on the plane X i, X i+1 passing from the origin can be described as an oriented straight segment starting from 0, 0 to the point m, n, where m and n are relatively prime integers. Because of the torus identifications each straight line identified by m, n does not close on T 2, but wraps the manifold multiple times [33] as in Figure Consider now two line segments identified by m 1, m 2 and n 1, n 2, even though they never intersect on the entire plane apart from the origin, they may intersect on the torus a number of times given by: m1 n m 1 n 2 m 2 n 1 = det 1. m 2 n 2 28

29 Riccardo Finotello 2.4 Physics in 4 dimensions and family replication X i+1 1 3, X i Figure 2.4.1: The identifications X i X i + 1 and X i+1 X i are such that the line 3, 1 wraps T 2 multiple times. Labeling m a 1, m a 2 and n a 1, n a 2 the lines on each torus Ta 2, a = 1, 2, 3, then the total number of intersections is: 3 N I = m a 1n a 2 m a 2n a a=1 This procedure can be generalised to a non factorized torus introducing homology classes of cycles, but the key idea would appear less evident. This analysis might seem obvious but it is fundamental to the construction of SM like scenarios in String Theory. One of the properties of the SM of particle physics is the family replications, that is there are three copies generations of particles with identical charges, but different masses. For example in the left chirality sector we have: ul cl tl Q L =,, ; d L ν e l L = L, e L s L ν µ L, µ L b L ν τ L Since chiral fermions arise at each intersection of D-branes, we could use the multiple intersections of the line on the torus to explain the family replication of particles. After the compactification, the open string light spectrum between two stacks of N a and N b D-branes is in fact made of: τ L. 4-dimensional U N a and U N b gauge bosons, 6 real adjoint scalars and 4 adjoint fermions filling a vector supermultiplet of the supersymmetry preserved by the corresponding D-branes, N I generations of chiral left handed fermions in the bifundamental representation N a, N b, where NI is defined in

30 Riccardo Finotello 2.5 Yukawa couplings X l+1 1 f 3,l A ijk f 2,l f 1,l 1 X l Figure 2.5.1: In the target space, the 3 intersecting D-branes locate a triangle inside which the strings stretched between the branes are free to move. In the case T 6 = 3 Tl 2, A ijk in is exactly the area of planar triangle on the torus T 2 l. l=1 2.5 Yukawa couplings Yukawa couplings are possible interaction between fermions and scalar particles of the form [41]: L Y ukawa = y ijk ψ i ψ j φ k, where ψ i and ψ j are 4-dimensional fermions and φ k is a scalar particle. The coupling constant y ijk is an adimensional parameter in natural units analogous to the charge of the electron in QED. They were first introduced by Yukawa to describe the interaction between nucleons fermions and pions scalars. The modern SM contains Yukawa couplings between chiral fermions quarks and leptons and the Higgs field. In the context of String Theory, and specifically intersecting D-brane models, chiral fermions and scalar fields may give rise to Yukawa coupling terms. As already stated, we are usually interested in supersymmetric models, where the Yukawa coupling are generated by the cubic terms in the superpotential [42, 43, 44]. We look in particular at N = 1 theories where such superpotential is generated non perturbatively by worldsheet instantons [45, 46]. Their computation involves the twist fields at the intersection of the branes and the chiral fermions and bosons. In fact, we need the amplitude Vψi x 1 V ψj x 2 V φk x 3,