F-theory Yukawa Couplings and Supersymmetric Quantum Mechanics
aa r X i v : . [ h e p - t h ] N ov F-theory Yukawa Couplings and Supersymmetric QuantumMechanics
V.K.Oikonomou ∗ Technological Education Institute of Serres,Department of Informatics and Communications 62124 Serres, GreeceandDepartment of Theoretical Physics Aristotle University of Thessaloniki,Thessaloniki 541 24 GreeceNovember 3, 2018
Abstract
The localized fermions on the intersection curve Σ of D7-branes, are connected toa N = 2 supersymmetric quantum mechanics algebra. Due to this algebra the fieldsobey a global U (1) symmetry. This symmetry restricts the proton decay operatorsand the neutrino mass terms. Particularly, we find that several proton decay operatorsare forbidden and the Majorana mass term is the only one allowed in the theory. Aspecial SUSY QM algebra is studied at the end of the paper. In addition we studythe impact of a non-trivial holomorphic metric perturbation on the localized solutionsalong each matter curve. Moreover, we study the connection of the localized solutionsto an N = 2 supersymmetric quantum mechanics algebra when background fluxes areturned on. Introduction
F-theory [1–84], has received a prominent role lately, due to the fact that GUTs can beconsistently constructed and well founded, within F-theory’s wide theoretical framework.It is an 12-dimensional theory that consists of toroidal elliptic fibrations over Calabi-Yaumanifolds. D7 branes are essential to the theory, since the D7 branes are located onthe T fiber. The modulus of the torus is a varying parameter and is related to the axio-dilaton. Thereupon, we can say that F-theory is an UV completion of type IIB superstringtheory with 7-branes. For comprehensive reviews on the formulation of F-theory GUT’ssee [6–10, 20, 23]One of it’s most interesting outcomes, is that within F-theory we can produce many phe-nomenological features of GUTs with gravity being excluded from the theoretical appara-tus (for recent work on realistic F-theory GUTs models see [4–9,11–67,70–73]). Moreover, ∗ [email protected] H × M × M for SU (5), or the spinor 16 representation of SO (10), now can be consistently incorporatedto the phenomenological outcomes of the theory.Complex manifolds singularities play a critical role in F-theory phenomenology [1, 4, 6–12], with gauge groups realized by the geometry of singularities. Additionally, N = 1, D = 4 supersymmetric gauge theories arise when F-theory is compactified on Calabi-Yaufourfolds [3, 4, 9, 12].One of the most important features that a UV completion of the Standard Model mustsomehow explain is the hierarchical structure of fermion masses and mixings. In F-theoryGUTs much work has been done towards this direction [4, 12, 16, 24, 27, 38, 49, 79, 80] andalso in order to explain the neutrino sector, see [81] and also [49,51,80,82,83]. Yukawa cou-plings in F-theory are obtained by calculating overlapping integrals of three matter curveswave functions over a complex surface S . Hence Yukawas depend drastically on the localstructure of the theory, near the intersection point of the three matter curves (nonethelessthe global structure of the theory affects the normalization of the wave functions).In this work we shall consider the localized fields that are generated on the intersectioncurve Σ of D7-branes without external gauge fluxes, and also the Yukawa couplings gen-erated by the intersection of three matter curves. The fields that have localized solutionsalong the matter curve Σ, are connected [89] to an N = 2 supersymmetric quantum me-chanics algebra [85, 86] and the number of the zero modes is connected to the Wittenindex of the susy algebra. We find that every localized field on the matter curve obeysa hidden global U (1) symmetry. We shall require that this symmetry holds even at theintersection point of three matter curves. The conditions that must hold in order this tohappen, pose some restrictions on various proton decay operators and on the operatorsthat give masses to the neutrinos. Furthermore, we study the impact of a certain typeof susy quantum algebra on the Yukawa coupling that gives mass to the top quark. Theresults are interesting, since, the imposed conditions result to a form of wave functionswith de-localized Higgs. Moreover, we shall include the effects of a gravitational backre-action on the complex surface S , in terms of linear perturbations of the Euclidean metric.We conclude that the spectral problems of the perturbed and unperturbed system areidentical, due to the topological invariance of the index of the corresponding operators.We shall examine all the matter curves. Finally, we shall check whether we can relate aSUSY QM algebra to the fermions localized along the three matter curves, in the case weintroduce constant background gauge fluxes.This paper is organized as follows: In section 1 we describe in brief the F-theory setupwe shall use, that is, D7-branes intersections, matter curves and the eight dimensionalSuper-Yang-Mills (SYM) theory. In section 2 we give in short, a self-contained reviewof the supersymmetric quantum mechanics algebra. In section 3 we connect the localizedsolutions of the BPS equations of motion, to an N = 2 supersymmetric quantum mechanicsalgebra and also we study the impact of a non-trivial linear perturbation of the metric onthe localized solutions. In section 4 we examine the localized solutions of the fermionicsystem under the influence of background gauge fluxes. In section 5 we study the impact ofa certain type of susy quantum algebra on the top quark Yukawa coupling. In section 6 we2tudy the U (1) symmetries and the restrictions these imply to the proton decay operatorsand to the neutrino mass operators. Finally in section 7 we present the conclusions. We shall consider F-theory compactifications on a Calabi-Yau fourfold. This manifold is anelliptic K S . Locally the theory can be de-scribed by the worldvolume of an ADE type D7 brane wrapping R , × S over the Calabi-Yau fourfold. The resulting d = 4 theory is an N = 1 su-persymmetric theory [4, 12]. Our analysis is based mostly on references [4, 12].The physics of the D7-branes wrapping S can be described in terms of an D = 8 twistedSuper Yang-Mills on R , × S . The supersymmetric multiplets contain the gauge field plusa complex scalar ϕ and the set of adjoint fermions η, ψ, χ . We parameterize the complexsurface S using the local coordinates ( z , z ). Then the supermultiplets are: A = A µ d x µ + A m d z m + A ¯ m d¯ z m , ϕ = ϕ d z ∧ d z (1)and additionally, ψ a = ψ a ¯1 d¯ z + ψ a ¯2 d¯ z , χ a = χ a d z ∧ d z (2)with a = 1 , m = 1 , A µ , η ) together with the chiral multiplets ( A ¯ m , ψ ¯ m ) and ( ϕ , χ ),plus their complex conjugates constitute the N = 1, D = 4 supersymmetric theory.Omitting the kinetic terms (we shall use the kinetic terms later on in this article), thebilinear in fermions part of the action is, I F = Z R , × S d x Tr (cid:16) χ ∧ ∂ A ψ + 2 i √ ω ∧ ∂ A η ∧ ψ + 12 ψ ∧ [ ϕ, ψ ] + √ η [ ¯ ϕ, χ ] + h . c . (cid:17) (3)with ω is the fundamental K¨ahler form of the complex surface S . The variation of η , ψ and χ , yields the equations of motion [4, 12]: ω ∧ ∂ A ψ + i φ, χ ] = 0 (4)¯ ∂ A χ − i √ ω ∧ ∂η − [ ϕ, ψ ] = 0¯ ∂ A ψ − √ ϕ, η ] = 0Before we proceed in details on how to find zero modes, we review in brief some issues,regarding the supersymmetric quantum mechanics algebra which we shall frequently usein the subsequent sections. N = 2 Supersymmetric Quantum Mechanics Algebra
Consider a quantum system, described by a Hamiltonian H and characterized by the set { H, Q , ..., Q N } , with Q i self-adjoint operators. The quantum system is called supersym-metric, if, { Q i , Q j } = Hδ i j (5)3ith i = 1 , , ...N . The Q i are the supercharges and the Hamiltonian “ H ” is called SUSYHamiltonian. The algebra (5) describes the N-extended supersymmetry with zero centralcharge. Owing to the anti-commutativity, the Hamiltonian can be written as, H = 2 Q = Q = . . . = 2 Q N = 2 N N X i =1 Q i . (6)A supersymmetric quantum system { H, Q , ..., Q N } is said to have unbroken supersym-metry, if its ground state vanishes, that is E = 0. In the case E >
0, that is, for apositive ground state energy, susy is said to be broken.In order supersymmetry is unbroken, the Hilbert space eigenstates must be annihilatedby the supercharges, Q i | ψ j i = 0 (7)for all i, j .The N = 2 algebra (“ N = 2 SUSY QM”, or “SUSY QM” thereafter) consists of twosupercharges Q and Q and a Hamiltonian H , which obey the following, { Q , Q } = 0 , H = 2 Q = 2 Q = Q + Q (8)We use the complex supercharge Q and it’s adjoint Q † defined as, Q = 1 √ Q + iQ ) Q † = 1 √ Q − iQ ) (9)which satisfy the following equations, Q = Q † = 0 (10)and also are related to the Hamiltonian as, { Q, Q † } = H (11)A very important operator that is inherent to the definition of a SUSY QM system is theWitten parity, W , which, for a N = 2 algebra, is defined as, { W, Q } = { W, Q † } = 0 , [ W, H ] = 0 (12)and satisfies, W = I (13)The main and important use of the operator W is that, by using it, we can span the Hilbertspace H of the quantum system to positive and negative Witten parity spaces, defined as, H ± = P ± H = {| ψ i : W | ψ i = ±| ψ i} . Therefore, the quantum system Hilbert space H isdecomposed into the eigenspaces of W , hence H = H + ⊕ H − . Since each operator actingon the vectors of H can be represented by 2 N × N matrices, we use the representation: W = (cid:18) I − I (cid:19) (14)4ith I the N × N identity matrix. Recalling that Q = 0 and { Q, W } = 0, the superchargestake the form, Q = (cid:18) A (cid:19) , Q † = (cid:18) A † (cid:19) (15)which imply, Q = 1 √ (cid:18) AA † (cid:19) , Q = i √ (cid:18) − AA † (cid:19) (16)The N × N matrices A and A † , are generalized annihilation and creation operators with A acting as A : H − → H + and A † as, A † : H + → H − . In the representation (14), (15),(16) the quantum mechanical Hamiltonian H , can be cast in a diagonal form, H = (cid:18) AA † A † A (cid:19) (17)We denote n ± the number of zero modes of H ± . The Witten index for Fredholm operatorsis defined as, ∆ = n − − n + (18)When the Witten index is non-zero integer, supersymmetry is unbroken and in the casethe Witten index is zero, if n + = n − = 0 supersymmetry is broken, while if n + = n − = 0supersymmetry is unbroken.The Fredholm index of the operator A and the Witten index are related as,∆ = ind A = dim ker A − dim ker A † = (19)dim ker A † A − dim ker AA † = dim ker H − − dim ker H + We shall consider only Fredholm operators.
The localized fields on each matter curve on S are related to a SUSY QM algebra, as shownin [89]. In order to make this article self contained we review the basic facts (for detailssee [89]). The localized fermion fields exist on a matter curve Σ which is the intersectionof the complex surfaces S and S ′ . In order to preserve N = 1 supersymmetry in D = 4,the theory defined on R , × Σ must be D = 6 twisted super Yang-Mills [4, 12].Solving the D = 8 equations of motion for the twisted fermions we find how localizedfermion matter on Σ results from zero modes of the D = 8 bulk theory. Consider threematter curves denoted as Σ i , with i = 1 , ,
3. Each matter curve has a group G i , that onthe intersection point further enhances to a higher group G p . A non-trivial backgroundfor the adjoint scalar is required in order to extract the localized fermionic solutions ofthe eight dimensional theory on S [4, 12], which is equal to [4, 12]: h ϕ i = m z Q + m z Q (20)5n the above, Q and Q are the U (1) generators that are included in the enhancementgroup G p at the intersection point, and “ m ” and “ m ” are mass scales related to theF-theory scale M ∗ . Taking m = m = m will simplify things but will not change theresults.The three matter curves can intersect at a point which is ( z , z ) = (0 , G p singularity at the intersection point. Thethree different curves Σ , Σ , Σ are defined by the loci z = 0, z = 0 and z + z = 0respectively. Note that each curve represents a fermion under the U (1) charges, the curvescan be classified according to the table, matter curve ( q , q ) surface locus Σ ( q , z = 0Σ (0 , q ) z = 0Σ ( − q , − q ) z + z = 0 Table 1: Charge Classification of the three matter curves
We assume that the K¨ahler form of S is the canonical form, ω = i z ∧ d¯ z + d z ∧ d¯ z ) (21) The coordinates z and z that parameterize S , describe the intersection Σ in transverseand tangent directions respectively. With ω as in (21) and neglecting the z derivatives,the equations of motion can be written as [4, 12]: √ ∂ η − m z q ψ ¯2 = 0 ∂ ψ ¯1 − m ¯ z q χ = 0 (22) ∂ ψ ¯2 − √ m z q η = 0 ¯ ∂ χ − m z q ψ ¯1 = 0where ( q , q ) are the U (1) charges of the fermions belonging to an irreducible represen-tation ( R, q , q ) of G S × U (1) × U (1) (note that Q is the U (1) generator and Q isthe U (1) generator). Taking the adjoint vacuum expectation value (20) the equations ofmotion can be cast as: ∂ ψ ¯2 + ∂ ψ ¯1 − m (¯ z q + ¯ z q ) χ = 0 (23)¯ ∂ χ − m ( z q + z q ) ψ ¯1 = 0¯ ∂ χ − m ( z q + z q ) ψ ¯2 = 0 z = 0 The curve Σ , corresponds to q = 0. The fermions localized at z = 0 are obtained by(23) and are equal to [12]: ψ ¯2 = 0 , χ = f ( z ) e − q m | z | , ψ ¯1 = − χ. (24)6ith f ( z ) a z -dependent holomorphic function. We can connect a N = 2 SUSY QMalgebra to this matter curve. Indeed, we can define the matrix D and also D † as follows, D = (cid:18) ∂ − m ¯ z q − m z q ¯ ∂ (cid:19) (25)and, D † = (cid:18) ¯ ∂ − m ¯ z q − m z q ∂ (cid:19) (26)acting on, (cid:18) ψ ¯1 χ (cid:19) (27)The solutions of the equations of motion (23) with ψ ¯1 and χ ¯1 the zero modes of D . TheFredholm index I D , of the operator D , is equal to,indI D = dim ker( D † ) − dim ker( D ) (28)which is equal to the number of zero modes of D minus the number of zero modes of D † .Using D we can define the N = 2 supersymmetric quantum mechanical system by definingthe supercharges Q and Q † , Q = (cid:18) D (cid:19) Q † = (cid:18) D † (cid:19) (29)Also the Hamiltonian of the system can be written, H = (cid:18) D D † D † D (cid:19) (30)The above matrices obey, { Q, Q † } = H , Q = 0, Q † = 0. Like so, the Witten index ofthe N = 2 supersymmetric quantum mechanics system, is related to the index I D of theoperator D . Indeed we have I D = − ∆, because, I D = dim ker D † − dim ker D = dim ker D D † − dim ker D † D = − ind D = − ∆ = n − − n + (31)with n − and n + defined in the previous section. Accordingly, the zero modes of theoperators D and D † are related to the zero modes of the operators D D † and D † D .Additionally, the zero modes of the operators D D † and D † D can be classified to paritypositive and parity negative solutions according to their Witten parity.Note that the SUSY QM structure exists if ψ ¯2 = 0 on this matter curve. Moreover, SUSYis unbroken, since I D = 0 (the operator D † has no localized zero modes).7 .2 Localized fermion around z = 0 Along the curve Σ , we have q = 0 and the fermions are peaked around z = 0. Thelocalized solutions to the equations of motion (23) read: ψ ¯2 = − χ, χ = g ( z ) e − q m | z | , ψ ¯1 = 0 . (32)with g ( z ) an arbitrary holomorphic function of z . The N = 2 SUSY QM algebra can bedefined in terms of the D matrix, which is equal to: D = (cid:18) ∂ − m ¯ z q − m z q ¯ ∂ (cid:19) (33)acting on (cid:18) ψ ¯2 χ (cid:19) (34) z + z = 0 The matter curve Σ , corresponds to generic charges q and q . Performing the transfor-mations: w = z + z , ψ ¯ w = 12 ( ψ ¯1 + ψ ¯2 ) (35) u = z − z , ψ ¯ u = 12 ( ψ ¯1 − ψ ¯2 )the equations of motion (23) can be written:2 ∂ w ψ ¯ w + 2 ∂ u ψ ¯ u − m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) χ = 0 (36)2 ¯ ∂ ¯ w χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ w = 02 ¯ ∂ ¯ u χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ u = 0When ψ ¯ u = 0, an N = 2 SUSY QM algebra underlies the fermion system, defined in termsof the matrices D and D † as: D = ∂ w − m (cid:16) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:17) − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ∂ w (37)and, D † = ∂ w − m (cid:16) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:17) − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ∂ w (38)8cting on, (cid:18) ψ ¯ w χ w (cid:19) (39)Then, the fermionic localized solutions to the new equations of motion (36) around z + z =0 are: ψ ¯ w = 1 √ χ, χ w = g ( u ) e − q m √ | w | , ψ ¯ u = 0 . (40)We therefore conclude that each matter curve corresponds to an underlying N = 2 SUSYQM algebra. In turn, each SUSY algebra can be constructed using the operators D , D and D respectively, the zero modes of which correspond to the solutions of (23). In the previous section we chose the canonical form for the metric that describes S . How-ever the surface S is more like a base space of the Calabi-Yau threefold and not a divi-sor [24]. Therefore there is no way to know what metric describes precisely the base space S , hence there is some freedom in the choice of the metric on S . The metric adopted inthe previous section is the simplest case and describes perfectly the case for which thesystem is fully described by an Super Yang-Mills theory, and gravity is decoupled, as wepreviously noted. However we are free to choose another metric that incorporates thegravitational backreaction of the surface S on the system. Note that the volume of S gives the gauge coupling of the effective four-dimensional GUT [92]. In this section weshall put the previous section’s index problem, into a different context, by perturbing themetric of the complex surface S in the following way:d s = (cid:0) ǫf ( z ) (cid:1) d z ⊗ d¯ z + (cid:0) ǫf ( z ) (cid:1) d z ⊗ d¯ z (41)Using the above metric, the K¨ahler form is written as follows, ω = i (cid:0) ǫf ( z ) (cid:1) d z ∧ d¯ z + i (cid:0) ǫf ( z ) (cid:1) d z ∧ d¯ z (42)The corresponding equations of motion for the fermionic fields are: (cid:0) ǫf ( z ) (cid:1) ∂ ψ ¯2 + (cid:0) ǫf ( z ) (cid:1) ∂ ψ ¯1 − m (¯ z q + ¯ z q ) χ = 0 (43)¯ ∂ χ − m ( z q + z q ) ψ ¯1 = 0¯ ∂ χ − m ( z q + z q ) ψ ¯2 = 0By looking at the equations of motion (43), we can generally say that the form of thelocalized solutions along each matter curve will have a more evolved dependence on allthe local coordinates that parameterize the complex surface S . By looking equation (41)we can see that the functions f , f have a holomorphic dependence on their coordinates.There is a particular reason for using holomorphic functions, which is the fact that thesolutions of the equations of motions (wave functions) are the sections of holomorphic9ine bundles along the loci z = 0, z = 0 and z + z = 0 [24]. In this section weshall study if the holomorphic linear perturbation of the metric (41) modifies the spectralproblem of the operator corresponding to each matter curve. However we shall not beinterested in the particular form of the localized wave functions that solve the equationsof motion. Additionally, due to the lack of knowledge of the global geometry that describesthe compact complex threefold (also since the local geometry around the singularity affectsthe Standard Model physics), and in order to avoid theoretical inconsistencies, we assumethat the functions f and f are decreasing functions of their arguments. z = 0Let us start with the matter curve z = 0, which means that q = 0. By using theholomorphic perturbation of the metric (41) we can see that, the whole problem is aperturbation of the one that corresponds to the canonical metric. Indeed, as can be easilychecked, localized solutions can exist if ψ ¯2 = 0 (the situation is similar to the un-perturbedcase). Then, by setting q = 0, the equations of motion corresponding to the matter curve z = 0, are: (cid:0) ǫf ( z , ¯ z ) (cid:1) ∂ ψ ¯1 − m ¯ z q χ = 0 (44)¯ ∂ χ − m z q ψ ¯1 = 0which can be recast as, ∂ ψ ¯1 − m ¯ z q (cid:0) ǫf ( z , ¯ z ) (cid:1) χ = 0 (45)¯ ∂ χ − m z q ψ ¯1 = 0Performing a perturbation expansion and keeping terms linear to the expansion parameter ǫ we obtain: ∂ ψ ¯1 − m ¯ z q (cid:0) − ǫf ( z , ¯ z ) (cid:1) χ = 0 (46)¯ ∂ χ − m z q ψ ¯1 = 0Clearly, the zero modes of the above equation (54) correspond to the zero modes of thematrix: D ǫ = (cid:18) ∂ − m ¯ z q (cid:0) − ǫf ( z , ¯ z ) (cid:1) − m z q ¯ ∂ (cid:19) (47)We can write D ǫ = D + C , with D as in equation (25) and C being the matrix: C = (cid:18) m ¯ z q ǫf ( z , ¯ z )0 0 (cid:19) (48)There exists a theorem in the mathematical literature that guarantees invariance of theindex of Fredholm operators under odd perturbations of Fredholm type [87, 88, 91]. Par-ticularly the theorem states: 10 Let Q be a Fredholm operator and C be an odd operator. Then, Q + C is a Fredholmoperator then the indices of the two operators are equal, i.e. :ind( D + C ) = ind D ǫ (49)We must note that an odd operator is defined as a matrix that anti-commutes with theWitten operator, W , that is { W, C } = 0. Using the notation we introduced in section 3,the matrix W is equal to: W = (cid:18) − (cid:19) (50)It can be easily seen that the matrix C , defined in equation (62) is odd (using the termi-nology of the theorem), since it anti-commutes with W . Therefore the indices of the twomatrices D + C and D are equal, that is,ind( D ǫ + C ) = ind D ǫ (51)As a consequence of the aforementioned results, the Witten index of the composite operator D + C is equal to the Witten index of the operator D . A direct implication of the equalityof the two indices is that the spectral problem of the two operators is the same. This doesnot necessarily imply that the zero modes of D is equal to the zero modes of the operator D + C , but it certainly implies that the net number of the zero modes corresponding tothe operators and their adjoint are equal. This is of particular importance since it givesus the opportunity to study more evolved cases and investigate more difficult aspects ofthese problems, such as the spectral asymmetry of the operators.The above result does not change if we include higher orders of ǫ in the matrix C . Indeed,the matrix C would then be: C = (cid:18) m ¯ z q ǫf ( z , ¯ z ) − m ¯ z q ǫ f ( z , ¯ z ) + ... (cid:19) (52)which still satisfies the theorem above.Note that the situation we studied in this section can be much more difficult in the casea background flux is turned on. In that case, the restrictions on K¨ahler form are morestringent, since the K¨ahler form must satisfy the D-term equation: i [ φ, ¯ φ ] + 2 ω ∧ F , + ∗ sD = 0 (53)where in the above F , stands for the flux. z = 0In the case of the z = 0 matter curve, we have q = 0. As a result of the holomorphicityof the function f ( z ), in order to solve the equations of motion, we must set ψ ¯1 = 0 justin the non-perturbed case. Then, the equations of motion are written, ∂ ψ ¯2 − m ¯ z q (cid:0) − ǫf ( z , ¯ z ) (cid:1) χ = 0 (54)¯ ∂ χ − m z q ψ ¯2 = 011s in the z = 0 case, we can write D ǫ = D + C , with D ǫ being, D ǫ = (cid:18) ∂ − m ¯ z q (cid:0) − ǫf ( z , ¯ z ) (cid:1) − m z q ¯ ∂ (cid:19) (55)and D as in equation (33). In this case the matrix C is equal to: C = (cid:18) m ¯ z q ǫf ( z , ¯ z )0 0 (cid:19) (56)Both the matrices C and D satisfy the requirements of the theorem we used previously,therefore we also have in this case:ind( D + C ) = ind D ǫ (57) z + z = 0The case z + z = 0 is much more evolved than the previous two cases. Using thetransformations (35), equation (43) can be cast as:(2 + ǫf + ǫf ) ∂ w ψ ¯ w + (2 + ǫf + ǫf ) ∂ u ψ ¯ u (58)+ ( ǫf − ǫf )( ∂ w ψ u + ∂ u ψ w ) − m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) χ = 02 ¯ ∂ ¯ w χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ w = 02 ¯ ∂ ¯ u χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ u = 0In this case the theorem we presented previously does not find application, since thecomplex derivatives are interrelated. The only case that the theorem can find application iswhen f = f . Nevertheless, the last case corresponds to a trivial (coordinate independent)deformation of the metric, thus it is a perturbative constant shift. In the same way as inthe un-perturbed z + z = 0 case, when ψ ¯ u = 0 and f = f = f , the above equation canbe cast as: (2 + 2 ǫf ) ∂ w ψ ¯ w − m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) χ = 0 (59)2 ¯ ∂ ¯ w χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ w = 0Following the same steps as previously, we obtain:2 ∂ w ψ ¯ w − (1 − ǫf ) m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) χ = 0 (60)2 ¯ ∂ ¯ w χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ψ ¯ w = 0The zero modes of the above equation are the zero modes of the matrix: D wǫ = (cid:18) ∂ w − m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) (1 − ǫf ) − m z q ¯ ∂ (cid:19) (61)12ikewise, we can write D wǫ = D w + C w , with D w as in equation (37) and C w : C w = (cid:18) m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) ǫf (cid:19) (62)Therefore applying the theorem for the two matrices, we have:ind( D w + C ) = ind D wǫ (63)Hence, the indices of the two operators are equal.The results of this section are very important since, in virtue of the theorem, the netnumber of the zero modes of the metric-perturbed fermionic system is equal to the netnumber of the zero modes that the Euclidean metric-fermionic system has. Neverthelesswe know that the solutions exist, but this theorem tells us nothing on how these perturbedsolutions behave. Before we close this section, we must note that in the case we performa non-holomorphic perturbation of the Euclidean metric, the solutions of the equation ofmotion are not the ones that appeared in this section. Indeed, let us take for examplethe matter curve z = 0, for which a non holomorphic perturbation of the metric wouldresult to three wave functions-solutions to the equation of motion, namely χ , ψ ¯1 and ψ ¯2 .The solutions ψ ¯1 and ψ ¯2 are given as functions of χ , which in turn is a perturbation of thegaussian profile solution. For a specific example of this type, see for example reference [92]. The situation of the fermionic system without background gauge fluxes is very useful butwe can get only one non-trivial Yukawa coupling [12]. In order to obtain the hierarchiesof the quark masses and the appropriate mixing of the quark and lepton matter fields, thewave functions we found in section 3 must be appropriately distorted [12]. This distortioncan be caused by the appearance of background gauge fields. It is proven that whenthe gauge fluxes are field dependent, then reasonable agreement with the observed masshierarchies and mixings can be achieved [12]. In this section we shall add non-trivialbackground gauge fluxes and study whether the resulting localized fields on each mattercurve on S are related to an N = 2 SUSY QM algebra. We shall follow reference [12].Trying to find localized solutions along the matter curve, when the gauge fields have alocal coordinate dependence can be quite difficult. We shall confine ourselves to the casewhere the gauge fields are constant and independent from the coordinates z , z .In general, the total flux can be written as follows [12]: F = F Q + F (1) Q + F (2) Q (64)In the above equation, F is the total flux, F is the U (1) bulk gauge flux, with generator Q , and F (1) , F (2) are the fluxes along the matter curves z and z respectively (with13enerators Q and Q as we saw in section 3). The corresponding gauge potentials are A , A and A (1) , A (2) , respectively, with, A = qA + q A (1) + q A (2) (65)In the above, q stands for the total U (1) bulk charge, q is the U (1) charge along thematter curve z and the and q is the U (1) charge along the matter curve z . The bulkflux breaks the initial G s gauge symmetry to Γ s × U (1), and the fermions transform to arepresentation R which a direct sum of irreducible representations labelled as ( q, q , q ).In the general case, and if we consider only diagonal components of the gauge flux, thebulk flux can be written [12]: F = F d z ∧ d¯ z + F d z ∧ d¯ z (66)and the U (1)’s along the matter curves are taken to be: F (1) = F (1)2¯2 d z ∧ d¯ z , F (2) = F (2)1¯1 d z ∧ d¯ z (67)Hence, if the adjoint vacuum expectation value h φ i is the same as in equation (20), theequations of motion for the charged fermionic fields are [12]:( ∂ − iA ) ψ ¯2 + ( ∂ − iA ) ψ ¯1 − m (¯ z q + ¯ z q ) χ = 0 (68)( ∂ − iA ¯1 ) χ − m ( z q + z q ) ψ ¯1 = 0( ∂ − iA ¯2 ) χ − m ( z q + z q ) ψ ¯2 = 0In the constant gauge flux case, we take: F = 2 iM d z ∧ d¯ z + 2 iN d z ∧ d¯ z (69)with M , N , real constants. The fluxes along the matter curves are then equal to: F (1) = 2 iN (1) d z ∧ d¯ z , F (2) = 2 iM (2) d z ∧ d¯ z (70)where N (1) and M (2) real constants. Therefore, the gauge potentials are equal to: A = iM ( z d¯ z − ¯ z d z ) + iN ( z d¯ z − ¯ z d z ) (71) A (1) = iN (1) ( z d¯ z − ¯ z d z ) A (2) = iM (2) ( z d¯ z − ¯ z d z )Consequently, the total gauge potential is equal to: A = i ( qM + q M (2) )( z d¯ z − ¯ z d z ) + i ( qN + q N (1) )( z d¯ z − ¯ z d z ) (72)Performing a suitable gauge transformation of the form, A = b A + dΩ (73)14e can set A ¯1 =0 and A ¯2 =0 in equation (68) and work with the hatted fields. Indeed,equation (68) can simplified to:( ∂ − i b A ) b ψ ¯2 + ( ∂ − i b A ) b ψ ¯1 − m (¯ z q + ¯ z q ) b χ = 0 (74)¯ ∂ b χ − m ( z q + z q ) b ψ ¯1 = 0¯ ∂ b χ − m ( z q + z q ) b ψ ¯2 = 0with χ = e i Ω b χ , ψ ¯1 = e i Ω b ψ ¯1 and ψ ¯2 = e i Ω b ψ ¯2 . Supposing that the gauge field is coordinateindependent, the total gauge potential reads: b A = − iM ¯ z d z − iN ¯ z d z (75)and b A = − iM ¯ z and b A = − iN ¯ z . The gauge parameter Ω is in this case:Ω = i ( M | z | + N | z | ) (76)Working in the gauge we chose above makes the calculation of the wave functions (andhence of the corresponding gauge invariant properties such as Yukawa couplings) simpler[12]. This gauge is referred to as holomorphic gauge [12]. z = 0 Let us study here the first matter curve z = 0. By taking q = 0, the localized solutionsin this case are [12]: b ψ ¯1 = − λ q m b χ, b χ = g ( z ) e − λ m | z | , b ψ ¯2 = 0 . (77)with λ equal to: λ = − M + q m s M q m (78)The above solutions correspond to the following equations of motion:( ∂ − i b A ) b ψ ¯1 − m ¯ z q b χ = 0 (79)¯ ∂ b χ − m z q b ψ ¯1 = 0It is very easy to prove that we can associate an N = 2 SUSY QM algebra correspondingto the equations of motion (79). Indeed, following the steps of section 3 we define thematrix D A and also D † A as follows, D A = (cid:18) ∂ − i b A − m ¯ z q − m z q ¯ ∂ (cid:19) (80)and, D † A = ¯ ∂ + i b ¯ A − m z q − m ¯ z q ∂ ! (81)15cting on, (cid:18) b ψ ¯1 b χ (cid:19) (82)In the above b A = − iM ¯ z . The matrix D † A has no zero modes, while the matrix D A has solutions the functions of equation (77). Therefore the Fredholm index I D A , of theoperator D A , is equal to,indI D A = dim ker( D † A ) − dim ker( D A ) (83)for which clearly indI D A = 0. From the last we conclude that SUSY is unbroken.The N = 2 supersymmetric quantum mechanical system can be defined using the super-charges Q A and Q † A , Q = (cid:18) D A (cid:19) , Q † = (cid:18) D † A (cid:19) (84)Furthermore, the Hamiltonian can be written as: H = (cid:18) D A D † A D † A D A (cid:19) (85)Finally, the Witten index of the SUSY QM algebra, is I D A = − ∆.We can see that the constant background gauge fluxes do not spoil the SUSY QMalgebra that underlies the fermionic system of the flux-less case. The algebra itself isof-course different but still SUSY is unbroken. z = 0 In the case of the z = 0 matter curve, the equations of motion are (for q = 0):( ∂ − iA ) b ψ ¯2 − m ¯ z q b χ = 0 (86)¯ ∂ b χ − m z q b ψ ¯2 = 0with A = − iN ¯ z . The localized solutions are: b ψ ¯2 = − λ q m b χ, b χ = g ( z ) e − λ m | z | , b ψ ¯1 = 0 . (87)with λ equal to: λ = − N + q m s N q m (88)The N = 2 SUSY QM algebra is built on the matrices: D A = (cid:18) ∂ − iA − m ¯ z q − m z q ¯ ∂ (cid:19) (89)16nd, D † A = (cid:18) ¯ ∂ + i ¯ A − m z q − m ¯ z q ∂ (cid:19) (90)acting on, (cid:18) b ψ ¯2 b χ (cid:19) (91)We shall not pursuit this case further, since it is identical with the previous z = 0. Theresult is that a N = 2 unbroken SUSY QM algebra underlies the system.As for the z + z = 0 case, it is much more difficult to handle, compared to the othertwo cases. Performing the transformation (35), the equations of motion (79) can be castin the form:2 ∂ w b ψ ¯ w + 2 ∂ u b ψ ¯ u + (cid:0) − N ( ¯ w − ¯ u ) − M ( ¯ w + ¯ u ) (cid:1) b ψ ¯ w (92)+ (cid:0) N ( ¯ w − ¯ u ) − M ( ¯ w + ¯ u ) (cid:1) b ψ ¯ u − m (cid:0) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:1) b χ = 02 ¯ ∂ ¯ w b χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) b ψ ¯ w = 02 ¯ ∂ ¯ u b χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) b ψ ¯ u = 0It is not easy to relate the above fermionic system to an N = 2 SUSY QM algebra.Perhaps central charges must be included to this N = 2 algebra. Such a behavior kindof surprised us, because we expected all the localized fermion solutions to have the same,central charge free, N = 2 SUSY QM algebra. It seems that this is not the case. We shallnot pursuit this issues further. N = 2 SUSY QM algebra.
In the previous we found that when a matter curve has localized zero modes, we can builta N = 2 SUSY QM algebra from the system. In most cases localization occurs when, oneof the fields that exist on the D7 brane intersection vanishes. By looking the equations ofmotion (36), it is natural to make the equations of motion look like the following, (cid:0) ∂ w + 2 ∂ u (cid:1)(cid:0) ψ ¯ w + ψ ¯ u (cid:1) − m (cid:16) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:17) χ = 0 (93) (cid:0) ∂ ¯ w + 2 ¯ ∂ ¯ u (cid:1) χ − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17)(cid:0) ψ ¯ w + ψ ¯ u (cid:1) = 0This is clarified by looking the χ derivative, which is 2 ¯ ∂ ¯ w + 2 ¯ ∂ ¯ u . In the equations ofmotion (93), the derivative that acts on χ is the conjugate derivative of the one that actson ψ ¯ w + ψ ¯ u . Let us see when this is possible and what would be the implications of thisconstruction. Recall the SUSY QM algebras for the matter curves z = 0 and z = 0 N = 2 SUSY QM algebra can be built based on (93), by using the matrix: D = ∂ w + 2 ∂ u − m (cid:16) ¯ w ( q + q ) + ¯ u ( q − q ) (cid:17) − m (cid:16) w ( q + q ) + u ( q − q ) (cid:17) ∂ w + 2 ¯ ∂ u (94)acting on : (cid:18) ψ ¯ w + ψ ¯ u χ ¯ w (cid:19) (95)It is obvious that, by using (94), we can construct the matrix D † and the rest of the SUSYalgebra, such as the Hamiltonian and so on.In order the equations of motion (36) to be identical to (93) the following conditionmust be imposed on the fields ψ ¯ u and ψ ¯ w , ∂ w ψ ¯ w = ∂ u ψ ¯ u (96)The implications of the above condition to the case of the three matter curves are quiteinteresting. For the Σ ( z = 0) matter curve, since ψ ¯2 = 0, relation (96) would imply ∂ ψ ¯1 = 0. The curve Σ ( z = 0) has localized solutions when ψ ¯1 = 0, and in conjunctionwith (96) we get the condition ∂ ψ ¯2 = 0. In the same vain, one has for the Σ ( z + z = 0)curve ∂ w ψ ¯ w = 0. The conditions ∂ ψ ¯2 = 0 and ∂ ψ ¯1 = 0 imply that ψ is a function onlyof z (thus has no z dependence) and ψ is a function only of z . This in turn wouldimply that the functions f ( z ) and g ( z ) defined in relations (24), (32) and (40) areconstant functions, that is, f ( z ) = c and g ( z ) = c , with c and c arbitrary constants.Furthermore, the condition ∂ w ψ ¯ w = 0, implies that ψ ¯ w is a constant function, say ψ ¯ w = c .We summarize: Matter Curve z = 0 → f ( z ) = c (97) ∂ w ψ ¯ w = ∂ u ψ ¯ u ⇒ Matter Curve z = 0 → g ( z ) = c Matter Curve z + z = 0 → ψ ¯ w = c The three conditions we just presented, are very much related to the calculation of Yukawacouplings, when we have constant Higgs wave function and absence of non-constant fluxes[12].As we saw earlier, the Yukawa coupling, in terms of the three matter wave functionsreads: Y = M ∗ Z S d z d z ψ ψ φ (98)In the presence of background fluxes, the Yukawa coupling is given by the overlappingintegral [81], Y ij ∼ Z S z − i z − j e M k ¯ l z k ¯ z l ˙ f ( z , z ) (99)with f ( z , z ) containing the gaussian profiles of the localized fermions along the mattercurves. When the M i,j is constant or zero, there is a U (1) × U (1) symmetry, under whichthe coordinates are invariant, z → e ia z z → e ia z (100)18n that case, all Yukawas other than the Y , vanish. These gauge symmetries are brokenwhen M i,j has a non trivial gauge dependence, which happens when background fluxes areturned on. This case is particularly interesting, since in this way a hierarchical fermionYukawa matrix is obtained but we shall not pursuit these issues further.In the absence of fluxes, the fermionic matter functions are equal to: ψ = f ( z ) e − q m | z | (101) ψ = g ( z ) e − q m | z | In the special case that φ = const, the wave function of the Higgs field is unlocalized,which means that the Higgs field lives in the bulk rather than localized on a matter curve.In the absence of non-constant fluxes, we have f ( z ) = g ( z ) = 1 and it can be provedthat in this case, the only non-vanishing Yukawa is the Y . This coupling gives mass tothe heaviest lepton and quark generations, and is equal to, Y ∼ M ∗ Z S d z d z e − q m | z | e − q m | z | . (102)Consequently, we see that the conditions (97), imposed by the N = 2 SUSY QM of thesystem described by the equations of motion (93), are the same with the conditions thatthe matter curves wave functions satisfy, in order to built Yukawa couplings in the absenceof non-constant fluxes with non-localized Higgs. Indeed the two cases are identical when c = c = 1. This type of Yukawa couplings is usually found in type IIB and F-theorycompactified on non del-Pezzo surfaces [12].Before closing this section we discuss an important issue. Spacetime supersymmetry andsupersymmetric quantum mechanics are not the same, nevertheless the connection is pro-found, since extended (with N = 4 , ... ) supersymmetric quantum mechanics models de-scribe the dimensional reduction to one (temporal) dimension of N = 2 and N = 1Super-Yang Mills models [90]. A serious question rises at this point. By looking the N = 2 SUSY QM algebra supercharges (94), one could thing that it is intended to embedany intersection curve in the same sort of N = 2 SUSY QM algebra. In general, a sortof N = 2 supersymmetry is rather unexpected, since the intersection of three D7-branes,breaks four-dimensional supersymmetry down to N = 1, and so the N = 2 structurementioned above is lost. However, this is not true, since supersymmetry and supersym-metric quantum mechanics is not the same. Indeed, the N = 2 SUSY QM superchargesdo not generate spacetime supersymmetry. By the same token, the supersymmetry insupersymmetric quantum mechanics, does not relate fermions and bosons. The SUSYQM supercharges do not generate transformations between fermions and bosons. Thesesupercharges generate transformations between two orthogonal eigenstates of a Hamilto-nian, eigenstates that are classified according to their Witten parity. Hence, this flow ofthe N = 2 breaking argument is not true, due to the non-spacetime structure of SUSYQM. 19 Global U (1) Symmetries Along Matter Curves, YukawaCouplings, Proton Decay Operators and Neutrino MassOperators
The N = 2 supersymmetric quantum mechanics algebra is invariant under an R-symmetry.Likewise, the Hamiltonian is also invariant under this symmetry [85]. Actually, the super-algebra (5) and (6) is invariant under the transformation, (cid:18) Q ′ Q ′ (cid:19) = (cid:18) cos a sin a − sin a cos a (cid:19) · (cid:18) Q Q (cid:19) (103)with a an arbitrary constant. Furthermore, the complex supercharges Q and Q † aretransformed under a global U (1) transformation: Q ′ = e ia Q, Q ′ † = e − ia Q † (104)This R-symmetry is also a symmetry of the Hilbert states corresponding to the subspaces H + and H − . Thus, the eigenfunctions of H + = A A † and H − = A † A , are invariant underthis U (1)-symmetry, namely, | ψ ′ + i = e iβ + | ψ + i , | ψ ′ − i = e iβ − | ψ − i (105)It is clear that the parameters β + and β − are global parameters with β + = β − . Consistencywith relation (104) requires that a = β + − β − . For our purposes we shall use only thesymmetry | ψ ′ + i = e iβ + | ψ + i . The implications of this symmetry are quite interesting,since this implies that the localized fields on each matter curve are invariant under thissymmetry. Let us see this for the z = 0 matter curve. Due to this U (1) symmetry, the Q and Q † supercharges of equation (29) are invariant under the transformation (104).Consequently the eigenfunctions of D , (cid:18) ψ ¯1 χ (cid:19) (106)are invariant under the following transformation, (cid:18) ψ ¯1 χ (cid:19) ′ = e ib + (cid:18) ψ ¯1 χ (cid:19) (107)In the same way, the localized fields on the second fermion matter curve z , are invariantunder, (cid:18) ψ ¯2 χ (cid:19) = e ib (cid:18) ψ ¯2 χ (cid:19) ′ (108)Finally, the localized fields on the Higgs matter curve z + z = 0, are invariant under, (cid:18) ψ ¯ w χ w (cid:19) ′ = e ib (cid:18) ψ ¯ w χ w (cid:19) (109)Note that the fields on each matter curve have the same transformation properties. Forconvenience, we gather the results in the following table,20 (1) matter curve z = 0 matter curve z = 0 matter curve z + z = 0 e ib + χ and ψ ¯1 e ib χ and ψ ¯2 e ib χ w and ψ ¯ w Table 2: U (1) -Classification of the Localized Fields on the Three MatterCurves The localized fields on each matter curve are invariant under this U (1) symmetry presentedabove. However, certain conditions must hold. Indeed, if this symmetry is an actualsymmetry of the localized fields, then the action (plus kinetic terms) for the localizedfields must be invariant under this U (1) symmetry. Let us examine for example thematter curve z = 0. The localized fields action reads (recall η = ψ = 0), I L = Z R , × S d x Tr (cid:16) χ ∧ ∂ A ψ ¯1 + 12 ψ ¯1 ∧ [ ϕ, ψ ¯1 ] + h . c . (cid:17) (110)In addition the kinetic terms are of the form, Z R , × S ψ † ¯1 ψ ¯1 d x , Z R , × S χ † χ d x (111)It is obvious that under the U (1) transformation, χ ′ = e ib + χ , ψ ′ ¯1 = e ib + ψ ¯1 (112)the kinetic terms (111) are invariant. Still, the action (110) cannot be invariant unless, e ib + = 1 (113)Also, we suppose that the field φ is not affected by the U (1)-symmetry. The condition(113) implies that b + = πn , with n = 0 , , , ... But the fields χ and φ belong to the same susy multiplet, thus we would expect that this U (1)-symmetry should be a symmetry of the whole action. It turns out that in order thelocalized fields have this U (1)-symmetry, the φ field must not transform under this symme-try. The implications of this symmetry are quite interesting, at least phenomenologically,as we shall see. Let us recall how Yukawas are constructed within F-theory [4, 9, 12, 14, 16, 24, 51].In F-theory, Yukawa couplings are considered to be overlapping integrals of the threematter curves wave functions over S . The matter curves are the two fermionic and theone corresponding to the Higgs. Let the wave functions that describe each fermionic mattercurve be, ψ , ψ describing Σ and Σ respectively. Owing to the N = 1 supersymmetryof the four dimensional theory, the wave function of the Higgs curve, φ , is the same with21he function ψ w , corresponding to the z + z = 0 curve, as we saw earlier. Then, in termsof the three wave functions, the Yukawa coupling reads: Y = M ∗ Z S d z d z ψ ψ φ (114)The Yukawa couplings give masses to fermions, therefore these couplings are most wel-come in the theoretical setup of the local model. As we saw, each localized fermion fieldcorresponding to the two matter curves z and z , obeys a SUSY QM U (1)-symmetry,different for each matter curve (see table 2). Yukawa couplings describe couplings be-tween a quark and a righthanded quark or between a lepton and a righthanded lepton.We denote the lepton fields with the field operators L = ( N, E ) and also with E c theright handed one. Additionally, the quark fields are represented by Q = ( U, D ) and theirrighthanded counterparts, U c , D c . The Yukawa’s stem from the superpotential and are ofthe form [12, 16], W Y uk = Y U QU c H u + Y D QD c H d + Y L LE c H d (115)We require the wave functions ψ and ψ in equation (114) to describe a lepton field andit’s righthanded field, or a quark field and it’s righthanded field, respectively. This meansthat leptons and quarks must be assigned to different matter curves. This situation cannotbe true in all local geometrical GUT setups, like in the case of SU (5), but can be true insome cases, like in the flipped SU (5) [49] construction. Let the transformations of ψ and ψ be that of table 2, that is, ψ ′ = e ib + ψ , ψ ′ = e ib ψ (116)Due to equation (113), the parameter b + is equal to b + = πn , with n = 0 , , , ... andsimilarly, b = πm , with m = 0 , , , ... In order the Yukawa coupling to be invariant under this combined action of the U (1)’s,we easily find that the parameters b + and b must be related as follows, b + = − b (117)On that account, we conclude that fermions belonging to a quark or lepton family andtheir righthanded fermions (corresponding to different matter curves), must have oppositetransformation properties under the SUSY QM- U (1) symmetry, in order the Yukawa cou-plings are invariant under this symmetry. Note that this U (1)-symmetry is not a resultcoming from the local geometric features of the surface S . It comes from the SUSY QMalgebra that the localized fields obey. The outcomes of the two conditions (113) and (117),are quite interesting phenomenologically. We consider first proton decay operators. Theproton decay operators are unwanted terms coming from the action. The 4-dimensionalproton decay operators are, W a ∼ LLE c W b ∼ QD c L, W c ∼ U c D c D c (118)22oreover, the 5-dimensional proton decay operators are, W a ∼ M QQQL W b ∼ M LLH u H u , W c ∼ M U c U c D c E c (119)The SUSY QM U (1)-symmetry restricts the proton decay operators, as is obvious bylooking the constraints (113) and (117). Let us see which operators are allowed subject tothe SUSY QM U (1)-symmetry in detail. We study first the dimension-4 operators. Theoperator W a (see relation (118)) is not allowed since although the LL part is invariant(same fermions, see (113)), the E c gives a total e ia factor to the term. Likewise, W b is not invariant, since, although the QD c part is invariant (fermion and correspondingrighthanded fermion, see 117)), the leptons L have different transformation propertiesfrom the quarks. The term W c is not allowed because, although the D c D c is invariant,the U c gives an overall exponential factor to the term. Hence, the dimension-4 protondecay operators of relation (118) are not allowed in the theory, if the SUSY QM U (1)-symmetry is obeyed by the fermion fields localized on the matter curves.Let us now check the dimension-5 operators. The operator W a is not invariant underthe U (1). Indeed, although the QQ part is invariant (same fermions) the QL part is notinvariant since the first fermion is a quark and the second is a lepton. For the same rea-soning the operator W c is not invariant under the U (1) so it cannot appear in the action.On the contrary, the operator W b is invariant, and thus can affect the phenomenologicaloutcomes off the model. U (1) -Invariance The minimal SU (5) F-theory GUT predicts Dirac and Majorana neutrino masses [81].Indeed, by integrating out massive Kaluza-Klein modes, generates higher dimensionaloperators that give phenomenologically acceptable masses for neutrinos. Particularly, theMajorana mass F-term is of the form [81], Z d θ H u LH u L Λ UV (120)When the Higgs field develops a vacuum expectation value, the above term yields a Ma-jorana mass for the neutrinos. The Majorana mass term (120) is clearly invariant underthe SUSY QM U (1) symmetry because the term LL is invariant (same fermions) and theHiggs fields are not affected at all.In the Dirac scenario, the D-term, generated by integrating out massive Kaluza-Kleinmodes on the Higgs curves, Z d θ H † d LN R Λ UV (121)gives a Dirac mass to the neutrinos. The field N R describes the right-handed neutrino.The peculiarity of N R is due to that, the right handed neutrino localizes on curves normalto the GUT-seven brane [81], a fact that put’s in question the local description concept ofF-theory GUTs. Still, normal curves can form part of a consistent local model [81]. The23irac mass term (121) is not invariant under the SUSY QM U (1), since from the term H † d LN R , only the field L is transformed under the U (1). Thus we can see that only theMajorana mass terms is favored in the scenario we presented. In this article we found that the fields localized at D N = 2 SUSY QM algebra. Particularly, each matter curve corresponds toa different algebra and due to this algebra, a global U (1)-symmetry underlies the system.In view of this symmetry, the localized fields on each matter curve satisfy certain conditionswhich we classified in Table 2. Furthermore, since the Yukawa couplings are importantto GUT phenomenology, they must be invariant under this U (1). This condition, inconjunction with the table 2 transformations, classifies the fermion transformations as inthe following table, Term e ib e ib + e ia e ia + U (1) − InvariantL e ia + U (1) − InvariantLL U (1) − Invariant E c e ia Q e ib + QQ U (1) − Invariant D c e ib U c e ib L E c U (1) − InvariantQ D c U (1) − InvariantQ U c U (1) − Invariant U c U c U (1) − Invariant D c D c U (1) − Invariant
Table 3: U (1) -Classification of various terms Owing to the above transformation properties, we found restrictions on the proton decayoperators, many of which are not allowed. Moreover, this U (1) SUSY QM symmetryrestricts the neutrino mass operators. Particularly we found that only the Majorana massterms are allowed in our scenario.We must mention that there are much more elaborated and geometry inspired techniquesto restrict proton decay operators (see for example [6–9, 20, 23]), such as monodromies,but we do not discuss these here.Moreover, the requirement of an N = 2 SUSY QM on a special system leads to specificconditions on the fermion fields and (due to supersymmetry) on the Higgs boson. As wefound, these conditions are met in the construction of the Yukawa coupling that givesmass to the top quark, with a delocalized bulk Higgs.In order to obtain the correct hierarchies and mixing of the matter fields, externalnon-trivial background fluxes must be turned on. We examined if the N = 2 SUSY QM24lgebra still underlies the localized fermionic solutions. We studied the constant flux caseand we found that the SUSY algebra still holds for the two matter curves, namely z = 0and z = 0. However, for the Higgs curve, namely z + z = 0, things are different. Itseems that the algebra is not a N = 2 without central charge SUSY QM algebra. Wehope to address this problem in the future, but it kind of surprised us. The surprise isdue to the fact that the adjoint vacuum expectation value < φ > remains the same as inthe flux-less case, so we did not expect things to change so drastically.Finally we performed a holomorphic perturbation of the metric that describes the complexsurface S and we studied how the perturbation modifies the net number of the zero modesthat the un-perturbed system has. We found that, due to a theorem characteristic forFredholm operators, the operators that describe the perturbed and un-perturbed systemshave equal indices. We checked the validity of the theorem, for every matter curve andHiggs curve. Unfortunately, this theorem does not gives us information on the specificform of the wave functions. Acknowledgments
V. Oikonomou is indebted to Prof. G. Leontaris for useful discussions on F-theory GUTsand related issues. Also the author would like to thank the anonymous referee of NPBwho’s questions and comments motivated section 5 of this article.
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