Aspects of Flavour and Supersymmetry in F-theory GUTs

0.25 MB·English

by  Joseph P. Conlon

OUTP-09/25P,CPHT-RR107.1009
PreprinttypesetinJHEPstyle-HYPERVERSION
0 Aspects of Flavour and Supersymmetry in F-theory
1
0 GUTs
2

n
a
J

1
1

Joseph P. Conlon 1, Eran Palti 2
]
h
t 1: Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
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p 2: Centre de Physique Theorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France.
e
h
[ E-mail: [email protected],[email protected]

2
v
3
1 Abstract: We study the constraints of supersymmetry on flavour in recently proposed
4
models of F-theory GUTs. We relate the topologically twisted theory to the canonical
2
. presentation of eight-dimensional super Yang-Mills and provide a dictionary between the
0
1 two. We describe the constraints on Yukawa couplings implied by holomorphy of the
9
superpotentialintheeffective 4-dimensionalsupergravitytheory, includingthescalingwith
0
: αGUT. Taking D-terms into account we solve explicitly to second order for wavefunctions
v
i and Yukawas due to metric and flux perturbations and find a rank-one Yukawa matrix
X
with no subleading corrections.
r
a
Contents
1. Introduction 1
2. Local models of intersecting branes in F-theory 2
2.1 Effective theory for intersecting 7-branes 3
2.2 Zero modes and Yukawa couplings 4
3. Relationship to Super Yang-Mills 6
3.1 Equations of Motion 7
3.2 Yukawa Couplings 13
3.3 Holomorphy Constraints on the Effective Theory 16
4. Wavefunctions and Yukawas with background fluxes 18
4.1 Flux solutions to D-terms 18
4.2 Matter wavefunctions 20
4.3 Higgs wavefunctions 21
4.4 Yukawa couplings 22
5. Summary and discussion 24
1. Introduction
The flavour mass hierarchies of the standard model are a tantalising hint towards deeper
andmorefundamentalstructuresintheoreticalphysics. Asacandidatefundamentaltheory
string theory is a naturalarena inwhich to study this topic. Therehas been recent interest
in F-theory model building, in part motivated by the ability to obtain an (1) top Yukawa
O
coupling. These models were originally proposed in [1,2] (see [3–28] for some further
developments). Flavour physics in these models has been studied in [9,10,13,17–19,27,
30]. The models describe locally intersecting 7-branes within a Calabi-Yau four-fold by a
Higgsed twisted 8-dimensional gauge theory. The Higgsing induces localised zero modes
along curves in the four-dimensional manifold S wrapped by the 7-brane. In the presence
of flux, thesezeromodes correspondto chiralmatter fieldsin thefour-dimensionaleffective
– 1 –
GUT. The existence of a tree-level top Yukawa coupling requires the gauge theory to have
an underlying exceptional gauge group, thereby enforcing the non-perturbative F-theory
approach.
Following the initial work an attractive proposal was made in [10] where the hierar-
chical structure of the Yukawas was proposed to arise from a rank one Yukawa matrix via
perturbative corrections in powers of flux. Furthermore the perturbative parameter was
argued to be related to the GUT gauge coupling. The resulting CKM matrix, following
some assumptions regarding order one factors and the loci where up-type and down-type
Yukawas where generated, took a phenomenologically attractive form. This proposal was
studied in more detail in subsequent papers [18,24,27]. These papers, as initial studies of
flavour, did not deeply explore the effects and constraints of supersymmetry. The aim of
this article is to study the connection between these proposed models of flavour and the
constraints arising from supersymmetry.
This paper is organised as follows. In section 2 we review the topologically twisted
theory as described in [2]. In section 3 we relate this theory to dimensional reduction
of the canonical presentation of 8d super Yang-Mills. We show how the the equations
of motion and Yukawa coupling take an identical form in the two approaches and give a
dictionary between the two formalisms. We show how solutions to the equations of motion
are unaltered underoverall metric rescalings. Thisis important sincethevolume of S gives
the gauge coupling of the effective four-dimensional GUT. We relate this to the constraints
requiredbyholomorphyoftheeffective 4dsupergravitytheoryandexplainwhyα must
GUT
appear universally in the Yukawa couplings.
In section 4 we study more specifically the equations of motion. In particular we study
the effect that solving the D-terms has on the form of the localised matter wavefunctions.
For general flux a flat metric on S no longer solves the D-term equations. In order to
maintain supersymmetry the metric is deformed, modifying the equations of motion and
subsequentlythematterwavefunctions. Wesolveforthecorrectedwavefunctionsatleading
orderintheperturbations. Wefindthatthemetricperturbationsdominateover thefluxin
the wavefunctions and give the leading corrections to the fluxless flat-metric case. Finally
we use the resulting wavefunctions to calculate the resulting Yukawa couplings. We find
that, other than the tree-level top Yukawa, all the Yukawa couplings vanish exactly.
Note added
While this manuscript was in preparation the paper [30] appeared studying similar is-
sues. There the vanishing of the Yukawa couplings was demonstrated and explained at a
deeper level than that which appears in this paper. We refer the reader to [30] for a more
fundamental explanation for the vanishing of Yukawa couplings.
2. Local models of intersecting branes in F-theory
In this section we review the topologically twisted theory of [2] that describes local models
– 2 –
of intersecting 7-branes in F-theory. In section 3 we shall relate this theory to the direct
dimensional reduction of 8-dimensional Super-Yang-Mills (SYM) and provide a dictionary
between the two formulations.
2.1 Effective theory for intersecting 7-branes
The relevant theory is an 8-dimensional twisted (off-shell supersymmetric) gauge theory,
with gauge group G, describing a 7-brane wrapping a 4-dimensional Ka¨hler hypersurface
S. For a local model S is a shrinkable manifold (more formally it has an ample normal
bundle) within a Calabi-Yau (CY) 4-fold X. The 8-dimensional fields are given in terms
of adjoint valued, S-valued, 4-dimensional N = 1 multiplets
A = (A ,ψ ,G ) , (2.1)
m¯ m¯ m¯ m¯
Φ = (ϕ ,χ ,H ) , (2.2)
mn mn mn mn
V = (η,A ,D) . (2.3)
µ
The indices on the fields denote their form-values on S. So for example A Ω¯1 ad(P)
m¯ ∈ S ⊗
p
where Ω denotes holomorphic p-form on S and P is the principle bundle (in the adjoint
S
representation) associated to the gauge group G. Here A and Φ are chiral multiplets with
respective F-terms G and H. V is a vector multiplet with D-term D. A and ϕ are
m¯ mn
complex scalars while ψ , χ , and η are fermions.
m¯ mn
The action for the effective theory was given in [2]. Setting 4-dimensional variations
of the fields to zero, the equations of motion that follow are
H F(2,0) = 0 , (2.4)
−
i[ϕ,ϕ¯]+2ω F(1,1) +⋆ D = 0 , (2.5)
S
∧
2iω G¯ ∂¯ ϕ = 0, (2.6)
A
∧ −
∂H¯ +2ω ∂¯D+G¯ ϕ¯ χ¯ ψ¯ i2√2ω η ψ = 0, (2.7)
− ∧ ∧ − ∧ − ∧ ∧
i
ω ∂ ψ+ [ϕ¯,χ] = 0 , (2.8)
A
∧ 2
∂¯ χ 2i√2ω ∂ η [ϕ,ψ] = 0 , (2.9)
A A
− ∧ −
∂¯ ψ √2[ϕ¯,η] = 0, (2.10)
A
−
1
√2[η¯,χ¯] ∂¯ G [ψ,ψ] = 0 . (2.11)
A
− − − 2
Thesearetheequations thatweworkwithinthispaperandthey applytoageneralKa¨hler
manifold S of large enough volume to neglect α′ corrections. Equation (2.4) imposes that
the flux must be of type (1,1). Equation (2.5) is the D-term equation that we discuss
in detail in section 4.1. In this paper we are concerned with vacua where the vacuum
expectation value (vev) of ϕ, denoted ϕ , and that of A , which is responsible for the
m¯
h i
flux, take values in the Cartan of G. Therefore equation (2.6) just imposes that ϕ is
h i
holomorphic
∂¯ ϕ = ∂¯ ϕ +[A, ϕ ]= ∂¯ ϕ = 0 . (2.12)
A
h i h i h i h i
– 3 –
We are interested in solving the equations on a vacuum with
χ = ψ = η = 0 , (2.13)
h i h i h i
which means that, using equations (2.4-2.6) equations (2.7) and (2.11) are satisfied. Equa-
tion (2.10) follows from (2.9) once (2.13) is imposed.
We take the manifold S to be Ka¨hler and spanned by two complex coordinates z and
1
z . We will also restrict to the metric ansatz
2
ds2 = h (z ,z¯ ) dz dz¯ +h (z ,z¯ ) dz dz¯ , (2.14)
1 1 ¯1 1 ¯1 2 2 ¯2 2 ¯2
⊗ ⊗
where h and h are general functions of z and z respectively (this is the form of the
1 2 1 2
metric we will usewhen solving for explicit wavefunctions in section 4). Thecorresponding
Ka¨hler form is given by
i i
ω = h dz dz¯ + h dz dz¯ . (2.15)
2 1 1∧ ¯1 2 2 2 ∧ ¯2
Putting this into (2.4-2.11) the relevant equations read
h ∂ ψ +h ∂ ψ [ϕ¯,χ]= 0 , (2.16)
1 A2 ¯2 2 A1 ¯1
−
∂¯ χ [ϕ,ψ ]= 0 , (2.17)
A¯1 ¯1
−
∂¯ χ [ϕ,ψ ]= 0 , (2.18)
A¯2 ¯2
−
where by abuse of notation we have relabeled ϕ ϕ and χ χ. Apart from these
12 12
→ →
equations we also have the D-term equation (2.5).
2.2 Zero modes and Yukawa couplings
Consider turning off the flux so that there is no gauge field background. We then turn on
a vev for ϕ given by
ϕ = m2z Q +m2z Q . (2.19)
h i 1 1 1 2 2 2
Here Q and Q are elements in G. In this paper we performthe simplification m = m =
1 2 1 2
m and define
1
v . (2.20)
≡ m2
Since there is no flux the D-term equation is solved identically and we are free to consider
a flat metric background h = h = 1. The equations of motion to be solved then read
1 2
1
∂ ψ +∂ ψ (z¯ q +z¯ q )χ = 0 , (2.21)
2 ¯2 1 ¯1− v 1 1 2 2
1
∂¯ χ (z q +z q )ψ = 0 , (2.22)
¯1 − v 1 1 2 2 ¯1
1
∂¯ χ (z q +z q )ψ = 0 . (2.23)
¯2 − v 1 1 2 2 ¯2
Here q and q are the charges of the fields under Q and Q . For simplicity consider the
1 2 1 2
case q = 1 and q = 0. Then the equations have a localised solution given by
1 2
χ = f(z2)e−|z1v|2 . (2.24)
– 4 –
Here and in section 4, when presenting a solution we only present the expression for χ
since the expressions for ψ and ψ can be determined directly from it using (2.22) and
¯1 ¯2
(2.23). This is a localised zero mode which can be thought of as strings stretching between
two 7-branes that are intersecting along z = 0. If we decompose the 8-dimensional fields
1
into externalspace-time4-dimensionalcomponents andinternal4-dimensionalcomponents
along S we see that the equations imply that the external 4-dimensional components of χ,
ψ and ψ are all equal and they count as a single 4-dimensional zero mode.
¯1 ¯2
If flux is turned on there are two changes: the form of the internal wavefunctions
changes and the number of zero modes can change. The change in the form of the wave-
functions is a local effect that we calculate explicitly in section 4. The replication of zero
modes is of course the family index and depends on the global properties of the flux. In a
local model we decouple the connection between the flux and the zero mode replication.
TheYukawacouplingsareassociatedtowavefunctionoverlapsonapointofintersection
of three curves where the gauge symmetry is enhanced. For example the bottom type
Yukawas are associated to a point of SO(12) gauge symmetry. The matter arises from the
decomposition of the adjoint
SO(12) SU(5) U(1) U(1) , (2.25)
a b
⊃ × ×
66 24(0,0) 1(0,0) 1(0,0) (5 ¯5)(−1,0) (5 ¯5)(1,1) (10 1¯0)(0,1) ,
→ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
wherethesuperscriptsdenotethechargesofthe5and10. Thenweseethatthe5¯ 5¯ 10
H M M
Yukawa, corresponds to curves of charges ( 1, 1), (1,0) and (0,1) respectively. We label
− −
the matter from each of the three curves with the index λ = 1,2,3 corresponding to 5¯ ,
H
5¯ and 10 respectively. The fields are therefore the sum over the localised solutions
M M
χ = χλ , ψ = ψλ . (2.26)
λ λ
X X
Sincetheequations of motion arelinear they exactly decoupleandcan besolved separately
for each component.
Thereisalsoafamilyindexa = 1,2,3whichcountscopiesofthezeromodesassociated
to global properties of the flux. A four-dimensional φ field is then labeled by
φ(λ,a) : χ(λ,a),ψ(λ,a),ψ(λ,a) . (2.27)
¯1 ¯2
n o
Since there is only one Higgs generation for λ = 1 we only have a = 1.
The relevant Yukawa coupling are the sum over all the terms arising from
A A Φ = ψˆiψˆjχˆk 2Tr([t ,t ]t )dz dz¯ dz dz¯ . (2.28)
∧ ∧ − ¯1 ¯2 12 i j k 1∧ ¯1∧ 2∧ ¯2
ZS ZS
Here t are generators of the point gauge group SO(12). Under the decomposition these
i
correspond to curve indices, λ in (2.27), with
λ = 1: 5 , λ = 2 : 5 , λ = 3: 10 . (2.29)
H M M
– 5 –
We therefore obtain the Yukawas
Yab = a ψˆ1ψˆ3,bχˆ2,a+ψˆ2,aψˆ1χˆ3,b ψˆ2,aψˆ3,bχˆ1 ψˆ1ψˆ2,aχˆ3,b
¯1 ¯2 ¯1 ¯2 − ¯1 ¯2 − ¯1 ¯2 −
ZS"
ψˆ3,bψˆ1χˆ2,a+ψˆ3,bψˆ2,aχˆ1 ǫ . (2.30)
¯1 ¯2 ¯1 ¯2
#
Here ǫ is the canonical volume form as in (2.28). The constant a is a normalisation factor
which is used to normalise the top Yukawa to 1.
3. Relationship to Super Yang-Mills
Above we have reviewed, following [2], the topological equations for the dynamics of the
8-dimensional field theory. In this section we show how these equations arise from direct
analysis of the canonical 8d SYM Lagrangian and provide a dictionary that translates
betweenthefieldsofthecanonicalpresentationof8dSYMandthefieldsofthetopologically
twisted theory used in [2]. While ultimately the equations are identical, this provides an
alternative perspective on the topological theory of [2] and may make certain properties of
the theory more intuitive.
8dsuperYang-Mills ismosteasilyobtainedbydimensionalreductionofthe10dtheory.
The 10d action is
1 1
√gd10x Tr F FMN Tr λ¯ΓM λ , (3.1)
MN M
−4 − 2 D
Z (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
where λ is a Majorana-Weyl spinor and X = ∂ X i[A ,X]. Dimensional reduction
M m m
D −
gives the 8-dimensional action,
1 1 1 i
d8x√g Tr(F FMN) Tr( φ Mφ∗)+quartics Tr(λ¯ΓmD λ)+ λ¯Γr[φ ,λ]
MN M m r
−4 − 4 D D − 2 2
Z (cid:18) (cid:19)
(3.2)
We drop quartic scalar interactions as they are not relevant for our purposes.
Let us now describe λ. It is convenient to view λ as a 10-dimensional Majorana-Weyl
spinor with 8 complex degrees of freedom (before imposing equations of motion). We
follow [29] to write the flat space gamma matrices as (µ = 0,1,2,3,m = 4,5,6,7,r = 8,9)
Γµ = γµ I I, Γm = γ5 γ˜m I, Γr = γ5 γ˜5 τr,
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
with
0 I 0 σ 0 σ 0 σ 1 0
γ0 = − , γ1 = x , γ2 = y , γ3 = z , γ5 = ,
I 0 σ 0 σ 0 σ 0 0 1
! x ! y ! z ! !
−
0 iI 0 σ 0 σ 0 σ I 0
γ˜1 = − , γ˜2 = z , γ˜3 = x , γ˜4 = y , γ˜5 = ,
iI 0 σ 0 σ 0 σ 0 0 I
! z ! x ! y ! !
– 6 –
and
0 1 0 i
τ8 = σ = , τ9 = σ = − .
x y
1 0 i 0
! !
Using u to denote the complexified 8,9 directions, we also note
0 1 0 0
τu σ +iσ = 2 , τu¯ σ iσ = 2 ,
x y x y
≡ 0 0 ≡ − 1 0
! !
Spinors can be decomposed by their chiralities under each of the three (4d + 4d + 2d)
components of the gamma matrices. The Majorana condition can be written as λ∗ = Bλ,
where B = Γ2Γ4Γ7Γ9.
As in [29] it can be shown that a general Majorana-Weyl spinor can be written as
λ = (λ +λ ) (λ +λ ) (3.3)
1 4 2 3
⊕
where (λ ,λ ,λ ,λ )= λabα,λa ,λ α,λ b ). Super/sub scripts indicate positive or neg-
1 2 3 4 { bα ab a α
ative chirality inthatdirection. Thenotation (λ +λ )indicates thatthecomponentsof λ
1 4 1
and λ are not independent: they are related by the Majorana condition. More precisely,
4
we have
ξ ψ θ 0 iσ ψ∗ 0
λ +λ = 1 1 1 + − y 1 ,
1 4 0 ⊗ 0 ⊗ 0 σ ξ∗ ⊗ 0 ⊗ iθ∗
! ! ! − y 1! ! − 1!
ξ 0 0 0 0 iθ∗
λ +λ = 2 + 2 . (3.4)
2 3 0 ⊗ ψ ⊗ θ σ ξ∗ ⊗ iσ ψ∗ ⊗ 0
! 2! 2! − y 2! − y 2! !
λ and λ correspond to the CPT conjugates of λ and λ respectively, and thus do not
4 3 1 2
represent physically distinct degrees of freedom.
For the extra-dimensional spinor components we will also often write
ψ
1,1
ψ ψ
1  1,2. (3.5)
ψ → ψ
2! 2,1
 
ψ 
2,2
 
 
3.1 Equations of Motion
We need to account for the non-trivial metric on the surface wrapped by the branes. For
this we suppose we start with a metric of the factorised form
ds2 = 4h (z,z¯)dzdz¯+4h (w,w¯)dwdw¯, (3.6)
1 2
so g = 2h (z,z¯),g = 2h (w,w¯),gzz¯ = 1 ,gww¯ = 1 . The spin connection is
zz¯ 1 ww¯ 2 2h1(z,z¯) 2h2(w,w¯)
given by
1 1 1
ωab = eaν ∂ eb ∂ eb ebν ∂ ea ∂ ea eψaeσb(∂ e ∂ e )ec. (3.7)
µ 2 µ ν − ν µ − 2 µ ν − ν µ − 2 ψ σc − σ ψc µ
(cid:16) (cid:17)
(cid:0) (cid:1)
– 7 –
Here a,b are thevielbein indices with a flat metric whileµ,ν are thespacetime indices that
run over z,z¯,w,w¯. As a vielbein we take
1 1
e1 = h2, e1 = h2, e1 = 0, e1 = 0.
z 1 z¯ 1 w w¯
1 1
e2 = ih2, e2 = ih2, e2 = 0, e2 = 0.
z 1 z¯ − 1 w w¯
1 1
e3 = 0, e3 = 0, e3 = h2, e3 = h2.
z z¯ w 2 w¯ 2
1 1
e4 = 0, e4 = 0, e4 = ih2, e4 = ih2.
z z¯ w 2 w¯ − 2
From these we can compute the non-vanishing elements of the spin connection to be
i i i i
ω12 = ∂ h , ω12 = ∂ h , ω34 = ∂ h , ω34 = ∂ h .
z −2h z 1 z¯ 2h z¯ 1 w −2h w 2 w¯ 2h w¯ 2
1 1 2 2
We also need to determine the gamma matrices. We have
−1/2 −1/2
h h
γ˜z = ez1γ˜ +ez2γ˜ = 1 (γ˜ iγ˜ ), γ˜z¯ =ez¯1γ˜ +ez¯2γ˜ = 1 (γ˜ +iγ˜ ), (3.8)
1 2 1 2 1 2 1 2
2 − 2
−1 −1
h 2 h 2
γ˜w = 2 (γ˜ iγ˜ ), γ˜w¯ = 2 (γ˜ +iγ˜ ). (3.9)
3 4 3 4
2 − 2
Therefore we can write
0 0 2i 0 0 0 0 0
−1 − −1
h 2 0 0 0 0 h 2 0 0 0 2i
γ˜z = 1  , γ˜z¯ = 1  − ,
2 0 0 0 0 2 2i 0 0 0
   
0 2i 0 0  0 0 0 0 
   
   
0 0 0 0 0 0 0 2
−1 −1
h 2 0 0 2 0 h 2 0 0 0 0
γ˜w = 2  , γ˜w¯ = 2 =  . (3.10)
2 0 0 0 0 2 0 2 0 0
   
2 0 0 0 0 0 0 0
   
   
σ 0
For the spin connection we need 1γMω γα,γβ . As γ1,γ2 = 2i z ,
8 M,αβ − 0 σ
z!
−
(cid:2) (cid:3) (cid:2) (cid:3)
σ 0
γ3,γ4 = 2i z , we have
0 σ
z!
(cid:2) (cid:3)
1 ∂ h σ 0 1 ∂ h σ 0
ω γα,γβ = z 1 z , ω γα,γβ = z¯ 1 z .
zαβ z¯αβ
8 − 4h 0 σ 8 4h 0 σ
1 z! 1 z!
h i − h i −
1 ∂ h σ 0 1 ∂ h σ 0
ω γα,γβ = w 2 z , ω γα,γβ = w¯ 2 z .
wαβ w¯αβ
8 4h 0 σ 8 4h 0 σ
2 z! 2 z!
h i h i
– 8 –
Putting these together we obtain
0 0 i ∂ + ∂zh1 1 ∂ + ∂w¯h2
−h21 z 4h1 h21 w¯ 4h2
 1 (cid:16) (cid:17) 2 (cid:16) (cid:17)
0 0 1 ∂ + ∂wh2 −i ∂ + ∂z¯h1
γM∂M+1γMωMαβγαβ =  h212 (cid:16) w 4h2 (cid:17) h121 (cid:16) z¯ 4h1 (cid:17) ,
4  i ∂ + ∂z¯h1 1 ∂ + ∂w¯h2 0 0 
 hh112112 (cid:16)∂wz¯+ ∂44whhh122(cid:17) hh212i21(cid:16)∂w¯z + ∂44zhhh121 (cid:17) 0 0 
 2 (cid:16) (cid:17) 1 (cid:16) (cid:17) 
 (3.11) 
where γαβ 1[γα,γβ]. We need to twist this operator to account for the nontrivial metric.
≡ 2
Thetwistingcorrespondstoanadditionalgaugefieldinthecanonicalbundle. Todetermine
this, let us first recall that a metric
ds2 = 4h (z,z¯)dzdz¯+4h (w,w¯)dwdw¯, (3.12)
1 2
has
∂ h ∂ h ∂ h ∂ h
Γz = z 1, Γz¯ = z¯ 1, Γw = w 2, Γw¯ = w¯ 2.
zz h z¯z¯ h ww h w¯w¯ h
1 1 2 2
The twisting corresponds to a gauge field in the canonical bundle. A gauge field with M
units of flux in the zz¯ direction and N units of flux in the ww¯ direction has
iM∂ h iM∂ h iN∂ h iN∂ h
z 1 z¯ 1 w 2 w¯ 2
A = , A = , A = , A = .
z z¯ w w¯
− 4h 4h − 4h 4h
1 1 2 2
The factor of i/4 can be determined by reference to the case of P1 P1, for which z and w
×
parameterise the two separate P1s. This gives
1
γM∂ + γMω γαβ iγMA = (3.13)
M Mαβ M
4 −
0 0 −i ∂ + ∂zh1(1 M) 1 ∂ + ∂w¯h2 (1+N)
h21 z 4h1 − h21 w¯ 4h2
 1 (cid:16) (cid:17) 2 (cid:16) (cid:17)
0 0 1 ∂ + ∂wh2(1 N) −i ∂ + ∂z¯h1(1+M)
 h21 w 4h2 − h21 z¯ 4h1 
 2 (cid:16) (cid:17) 1 (cid:16) (cid:17) 
 i ∂ + ∂z¯h1(1+M) 1 ∂ + ∂w¯h2(1+N) 0 0 
 hh112121 (cid:16)∂wz¯ + ∂44whh1h22(1−N)(cid:17) hh2i1212 (cid:16)∂wz¯+ ∂44zhhh121(1−M)(cid:17) 0 0 
 2 (cid:16) (cid:17) 1 (cid:16) (cid:17) 
 
Asdiscussedin[2,29], theleft-andright-handedspinorsψ andψ haveoppositeR-charges
1 2
and so should be twisted in opposite directions. To accomplish this we modify the Dirac
operator (3.14) by
(M,N) (M,N)+(1,1) for (ψ ,ψ ),
1,1 1,2
→
(M,N) (M,N) (1,1) for (ψ ,ψ ).
2,1 2,2
→ −
The twisted Dirac operator is then
1
γM∂ + γMω γαβ iγMA = (3.14)
M Mαβ twisted,M
4 −
– 9 –