1. Fermionic Strings
Highlights 2015
Roa, F. J. P.
Topics:
°1 Two-dimensional Fermionic Action
°2 Energy-Momentum Tensor for Fermions
°3 Equations of Motion for Majorana Fields
Author’s note: This document is still in the process of typing. The posting of the
complete draft to cover the outlined topics above is in an installment basis.
°1 Two-dimensional Fermionic Action
Basically I start with an action for a two-component Majorana field µ
ψ on a two-
dimensional bulk with flat Minkowski’s metric as background. The said action is given in
index form
(1)
∫ ∂−= µ
µ
ψρψη
π
k
k
yd
i
S 2
2
In this flat two-dimensional bulk I ascribe to a point the set of two coordinates
),( 21
yyyi
= with distinguishing Latin indices (e. g., i = 1, 2). The bulk metric as it is
two-dimensional Minkowski is in the diagonal form
(2)
)1,1(−= diagjiη
also with distinguishing Latin indices. The contraction k
k
∂ρ means contraction with this
given flat metric

2. (3)
jk
jkjk
jkk
k
∂=∂=∂ ρηρηρ
It is implied in this contraction that I lower or raise a Latin index with the components of
the flat metric. I won’t attach as yet the specified forms of the rho matrices ρ , which are
here two-dimensional analogs of the Dirac-Gamma matrices in the 3 + 1 Fermionic field
theory. The contraction µ
µ
ψψ k∂K I shall specify as a contraction in a D-dimensional
ambient spacetime so the contraction index mu runs from zero to (D – 1 ).
We can re-parametrize this action to a general non-flat two-dimensional spacetime of a
bulk with a set of coordinates ),( 21
σστσσ α
=== . These new coordinates are
parametric functions of the original coordinates
(4)
)( ii
yy αα
σσ =→
With this parametrization we can assume that there exists a set of veilbein bases defined
by
(5)
αα
σ∂
∂
=
i
i y
e
so that from the original flat metric (2) we can switch to the induced metric βαh
(6)
ji
ji
ji eeh ηη βαβα =→
Taking note that given this re-parametrization, from which follows that
(7.1)
ηη α
−=−→−
−1
ieh
(7.2)
∫ ∫ ∫=→ α
σ ieyddyd 222

3. the original action (1) can be equivalently written in its “re-parametrized” form
(8)
∫ ∂−= µ
βα
βα
µ
ψρψσ
π
hhd
i
S 2
2
Here, ]det[]det[ βα
βα hhh ≠= , so from this we can see that this new equivalent action
(8) is seemingly not invariant if we Weyl transform the new metric
(9)
βα
φ
βαβα η heh −
=→
where βαη is a Weyl transformed metric that goes with conformal factor φ
e . We take
that this Weyl transformed metric is in diagonal form
(10)
)1,1(−= diagβαη
although we must distinguish it from the flat metric form (2). So as we Weyl transform
the metric through (9) we also transform action (8) into
(11.1)
∫ ∂−= µ
βαφ
βα
µ
ψρηψησ
π
22
2
ed
i
S
Now, η here means ]det[ βαηη = .
Action (8) is seemingly not invariant under (9) because of the presence of the extra factor
φ2
e in Weyl transformed action (11.1) unless the Majorana field itself also transforms in
a way that cancels this extra factor.
We can immediately remedy this short-coming in (8) say without having to resort to
transforming our Majorana field so as to cancel the extra factor. We re-write that action
simply as
(11.2)

4. ∫ ∂−= µβα
βαµ
ψρψσ
π
hhd
i
S 2
2
by adhocally assuming that the following holds
(11.3)
µ
βα
βαµβα
βα ψρψρ ∂=∂ hh
This re-written action is conformally invariant under (9) since
(11.4)
βαβα
ηη−=− hh
where ][det][det βα
βα hhh ≠= and ][det][det βα
βα ηηη ≠= thus, cancellation of the
extra factors occurs. By conformal transformation we write (11.2) as
(11.5)
∫ ∂−= µβα
βαµ
ψρηψησ
π
2
2
d
i
S
Let’s digress briefly to justify (11.3). Through a set }{ i
eα of veilbein bases we can
express the induced metric βαh in terms of a two-dimensional flat metric jiη via (6).
Say we construct the contraction
(11.6)
α
βα
α
βα ρηρ ji
ji
eeh =
and invoke that we can turn an upper Greek index into an upper Latin index with the use
of this veilbein basis. That is,
(11.7)
ii
e ρρα
α =
In addition, we can do the usual lowering of an index in flat space,
(11.8)

5. j
i
ji ρρη =
We then write (11.6) as
(11.9)
j
j
eh ρρ β
α
βα =
and also invoke here that we turn a lower Latin index into a lower Greek index with the
vielbein so that from (11.9) we have
(11.10)
β
α
βα ρρ =h
Comparing (11.10) and (11.8) we see that as we can lower an upper Latin index with the
flat metric we can also lower an upper Greek index with the induced metric. This is so by
invoking (11.7) that also goes along with turning a lower Latin index into a lower Greek
index with the vielbein basis.
With the use of the set }{ α
θ i of duals to the veilbeins we can express the inverse induced
metric βα
h in terms of the inverse flat metric
(11.11)
ji
jih ηθθ βαβα
=
and contract this with αρ
(11.12)
α
βα
α
βα
ρηθθρ ji
jih =
We can also turn an index here with the use of dual bases say, for example, a lower Greek
index into a lower Latin index,
(11.13)
ii ρρθ α
α
=
and that we can raise Latin index with the flat metric. Thus, writing (11.12) as
(11.14)

6. j
jh ρθρ β
α
βα
=
in which we also turn upper Latin index into upper Greek index with the dual basis and
following this, from (11.14) we have
(11.15)
β
α
βα
ρρ =h
In contrast to (11.10), with (11.15) we can raise a Greek index with the inverse induced
metric.
Say we contract (11.7) with the dual basis β
θ i ,
(11.16)
α
α
ββ
ρθρθ i
i
i
i e=
We invoke that we can turn an upper Latin index into upper Greek index, and write
(11.16) as
(11.17)
αβ
α
α
α
ββ
ρδρθρ == i
i e
What we will underline in (11.17) is the Kronecker-Delta tensor expressed as a
contraction of a dual basis with a corresponding veilbein basis,
(11.18)
β
αα
β
δθ =i
i e
The dummy index in that is a Latin index, while the lower and upper indices on the Delta
are Greek indices.
To arrive at an alternative to (11.18), say we form the contraction of βρ with a given
dual α
θ i , where
(11.19)
j
j
e ρρ ββ =

7. in consequence to transforming a Latin lower index into a Greek lower index with the
veilbein basis. The contraction is then given by
(11.20)
j
j
ii e ρθρθ β
β
β
β
=
in which the left-hand side invokes transformation of a Greek lower index into a Latin
lower index so that we write (11.20) as
(11.21)
j
j
ij
j
ii e ρδρθρ β
β
==
from which we read off
(11.22)
j
i
j
i e δθ β
β
=
and this time, our upper and lower indices on the Delta are Latin, while the dummy index
on the right-hand side of (11.22) is a Greek index.
We can go on with these initial results to consider the contraction
(12.1)
βα
βα ρ ∂h
which we shall write, upon noting (6), as
(12.2)
βα
βα
βα
βα ρηρ ∂=∂ ji
ji
eeh
By applying the transformation of Greek upper indices into Latin upper indices with the
use of veilbeins, we can then write (12.3) in the form
(12.3)
ji
jih ∂=∂ ρηρ βα
βα

8. This is just an invariance in the contraction, from the non-flat induced metric into a bulk
with a flat metric background. From this we shall note the raising of Latin indices with
the flat metric so that we can actually write the preceding equation as
(12.4)
lk
ljki
jih ∂=∂ ρηηηρ βα
βα
and recall from our Tensor Analysis that the Kronecker Delta tensor can be written as a
contraction between a given flat metric with its inverse
(12.5)
k
j
ki
ji δηη =
Thus, from (12.4) we have
(12.6)
lk
lk
lk
ljk
jh ∂=∂=∂ ρηρηδρ βα
βα
Let us contract (11.11) with veilbein basis, and take note of (11.22) to write this
contraction as
(12.7)
lklk
hee ηβα
βα =
which we consequently substitute in (12.6) to get, after transforming Latin lower indices
into Greek lower indices with the veilbeins,
(12.8)
βα
βαβα
βα ρρ ∂=∂ hh
This is the needed justification for (11.3).
°2 Energy-Momentum Tensor for Fermions

9. Let us take action (11.2) to start with and since the induced metric as well as the inverse
induced metric is symmetric over the interchange of the Greek indices, we take note of
the symmetrization
(13.1)
( )µ
αβ
µ
βα
βαµ
βα
βα
ψρψρψρ ∂+∂=∂ hh
2
1
Expressing the variation of (11.2) in terms of the variation βα
δ h , we have
(13.2)
( )∫ +−= αββα
βα
δσ
π
δ FFhhd
i
S 2
4
where
(13.3)
)( µλφ
µλφ
βαµαβ
µ
µβα
µ
αββα ψρψψρψψρψ ∂−∂+∂=+ hhFF
Say take,
(13.4)
∫ −= βα
βα
δσ
π
δ Fhhd
i
S 2
1
4
and using (11.11), we express the variation βα
δ h in terms of the variation of the duals
(13.5)
ji
ji
ji
jih ηθδθηθθδδ βαβαβα
+=
We can then write
(13.6)
( )αββα
βα
βα
βα
ηθθδδ FFFh ji
ji +=
Noting (11.11) again and using (11.18), we obtain the following identities

10. (13.7)
ji
j
i
he ηθ ββλ
λ =
(13.8)
i
i
i
i ee λ
α
λ
α
δθθδ −=
Using the results in (13.4), we are led to write this variation as
(13.9)
∫ +−−= )(
4
2
1 αββα
βλ
λ
α
δθσ
π
δ FFhehd
i
S i
i
Next, let’s take the second variation
(13.10)
∫ −= αβ
αβ
δσ
π
δ Fhhd
i
S 2
2
4
and upon noting the symmetry ji
jihh ηθθ βαβααβ
== , over the indices α and β , we
see that this second variation is equivalent to 1Sδ .(To be continued.)
[stop. pp. FS26, cited notes]
Ref’s
[1]Gerard ’t Hooft, INTRODUCTION TO STRING THEORY,
http://www.phys.uu.nl/~thooft/
[2] Joseph Polchinski, WHAT IS STRING THEORY?, arXiv:hep-th/9411028v1