Arnold Sommerfeld Center
Dr. Erik Plauschinn
LMU Munich
SoSe 2017
String Theory II (TMP-TD2)
Problem Set 1
due on 03.05.17
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Spinors in two dimensions
In the RNS-formulation (Ramond-Neuveu-Schwarz) of superstring theory, the embedding maps X µ are augmented with fermionic, spinorial degrees of freedom µ . In this
problem we want to discuss some properties of such anti-commuting fields.
The Cli↵ord algebra in Lorentzian signature is generated by matrices ⇢↵ that obey the
following anti-commutation relations
{⇢↵ , ⇢ } = 2⌘ ↵ ,
(1)
where ds2 = dt2 + dx2 and ⌘ is the usual metric in Minkowski space. The chirality
operator is = ⇢0 ⇢1 , and it satisfies the usual properties 2 = 1 and { , ⇢↵ } = 0. The
minimal complex representation of the Cli↵ord algebra is two-dimensional and, hence, a
Dirac spinor in two-dimensions contains four real degrees of freedom.
(i) Verify that the choice ⇢0 = i 2 and ⇢1 = 1 , where
provides a Dirac representation of the Cli↵ord algebra.
i
is the ith Pauli matrix,
(ii) The representation described in the previous problem are entirely real. This means
that we can impose a reality condition = ⇤ , where the ⇤ denotes ordinary complex conjugation. We obtain Majorana spinors. Show that the Majorana condition
is compatible with the Weyl condition, i.e. that has only real eigenvalues.
(iii) Verify that the pairing ( , ) = † ⇢0 = ¯ = ( ⇤ )Ā (⇢0 )ĀB B is invariant under
the Cli↵ord algebra, i.e. (⇢↵ , ) + ( , ⇢↵ ) = 0. Conclude that this product does
only pair spinors of di↵erent chirality. The uppercase latin indices Ā, B denote
spinor indices.
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(iv) Introduce light-cone coordinates x± in which ds2 = 2dx+ dx . Show that if is a
Weyl spinor of positive (negative) chirality, that ⇢
= 0 (⇢+ = 0). Deduce that
the Dirac kinetic term for a massless Majorana spinor can be written as
Z
Z
1
i
SD =
d 2 x i ¯ ⇢↵ @ ↵ =
d2 x ( + @ + +
@+ ) ,
(2)
2⇡
⇡
where ± denote the positive/negative chirality components of
equations of motion?
. What are the
(v) Recall that boosts are generated by the matrix J ↵ = 4i [⇢↵ , ⇢ ] = ✏↵ J. Relate
the eigenvalues of J to the chirality operator . What is the interpretation of these
eigenvalues after a Wick-rotation to imaginary time ⌧ = it?
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