Lectures on String Theory

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One can check that these objects are charges which correspond to new conserved
currents
J B + = 2c + (T X ++ + 1 2 T gh
++)
J + = c + b ++ .
There is new and more fundamental fermionic symmetry present – it is BRST
symmetry. Let λ be a grassman (anticomuting) parameter. The BRST transformation
is defined as
δY = [λQ, Y ]
It is given by
δX µ = λc + ∂ + X µ + λc − ∂ − X µ ,
δc + = λc + ∂ + c + ,
δb ++ = 2iλT ++ ,
δT ++ = 0 .
The ghost number operator
U = 1 2 (c 0b 0 − b 0 c 0 ) +
∞∑
(c −n b n − b −n c n )
n=1
Here c n , b n for n > 0 are annihilation operators.
Zero-modes require special treatment. We have
c 2 0 = b 2 0 = 0 , {c 0 , b 0 } = 1
There is a two-dimensional representation of this relations:
c 0 | ↓〉 = | ↑〉 , b 0 | ↑〉 = | ↓〉
c 0 | ↑〉 = 0 , b 0 | ↓〉 = 0 .
The ghost numbers are U ↓ = −1/2 and U ↑ = 1/2.
Physical states should have the ghost number −1/2. They are annihilated by b 0 .
Indeed, consider
c n |χ〉 = b n |χ〉 = 0 , n > 0 and b 0 |χ〉 = 0 .
The condition of the BRST invariance reduces to
(
0 = Q|χ〉 = c 0 (L 0 − 1) + ∑ c −n L n
)|χ〉 .
n>0
We thus reproduced the conditions for a physical state obtained in the old covariant
quantization approach. Physical states of bosonic string are cohomology classes of
the BRST operator with the ghost number −1/2.