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Copyright 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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side effect of this effort will be that the critical dimension drops from 26 to 10. If you aren t familiar with the description of fermionic elds (and supersymmetry) in quantum theory, try reviewing your favorite quantum eld theory book before tackling this chapter (Quantum Field Theory Demysti ed provides a relatively painless introduction).
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We can proceed with a straightforward modi cation of the theory to include fermions using an approach called the Ramond-Neveu-Schwarz (RNS) formalism. This approach is supersymmetric on the worldsheet. Later we consider the GreenSchwarz formalism, which is supersymmetric in space-time. When the number of space-time dimensions is 10, these two approaches are equivalent. The program we will follow can be done using basically the same which was applied in the bosonic case: introduce an action, nd the equations of motion, and quantize the theory. However, this time we are going to include fermionic elds on the worldsheet. We start with the Polyakov action, rst described in Eq. (2.27) and reproduced here in the conformal gauge: S= T 2 d X X 2 (7.1)
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To include free fermions in the theory using the RNS formalism, we add a kinetic energy term for a Dirac eld to the lagrangian. That is, we include D free fermionic elds to the action, so that it assumes the form S= T 2 d X X i 2
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(7.2)
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Again, if you are not familiar with Dirac elds, consult Quantum Field Theory Demysti ed or your own favorite quantum eld theory text. The are Dirac matrices on the worldsheet. Since the worldsheet has 1 1 dimensions, the are Dirac matrices in 1 1 dimensions. Hence there are two such 2 2 matrices, which can be written in the form 0 i 0 = i 0 using an appropriate choice of basis. 0 i 1 = i 0 (7.3)
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CHAPTER 7 RNS Superstrings
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EXAMPLE 7.1 Show that the Dirac matrices on the worldsheet obey an anticommutation relation known as the Dirac algebra { , } = 2 by explicit computation. SOLUTION This result is easy to verify. Since 1 0 = 0 1 The Dirac algebra will be satis ed by the if the following relations hold: { 0 , 0 } = 0 0 + 0 0 = 2 0 0 = 2 00 = 2 I { 1 , 1} = 1 1 + 1 1 = 2 1 1 = 2 11 = 2 I { 0 , 1} = 0 1 + 1 0 = 2 01 = 0 and likewise for { 1 , 0 }. Now, 0 i 0 i 0 0 + ( i ) i 0 ( i ) + ( i ) 0 1 0 = = =I 0 0 = i ( i ) + 0 0 0 1 i 0 i 0 i 0 + 0 i Hence the rst relation { 0 , 0 } = 2 I is satis ed. We verify that the second relation { 1 , 1} = 2 I is also satis ed: 0 i 0 i 0 0 + i i 0 i + i 0 1 0 = = = I 1 1 = i 0 i 0 i 0 + 0 i i i + 0 0 0 1 Finally, noting that 0 i 0 i 0 0 + ( i ) i 0 i + ( i ) 0 1 0 0 1 = = = i 0 i 0 i 0 + 0 i i i + 0 0 0 1 0 i 0 i 0 0 + i i 0 ( i ) + i 0 1 0 1 0 = = = i 0 i 0 i 0 + 0 i i ( i ) + 0 0 0 1 we see that { 0 , 1} = { 1 , 0 } = 0 . (7.4)
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