University:
Massachusetts Institute of Technology
Course:
8.821 | String Theory
Academic year:

2020
Views:

260

Pages:

8
Author:

Leyla Diaz

8.821 Problem Set 2 Solutions This problem set contains a high number density of fermions and their associated minus signs. While everything here appeared to work out, really that only means that I succeeded in making an even number of mistakes, so please treat all formulas here with caution. I. HOW TO REMEMBER THAT N = 4 ACTION The 10d action that we are going to dimensionally reduce is as follows � � � 1 S10d = − d10 x 2 tr FM N F M N + 2iλ̄ΓM DM λ 4gYM (1) Here M, N are 10d indices, µ, ν 4d indices, and i, j run over the six dimensional space we are reducing on torii. We will do this problem in various increasingly detailed stages. Depending on how much you like dimensional reduction and spinor representation theory, you can stop reading whenever you want. 1. No work at all: This 10d theory has 16 supercharges and no ﬁelds higher than spin 1. Its dimensional reduction to four dimensions also has 16 supercharges and no ﬁelds higher than spin 1. The only such theory in four dimensions is N = 4 SYM, whose Lagrangian is completely determined by its symmetries. Therefore it must work out. 2. Bosons only: Just for peace of mind we might as well work out the bosonic sector. Assuming that nothing depends on xi , we can write down the components of FM N : Fij = −i[Ai , Aj ] Fµi = ∂µ Ai − i[Aµ , Ai ] Fµν = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ] (2) FM N F M N = Fµν F µν + 2(∂µ Ai − i[Aµ , Ai ])(∂ µ Ai − i[Aµ , Ai ]) − [Ai , Aj ][Ai , Aj ] (3) From here, the ﬁrst kinetic term from (1) becomes The components Ai are really just scalars in the remaining 4d space, so from now on we will call them Xi instead. Noting also that the middle term is really the covariant derivative of a scalar in the adjoint of the gauge group, we obtain � FM N F M N = Fµν F µν + 2(Dµ Xi )2 − 2 [Ai , Aj ]2 (4) i

8.821 Problem Set 2 Solutions

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