Spinor representations decomposed under subgroups in Joe's big book

[whitejet]

The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book volume II. For a particular decomposition, SO(2k+1,1) → SO(2l+1,1) × SO(2k-2l) (B.1.43), the Weyl spinors decompose as the formula (B.1.44), 2^k → (2^l, 2^k-l-1)+(2'^l,2'^k-l-1) and 2'^k → (2'^l, 2^k-l-1)+(2^l,2'^k-l-1), where 2^k and 2'^k are the Weyl representations of Lorentz group SO(2k+1,1) with chirality +1 and -1 respectively.

Specifically on the case SO(9,1)→ SO(5,1)× SO(4) with decomposetions 16 → (4,2)+(4',2'), which appears at (B.6.3). My question is **the contradiction with minimum representations**. By checking Majorana and Weyl conditions, the minimum spinors for d=6 and d=4 have 8 and 4 components respectively. (Ref. table B.1 Polchinski) So How can you find (4,2) representation for SO(5,1)× SO(4)?

Also, I'm very interested in the prove of (B.1.44) under (B.1.43)? How is it proved by comparing the eigenvalues of $\Gamma^{+}\Gamma^{-}-\frac{1}{2}$ as claimed by Polchinski?

 

[fzero]

The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book volume II. For a particular decomposition, SO(2k+1,1) → SO(2l+1,1) × SO(2k-2l) (B.1.43), the Weyl spinors decompose as the formula (B.1.44), 2^k → (2^l, 2^k-l-1)+(2'^l,2'^k-l-1) and 2'^k → (2'^l, 2^k-l-1)+(2^l,2'^k-l-1), where 2^k and 2'^k are the Weyl representations of Lorentz group SO(2k+1,1) with chirality +1 and -1 respectively.

Specifically on the case SO(9,1)→ SO(5,1)× SO(4) with decomposetions 16 → (4,2)+(4',2'), which appears at (B.6.3). My question is **the contradiction with minimum representations**. By checking Majorana and Weyl conditions, the minimum spinors for d=6 and d=4 have 8 and 4 components respectively. (Ref. table B.1 Polchinski) So How can you find (4,2) representation for SO(5,1)× SO(4)?

That table lists the Dirac spinors, but in the text it is described how in even dimensions we can apply a Weyl condition to split the Dirac spinor into a pair of Weyl spinors of half the dimension. B.6.3 and B.1.44 are both considering the Weyl spinors.

Also, I'm very interested in the prove of (B.1.44) under (B.1.43)? How is it proved by comparing the eigenvalues of $\Gamma^{+}\Gamma^{-}-\frac{1}{2}$ as claimed by Polchinski?

Remember that the $S_a$ are written in terms of raising and lowering operators, so comparing the representation on the left with the product representations on the right is a generalization of the same method used to compute Clebsch-Gordan coefficients for SU(2). The pairings on the RHS of B.1.44 work out because the unprimed Weyl spinor has positive eigenvalue under $\Gamma = 2^{k+1} S_0 \cdots S_k$ (B.1.13), while the primed spinor has negative eigenvalue. So B.1.44A contains the even pairings, while B.1.44B has the odd ones.

 

[whitejet]

It makes sense that Polchinski is putting Weyl spinors in B.1.44 and B.6.3. As fzero pointed out by checking the chirality. However, I was told the number in formula like B.1.44 refers to the dimensions of real irreducible representation for the corresponding algebra/subalgebra. Since so(5,1) has the minimum 8 by 8 spinor representation, there are no 4 and 4' Weyl spinors for it. I think (4,2) in B.6.3 refers to representations other than spinors with dimensions 4 and 2 for so(5,1) and so(4) respectively. The following problem is what are these 4 and 2?

Taking another example so(9,1)→so(3,1)×so(6) as in the appendix A of http://arxiv.org/abs/hep-th/0205185v4, B.1.44 than states 16→(4,2)+(4',2'). (I think the authors might ignore the time in their notation.) Because of the accidental symmetry, so(6)≈su(4) and so(3)≈su(2), the 4 and 2 are realized as the fundamental representations of su(4) and su(2) here. Even if they didn't miss the time and were actually talking about so instead of lorentz, the branching rules might be different, but my point is still made, that B.1.44 is not using Weyl spinors.

Nevertheless, by using Clesch-Gordon decomposition to calculate B.1.44 explicitly, there shouldn't be any more doubts. Begging for a reference with explicit examples. All the books I have only consider tensor product of different representations under the same group, but not the tensor product of different groups.

 

[fzero]

It makes sense that Polchinski is putting Weyl spinors in B.1.44 and B.6.3. As fzero pointed out by checking the chirality. However, I was told the number in formula like B.1.44 refers to the dimensions of real irreducible representation for the corresponding algebra/subalgebra. Since so(5,1) has the minimum 8 by 8 spinor representation, there are no 4 and 4' Weyl spinors for it. I think (4,2) in B.6.3 refers to representations other than spinors with dimensions 4 and 2 for so(5,1) and so(4) respectively. The following problem is what are these 4 and 2?

The text is being sloppy: "smallest representation" in Table B.1 refers to the Dirac spinors. I think these are irreducible when considered as representations of the Clifford algebra. Viewed as representations of the spin algebra, they are reducible to Weyl representations in even dimensions.

Presumably representations of $so(4) =su(2) \oplus su(2)$ are already very familiar to you from QFT. The spinors are in the $(\mathbf{2},\mathbf{1})$ or $(\mathbf{1},\mathbf{2})$ corresponding to the left- and right-handed Weyl spinors. For $so(5,1)$, the spinors are constructed in B.6.4.

Taking another example so(9,1)→so(3,1)×so(6) as in the appendix A of http://arxiv.org/abs/hep-th/0205185v4, B.1.44 than states 16→(4,2)+(4',2'). (I think the authors might ignore the time in their notation.) Because of the accidental symmetry, so(6)≈su(4) and so(3)≈su(2), the 4 and 2 are realized as the fundamental representations of su(4) and su(2) here. Even if they didn't miss the time and were actually talking about so instead of lorentz, the branching rules might be different, but my point is still made, that B.1.44 is not using Weyl spinors.

I explained the spinors of $so(4)$ above. You might want to refer to your favorite QFT book to refresh your memory.

Nevertheless, by using Clesch-Gordon decomposition to calculate B.1.44 explicitly, there shouldn't be any more doubts. Begging for a reference with explicit examples. All the books I have only consider tensor product of different representations under the same group, but not the tensor product of different groups.

It's not exactly the same thing, since one is about products of representations of the same group (decomposition of reducible representations) and the other is about decomposing a irreducible representation of a group in terms of irreps of some subgroups (called branching rules for representations). The logic is still the same, we will identify highest weight states, use the raising and lowering operators to generate the complete representation, etc. Any book on representations of lie algebras will discuss this formalism, if not all aspects. For a hep-oriented physicist, the books by Georgi or Cahn would be reasonable choices.

 

[blue_raver22]

I find this question very interesting and relevant to my field of study. Spinor representations and their decompositions under subgroups are crucial in understanding the symmetries and structures of physical systems, particularly in string theory.

Firstly, let's address the issue of the contradiction with minimum representations. The minimum spinors for d=6 and d=4 have 8 and 4 components respectively, as stated in Polchinski's book. However, the decompositions (4,2) and (4',2') in the case of SO(9,1)→ SO(5,1)× SO(4) have 8 and 4 components respectively. This may seem contradictory at first, but it is important to note that these decompositions are not minimum representations. They are simply representations of the original spinor representations under the subgroup SO(5,1)× SO(4). Therefore, the number of components in these decomposed representations may differ from the minimum representations.

Moving on to the proof of (B.1.44) under (B.1.43), it is indeed a complex and technical process. The proof involves comparing the eigenvalues of $\Gamma^{+}\Gamma^{-}-\frac{1}{2}$ for the original spinor representations and the decompositions under the subgroup. This comparison shows that the decomposed representations indeed satisfy the necessary conditions for being spinors under the subgroup. However, I would recommend consulting other resources or consulting with experts in this field for a more detailed and thorough understanding of the proof.

In conclusion, the spinor representation decompositions under subgroups are a powerful tool in understanding the symmetries and structures in string theory. The apparent contradiction with minimum representations can be resolved by understanding the difference between the two and the proof of (B.1.44) under (B.1.43) can be understood through careful analysis of the eigenvalues. I hope this response has provided some clarification and insight into the topic.