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Circle and Torus Compactifications

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Six-Dimensional Superconformal Field Theories and Their Torus Compactifications

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Abstract

In Chap. 2, we have reviewed some basic properties of some examples of 6d SCFTs. In this chapter, we would like to investigate torus compactifications of the theories which appeared in the previous chapter.

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Notes

  1. 1.

    \(\mathscr {N}{=}\,4\) SYM is not self-dual under the S-duality even when \(G=\mathrm {SU}(N)\) since its Langlands dual is \(\mathrm {SU}(N)/\mathbb {Z}_N\). The global data depends on choice of basis of cycle, and this is because the “meta”-ness of the \(A_{N-1}\) \(\mathscr {N}{=}\,(2,0)\) theory [1]. This subtlety exists also for \(\mathscr {N}{=}\,(1,0)\) theories which is not very-higgsable though we will not study further in this direction.

  2. 2.

    Here, we focus on the case where we can go to the root to \(\mathscr {N}{=}\,(2,0)\) theory by recursively shrinking tensor vev \(a^k\) with \(\eta ^{kk}=1\). In other words, the root to \(\mathscr {N}{=}\,(2,0)\) is the endpont. A counterexample of this restriction is \(\mathscr {T}^{({\mathfrak {usp}},{\mathfrak {usp}})}_{N}\).

  3. 3.

    Here we do not introduce Wilson lines along the torus. When generic Wilson lines are turned on, the situation is different [3].

  4. 4.

    Instead, if we allow ourselves to turn on Wilson lines as we discussed in Sect. 2.4.4 for \(\mathscr {T}^{({\mathfrak {g}},{\mathfrak {g}})}_{N}\), the two properties are satisfied when compactified further to 4d, since the affine quiver is conformal in 4d. In fact the generalization to compactification by general Riemann surfaces with nontrivial flavor bundles gives 4d \(\mathscr {N}{=}\,1\) SCFTs [4], and \({\mathfrak {g}}=A_{k-1}\) case which is called class \(\mathrm {S}_k\) is somewhat extensively studied [3, 5].

  5. 5.

    The Higgs branch is robust under the compactification thanks to eight supercharges, thus u does not mix with Higgs scalars.

  6. 6.

    Instead, asymptotic behavior (3.6) is enough to constrain the special geometry as said in [6].

  7. 7.

    This number is related to the fact that an O7\({}^-\) is actually a non-pertubative bound-state of (1, 1) and \((1,-1)\) 7-branes and thus there are 12 branes in the left of Fig. 3.1. We are going to heavily use this fact in Sect. 3.2.

  8. 8.

    In fact, in general A(u) though to be equal to \((\frac{\partial u}{\partial a})^{\frac{1}{2}}\). The later calculation will be simplified when this formula is assumed [9].

  9. 9.

    The method here is never independent of the method of [9]. This is just a consistency check.

  10. 10.

    The paper [24] coincidently appeared on arXiv with [23]. The basic strategy is almost the same, and the former covers more general cases than the latter.

  11. 11.

    When \(u_i=1\) for \(i=2,\ldots ,N\), \(\mathscr {T}^\text {6d}\{u_i=1\}\) is the rank N E-string theory plus a decoupled hyper, and the corresponding theory is \(\widehat{\mathsf {T}}_{6N}\{[N^5,N-1,1],[2N,2N,2N],[3N,3N]\}\), which was firstly observed by the index calculation [26].

  12. 12.

    Our normalizations for central charges and anomaly polynomial are those of [6, 36]

  13. 13.

    The formulas below are valid only when \(\sum _ip^{(i)}_k\ge 2k-1\). When \(u_i=0\) which corresponds to the higher rank \(E_8\) Minahan–Nemeschansky theory, the pole structure for the class S description violates this bound. That case was studied well in [25] as already mentioned.

  14. 14.

    This condition is the same as the conformality condition of 4d \(\mathscr {N}{=}\,2\) quiver theory with \({\mathfrak {su}}\) gauge algebras. Intuitive understanding of this coincidence seems to be absent.

  15. 15.

    Since the 6d theory has the Higgs branch on which the theory flows to the \(\mathscr {N}{=}\,(2,0)\) theory along \(\mathscr {C}_\mathrm {T}\), there is also a subspace of the 5d/4d Coulomb branch where the corresponding branch opens. This clearly defines the subspace \(\mathscr {C}_\mathrm {T}\) in 5d/4d.

  16. 16.

    There, it was shown that the \(T^2\) compactification of very higgsable theory is a 4d SCFT, and the structure of the singularities on its Coulomb branch was also completely fixed. Taking the limit of very thin \(T^2\), we can obtain the singularity structure of the Coulomb branch of the 5d theory, which shows that the origin of the 5d theory is superconformal.

  17. 17.

    See the last equation in Sect. 3.4 of [45]. The \(\mathfrak {m}_\pm \) in that paper is taken to be \(m_G\) here, and \(H_i\) there is \(\frac{1}{2}H^i\) here.

  18. 18.

    The full answer for \({\mathfrak {g}}=D_k\) case was obtained after publishing [36], and appears nowhere in the literature.

  19. 19.

    Note that we have \({{\mathfrak {g}}}_T={\mathfrak {g}}_B={{\mathfrak {g}}}_L={\mathfrak {g}}_R={\mathfrak {g}}\) here. The subscripts are there to distinguish various factors.

  20. 20.

    A puncture of class S of type G theory can be twisted by a nontrivial outer-automorphism of G.

  21. 21.

    When \(N=2\), since the puncture given by colliding [2, 2] and \(\underline{T\!M}\) is \(\underline{[2^2,1]}\) in the notation of [56] which is smaller than the twisted full puncture \(\underline{T\!F}\), \(\underline{\mathscr {O}_k}\) = \(\underline{[2^2,1]}\). When \(N\ge 3\) the puncture arising from [2, 2] and \(\underline{T\!M}\) is the twisted full puncture \(\underline{T\!F}\), so the statement of the main text is correct.

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  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA

    Kantaro Ohmori

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Ohmori, K. (2018). Circle and Torus Compactifications. In: Six-Dimensional Superconformal Field Theories and Their Torus Compactifications. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-13-3092-6_3

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