Geometric engineering, mirror symmetry and 6 d 1 0 → 4 d N = 2 $$ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $$

Abstract

Abstract:We study compactification of 6 dimensional (1,0) theories on T 2 . We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N = 2 geometry for a large number of the (1, 0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N = 2 SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, Z) duality symmetry inherited from global diffeomorphisms of the T 2 . This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T 2 . Among the resulting 4d N = 2 CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S with punctures from toroidal compactification of (1, 0) SCFTs where the curve of the class S theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S theories with no punctures on arbitrary genus Riemann surface.

6d N = 1 , 0 $$ \mathcal{N}=\left(1,\;0\right) $$ theories on S 1 /T 2 and class S theories: part II

“…In this way, we get two theories; one is a linear SU(k) N −1 quiver with the gauge coupling determined by the vev of Φ SU(N ) , and the other is a necklace SU(N ) k quiver. These are the theories discussed in [8]. Now we can see that these two theories flow from the single 4d theory (4.10) which has manifest SL(2, Z) S-duality and SU(k) L × SU(k) R flavor symmetry.…”

“…As is well known, the 5d SU(2) theory with four flavors in the strongly-coupled limit has the E 4+1 = SO (10) The enhancement from SO(8) that acts in the vector representation on the original four flavors to SO(10) is realized by adding a spinor representation of the SO (8). In (A.6), the SO(8) is broken to SO(7) in such a way that this spinor is decomposed as 8 = 7+1.…”

“…Before proceeding, we note that there recently appeared a paper [8] where the T 2 compactification of many classes of 6d N =(1, 0) theories were studied in terms of their Seiberg-Witten solutions, and many of the theories we discuss have already been analyzed there. We believe our paper provides complementary information about them, as our approach does not directly use the stringy features of F-theory as they did, and our focus is about the most singular point in the Coulomb branch whereas they mainly considered less singular points.…”

“…When g = E 7 and E 8 : the class S data for g = E 7 and E 8 are not yet available. Nonetheless, we consider the agreement we found so far is convincing enough that this correspondence works for all g. This can also be considered as a prediction for the repeated collision of the simple punctures in the class S theory of type E 7 and E 8 .…”

“…Given this statement, the two models above are clearly excluded, since so(7) V or so (8) under which the original four flavors transform as a vector representation is not a subgroup of so(7) S . Now let us see how this statement can be derived.…”