D-type fiber-base duality

Abstract

M5 branes probing D-type singularities give rise to 6d (1,0) SCFTs with SO × SO flavor symmetry known as D-type conformal matter theories. Gauging the diagonal SO-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits.

M5 branes and theta functions

“…We end this Section by proving that the Seiberg-Witten curve given in (3.83) matches exactly the result obtained in [29] where different methods were used. To see this, note that the Seiberg-Witten curve there was given by 0 = θ 3 (τ, 0) 2 θ 2 (τ, 0) 2 4η(τ ) 6 · 4 n=0 a n X(ρ, z 2 ) n − θ 3 (τ, 0) 4 + θ 2 (τ, 0) 4 12 + X(τ, z 1 ) 4 · c 0 64 Y (ρ, z 2 ) 2 ,…”

“…Another aspect is that everything we have done in this paper is within a specific limit of the Omega background, namely we put our M5 branes on T 2 × 1 , 2 R 4 and take the limit 1 = − 2 → 0. Indeed this is the same limit taken in [19,29] when deriving the Seiberg-Witten curves using the formalism of the "thermodynamic limit". It would be very interesting to repeat our analysis for the case of the full Omega background, i.e.…”

“…In this Section we want to construct wave functions for M5 branes probing D-type singularities. Here our discussion will be enriched by a fundamental new symmetry, namely the Z 2 reflection symmetry of the root lattice of D n as already observed in [29], which acts on our wave functions and under which they have to be invariant. Consider in the following the LST arising from a single M5 brane probing a singularity of type D 4 , i.e.…”

“…Putting the two duality frames together we again see that our wavefunctions must be sections of line bundles on T Ω = T τ × T ρ where this time the off-diagonal components of Ω are zero as there is no mass-deformation in the D-type theory. The action of Z 2 is then nothing else than the reflection [29] (−1) *…”

Blowup equations for 6d SCFTs. Part I

“…We trace this symmetry to fiber-base duality of D-type theories [45]. This can be understood as follows.…”