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Testing 5d-6d dualities with fractional D-branes

Youngbin Yun

TL;DR

This work tests a proposed 5d-6d duality for 6d $ ext{N}=(1,0)$ SCFTs with $Sp(N)$ gauge symmetry and $N_f=2N+8$ fundamentals by comparing 6d self-dual string elliptic genera with 5d instanton partition functions in the dual $Sp(N+1)$ theories. It highlights the need to treat fractional D-brane sectors and crucial background Wilson lines to align the two pictures, establishing a precise fugacity map between 5d and 6d data. The authors compute explicit one-, two-, and three-string elliptic genera in 6d and match them to corresponding 5d indices (perturbative plus instanton) up to specified orders, confirming the proposed dualities and clarifying parameter identifications. The results show how 5d instantons correspond to KK momenta along the 6d circle and underscore the importance of the $O(k)$ ADHM structure and Wilson lines in achieving agreement, with implications for related dualities (e.g., $Sp(N+1)$ vs. $SU(3)$) and the interpretation of stringy degrees of freedom.

Abstract

6d SCFTs compactified on a circle can often be studied from nonperturbative 5d super-Yang-Mills theories, using instanton solitons. However, the 5d Yang-Mills theories with 6d UV fixed points frequently have too many hypermultiplet matters, which makes it difficult to use the ADHM techniques for instantons. With the examples of 6d $\mathcal{N}=(1,0)$ SCFTs with $Sp(N)$ gauge symmetry and $2N+8$ fundamental hypermultiplets, we show that one can still make rigorous studies of these 5d-6d relations in the `fractional D-brane sectors'. We test the recently proposed 5d duals given by $Sp(N+1)$ gauge theories, and compare their instanton partition functions with the elliptic genera of 6d self-dual strings.

Testing 5d-6d dualities with fractional D-branes

TL;DR

This work tests a proposed 5d-6d duality for 6d SCFTs with gauge symmetry and fundamentals by comparing 6d self-dual string elliptic genera with 5d instanton partition functions in the dual theories. It highlights the need to treat fractional D-brane sectors and crucial background Wilson lines to align the two pictures, establishing a precise fugacity map between 5d and 6d data. The authors compute explicit one-, two-, and three-string elliptic genera in 6d and match them to corresponding 5d indices (perturbative plus instanton) up to specified orders, confirming the proposed dualities and clarifying parameter identifications. The results show how 5d instantons correspond to KK momenta along the 6d circle and underscore the importance of the ADHM structure and Wilson lines in achieving agreement, with implications for related dualities (e.g., vs. ) and the interpretation of stringy degrees of freedom.

Abstract

6d SCFTs compactified on a circle can often be studied from nonperturbative 5d super-Yang-Mills theories, using instanton solitons. However, the 5d Yang-Mills theories with 6d UV fixed points frequently have too many hypermultiplet matters, which makes it difficult to use the ADHM techniques for instantons. With the examples of 6d SCFTs with gauge symmetry and fundamental hypermultiplets, we show that one can still make rigorous studies of these 5d-6d relations in the `fractional D-brane sectors'. We test the recently proposed 5d duals given by gauge theories, and compare their instanton partition functions with the elliptic genera of 6d self-dual strings.

Paper Structure

This paper contains 21 sections, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. E-strings and their $Sp(N)$ generalizations
  3. The elliptic genera of self-dual strings
  4. One-string
  5. Two-strings
  6. 5d SYM and instanton partition functions
  7. One-instanton: $N=0$
  8. Perturbative part: $N=0$
  9. 6d SCFT with $Sp(1)$ gauge symmetry
  10. 6d index
  11. One-string
  12. Two-strings
  13. Three-strings
  14. Perturbative index
  15. 5d index
  16. ...and 6 more sections

Figures (4)

  • Figure 1: type IIA brane system for 6d $\mathcal{N}=(1,0)$$Sp(N)$ gauge theory with $N_f=2N+8$ fundamental hypermultiplets. $n$ D2 branes engineer $n$ self-dual strings.
  • Figure 2: 2d ADHM quiver diagram for the self-dual strings
  • Figure 3: type IIB brane diagram for the 5d $\mathcal{N}=1$$Sp(2)$ gauge theory with $N_f=10$ hypermultiplets. The figure shows the covering space of $\mathbb{Z}_2$ quotient by O7 (the cross in the figure). The blue dots denote 7-branes on which vertical 5-branes can end. Half-D1 brane is stuck to the O7$^-$-plane.
  • Figure 4: type IIB brane diagram for the 5d $\mathcal{N}=1$$Sp(2)$ gauge theory with $N_f=8$ hypermultiplets.