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Classical and quantum curves of 5d Seiberg's theories and their 4d limit

Oleg Chalykh, Yongchao Lü

Abstract

In this work, we examine the classical and quantum Seiberg-Witten curves of 5d N = 1 SCFTs and their 4d limits. The 5d theories we consider are Seiberg's theories of type $E_{6,7,8}$, which serve as the UV completions of 5d SU(2) gauge theories with 5, 6, or 7 flavors. Their classical curves can be constructed using the five-brane web construction [1]. We also use it to re-derive their quantum curves [2], by employing a q-analogue of the Frobenius method in the style of [3]. This allows us to compare the reduction of these 5d curves with the 4d curves, i.e. Seiberg-Witten curves of the Minahan-Nemeschansky theories and their quantization, which have been identified in [4] with the spectral curves of rank-1 complex crystallographic elliptic Calogero-Moser systems.

Classical and quantum curves of 5d Seiberg's theories and their 4d limit

Abstract

In this work, we examine the classical and quantum Seiberg-Witten curves of 5d N = 1 SCFTs and their 4d limits. The 5d theories we consider are Seiberg's theories of type , which serve as the UV completions of 5d SU(2) gauge theories with 5, 6, or 7 flavors. Their classical curves can be constructed using the five-brane web construction [1]. We also use it to re-derive their quantum curves [2], by employing a q-analogue of the Frobenius method in the style of [3]. This allows us to compare the reduction of these 5d curves with the 4d curves, i.e. Seiberg-Witten curves of the Minahan-Nemeschansky theories and their quantization, which have been identified in [4] with the spectral curves of rank-1 complex crystallographic elliptic Calogero-Moser systems.

Paper Structure

This paper contains 28 sections, 1 theorem, 81 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. 4d classical curves
  3. Case $m=3$, type $E_6$ :
  4. Case $m=4$, type $E_7$ :
  5. Case $m=6$, type $E_8$ :
  6. 5d classical curves and 4d limits
  7. 5d classical curves
  8. Case $m=3$, type $E_6$:
  9. Case $m=4$, type $E_7$:
  10. Case $m=6$, type $E_8$:
  11. 4d limit
  12. Case $m=3$, type $E_6$:
  13. Case $m=4$, type $E_7$:
  14. Case $m=6$, type $E_8$:
  15. 4d quantum curves
  16. ...and 13 more sections

Key Result

Proposition 5.1

For a difference operator $D = \sum_{i=0}^d x^iA_i(y)$, we have $(1)$$D$ has non-logarithmic singularities at $x = 0$ with $y = a,qa,\dots, q^{m-1}a$ iff $A_i(y)\propto \prod_{j=0}^{m-i-1}(y-q^ja)$ for $0\le i\le m-1$, $(2)$$D$ has non-logarithmic singularities at $x = \infty$ with $y = a,q^{-1}a,\d

Figures (2)

  • Figure 1: Shown above are the five-brane webs corresponding to 5d Seiberg's theories. The internal part of each diagram is schematically represented by a central circle, with the external legs illustrated in detail. The black dots represent seven-branes, and the lines represent five-branes.
  • Figure 2: Elliptic pencils. We use black dots to represent simple base points, 2-crosses for double points, and 3-crosses for triple points. In homogeneous coordinates, $x=U/W$, $y=V/W$ and the lines are $l_0: W=0$, $l_1: U=0$, $l_2: U-W=0$. The positions of the points are related to the mass parameters of the SCFT.

Theorems & Definitions (4)

  • Proposition 5.1: Proposition 3.1, MoriyamaY2021_AffineWeylQuantumCurve
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4