Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On holographic duals of certain isolated weighted Gorenstein cDV singularities

Yuanyuan Fang, Zekai Yu

TL;DR

<3-5 sentence high-level summary> This paper develops a holographic no-go framework for a class of isolated weighted Gorenstein cDV singularities by combining K-stability with a mirror-symmetry–driven Hochschild-cohomology computation of LG mirrors. The approach uses symplectic cohomology as a diagnostic for crepant resolutions and NCCRs, and then cross-validates with a physics-side enumeration of large-N 4d N=1 superconformal quiver theories. For the cE6 and cE7 families, the authors compute Hochschild cohomology on Berglund– Hübsch–Krawitz mirrors, showing vanishing of HH^2 and stabilization of negative-degree SH only in exceptional k, thereby ruling out crepant resolutions in most cases and hence the corresponding holographic backgrounds. They further implement a physics-based search to enumerate potential dual quivers and confirm there are no consistent large-N duals, reinforcing the no-go result and illustrating a practical workflow for testing holographic viability of singularities via mirror-symmetric invariants.

Abstract

We employ a novel approach,based on homological mirror symmetry for Landau-Ginzburg models,to demonstrate the non-existence of crepant resolutions for certain weighted homogeneous Gorenstein compound Du Val singularities.Physically,this implies that such singularities cannot serve as holographic backgrounds for four dimensional N=1 superconformal quiver gauge theories realized on the worldvolume of a large number of D3 branes placed at the singular locus.This is confirmed by enumerating all consistent quiver gauge theories.

On holographic duals of certain isolated weighted Gorenstein cDV singularities

TL;DR

<3-5 sentence high-level summary> This paper develops a holographic no-go framework for a class of isolated weighted Gorenstein cDV singularities by combining K-stability with a mirror-symmetry–driven Hochschild-cohomology computation of LG mirrors. The approach uses symplectic cohomology as a diagnostic for crepant resolutions and NCCRs, and then cross-validates with a physics-side enumeration of large-N 4d N=1 superconformal quiver theories. For the cE6 and cE7 families, the authors compute Hochschild cohomology on Berglund– Hübsch–Krawitz mirrors, showing vanishing of HH^2 and stabilization of negative-degree SH only in exceptional k, thereby ruling out crepant resolutions in most cases and hence the corresponding holographic backgrounds. They further implement a physics-based search to enumerate potential dual quivers and confirm there are no consistent large-N duals, reinforcing the no-go result and illustrating a practical workflow for testing holographic viability of singularities via mirror-symmetric invariants.

Abstract

We employ a novel approach,based on homological mirror symmetry for Landau-Ginzburg models,to demonstrate the non-existence of crepant resolutions for certain weighted homogeneous Gorenstein compound Du Val singularities.Physically,this implies that such singularities cannot serve as holographic backgrounds for four dimensional N=1 superconformal quiver gauge theories realized on the worldvolume of a large number of D3 branes placed at the singular locus.This is confirmed by enumerating all consistent quiver gauge theories.

Paper Structure

This paper contains 24 sections, 82 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General strategies
  3. $K$-stability of the singularities
  4. Reduction of the problem
  5. Resolutions via semi-universal unfolding.
  6. Topology of the link and number of exceptional curves.
  7. Symplectic cohomology
  8. Hochschild cohomology: an introduction
  9. Berglund-Hübsch-Krawitz mirror symmetry for Landau-Ginzburg models
  10. The example of $cA_1$:
  11. Computation of symplectic cohomology of $cE_6$ and $cE_7$ singularities
  12. $cE_6$ singularities
  13. Example $k=8$:
  14. General values of $k$:
  15. $cE_7$ singularities
  16. ...and 9 more sections

Figures (3)

  • Figure 1: The possible shape of the quivers corresponding to the (one-node-deleted) NCCR of the cDV singularities. august2020tilting The dotted arc represents that there may be an adjoint chiral or may be not.
  • Figure 2: There are at most two adjoints on the distinguished node $N_1$. There may be a pair of bifundamental chirals between all the other nodes and the distinguished node.
  • Figure 3: Two quiver gauge theories in which every gauge node is an unitary gauge group. Both theories have $TrR=0$. $TrR^3$ are all $\frac{2560 N^2}{81}$. Furthermore, we have checked that their superconformal indices coincide in the large $N$ limit.

Theorems & Definitions (3)

  • Claim 1.1
  • Conjecture 2.2
  • Conjecture 2.3