Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On square-zero elements in the IIB supersymmetry algebra

Fabian Hahner

TL;DR

The paper analyzes the nilpotence variety $Y$ of odd square-zero elements in the ten-dimensional $\mathcal N=(2,0)$ super Poincaré algebra and classifies its orbits under $G=\mathrm{Spin}(10)\times \mathrm{O}(W)\times \mathbb C^\times$, revealing a rich stratification with two holomorphic twisting orbits and several mixed topological-holomorphic twists for type IIB supergravity in flat backgrounds. It uses a rank-based approach combined with the intersection pattern $\mathbb P(S_Q)\cap \mathcal P$ (where $\mathcal P$ is the pure-spinor variety $\mathrm{OG}(5,10)$) to distinguish rank-one and rank-two cases, and computes stabilizers and geometric backgrounds for each orbit (e.g., pure-isotropic, pure-nonisotropic, impure spinor, and four rank-two subcases). Key findings include the identification of two distinct holomorphic twisting orbits whose stabilizers differ by a $\mathbb C^\times$ factor, as well as multiple mixed twists with varying holonomy data, such as $\mathrm{SL}(3)\times\mathrm{SL}(2)$, $\mathrm{SL}(4)$, and $\mathrm{Sp}(4)$-type structures. The results provide a precise atlas of admissible twists for type IIB supergravity in flat space and illustrate the finer orbit stratification in ten dimensions compared to lower-dimensional cases, with implications for twisted supersymmetric theories and holography.

Abstract

We describe the variety of square-zero elements in the (2,0) super Poincaré algebra in ten dimensions, together with its orbit stratification induced by the action of the spin group and R-symmetry. This provides a classification of all possible twisting supercharges of type IIB supergravity in a flat background. Compared to other varieties of square-zero elements in super Poincaré algebras, we find that the orbit structure is more involved and exhibits several novel features. Among other things, we show that there are two distinct orbits of holomorphic twisting supercharges and a number of mixed topological-holomorphic twists.

On square-zero elements in the IIB supersymmetry algebra

TL;DR

The paper analyzes the nilpotence variety of odd square-zero elements in the ten-dimensional super Poincaré algebra and classifies its orbits under , revealing a rich stratification with two holomorphic twisting orbits and several mixed topological-holomorphic twists for type IIB supergravity in flat backgrounds. It uses a rank-based approach combined with the intersection pattern (where is the pure-spinor variety ) to distinguish rank-one and rank-two cases, and computes stabilizers and geometric backgrounds for each orbit (e.g., pure-isotropic, pure-nonisotropic, impure spinor, and four rank-two subcases). Key findings include the identification of two distinct holomorphic twisting orbits whose stabilizers differ by a factor, as well as multiple mixed twists with varying holonomy data, such as , , and -type structures. The results provide a precise atlas of admissible twists for type IIB supergravity in flat space and illustrate the finer orbit stratification in ten dimensions compared to lower-dimensional cases, with implications for twisted supersymmetric theories and holography.

Abstract

We describe the variety of square-zero elements in the (2,0) super Poincaré algebra in ten dimensions, together with its orbit stratification induced by the action of the spin group and R-symmetry. This provides a classification of all possible twisting supercharges of type IIB supergravity in a flat background. Compared to other varieties of square-zero elements in super Poincaré algebras, we find that the orbit structure is more involved and exhibits several novel features. Among other things, we show that there are two distinct orbits of holomorphic twisting supercharges and a number of mixed topological-holomorphic twists.

Paper Structure

This paper contains 15 sections, 10 theorems, 27 equations, 1 table.

Table of Contents

  1. Introduction and Summary
  2. Supersymmetry in ten dimensions and spinor orbits
  3. Spinor orbits of $\mathop{\mathrm{Spin}}\nolimits(10)$Spinor orbits of $\mathop{\mathrm{Spin}}\nolimits(10)$
  4. R-symmetry orbitsR-symmetry orbits
  5. Summary of the results
  6. Classification of orbits
  7. Rank one orbits
  8. The pure-isotropic orbitThe pure-isotropic orbit
  9. The pure-nonisotropic orbitThe pure-nonisotropic orbit
  10. The impure spinor orbitThe impure spinor orbit
  11. Rank two elements
  12. $\mathcal{P} \cap \mathbb P(S_Q) = \mathbb P(S_Q)$$\mathcal{P} \cap \mathbb P(S_Q) = \mathbb P(S_Q)$
  13. $\mathcal{P} \cap \mathbb P(S_Q) = \{p_1, p_2\}$$\mathcal{P} \cap \mathbb P(S_Q) = \{p_1, p_2\}$
  14. $\mathcal{P} \cap \mathbb P(S_Q) = \{p\}$$\mathcal{P} \cap \mathbb P(S_Q) = \{p\}$
  15. $\mathcal{P} \cap \mathbb P(S_Q) = \emptyset$$\mathcal{P} \cap \mathbb P(S_Q) = \emptyset$

Key Result

Theorem 1.2

The variety of square-zero supercharges for ten-dimensional $\mathcal{N}=(2,0)$ supersymmetry decomposes into the following orbits.

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 10 more