We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in Supersymmetric Yang Mills (SYM) . In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in . We verify the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study.
Publié le :
DOI : 10.4171/aihpd/133
Mots-clés : SYM $N=4$, positive Grassmannians, Deodhar decomposition
@article{AIHPD_2023__10_1_159_0,
author = {Agarwala, Susama and Fryer, Si\^an Zee},
title = {A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of {Wilson} loop diagrams and {SYM} $N=4$},
journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
pages = {159--204},
volume = {10},
number = {1},
year = {2023},
doi = {10.4171/aihpd/133},
mrnumber = {4548773},
zbl = {1520.81074},
language = {en},
url = {http://archive.numdam.org/articles/10.4171/aihpd/133/}
}
TY - JOUR
AU - Agarwala, Susama
AU - Fryer, Siân Zee
TI - A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$
JO - Annales de l’Institut Henri Poincaré D
PY - 2023
SP - 159
EP - 204
VL - 10
IS - 1
UR - http://archive.numdam.org/articles/10.4171/aihpd/133/
DO - 10.4171/aihpd/133
LA - en
ID - AIHPD_2023__10_1_159_0
ER -
%0 Journal Article
%A Agarwala, Susama
%A Fryer, Siân Zee
%T A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$
%J Annales de l’Institut Henri Poincaré D
%D 2023
%P 159-204
%V 10
%N 1
%U http://archive.numdam.org/articles/10.4171/aihpd/133/
%R 10.4171/aihpd/133
%G en
%F AIHPD_2023__10_1_159_0
Agarwala, Susama; Fryer, Siân Zee. A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$. Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 1, pp. 159-204. doi : 10.4171/aihpd/133. http://archive.numdam.org/articles/10.4171/aihpd/133/
Cité par Sources :
