Improved methods for hypergraphs
(talk at Adventures in Superspace, 4/19/13)
Dharmesh Jain and Warren Siegel
Earlier work:
“Hypersymmetry”: N=2 supersymmetry — Fayet (’76)
⋮ (cf. “Bambi Meets Godzilla”)
Hypergraphs:
 Ivanov, Galperin, Ogievetsky, Sokatchev (’85)
 GonzalezRey, Roček, Wiles, Lindström, von Unge (’978)
 Jain, Siegel (’0912)
Background hyperfields: Buchbinder², Ivanov, Kuzenko, Ovrut, McArthur, Petrov (’97’02)
This paper does for N=2 supergraphs what was done for N=1 by ...
Improved methods for supergraphs
Marcus T. Grisaru, W. Siegel (Brandeis U.), M. Roček (Cambridge U.). Jun 1979. 32 pp.
Published in Nucl.Phys. B159 (1979) 429
Cited by 742 records (INSPIRE)

⇐ Actual experimental data related to supersymmetry
⇓ 
Cited by 910 (Google scholar)
Background field formalism


N = 1 

N = 2 (6D N=1) 
quantum superfield 

scalar 

scalar 
superspace 

x, full θ 

x, analytic (½) θ, (internal) y 
representation 

chiral 

analytic 

background superfield 

spinor 

spinor 
superspace 

x, full θ 

x, full θ, no y 
representation 

real 

real 

nonrenormalization 

obvious* 

obvious* 
effective action 

x, θ 

x, θ (no internal) 
*As for N=1,
Quantum superfield is scalar prepotential of dimension 0;
background superfield is spinor (or maybe vector) potential with dimension > 0.
N=4 YangMills
1loop cancelations in N=4 YangMills as formulated in N=1 or N=2 superspace:


N = 1 

N = 2 
scalar multiplets 

3 

1 
FaddeevPopov ghosts 

2 

2 
NielsenKallosh ghosts 

1 

1 

total 

0 

0 

vector multiplets 

1 

1 
“extra” ghosts 

0 

12 

total 

1 

0* 
*Same propagator, different vertex ⇒ cancels only ydivergence δ(0).
In both cases, vector multiplets etc. contribute only @ 4point & higher,
scalar multiplets etc. also @ lowerpoint.
Equations
In case there’s too much time left, some actual equations:
scalar/FP/NK propagator: 

∇₁_{ϑ}⁴∇₂_{ϑ}⁴δ⁸(θ₁₂)

vector/XR propagator: 

∇₁_{ϑ}⁴δ⁸(θ₁₂)

scalar/FP/NK vertex: 

∫d⁴θ dy (◻̂◻₀)

vector vertex: 

∫d⁴θ dy y(◻̂◻₀)

XR vertex: 

∫d⁴θ d²y [1 + y₁δ(y₁₂)](◻̂◻₀)

Above are for just 1 loop (free quantum in background).
For vertices, use ∇_{ϑ}⁴ from propagator to make ∫d⁴θ ∇_{ϑ}⁴ = ∫d⁸θ.
Conclusions
 Same kind of simplifications for N=2 as for N=1 (1 loop & higher)
 Quantum field V (x, θ, y), where A_{ϑ} = 0;
background fields A_{θ}, A_{ϑ}, where A_{y} = 0, trivial dependence on y
 Classical action in analytic superspace d⁴x d⁴θ dy, nonlocal in y;
effective action in “full” superspace d⁴x d⁴θ d⁴ϑ, no y
 N=3 supergraphs (for N=4 YangMills): in progress
 Supergravity
 1stquantization?
That’s a good question!
Quantum vertices (background appears only through ∇
_{ϑ}⁴):
scalar: 

∫d⁴θ Ῡ(e^{V}1)Υ

vector: 

∫d⁴θ d⁴ϑ dⁿy
(e^{V₁}1)∙∙∙(e^{Vn}1) 
y₁₂y₂₃∙∙∙y_{n}₁ 

FP: 

∫d⁴θ (yb+b̄)L_{V/2}
coth(L_{V/2})
c
+
c+

Nonlocality in y gets no background covariantization, since A_{y} = 0.