Combinatorics in 𝒩=1 Heterotic Vacua

Table 2

The list of standard-like constraints on the 𝒩=1 heterotic vacua, each described by a Calabi-Yau threefolds 𝑋 and a 𝒢-bundle 𝑉 thereon; the integers 𝑏^𝑟_𝑖 and 𝑐^𝑟_𝑗 parameterise the bundle 𝑉 as in (3.1), where 𝑖=1,…,𝑟_𝑏,  𝑗=1,…,𝑟_𝑐, and 𝑟=1,…,ℎ^1,1(𝑋). The second column states the constraints on the background geometry, and the third column expresses the corresponding algebraic equations that the parameters 𝑏^𝑟_𝑖 and 𝑐^𝑟_𝑗 must obey, where 𝑑_𝑟𝑠𝑡=∫_𝑋𝐽_𝑟∧𝐽_𝑠∧𝐽_𝑡 are the intersection numbers for a given basis {𝐽_𝑟}_{1≤𝑟≤ℎ}^1,1_(𝑋) for 𝐻^1,1(𝑋,ℝ).

Physics origin	Background geometry	Algebraic constraints on 𝐛𝑖 and 𝐜𝑗
Gauge group	(a)𝒢=𝑆𝑈(𝑛),for(𝑛=4,5b)𝑋hasadiscretefreeaction𝐺	(a1)𝑟𝑏−𝑟𝑐=4or5(a2)∑^𝑟𝑏𝑖=1𝑏^𝑟𝑖=∑^𝑟𝑐𝑗=1𝑐^𝑟𝑗,forall𝑟
Anomaly	𝑐2(𝑇𝑋)−𝑐2(𝑉) is effective	𝑐2(𝑇𝑋)𝑟−(1/2)𝑑𝑟𝑠𝑡(∑^𝑟𝑐𝑗=1𝑐^𝑠𝑗𝑐^𝑡𝑗−∑^𝑟𝑏𝑖=1𝑏^𝑠𝑖𝑏^𝑡𝑖)>0,forall𝑟
Particle spectra	(1/2)∫𝑋𝑐3(𝑉)=−3|𝐺|	𝑑𝑟𝑠𝑡(∑^𝑟𝑏𝑖=1𝑏^𝑟𝑖𝑏^𝑠𝑖𝑏^𝑡𝑖−∑^𝑟𝑐𝑗=1𝑐^𝑟𝑗𝑐^𝑠𝑗𝑐^𝑡𝑗)=−18|𝐺|