Arith / CONCEPT

LatticeThetaTransformation

For actual LatticeData D, assume its real Gram matrix is positive definite, its integral Gram determinant is a unit, and its rank equals m+m for a natural m. For every complex tau with positive imaginary part, the ordinary charge theta terms have summable norms and latticeTheta D (-1/tau)=(-i tau)^m latticeTheta D tau. The integer Gram matrix acts bijectively on charges, with its integral matrix inverse providing the inverse map. This reindexes the inverse-form theta series; positivity and integral unimodularity give determinant one. Rank zero is included. This theta identity does not identify any operator trace or oscillator factor.