Thermofield Dynamics of the Heterotic String — Physical Aspects of the Thermal Duality —

The thermofield dynamics of the D = 10 heterotic thermal string theory is described in proper reference to the thermal duality symmetry as well as the thermal stability of modular invariance in association with the global phase structure of the D = 10 heterotic thermal string ensemble. Building up thermal string theories based upon the thermofield dynamics (TFD) [1] has gradually been endeavoured in leaps and bounds [2] – [9]. In the present communication, physical aspects of the thermal duality symmetry are commentarially exemplified à la recent publication of ourselves [8] through the infrared behaviour of the one-loop cosmological constant in proper reference to the thermal stability of modular invariance for the D = 10 heterotic thermal string theory based upon the TFD algorithm. The global phase structure of the D = 10 heterotic thermal string ensemble is then recapitulatively touched upon. Let us start with the one-loop cosmological constant Λ(β) as follows: Λ(β) = α 2 lim μ2→0 Tr [∫ μ2 ∞ dm ( ∆βB(p, P ;m ) + ∆βF (p, P ;m ) )] (1) at any finite temperature β = kT in the D = 10 heterotic thermal string theory based upon the TFD algorithm, where α means the slope parameter, p reads loop momentum, P I lie on the even self-dual root lattice L = Γ8×Γ8 for the exceptional group G = E8×E8 [14] and the thermal propagator ∆βB[F ](p, P ;m ) of the free closed bosonic [fermionic] string is written à la Leblanc [2] in the form ∆βB[F ](p, P ;m ) = ∫ π −π dφ 4π e( ∑16 I=1 (P )) × ( + [− ] ∫ 1 0 dx+ 1 2 ∞ ∑ n=0 δ[α/2 · p + α/2 ·m + 2(n− α)] eβ|p0| − [ + ] 1 ∮ c dx  ×x ′/2·p2+N−α+N̄−ᾱ+1/2· ∑16 I=1 (P I )2+α′/2·m2−1 ) , (2) where N [N̄ ] denotes the number operator of the right[left-] mover, the intercept parameter α [ᾱ] is fixed at α = 0 [ᾱ = 1] and the contour c is taken as the unit circle around the origin. We are then eventually led to the modular parameter integral representation of Λ(β) at D = 10 as follows [8]: 1