Thermal Duality and Hagedorn Transition from $p$-adic Strings

We develop the finite temperature theory of $p$-adic string models. We find that the thermal properties of these nonlocal field theories can be interpreted either as contributions of standard thermal modes with energies proportional to the temperature, or inverse thermal modes with energies proportional to the inverse of the temperature, leading to a thermal duality at leading order (genus one) analogous to the well-known $T$ duality of string theory. The $p$-adic strings also recover the asymptotic limits (high and low temperature) for arbitrary genus that purely stringy calculations have yielded. We also discuss our findings surrounding the nature of the Hagedorn transition.

Figure 1

Schematic description of string thermal modes. In string theory, the thermal computation can be understood by studying the propagation of fields in $R^{d-1}\times S^1$. The left plot shows the zero thermal state. In string theories, there are not only the standard thermal modes associated with the quantization of momentum, as the one in the middle ($n=11$), but also topological winding modes as the right one ($n_W=1$).

Figure 2

The one-loop diagram on the left does not contribute to the partition function. It shows that the free theory is empty. The two-loop diagram on the right constitutes the leading order contribution, first order in $\lambda$.

Figure 3

Partition function of the $p$-adic string model for $p=3$, $D=4$, $T_c=T_2$, $\lambda T_c^4=-0.04$, and $V=T_c^{-3}$. The symmetry of $Z_2(T)$ [solid (blue) line] with respect to $T_c$ (central vertical line) shows the exact realization of the thermal duality. However, the duality is broken at high temperatures by higher loop contributions [long dashed (green) line] or even at low energies by the addition of a self-energy counterterm [short dashed (red) line] [26].

Figure 4

Diagrams that contribute at second order in $\lambda$. The one on the left is the three-loop contribution to the necklace. The right one is the so-called sunset diagram.