the quark condensate is known to be temperature-independent [@large-N-condensate]. At present, one does not know how to compute higher-order $1/N_c$ terms in the bulk sector. Clearly further progress in this direction is needed for the approach to describe what happens in the vicinity of the chiral restoration, a matter of prime importance in the field.

Vector Manifestation
====================

HLS à la Harada and Yamawaki
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Since one cannot yet adequately exploit the dynamics of the infinite tower in confronting nature, we will rely on the HLS theory of Harada and Yamawaki (HY) [@HY:PR] which involves only the lowest members $\rho$ (and $\omega$) of the infinite tower. Let us call it HLS$_1$ in contrast to the infinite-tower HLS theory of Sakai and Sugmoto which we will call HLS$_\infty$. The HLS$_1$ theory can be interpreted as a truncated version of HLS$_\infty$ in the following sense. One picks the scale $\Lambda_M$ as a matching scale, integrates out all the members of the tower except the lowest member $V_1$ lying below $\Lambda_M$, writes HLS Lagrangian with the $V_1$ and the pion and Wilsonian-matches at $\Lambda_M$ the HLS correlators to the QCD correlators. One can obviously include in the theory other vector mesons as for 