  We present high-resolution observations (0".2-1".5) of multiple dense gas
tracers, HCN and HCO$^+$ ($J$ = 1-0, 3-2, and 4-3), HNC ($J$ = 1-0), and CS
($J$ = 7-6) lines, toward the nearby luminous infrared galaxy VV 114 with the
Atacama Large Millimeter/submillimeter Array. All lines are robustly detected
at the central gaseous filamentary structure including the eastern nucleus and
the Overlap region, the collision interface of the progenitors. We found that
there is no correlation between star formation efficiency and dense gas
fraction, indicating that the amount of dense gas does not simply control star
formation in VV 114. We predict the presence of more turbulent and diffuse
molecular gas clouds around the Overlap region compared to those at the nuclear
region assuming a turbulence-regulated star formation model. The intracloud
turbulence at the Overlap region might be excited by galaxy-merger-induced
shocks, which also explains the enhancement of gas-phase CH$_3$OH abundance
previously found there. We also present spatially resolved spectral line energy
distributions of HCN and HCO$^+$ for the first time, and derive excitation
parameters by assuming optically-thin and local thermodynamic equilibrium (LTE)
conditions. The LTE model revealed that warmer, HCO$^+$-poorer molecular gas
medium is dominated around the eastern nucleus, harboring an AGN. The HCN
abundance is remarkably flat ($\sim$3.5 $\times$ 10$^{-9}$) independently of
the various environments within the filament of VV 114 (i.e., AGN, star
formation, and shock).
