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The chiral QED$_3$--Gross-Neveu-Yukawa (QED$_3$-GNY) theory is a$2+1$-dimensional U(1) gauge theory with $N_f$ flavors of four-component Diracfermions coupled to a scalar field. For $N_f=1$, the specific chiral IsingQED$_3$-GNY model has recently been conjectured to be dual to the deconfinedquantum critical point that describes Neel--valence-bond-solid transition offrustrated quantum magnets on square lattice. We study the universal criticalbehaviors of the chiral QED$_3$-GNY model in $d=4-\epsilon$ dimensions for anarbitrary $N_f$ . We calculate the boson anomalous dimensions, inversecorrelation length exponent, as well as the scaling dimensions of nonsingletfermion bilinear in the chiral QED$_3$-GNY model. The Pad$\acute{e}$ estimatesfor the exponents are obtained in the chiral Ising-, XY- andHeisenberg-QED$_3$-GNY universality class respectively. We also establish thegeneral condition of the supersymmetric criticality for the ungaugedQED$_3$-GNY model. For the conjectured duality between chiral QED$_3$-GNYcritical point and deconfined quantum critical point, we find the inversecorrelation length exponent has a lower boundary $\nu^{-1}>0.75$, beyond whichthe Ising-QED$_3$-GNY--$\mathbb{C}$P$^1$ duality may hold.

Critical behavior of QED$_3$--Gross-Neveu-Yukawa Theory in an Arbitrary Gauge