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The particle origin of dark matter (DM) is still one of the main puzzles inmodern physics. One of the most promising search strategy to detect DM atlaboratories is through the indirect search of cosmic particles that areproduced from DM annihilation in space. In particular, the flux of cosmicpositrons has been measured with high precision by the AMS-02 experimentdemonstrating that an excess above 10 GeV, with respect to the secondaryproduction, is present. We study in this paper the possible DM origin of thepositron excess finding the values of the DM mass $M$ and annihilation crosssection $\langle \sigma v \rangle$ that are needed to fit high-energy positrondata. In particular, we find that for DM annihilating into $b\bar{b}$ it isrequired to have $M=43$ TeV and $\langle \sigma v \rangle = 10^{-21}$ cm$^3$/swhile for $\tau^+\tau^-$ $M=2$ TeV and $\langle \sigma v \rangle = 3\times10^{-23}$ cm$^3$/s. If DM produce positrons, they are expected to generategamma rays from the center of the Milky Way and around dwarf galaxy satellitesof the Galaxy. We thus combine the values for the DM mass and annihilationcross section obtained with the fit to AMS-02 positron data with the upperlimits derived with the non-detection of $\gamma$ rays with HESS in thedirection of the Galactic center and Fermi-LAT for the combined analysis ofdwarf galaxies. The main result of the paper is that only DM annihilating into$\mu^+ \mu^-$ with a mass around 500 GeV and $\langle \sigma v \rangle =4\times 10^{-24}$ cm$^3$/s can fit AMS-02 data and be compatible with the upperlimits found with $\gamma$ rays. As for the $\tau^+ \tau^-$ ($b\bar{b}$)channel, DM can contribute at most at a few tens $\%$ (a few \%) level.

Constraining the dark matter interpretation of the positron excess with $\gamma$-ray data