Perhaps you don't even need a negative voltage.

IF I understood correctly, you're trying to generate a bipolar signal (sine wave) using the Arduino, and a Unipolar DAC. You cannot produce DC negative voltages with this hardware. But you CAN produce AC audio signals regardless, and WITHOUT using bipolar supplies.

You have the Arduino which cannot output anything negative. You have your DAC (which by description is Unipolar). Research Bipolar DACs to understand the difference.

Either way, you cannot produce negative voltages directly BUT... suppose you want to produce a Sine Wave, and you decide that your resolution is +/- 127, meaning 7 bits with sign. Write your CODE such that in calculating Amplitude of the SINE, comes up with a 7 bit result (0-127, negative or positive) which is then simply ADDED to 128... Feed that to the DAC.

Follow me here. Your result +/- 127 is added to 128, producing a result which is between 1 and 255. This is in the range of the 8 bit DAC, representing a +/- excursion which is atypical of a bipolar output, but is otherwise a positive voltage.

Now AC couple this DAC output voltage to your audio amplifier. Capacitive coupling if you will. This is the answer without resorting to bipolar solutions; if you don't know what this means, do some research in AC coupling.

HTH, DHM