Mass-deformed super Yang-Mills theory on $$ {\mathbbm{T}}^4 $$: sum over twisted sectors, θ-angle, and CP violation

A bstract We study SU( N ) super Yang-Mills theory with a small gaugino mass m and vacuum angle θ on the four-torus $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> with ’t Hooft twisted boundary conditions. Introducing a detuning parameter ∆, which measures the deviation from an exactly self-dual $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> , and working in the limits mLN ≪ Λ LN ≪ 1 and $$ \frac{\left(N-1\right){m}^2{L}^2}{4\pi}\ll \Delta \ll 1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>≪</mml:mo> <mml:mtext>∆</mml:mtext> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:math> , where L is the torus size and Λ the strong-coupling scale, we compute the scalar and pseudo-scalar condensates to leading order in m 2 L 2 /∆. The twists generate fractional-charge instantons, and we show that summing over all such contributions is crucial for reproducing the correct physical observables in the decompactified strong-coupling regime. From a Hamiltonian perspective, the sum over twisted sectors, already at small torus size, projects in the m = 0 limit onto a definite superselection sector of the ℝ 4 theory. In the massless limit, we recover the exact value of the gaugino condensate |〈 λλ 〉| = 16 π 2 Λ 3 , and demonstrate how a spurious U(1) symmetry eliminates all $$ \mathcal{CP} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CP</mml:mi> </mml:math> -violating effects. Our results are directly testable in lattice simulations, and our method extends naturally to non-supersymmetric gauge theories.