Large θ angle in two-dimensional large N $$ {\mathbbm{CP}}^{N-1} $$ model

A bstract In confining large N theories with a θ angle such as four-dimensional SU( N ) pure Yang-Mills theory, there are multiple metastable vacua and it makes sense to consider the parameter region of “large θ of order N ” despite the fact that θ is a 2 π -periodic parameter. We investigate this parameter region in the two-dimensional $$ {\mathbbm{CP}}^{N-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>CP</mml:mi> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> model by computing the partition function on T 2 . When θ / N is of order $$ \mathcal{O}(0.1) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mn>0.1</mml:mn> </mml:mfenced> </mml:math> or less, we get perfectly sensible results for the vacuum energies and decay rates of metastable vacua. However, when θ / N is of order $$ \mathcal{O}(1) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:math> , we encounter a problem about saddle points that would give larger contributions to the partition function than the true vacuum. We discuss why it might not be straightforward to resolve this problem.