Instanton contributions to the ABJM free energy from quantum M2 branes
Matteo Beccaria, Simone Giombi, A.A. Tseytlin
Abstract
A bstract We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at large N and fixed level k . Using supersymmetric localization, such instanton contribution was found earlier to take the form $$ {F}^{inst}\left(N,k\right)=-{\left({\sin}^2\frac{2\pi }{k}\right)}^{-1}\exp \left(-2\pi \sqrt{\frac{2N}{k}}\right)+.\dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>F</mml:mi> <mml:mtext>inst</mml:mtext> </mml:msup> <mml:mfenced> <mml:mi>N</mml:mi> <mml:mi>k</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:msup> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> </mml:msqrt> </mml:mrow> </mml:mfenced> <mml:mo>+</mml:mo> <mml:mo>.</mml:mo> <mml:mo>…</mml:mo> </mml:math> The exponent comes from the action of an M2 brane instanton wrapped on S 3 /ℤ k , which represents the M-theory uplift of the ℂP 1 instanton in type IIA string theory on AdS 4 × ℂP 3 . The IIA string computation of the leading large k term in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor $$ {\left({\sin}^2\frac{2\pi }{k}\right)}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> is reproduced by the 1-loop term in the M2 brane partition function expanded near the S 3 /ℤ k instanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization.