Geometry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> Minkowski vacua of gauged <mml:math xmlns:mml="http://www.w3.org/1998/Math/Ma

Gauging isometries of four-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> supergravity theories yields an <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="script">N</d:mi><d:mo>=</d:mo><d:mn>2</d:mn></d:math> supersymmetric theory with a scalar potential. In this paper, we study the well-known constraints for four-dimensional <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="script">N</g:mi><g:mo>=</g:mo><g:mn>2</g:mn></g:math> Minkowski vacua of such theories. We propose that classically a projective special Kähler submanifold of the projective Kähler target space of the ungauged theory describes the moduli space of the complex scalar fields of massless vector multiplets for <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="script">N</j:mi><j:mo>=</j:mo><j:mn>2</j:mn></j:math> Minkowski vacua configurations, which then receives quantum corrections from integrating out massive fields. Subloci of projective special Kähler manifolds appear as supersymmetric flux vacua in the context of type IIB Calabi–Yau threefold compactifications with background fluxes as well. While these flux vacua equations arise from the critical locus of an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi mathvariant="script">N</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:math> superpotential, we show that these equations can also be obtained from the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi mathvariant="script">N</p:mi><p:mo>=</p:mo><p:mn>2</p:mn></p:math> supersymmetric Minkowski vacuum equations of gauged <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi mathvariant="script">N</s:mi><s:mo>=</s:mo><s:mn>2</s:mn></s:math> supergravity theories upon gauging suitable isometries in the semiclassical universal hypermultiplet sector of type IIB string Calabi–Yau threefold compactifications. Thus, we give an intrinsic <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi mathvariant="script">N</v:mi><v:mo>=</v:mo><v:mn>2</v:mn></v:math> supersymmetric interpretation to the flux vacua equations. Published by the American Physical Society 2024