Self-duality and the holomorphic <i>ansatz</i> in a generalized BPS Skyrme model

We propose a generalization of the Bogomol’ny–Prasad–Sommerfield Skyrme model [L. A. Ferreira, Exact self-duality in a modified Skyrme model, ] for simple compact Lie groups <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>G</a:mi> </a:math> that leads to Hermitian symmetric spaces. In such a theory, the Skyrme field takes its values in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>G</c:mi> </c:math> , while the remaining fields correspond to the entries of a symmetric, positive, and invertible <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>dim</e:mi> <e:mtext> </e:mtext> <e:mi>G</e:mi> <e:mo>×</e:mo> <e:mi>dim</e:mi> <e:mtext> </e:mtext> <e:mi>G</e:mi> </e:math> -dimensional matrix <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>h</g:mi> </g:math> . We also use the holomorphic map between <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>S</i:mi> <i:mn>2</i:mn> </i:msup> <i:mo stretchy="false">→</i:mo> <i:mi>G</i:mi> <i:mo>/</i:mo> <i:mi>H</i:mi> <i:mo stretchy="false">⊗</i:mo> <i:mi>U</i:mi> <i:mo stretchy="false">(</i:mo> <i:mn>1</i:mn> <i:mo stretchy="false">)</i:mo> </i:math> proposed in Ferreira and Livramento [Harmonic, holomorphic and rational maps from self-duality, ] to study the self-dual sector of the theory, which generalizes the holomorphic between <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>S</o:mi> <o:mn>2</o:mn> </o:msup> <o:mo stretchy="false">→</o:mo> <o:mi>C</o:mi> <o:msup> <o:mi>P</o:mi> <o:mi>N</o:mi> </o:msup> </o:math> proposed in Ioannidou [Low-energy states in the SU(N) Skyrme models, in International Meeting on Mathematical Methods in Modern Theoretical Physics (ISPM 98) (1998), pp. 91–123, ]. This is constructed using the fact that stable harmonic maps of the two <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msup> <r:mi>S</r:mi> <r:mn>2</r:mn> </r:msup> </r:math> spheres for compact Hermitian symmetric spaces are holomorphic or antiholomorphic [J. Eells and L. Lemaire, (World Scientific Publishing Company, Singapore, 1995)]. Apart from some special cases, the self-duality equations do not fix the matrix <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>h</t:mi> </t:math> entirely in terms of the Skyrme field, which is completely free, as it happens in the original self-dual Skyrme model for <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>G</v:mi> <v:mo>=</v:mo> <v:mi>S</v:mi> <v:mi>U</v:mi> <v:mo stretchy="false">(</v:mo> <v:mn>2</v:mn> <v:mo stretchy="false">)</v:mo> </v:math> . In general, the freedom of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>h</z:mi> </z:math> fields tend to grow with the dimension of <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>G</bb:mi> </bb:math> . The holomorphic enable us to construct an infinite number of exact self-dual Skyrmions for each integer value of the topological charge and for each value of <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mi>N</db:mi> <db:mo>≥</db:mo> <db:mn>1</db:mn> </db:math> , in case of the <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mi>C</fb:mi> <fb:msup> <fb:mi>P</fb:mi> <fb:mi>N</fb:mi> </fb:msup> </fb:math> , and for each values of <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mrow> <hb:mi>p</hb:mi> </hb:mrow> </hb:math> , <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mrow> <jb:mi>q</jb:mi> <jb:mo>≥</jb:mo> <jb:mn>1</jb:mn> </jb:mrow> </jb:math> in case of <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>p</lb:mi> <lb:mo>+</lb:mo> <lb:mi>q</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo>/</lb:mo> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>p</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo stretchy="false">⊗</lb:mo> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>q</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo stretchy="false">⊗</lb:mo> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mn>1</lb:mn> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> </lb:math> .