Jacobi-Lie T-plurality

Abstract

<jats:p>We propose a Leibniz algebra, to be called\nDD<jats:inline-formula><jats:alternatives><jats:tex-math>^+</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:msup><mml:mi /><mml:mo>+</mml:mo></mml:msup></mml:math></jats:alternatives></jats:inline-formula>,\nwhich is a generalization of the Drinfel’d double. We find that there is\na one-to-one correspondence between a DD<jats:inline-formula><jats:alternatives><jats:tex-math>^+</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:msup><mml:mi /><mml:mo>+</mml:mo></mml:msup></mml:math></jats:alternatives></jats:inline-formula>\nand a Jacobi–Lie bialgebra, extending the known correspondence between a\nLie bialgebra and a Drinfel’d double. We then construct generalized\nframe fields <jats:inline-formula><jats:alternatives><jats:tex-math>E_A{}^M\\in\\text{O}(D,D)\\times\\mathbb{R}^+</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:msup><mml:mrow /><mml:mi>M</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mtext mathvariant= normal >O</mml:mtext><mml:mo stretchy= false form= prefix >(</mml:mo><mml:mi>D</mml:mi><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy= false form= postfix >)</mml:mo><mml:mo>×</mml:mo><mml:msup><mml:mstyle mathvariant= double-struck ><mml:mi>ℝ</mml:mi></mml:mstyle><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>\nsatisfying the algebra <jats:inline-formula><jats:alternatives><jats:tex-math>\\hat{\\pounds}_{E_A}E_B = - X_{AB}{}^C\\,E_C</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mrow><mml:msub><mml:mover><mml:mi>£</mml:mi><mml:mo accent= true >̂</mml:mo></mml:mover><mml:msub><mml:mi>E</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>E</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow /><mml:mi>C</mml:mi></mml:msup><mml:mspace width= 0.167em /><mml:msub><mml:mi>E</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>,\nwhere <jats:inline-formula><jats:alternatives><jats:tex-math>X_{AB}{}^C</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow /><mml:mi>C</mml:mi></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>\nare the structure constants of the DD<jats:inline-formula><jats:alternatives><jats:tex-math>^+</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:msup><mml:mi /><mml:mo>+</mml:mo></mml:msup></mml:math></jats:alternatives></jats:inline-formula>\nand <jats:inline-formula><jats:alternatives><jats:tex-math>\\hat{\\pounds}</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mover><mml:mi>£</mml:mi><mml:mo accent= true >̂</mml:mo></mml:mover></mml:math></jats:alternatives></jats:inline-formula>\nis the generalized Lie derivative in double field theory. Using the\ngeneralized frame fields, we propose the Jacobi–Lie\n<jats:inline-formula><jats:alternatives><jats:tex-math>T</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mi>T</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-plurality\nand show that it is a symmetry of double field theory. We present\nseveral examples of the Jacobi–Lie <jats:inline-formula><jats:alternatives><jats:tex-math>T</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML display= inline ><mml:mi>T</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-plurality\nwith or without Ramond–Ramond fields and the spectator fields.</jats:p>