{"file_name": "0.png", "latex_formula": "\\[\\mathbb{C}^{4}\\stackrel{{\\pi_{1}}}{{\\longleftarrow}}\\mathcal{ F}\\stackrel{{\\pi_{2}}}{{\\rightarrow}}\\mathcal{PT},\\]"}
{"file_name": "1.png", "latex_formula": "\\[W^{*}_{Z}(x_{1},x_{2})=W_{f\\lrcorner Z}(y_{1},y_{2})=\\mathcal{P}\\exp\\left( \\int_{\\gamma}A_{\\mu}dx^{\\mu}\\right).\\]"}
{"file_name": "2.png", "latex_formula": "\\[G=W^{*}_{Z}(q,p)=\\tilde{H}H^{-1}\\]"}
{"file_name": "3.png", "latex_formula": "\\[H=W^{*}_{Z}(p,x),\\ \\ \\tilde{H}=W^{*}_{Z}(q,x).\\]"}
{"file_name": "4.png", "latex_formula": "\\[v\\cdot f^{*}A|_{x}=(f\\lrcorner Z)_{*}v\\cdot A|_{f\\lrcorner Z(x)},\\quad x\\in Z, \\ v\\in T_{x}Z.\\]"}
{"file_name": "5.png", "latex_formula": "\\[(f\\lrcorner Z)_{*}v\\cdot A|_{f\\lrcorner Z(x)}=v^{\\alpha\\dot{\\alpha}}\\Big{(} \\frac{\\partial y^{\\beta\\dot{\\beta}}}{\\partial x^{\\alpha\\dot{\\alpha}}}A_{\\beta \\dot{\\beta}}\\Big{)}\\Big{|}_{f\\lrcorner Z(x)},\\ x\\in Z,\\ v\\in T_{x}Z,\\]"}
{"file_name": "6.png", "latex_formula": "\\[\\{T_{i},T_{j}\\}=\\{\\tilde{T}^{i},\\tilde{T}^{j}\\}=0,\\ \\ \\{T_{i},\\tilde{T}^{j}\\}=2i \\delta^{j}_{i}D,\\]"}
{"file_name": "7.png", "latex_formula": "\\[(\\partial_{s},q_{i},\\tilde{q}^{k})\\rightarrow(D,M^{j}_{i}T_{j},\\tilde{M}^{k}_ {l}\\tilde{T}^{l}),\\]"}
{"file_name": "8.png", "latex_formula": "\\[M^{i}_{j}\\tilde{M}^{j}_{k}=\\delta^{i}_{k}.\\]"}
{"file_name": "9.png", "latex_formula": "\\[Q_{i\\alpha}=q_{i\\alpha}+\\omega_{i\\alpha},\\ \\tilde{Q}^{i}_{\\dot{\\alpha}}=q^{i}_{ \\dot{\\alpha}}+\\tilde{\\omega}^{i}_{\\dot{\\alpha}},\\ D_{\\alpha\\dot{\\alpha}}= \\partial_{\\alpha\\dot{\\alpha}}+A_{\\alpha\\dot{\\alpha}}.\\]"}
{"file_name": "10.png", "latex_formula": "\\[\\hat{f}(g,\\theta^{i\\alpha},\\tilde{\\theta}^{\\dot{\\alpha}}_{j})=(f(g),[V^{-1}]^ {\\alpha}_{\\beta}\\theta^{i\\beta},[\\tilde{V}^{-1}]^{\\dot{\\alpha}}_{\\dot{\\beta}} \\tilde{\\theta}^{\\dot{\\beta}}_{j}),\\ g\\in{\\cal G},\\]"}
{"file_name": "11.png", "latex_formula": "\\[v^{\\beta\\dot{\\beta}}V^{\\alpha}_{\\beta}\\tilde{V}^{\\dot{\\alpha}}_{\\dot{\\beta}} =((f\\lrcorner L_{0})_{*}v)^{\\alpha\\dot{\\alpha}},\\]"}
{"file_name": "12.png", "latex_formula": "\\[\\omega_{i\\alpha}=\\tilde{\\theta}^{\\dot{\\alpha}}_{i}h_{\\alpha\\dot{\\alpha}}(x^{ \\beta\\dot{\\beta}},\\tau^{\\beta\\dot{\\beta}}),\\ \\ \\tilde{\\omega}^{i}_{\\alpha}=\\theta^{i\\alpha}\\tilde{h}_{\\alpha\\dot{\\alpha}}(x^{ \\beta\\dot{\\beta}},\\tau^{\\beta\\dot{\\beta}}),\\]"}
{"file_name": "13.png", "latex_formula": "\\[\\begin{split}&\\lambda^{\\alpha}\\hat{f}^{*}\\omega_{i\\alpha}(z)= \\tilde{\\theta}^{\\dot{\\beta}}_{i}\\lambda^{\\alpha}\\left(V^{\\beta}_{\\alpha}h_{ \\beta\\dot{\\beta}}(x^{\\prime},\\tau^{\\prime})\\right),\\\\ &\\tilde{\\lambda}^{\\dot{\\alpha}}\\hat{f}^{*}\\tilde{\\omega}^{i}_{ \\dot{\\alpha}}(z)=\\theta^{i\\beta}\\tilde{\\lambda}^{\\dot{\\alpha}}\\left(\\tilde{V}^ {\\dot{\\beta}}_{\\dot{\\alpha}}\\tilde{h}_{\\beta\\dot{\\beta}}(x^{\\prime},\\tau^{ \\prime})\\right),\\end{split}\\]"}
{"file_name": "14.png", "latex_formula": "\\[A_{\\alpha\\dot{\\alpha}}=A_{\\alpha\\dot{\\alpha}}(x^{\\beta\\dot{\\beta}},\\tau^{ \\beta\\dot{\\beta}})\\]"}
{"file_name": "15.png", "latex_formula": "\\[D=\\lambda^{\\alpha}\\tilde{\\lambda}^{\\dot{\\alpha}}D_{\\alpha\\dot{\\alpha}}\\]"}
{"file_name": "16.png", "latex_formula": "\\[D=\\lambda^{\\alpha}\\tilde{\\lambda}^{\\dot{\\alpha}}\\partial_{\\alpha\\dot{\\alpha}}\\]"}
{"file_name": "17.png", "latex_formula": "\\[[v_{1}\\cdot D^{*},v_{2}\\cdot D^{*}]=0\\]"}
{"file_name": "18.png", "latex_formula": "\\[\\Phi_{A}=(\\omega_{i\\alpha},\\tilde{\\omega}^{i}_{\\dot{\\alpha}},A_{\\alpha\\dot{ \\alpha}})\\]"}
{"file_name": "19.png", "latex_formula": "\\[\\hat{f}:{\\cal F}^{6|4N}\\rightarrow{\\cal F}^{6|4N}\\]"}
{"file_name": "20.png", "latex_formula": "\\[\\sigma=(s,\\xi^{i},\\tilde{\\xi}_{j})\\in\\mathbb{C}^{1|2N}\\]"}
{"file_name": "21.png", "latex_formula": "\\[\\tau^{\\alpha\\dot{\\alpha}}(h_{\\alpha\\dot{\\alpha}}+\\tilde{h}_{\\alpha\\dot{\\alpha} })=0\\]"}
{"file_name": "22.png", "latex_formula": "\\[\\tau^{\\alpha\\dot{\\alpha}}\\rightarrow[V^{-1}]^{\\alpha}_{\\beta}[\\tilde{V}^{-1}]^{ \\dot{\\alpha}}_{\\dot{\\beta}}\\tau^{\\beta\\dot{\\beta}}\\]"}
{"file_name": "23.png", "latex_formula": "\\[\\tau^{\\beta\\dot{\\beta}}=\\sum_{i}\\theta^{i\\beta}\\tilde{\\theta}^{\\dot{\\beta}}_{i}\\]"}
{"file_name": "24.png", "latex_formula": "\\[\\theta^{i\\alpha}\\omega_{i\\alpha}+\\tilde{\\theta}^{i}_{\\dot{\\alpha}}\\tilde{ \\omega}^{\\dot{\\alpha}}_{i}=0\\]"}
{"file_name": "25.png", "latex_formula": "\\[\\tilde{T}^{i}=\\tilde{\\lambda}^{\\dot{\\alpha}}\\tilde{Q}^{i}_{\\dot{\\alpha}}\\]"}
{"file_name": "26.png", "latex_formula": "\\[\\tilde{T}^{i}=\\tilde{\\lambda}^{\\dot{\\alpha}}\\tilde{q}^{i}_{\\dot{\\alpha}}\\]"}
{"file_name": "27.png", "latex_formula": "\\[\\tilde{\\lambda}^{\\dot{\\alpha}}f^{*}A_{\\alpha\\dot{\\alpha}}=H^{-1}\\tilde{ \\lambda}^{\\dot{\\alpha}}\\partial_{\\alpha\\dot{\\alpha}}H\\]"}
{"file_name": "28.png", "latex_formula": "\\[\\tilde{q}^{i}=\\partial_{\\tilde{\\xi}_{i}}+i\\xi^{i}\\partial_{s}\\]"}
{"file_name": "29.png", "latex_formula": "\\[\\tilde{q}^{i}_{\\dot{\\alpha}}=\\frac{\\partial}{\\partial\\tilde{\\theta}^{\\dot{ \\alpha}}_{i}}+i\\theta^{i\\alpha}\\frac{\\partial}{\\partial x^{\\alpha\\dot{\\alpha}}}\\]"}
{"file_name": "30.png", "latex_formula": "\\[f\\lrcorner L(z)=\\pi_{1}\\circ f(z,\\lambda,\\tilde{\\lambda})\\ \\forall z\\in L\\]"}
{"file_name": "31.png", "latex_formula": "\\[q_{i\\alpha}=\\frac{\\partial}{\\partial\\theta^{i\\alpha}}+i\\tilde{\\theta}^{\\dot{ \\alpha}}_{i}\\frac{\\partial}{\\partial x^{\\alpha\\dot{\\alpha}}}\\]"}
{"file_name": "32.png", "latex_formula": "\\[q_{i}=\\partial_{\\xi^{i}}+i\\tilde{\\xi}_{i}\\partial_{s}\\]"}
{"file_name": "33.png", "latex_formula": "\\[v^{\\alpha\\dot{\\alpha}}=\\lambda^{\\alpha}\\tilde{\\lambda}^{\\dot{\\alpha}}\\]"}
{"file_name": "34.png", "latex_formula": "\\[z^{A}=(x^{\\alpha\\dot{\\alpha}},\\theta^{i\\alpha},\\tilde{\\theta}^{\\dot{\\alpha}}_{ j})\\]"}