The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory

We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the AKSZ construction, first implemented in this context by Pantev-Toën-Vaquié-Vezzosi, and show that it induces a family of symmetric monoidal functors from oriented stacks to symplectic stacks. Finally, we construct forgetful functors from the unoriented bordism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal infinity comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category to cospans of spaces, and from the oriented bordism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal infinity comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category to cospans of spaces equipped with an orientation; the latter combines with the AKSZ functors by viewing spaces as constant stacks, giving the desired field theories.