Non-semisimple topological field theory and $$\widehat{Z}$$-invariants from $$\mathfrak {osp}(1 \vert 2)$$

Abstract We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $$\mathfrak {osp}(1 \vert 2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>osp</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . More precisely, the quantum group depends on a root of unity $$q=e^{\frac{2 \pi \sqrt{-1}}{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msqrt> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> <mml:mi>r</mml:mi> </mml:mfrac> </mml:msup> </mml:mrow> </mml:math> , where r is a positive integer greater than 2, and the construction applies when r is not congruent to 4 modulo 8. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When r is congruent to $$\pm 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> or $$\pm 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> modulo 8, we relate the resulting 3-manifold invariants with physicists’ $$\widehat{Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Z</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> -invariants associated to $$\mathfrak {osp}(1 \vert 2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>osp</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Finally, we establish a relation between $$\widehat{Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Z</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> -invariants associated to $$\mathfrak {sl}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>sl</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\mathfrak {osp}(1 \vert 2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>osp</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> which was conjectured in the physics literature.