I will address the question of how background independent target space physics emerges in string theory. The point of view I will take is to identify the configuration space of target space with the space of 2d worldsheet QFTs. On-shell configurations are identified with c=0 worldsheet theories (i.e. a c=26 matter sector), and non-conformal QFTs correspond to generic off-shell configurations. I will demonstrate that a quantity built from the sphere partition function and the Zamolodchikov c-function has the correct properties to be a valid background-independent action on this configuration space, and is valid for all possible relevant and irrelevant deformations on the worldsheet (including non-minimally coupled and descendant operators). For the massless and tachyonic sectors in target space, this action is equivalent by field-redefinition to known actions developed by Tseytlin and collaborators in the 80s and 90s, constructed by taking derivatives with respect to the sphere partition function. This talk is based on recent work by Amr Ahmadain and Aron Wall (https://arxiv.org/abs/2410.11938).

In many situations in Differential or Algebraic Geometry, one forms moduli spaces $\cal M$ of geometric objects, such that $\cal M$ is a manifold, or something close to a manifold (a derived manifold, Kuranishi space, …). Then we can ask whether $\cal M$ is orientable, and if so, whether there is a natural choice of orientation.

  This is important in the definition of enumerative invariants: we arrange that the moduli space $\cal M$ is a compact oriented manifold (or derived manifold), so it has a fundamental class in homology, and the invariants are the integrals of natural cohomology classes over this fundamental class.

  For example, if $X$ is a compact oriented Riemannian 4-manifold, we can form moduli spaces $\cal M$ of instanton connections on some principal $G$-bundle $P$ over $X$, and the Donaldson invariants of $X$ are integrals over $\cal M$.

  In the paper arXiv:2503.20456, Markus Upmeier and I develop a theory of "bordism categories”, which are a new tool for studying orientability and canonical orientations of moduli spaces. It uses a lot of Algebraic Topology, and computation of bordism groups of classifying spaces. We apply it to study orientability and canonical orientations of moduli spaces of $G_2$ instantons and associative 3-folds on $G_2$ manifolds, and of Spin(7) instantons and Cayley 4-folds on Spin(7) manifolds, and of coherent sheaves on Calabi-Yau 4-folds. These have applications to enumerative invariants, in particular, to Donaldson-Thomas type invariants of Calabi-Yau 4-folds.

   All this is joint work with Markus Upmeier.

Feynman integrals are the essential building blocks of observables in perturbative Quantum Field Theories. As precision experiments in high-energy physics are becoming more common, understanding the structure of higher loop integrals has become very important from a phenomenology point of view. On the mathematical physics side, such investigations have led to profound connections to geometry. In particular, there is a correspondence between Feynman integrals and algebraic varieties and knowing what geometry a given Feynman integral corresponds to offers invaluable lessons in solving it. In this talk, I will start with a pedagogical review of modern methods to solve higher loop integrals. Then, with a simple example, I will show how one can infer the geometry associated with an integral and discuss some of the implications of this connection.