### Nonabelian Multiplicative Integration On Surfaces

#### Amnon Yekutieli

#### $56.00

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### Description

Nonabelian multiplicative integration on curves is a classical theory. This volume is about the 2-dimensional case, which is much more difficult. In our construction, the setup is a Lie crossed module: there is a Lie group *H*, together with an action on it by another Lie group *G*. The multiplicative integral is an element of *H*, and it is the limit of Riemann products. Each Riemann product involves a fractal decomposition of the surface into kites (triangles with strings connecting them to the base point). There is a twisting of the integrand, that comes from a 1-dimensional multiplicative integral along the strings, with values in the group *G*.

The main result of this work is the 3-dimensional nonabelian Stokes theorem. This result is new; only a special case of it was predicted (without proof) in papers in mathematical physics. Our constructions and proofs are of a straightforward nature. There are plenty of illustrations to clarify the geometric constructions.

Our volume touches on some of the central issues (e.g., descent for nonabelian gerbes) in an unusually down-to-earth manner, involving analysis, differential geometry, combinatorics and Lie theory — instead of the 2-categories and 2-functors that other authors prefer.

**Contents:**

- Introduction
- Polyhedra and Piecewise Smooth Geometry
- Estimates for the Nonabelian Exponential Map
- Multiplicative Integration in Dimension One
- Multiplicative Integration in Dimension Two
- Quasi Crossed Modules and Additive Feedback
- Stokes Theorem in Dimension Two
- Square Puzzles
- Stokes Theorem in Dimension Three
- Multiplicative Integration on Triangular Kites

**Readership:**Graduate students, algebraic topologists, mathematical physicists and theoretical physicists.

**Key Features:**

- The author is an expert on algebraic geometry and deformation quantization. In this volume, he takes a journey into another area of mathematics (differential geometry), and the result of this journey is a collection of original and deep constructions and theorems
- The volume contains an unusually detailed study of the nonabelian exponential map and of piecewise smooth differential forms
- The volume includes an edited version of lecture notes on the subject, that provides a quick and lucid overview of the main features