Positivity of the universal pairing in 3 dimensions

Introduction

The earliest objects of study in the theory of manifolds were the fundamental group, Poincaré duality, and by the 1930’s, characteristic classes. The main theme was classification: developing invariants to distinguish one manifold from another and trying to construct manifolds with prescribed invariants. In the mid 50’s to early 60’s, Thom, Milnor, and Smale shifted the emphasis to operations on manifolds such as cutting, gluing and especially surgery, using powerful new structural tools such as Morse theory and cobordism. The most spectacular successes of this theory were confined to sufficiently high dimensions. Especially in $3$ dimensions, arbitrary surgery or cutting and pasting is too disruptive, and incompatible with even the coarsest features of the classification theory such as the prime decomposition and JSJ theorems. Three manifold topology developed quite independently reaching a culmination, from the classification perspective, with Thurston and Perelman.

From a completely different direction, low-dimensional topology has been invigorated over the last two decades by ideas from physics, especially quantum field theory. Classically, one studies fields (e.g. functions, or sections of some bundle) and their dynamics on Euclidean space, or on some smooth manifold. From quantum physics one gets the key idea of superposition — that one should study complex linear combinations of fields. A combination of these two ideas — cobordism and superposition — unexpectedly resonates in low-dimensional topology, and gives rise to a host of beautiful and subtle invariants, such as the Jones polynomial, Reshetikhin-Turaev-Viro invariants, and Chern-Simons partition functions.

In fact, these invariants are perhaps too subtle. After twenty years of work, it is profoundly frustrating that we cannot say precisely what these invariants measure or what they distinguish, and despite some tantalizing hints (e.g. Kashaev’s conjecture), these invariants remain disconnected from the (hugely successful) Thurston theory of $3$-manifolds. There are two key questions: what information does $3$-dimensional quantum topology distinguish in principle, particularly in the physically motivated unitary case? and how does it relate in detail to the structure theory of $3$-manifolds revealed by the geometrization program?

This paper addresses both of these questions simultaneously. Firstly, we establish positivity for the universal manifold pairing in $3$-dimensions, an abstract “universal topological quantum field theory” which is, by construction, sensitive to all details of $3$-manifold topology. Three dimensions is the critical case here: in two dimensions and lower, such positivity is straightforward to establish; in four dimensions and higher, positivity fails badly (see [FKNSWW] and [Kreck_Teichner]). For instance, the partition function of a unitary $(3+1)$-dimensional TQFT is equal on $s$-cobordant $4$-manifolds regardless of their Donaldson invariants ([FKNSWW]). Secondly, the process of establishing positivity turns out to involve the construction of a complexity function on closed $3$-manifolds which involves input from every aspect of the geometric theory of $3$-manifolds, and obeys a (highly nontrivial) gluing axiom, which may be thought of as a kind of topological Cauchy-Schwarz inequality.

Given $A,B\in{\dot{\mathcal{M}}}(S)$, let $\overline{B}$ denote $B$ with the opposite orientation. Note that $\overline{B}\in{\dot{\mathcal{M}}}(\overline{S})$. By abuse of notation, we denote by $AB$ the result of gluing $A$ to $\overline{B}$ by the identity map on their boundaries.

Our central object of study is the complex sesquilinear pairing

This pairing is known as the universal pairing associated to $S$, and is denoted $\langle\cdot,\cdot\rangle_{S}$. A pairing on a vector space is positive if $\langle v,v\rangle=0$ if and only if $v=0$. With this terminology, our main theorem is the following.

For all closed, oriented surfaces $S$, the pairing $\langle{\cdot},{\cdot}\rangle_{S}$ is positive.

In a necessarily non-positive pairing, a vector $v$ satisfying $v\neq 0$ and $\langle v,v\rangle=0$ is said to be lightlike.

For some reason, it is common practice to use the letters $V$ and $Z$ to denote the image of the functor on an $(n-1)$-manifold and an $n$-manifold respectively, so that for instance $Z(M)\in V(S)$ for $M\in{\dot{\mathcal{M}}}(S)$. We typically do not use this convention when discussing abstract TQFTs, but we do sometimes when discussing specific TQFTs in order to be consistent with the wider literature.

In many interesting TQFT’s, the images $Z(A)$ span $Z(S)$. For example, if $Z$ is the $SU(2)$ Chern-Simons partition function at level $k$ where $k+2$ is prime, then Roberts [Roberts] has shown the images $Z(A)$ over all $A\in{\dot{\mathcal{M}}}(S)$ span $Z(S)$. In this case, there is a natural pairing

defined on generators by composition of cobordisms:

In physical quantum field theories, $Z(S)$ denotes the vector space of quantum fields on a spacelike slice $S$ of spacetime. The vector space $Z(S)$ is naturally a (positive definite) Hilbert space. This motivates an additional axiom for so-called unitary TQFT’s, that $Z(S)$ (now usually finite dimensional) should admit the natural structure of a Hilbert space, and for every nonzero $A\in{\mathcal{M}}(S)$, the pairing defined above should satisfy

It follows that studying positivity of the universal pairing (in a given dimension) is tantamount to studying what kind of topological information in principle might be extracted from unitary TQFT’s in that dimension, since lightlike vectors must map to zero in any unitary TQFT. As a specific and important example, given any compact Lie group $G$ and level $k>0$, the Reshetikhin-Turaev TQFT [Reshetikhin_Turaev], denoted by $V_{G,k}$, and as reconstructed by [BHMV] fits into the following diagram: