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This is a sub-entry of sigma-model. See there for background and context.
Low dimensional $\sigma$-models that describe the dynamics of particles – or generally branes – propagating in a target space $X$ subject to the forces exerted by a background field, such as discussed above, are just the first ingredient in a description of the quantum physics on target space $X$ : one is interested in describing a quantum field theory (or possibly some higher analog, such as a string field theory ) on $X$, such that the original branes described by the $\sigma$-model are quanta of the fields on $X$. The $\sigma$-model quantum field theory on the worldvolumes $\Sigma$ is supposed to induce, in turn, another quantum field theory, or something similar, but now on $X$. This process – or its idea – goes by the name second quantization . We shall try to indicate below in which sense this should indeed be a direct iteration of the (“first”) quantization of the original $\sigma$-model itself, as described above. But we will (have to) be more vague and schematic than before.
Assume now that we started with a classical $\sigma$-model and have quantized it to obtain the functorial QFT
which we take here to be 1-categorical (un-extended), not to overburden the discussion. Assume also for simplicity that the corresponding target space is of the form $X = Y \times \mathbb{R}$, where the second factor is the time-axis.
Using this, we define a space of states of the second quantization to be the Fock space
the free commutative tensor algebra over the vector space $\mathcal{V}$ of states of the given single particle, string or, generally, brane. This has a contribution for each equivalence class $[\Sigma_{n-1}]$ of connected shapes $\Sigma_{n-1}$ of this brane. For instance for $n = 1$ there is just the point $\Sigma_0 = *$, but for the string with $n = 2$ there is one contribution from the closed string $\Sigma_1 = S^1$ and one contribution from the open string $\Sigma_1 = [0,1]$; and in turn several contributions of this type if the open string carries boundary D-brane labels. (In the presence of fermions the vector spaces appearing here are super vector spaces and the Fock space construction is the corresponding super algebra version.)
This space of states is that of a quantum theory whose classical field configurations are given by many states of the original quantum $\sigma$-model. We therefore say that the original brane of shape $\Sigma_{n-1}$ is a quantum or a single excitation of the field theory that it induces on target space $X$: where the original $\sigma$-model describes the dynamics of a single brane on $X$, its second quantization describes the dynamics of many and interacting copies of that brane.
Of course $Sym^\bullet(-) : Vect \to Vect$ is a functor. This is traditionally stated as the notorious slogan “second quantization is a functor”, usually meant to state a contrast with first quantization. Notice however, firstly, that the (“first”) quantization of $\sigma$-models – at least for the version discussed above – also is a functor (what is not functorial is the quantization of a random Poisson manifold, but $\sigma$-models carry much more structure than that), and, secondly, that the $Sym^\bullet$-construction is at best half of what second quantization is about: the other half is the description of the interaction of the many branes:
for $X = Y \times (-t,t)$ a piece of target space, the would-be second quantized theory on $X$ should assign a morphism
between the space of states of the incoming space $Y$, to the outgoing space $Y$, here taken to be the same. Usually one assumes here $t = \infty$ and calls $S$ the S-matrix of the theory, where “S” is for scattering : for $(v^{in}_1 \otimes \dots v^{in}_r) \in Sym^r \mathcal{V}$ the state of $r$ incoming brane quanta and $(v^{out}_1 \otimes \dots v^{out}_s) \in Sym^s \mathcal{V}$ a state of $s$ outgoing brane quanta, the matrix element
(using an inner product space-structure on $\mathcal{V}$) is the probability amplitude for $r$-many branes in the given states propagating through $X$, interacting among each other, thereby transmuting into other states – hence scattering off each other – and eventually emerging again in the state $(v^{out}_1 \otimes \dots v^{out}_s)$.
The idea is that this scattering happens in all possible ways that it can happen, with each way weighted by the probability amplitude for it to happen as seen by the quantum $\sigma$-model $Z$: the S-matrix is taken to be given by an expression like
and called the brane perturbation series . For instance the Feynman perturbation series for $n = 1$, or the string perturbation series for $n = 2$: we sum over all possible $S$-structured cobordisms $\Sigma$ – each representing an interaction “channel” for branes of shape $\partial_{in} \Sigma$ to scatter into shape $\partial_{out} \Sigma$ – the linear maps $S(\Sigma) : \mathcal{V}_{\partial_{in} \Sigma} \to \mathcal{V}_{\partial_{out} \Sigma}$ weighted by some measure $d\mu(\Sigma)$, and all regarded as giving one single endomorphism on $Sym^\bullet(\mathcal{V})$ in the evident way.
Except for simple cases, plenty of technical subtleties may have to be taken care of here in order to get anything close to being well defined. For instance there is an integral over some suitably compactified moduli space of $S$-structured cobordisms involved. For sufficiently simple but still nontrivial $\sigma$-models this has been made fully precise (for instance for $n = 1$ in standard QFT perturbation theory with renormalization or for $n = 2$ in the example of Gromov-Witten theory, discussed above), and intuition and motivation is drawn from these cases, but in full generality it remains an open problem to fully realize this idea.
This S-matrix-construction provides at least the rudiments of a quantum field theory on target space $X$ obtained by second quantization of a $\sigma$-model describing brane dynamics on $X$.
In application to phenomenological physics we think of $X$ here as our spacetime. Everything that is measured in particle accelerator experiments is explained with such a construction for $n = 1$. Everything that perturbative string theory hypothesizes is a refinement of this theory relevant at energies not visible in current accelerator experiments is described with such a construction for $n = 2$. A little bit of investigation has gone into exploring the $n = 3$-case. In principle one could investigate this further for $n \geq 4$. but so far the jump in complexity given by the step from $n = 1$ to $n = 2$ has kept mankind busy enough. There is already an intricate interrelation of quantum field theories showing up at this level. For instance the second quantization of the 2-dimensional A-model string $\sigma$-model (see there) has be shown to be Chern-Simons theory, which we have seen, above, may itself be understood as a 3-dimensional $\sigma$-model. The second quantization of the non-topological string $\sigma$-model is more complicated but follows this pattern. It contains even Yang-Mills theory and gravity, and this is what has driven much of the interest in this structure.
Given the evident importance of the “brane perturbation series” or “second quantization” of $\sigma$-models it would be desireable to have a more general abstract and systematic description of it, in the spirit of the above discussion of general abstract first quantization. Here is an observation that might be suggestive:
we had amplified that the input data for a classical (“0-quantized”) $\sigma$-model is a background field on a target space $X$. At least in nice cases this background field is entirely encoded in its higher parallel transport and holonomy-assignment, which is a map
from $n$-dimensional bordisms in $X$ to n-vector spaces (we had mostly discussed an equivalent characteristic morphism $\nabla : X \to \mathbf{B}^n U(1)_{conn}$ and should eventually discuss in more detail how both perspectives are related…).
The process of (“first”) quantization of this $\sigma$-model involves some kind of extension of this functor through the projection to abstract cobordisms
given by the path integral
over all trajectories $\gamma : \Sigma \to X$ of given shape $\Sigma$. We saw that second quantization reads in this $Z_\nabla$, in turn, and from the perturbation series
produces a target space quantum field theory. Hence apparently there is a pattern of iterated path integrals, the first over morphisms in $Bord_n^S(X)$, the second over morphisms in $Bord_n^S$:
classical (0-quantized) $\sigma$-model (higher parallel transport of background gauge field):
quantum (1st quantized) $\sigma$-model (FQFT):
second quantized model (S-matrix):
Remember that these formulas are to be taken with a grain of salt. Quite some additional effort is in general needed to make them well-defined. Already in the well-understood case $n = 1$ the path integral in the expression for $Z_\nabla$ needs attention, then the single terms in the expression for $S$ may still need renormalization and after all that the sum still may need resummation, at least for models richer than for instance the $\infty$-Dijkgraaf-Witten theories, for which all integrals reduce to finite sums.