100 The Quantum Structure of Space and Time
This allows us to write
at = e--tH
with H the Hamiltonian. In the case of a particle on X the Hamiltonian is of course
simply given by (minus) the Laplacian H = -A. The composition property (1) is
a general property of quantum field theories. It leads us to Segal’s functorial view
of quantum field theory, as a functor between the categories of manifolds (with
bordisms) to vector spaces (with linear maps) [19].
In the supersymmetric case the Hamiltonian can be written as
H = -A = -(dd + dd)
Here the differentials d, d* play the role of the supercharges
The ground states of the supersymmetric quantum mechanics satisfy H!P = 0 and
are therefore harmonic forms
d9 = 0, d*!P = 0.
Therefore they are in 1-to-1 correspondence with the de Rham cohomology group
of the space-time manifold
9 E Harm(X) E H(X).
We want to make two additional remarks. First we can consider also a closed
1-manifold, namely a circle S1 of length t. Since a circle is obtained by identifying
two ends of an interval we can write
zS~ = TrEe-tH.
Here the partition function Z~I is a number associated to the circle S1 that encodes
the spectrum of the operator A. We can also compute the supersymmetric partition
function by using the fermion number F (defined as the degree of the corresponding
differential form). It computes the Euler number
Trx ((-l)Fe-tH) = dimHeUen(X) - dimHodd(X) = x(X).
4.1.3.2
We will now introduce our first deformation parameter a’ and generalize from point
particles and quantum mechanics to strings and conformal field theory.
A string can be considered as a parameterized loop. So, in this case we study
the manifold X through maps
Conformal field theory and strings