A ring is a collection of elements endowed with two binary operations, say + and *. With respect to +, the ring is a group; that is, for any three elements of the ring, a,b,c,
• (a+b) is also an element... [closure]
• there exists a ‘0’, such that (0+a) = a = (a+0)... [unit]
• there exists a ‘-a’, such that (-a+a) = 0 = (a+(-a))... [inverse]
• (a+(b+c)) = ((a+b)+c)... [associativity].
With respect to *, the ring is closed, i.e., (a*b) is also in the ring.
The typical example are (positive, negative and zero) integers.
Cohomology groups are collections of exterior forms, graded by their order (number of differentials, dz's) which are annihilated by the exterior derivative(s) and are taken modulo exterior derivatives of other forms; they are additive groups, i.e., groups with respect to addition. They may be made into rings by endowing them with a multiplication - which may, but need not be the standard ‘wedge’ product.
The cohomology ring (with an appropriately chosen multiplication) of Calabi-Yau varieties reflects the mirror & duality transforms when these varieties are used for compactifying string theories. These transformations then also induce the corresponding transformation on the moduli space geometry and parameter space dynamics, since the latter are locally approximated by the cohomology groups. This relation between cohomology groups and the moduli space is used to detect the critical regions of the latter, as in the framework of variable spacetime vacua, in which the string theory changes drastically as such critical regions are approached - in real spacetime.
The operator formed as the scalar product of the differential of the position vector and the gradient operator is called the exterior differential, and denoted ‘d’. The differentials of the position vector are always multiplied antisymmetrically (this is the so-called ‘wedge product’) so as to form the differential area,- volume,- etc. elements. (The determinant nature of the Jacobian forces these to be odd with respect to the exchange of ordering.) Since gradient operators commute, it follows that dd=0. Therefore, the equation dF=0 (“F is d-closed”) has a trivial solution, when F=dA (“F is d-exact”, i.e., it is an ‘exact differential’). It then makes good sense to look for solutions of the equation dF=0 up to d-exact terms. That is, two solutions are considered equivalent if their difference is d-exact; in particular, a d-exact solution is equivalent to zero. Finally, the solutions of the equation dF=0 may be graded by their order, ranging from 0th order differentials - scalars, through 1st order differentials - covariant vectors, ... through nth order differentials - antisymmetric rank-n covariant tensors.
© Tristan Hübsch, 2008