Critical manifolds

This shows that the Hessian of / is positive definite (resp. negative definite) on (resp. N ). Therefore / is non-degenerate in the sense of Bott, i.e. the set of critical points is a disjoint union of submanifolds of X, and the Hessian is non-degenerate in the normal direction at any critical point. We put = dim N = 2 dime N which is the index of / at the critical manifold Cj,. Note that the index is always even in this case. 

It is well known (see e.g., [1, p.537]) that there exists a partial ordering < on the index set of the critical manifolds with the property... 

Suppose that for c G E there exists only one critical manifold with fiCv) = c. This assumption is just for saving notations, and the argument for the general case is essentially... 

Remark 7.2. In the above argument, we assumed that the C -action on a holomorphic symplectic Kahler manifold X satisfies iploJc = tujc for t E C. This is possible only when X is non-compact. The reason is as follows. If X is compact there exists a critical manifold corresponding to the maximum of the Morse function for which the unstable manifold is open submanifold of X, but this contradicts the propostion which asserts that every unstable manifold is Lagrangian. 

As mentioned above, the matter is quite involved and we will only sketch the arguments. Discussing some low order terms we will demonstrate the additive renormalizations due to higher order interactions (Sect. 10.1). Then we will discuss the general ideas on the structure of the renormalization group, defining important concepts like relevance or irrelevance of interactions or the critical manifold (Sect. 10.2). Concerning the field theoretic realization of the RG, we will summarize some results (Sect. 10.3). 

This relation defines the critical manifold (often called critical surface), which is an (infinite dimensional) submanifold of parameter space. Since the macroscopic scale is invariant, under renormalization, scales as... 

We are now in the position to sketch the RG flow globally. A schematic picture of the flow in the critical manifold is shown in Fig. 10.2, where the manifold is approximated as a plane parameterized by wfy, wfy J. Assuming that is the irrelevant field of smallest fixed point dimension we note that... 

The flow outside the critical manifold is sketched in Fig. 10.3, where we schematically represent the critical manifold by Asymptotically the... 

Up to now we tacitly assumed that the single fixed point t> dominates the critical manifold. This is not the full story. In the applications of interest there are at least two fixed points located on the critical manifold. Besides t)J there is a fixed point vj at which an additional coordinate... 

So far we have considered the RG for a general critical system. Figure 10.3, in particular, sketches the flow for a typical phase transition problem, where the physical path truly intersects the critical manifold. This manifold... 

This structure of the critical manifold is valid for d < 4. For d —> 4 the nontrivial fixed point merges into the Gaussian fixed point. For d > 4 only the Gaussian fixed point is physical, governing the system for all ft > 0. In RG language this is the reason for the triviality of the results for d > 4, pointed out in Chap. 6. 

V - Op/A. The turning point of each diagram is special there are no steady states to the left of this point two steady states exist on the right. We say that the turning point is a critical point. If we consider the family of V - aP/A diagrams traced for different values of the additional parameter T, a surface is obtained. This is a two-dimensional manifold in the three-dimensional V - oP/A - T space. As each diagram has its own turning point defined by one additional equation, the locus of critical points is a line, a one-dimensional manifold in the V - cP/A - T space. It is easy to see that the dimension of the critical manifold is na - 1, where na is the number of parameters included in the analysis. 

The approach presented in Figure 9.9 can be generalized to problems involving any number of parameters. The idea is to measure, in the space of uncertain parameters, the distance between a candidate point of operation and the critical manifold. If this distance exceeds a certain predefined quantity, the operating point is robust with respect to parameter uncertainty. To define a distance in the space of parameters having different units, we introduce the scaled variables af ... 

Equations (9.18a) to (9.18c) define the manifold of turning points. Equations (9.18d) and (9.18e), allow calculation of the normal vector r and of the distance p from the operating point c o to the critical manifold (Figure 9.10). 

Figu re 9.10 Distance to the critical manifold, in the space of scaled uncertain parameters. 

Calculation of normal vector and the distance to the critical manifold can be included in an optimization methodology. Here, we are looking for the reactor of minimum volume that ensures feasible operation of the reactor/separation/recycle system, for temperatures in the range 268 5 K, and kinetic uncertainties in the range 1 0.5. Using the scaled variables, the optimization problem can be written as ... 

T — T, valid to leading order dose to the critical manifold. Thus the first scaling variable talces the form ft./ T—. The neglect of higher order terms in the expansion of about ft = 0, T = Tc is equivalent to... 

To calculate the scaling functions we now follow the flow up to some fixed surface chosen far from the critical manifold, such that i/ C- The resulting macroscopic Hamiltonian allows for a perturbative treatment, all critical effects being absorbed into the renormalization of the couplings. On the purely formal level the precise choice of tliis manifold where the scaling... 

As in the four-dimensional case, one could distinguish between orientable and nonorientable Hamiltonians H, We call a Hamiltonian orientable if all of its critical submanifolds (on are orientable, that is, there is not a single critical manifold... 

---