Introduction to Resurgence Note 1
Table of Content
- 1. Motivation
- 2. Analytic continuation and monodromy
- 3. Another example by Poincare
- 4. The differential algebra
1. Motivation
The following are some examples of potential applications of resurgence theory.
- Normal forms for dynamical systems
- Gauge theory of singular connections
- Quantization of symplectic and Poisson manifolds
-
Floer
homology andFukaya
categories - Knot invariants
- Wall-crossing and stability conditions in algebraic geometry
- Spectral networks
- WKB approximation in quantum mechanics
- Perturbative expansions in quantum field theory (QFT)
One of the most astonishing achievements of resurgence theory in QFT is that one can uncover the non-perturbative results from perturbative expansion alone! Typically, calculating non-perturbative results requires every possible resource you can have, such as the topology of the vacuum manifold, some real special cancellation, but perturbative calculation is much more straightforward, how can it contain the information that was so hardly revealed by all kinds of non-perturbative techniques? Nevertheless, resurgence theory enables us to derive non-perturbative results, such as instanton contribution, through the analytical continuation of perturbative data! On the one hand, it feels like black magic; on the other hand, perhaps I shouldn’t be so surprised, since if we could obtain the full perturbative expansions to all orders, we can essentially reproduce the equation of motion, the Lagrangian, which inherently includes all non-perturbative information. At least in theory then, perturbative expansions should be capable of yielding non-perturbative insights. However, this is only in theory. It’s still utterly astonishing to witness it happening in front of your eyes in practice. Rainbow doesn’t become less beautiful just because you’ve learnt the science behind it.
Actually, I would argue that it is impossible to make sense of perturbation theory without knowing at least some facets of resurgence theory. We include more and more perturbative terms in any perturbative power expansion, the power series eventually diverges, so what sense does it make to just consider the first few terms? Claiming that the first few terms of a divergent series give the dominant results would be ridiculous. However the miraculous agreement between perturbative QED and experiments clearly suggests that perturbation expansion makes sense, it is by no chance an accident. The answer to this question lies in resurgence theory.
H. Poincare mentioned “a kind of misunderstanding between geometers and astronomers about the meaning of the word convergence”, he proposed a simple example: consider the following series
Poincare said that for geometers, i.e. mathematicians in his time, the first one converges because at large
He then proposes to reconcile both points of view by clarifying the role that divergent series (in the sense of geometers) can play in the approximation of certain functions. This is the origin of the modern theory of asymptotic expansion.
In physics, the divergence of power series can be shown via different approaches:
- Dyson Freeman show it in a heuristic way, see his short paper;
- By combinatorial argument. The number of Feynman diagrams grows factorially as
, while the contribution of each diagram decreases as , then if there is no cancellation between different diagrams, the perturbative series grows as , which eventually diverges. Lipatov1 first show that in scalar QFT the series indeed grows as . Similar growth are found in quantum mechanics2 and matrix models3.
2. Analytic continuation and monodromy
In this note we shall focus on formal power series
, sometimes called the polynomial forms
. When regarded as formal power series with real coefficients of indeterminant
where the first one has infinite radius of convergence with respect to divergent series
will usually mean a formal power series with zero radius of convergence.
Formal power series
is a generalization of normal power series, or polynomial, in the sense that we consider a power series of infinite order as a formal object and don’t worry about evaluation, or if it is convergent. Given a ring
First we clarify some definitions and concepts that might be useful in the future.
We say a complex-valued function analytic
if
We say a complex-valued function holomorphic
iff it satisfied Cauchy-Riemann relation, which is equivalent to
A differential manifold is a topological space (given by closed sets and all that stuff) with differential structure, which is Hausdorff and second-separable. It practically means that you can find a way to do derivatives on the manifold. A homeomorphic (holomorphic)
.
An open disk
of radius
A open deleted disk of radius
a deleted disk is also called a punctured disk, since the origin
A complex atlas on a 2-dimensional manifold is a collection of charts that cover the manifold, where each chart maps a portion of the manifold to an open subset of the complex plane
-
Model: The (n+1)-tuple defines a vector in space . Define a equivalence relation between two vectors,
then
and the homogeneous coordinates
and is denoted by inhomogeneous coordinates
.
The Riemann sphere
-
Chart
: This chart covers the sphere minus the point at infinity. It maps each point in the complex plane to itself: Here, . -
Chart
: This chart covers the sphere minus the origin. It maps each point in the complex plane, excluding the origin, to its reciprocal: Here, .
The two charts overlap on
- From
to : - From
to :
Both transition functions are holomorphic, as the function
To say that two complex atlases on a manifold are analytically equivalent
means that the combined atlas they form (by taking all the charts from both atlases) is itself a complex atlas. This implies that the transition functions between the charts of one atlas and the charts of the other atlas are holomorphic wherever they overlap.
By a complex structure
on a manifold, we mean an equivalent class of analytically equivalent complex atlases on the manifold. If we give a manifold a complex atlas, then we have given it a complex structure.
A Riemann surface is a pair
Meromouphic functions
are functions holomorphic except at a discrete set of isolated poles.
A domain
in
Riemann Sphere
There are two ways to think of
- complex projective plane
. - a 2D sphere
. The coordinates is given by the stereographic projection, except for the north pole , .
The space Riemann sphere
. We can introduce two charts on the Riemann sphere,
where
Theorem. A function is meromorphic on
By the restriction of a function, we mean that to restrict the domain so that the it is well defined, no singularities. For example, the function
A germ
of functions at a point
Use
Intuitively, the germ of a function tells us how a function behaves locally at point
The Fundamental Uniqueness Theorem (FUT)
for holomorphic functions:
Theorem. If
The germ of
Analytic Continuation, Monodromy
First we introduce the analytic continuation in a different way, more formal and more mathematical. We begin by defining pairs
. The idea is that, a function is not only defined by the values but also the domain on which it is defined.
A pair
Another concept is adjacency, two pairs adjacent
if
If there is a finite
sequence of pairs
We can define a analytical continuation along a curve
Theorem. If the two path are homotopic, then the analytical continuation results to the same functions.
If in any doubt, just think of the
Linear Differential System
A linear system
is short for a complex linear ordinary differential system. A linear system of order p is a system with p first order ordinary differential equations.
where
We used
A fundamental solution
of
The Wronskian
for
Let
Recall a nice property of Wronskian: Let
- W is a fundamental solution of
for some for all
In other words, the Wronskian is either nonzero on the entire domain, or identically zero.
Differential Galois Theory
A differential field
differential homomorphism
A differential system
where
3. Another example by Poincare
To get some feeling about resurgence, let’s start with an example first given by Poincare. This example shows how an divergent series emerges from a function.
Fix
This series is uniformly convergent on
Hence the sum holomorphic
in
We now show how this function
One might be tempted to recombine the (convergent) Taylor expansion of
We see that
However, it turns out that this formal series is divergent!
To see this, make the substitution
and
There is an easy way to tell if a series has non-zero radius of convergence, by some kind of a “dominance criterion”. If the series of study
would have infinite radius of convergence.
This functions has finite radius of convergence, since the last expression in the equation above diverges at
Now the question is to understand the relation between Borel-Laplace
summation is a way of going from the divergent formal series asymptotic expansion
of
We can already observe that the moduli of the coefficients
for some 1-Gevrey estimates
for the formal series
We remark that, since the original function
Resurgence theory can also be used to study power series of the form
note that the variable
In the next note we will dive into the details of resurgence theory, beginning with the differential algebra
4. The differential algebra
It will be convenient for us to set formal power series
, i.e., polynomials in
This is a vector space with basis
The derivation
further makes it a differential algebra, which simply means
This is the derivative in terms of
Then, in mathematical terminology, there is an isomorphism of differential algebra between
The standard valuation
, or sometimes called the order
, on
defined by
For
This is the set of all the complex polynomials in
From the viewpoint of the ring structure,
It is obvious that
with equality iff there is no constant term.
With the help of the standard valuation, we can introduce the concept of distance
into the ring of the formal series. Define
as the distance between
With the definition of distance, complete metric space
. The topology induced by this distance is called the Krull topology
or the topology of the formal convergence
(or the
We mention that a sequence
-
Divergence of the Perturbation Theory Series and the Quasiclassical Theory, Published in: Sov.Phys.JETP 45 (1977), 216-223, Zh.Eksp.Teor.Fiz. 72 (1977), 411-427. ↩
-
J. Zinn-Justin, “Perturbation Series at Large Orders in Quantum Mechanics and Field Theories: Application to the Problem of Resummation”, Phys. Rept., vol. 70, p. 109, 1981. doi: 10.1016/0370-1573(81)90016-8;C. M. Bender and T. T. Wu, “Anharmonic oscillator. 2: A Study of perturbation theory in large order”, Phys. Rev., vol. D7, pp. 1620–1636, 1973. doi: 10.1103/PhysRevD.7.1620. ↩
-
M. Marino, R. Schiappa, and M. Weiss, “Nonperturbative E↵ects and the Large-Order Behavior of Matrix Models and Topological Strings”, Commun. Num. Theor. Phys., vol. 2, pp. 349–419, 2008. doi: 10.4310/CNTP.2008.v2.n2.a3. arXiv: 0711.1954 [hep-th]. ↩
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