Computation of SU(2)-instanton invariants

On this page we present a program written in Macaulay2 to compute two invariants of SU(2)-instantons over certain non-compact complex surfaces. Macaulay2 is a fantastic open-source computer algebra software specifically designed for commutative algebra and algebraic geometry (online documentation, book).

See below for an explanation of the mathematics.

The theoretical groundwork and the original program were developed by Irena Swanson and Elizabeth Gasparim. This website and enhancements of the program were made by Thomas Köppe.

The program

• InstantonInvariants.m2: Computes instanton invariants over an arbitrary base space Tot(O(−k)); the extension class is specified in variables (u, z).
• InstantonInvariants2.m2: Like above, but also computes the height of bundles over Tot(O(−1)⊕O(−1)), the simple flop, where the extension class is specified in variables (u1, u2, z).

Old versions

• o1xy.m2, base space Tot(O(−1)), extension class given in variables (x, y).
• o1uz.m2, base space Tot(O(−1)), extension class given in variables (u, z).
• o2uz.m2, base space Tot(O(−2)), extension class given in variables (u, z). Does not compute the instanton height and may be buggy.
• FlopInvariants.m2, base space Tot(O(−1)⊕O(−1)), extension class given in variables (u1, u2, z). The algorithm is non-trivial, but the width is always zero for mathematical reasons. The height computation is also available in InstantonInvariants2.m2.

Usage instructions

Download the desired program. Either run Macaulay2 with the program file name as an argument, i.e. macaulay InstantonInvariants.m2 or run Macaulay2 without any arguments and load the program from within: $ macaulay

Macaulay 2, version 0.9.95
etc. etc.

i1 : load "InstantonInvariants.m2" Now, define an extension class p as a polynomial in (u, z, z⁻¹). (In the program o1xy.m2, define p in terms of x and y. In the programs o?xy.m2, you may need to specify the base ring to which the variables belong, which is called Rbase.) i2 : p = u^3 * z + 3 * u^4 * z^3 Now you are ready to run the program. For the instanton width and height, respectively, with a given splitting type (e.g. j = 3) over a given base space Z_k (e.g. k=1) run the following: i3 : iWidth(1, p, 3)
...
o3 = {4, M}

i4 : iHeight(1, p, 3)
...
o4 = {3, gens, rels} The output of iWidth is a list containing the actual instanton width and a presentation of the module M. (In the old programs, do not specify the base space type, i.e. just iWidth(p, j).)
The output of iHeight is a list containing the actual instanton height, plus a list of generators and a list of relations. We believe that we usually have height = #gens − #rels, but to apply the relations and obtain a minimal set of generators and relations, use i5 : fixHeightRelations oo
o5 = {3, mingens, {}}

To compute the height of a bundle on the flop, use the same function iHeight, but specify an extension class from the ring RbaseW1: i6 : use RbaseW1
...
i7 : iHeight(z^-3 * u1 * u2 + z^-4 * u2, 6)
o7 = {25, {...}, {...}}
i8 : fixHeightRelations oo
o8 = {25, {...}, {}}

The mathematics

The setting: Z_k = Tot(O(−k))

We will denote by Z_k the total space of the holomorphic vector bundle O(−k) over CP¹. Thus the space Z_k is naturally a non-compact, complex 2-manifold. It is homotopic to its zero-section, which is an embedded projective line CP¹ with self-intersection number −k.

Thus the total space Z_k has the same charts as CP¹, namely two hemispheres times a line, with coordinates, say, (z, u) on the northern hemisphere and (1/z, z^ku) on the southern hemisphere, where z is the coordinate on the base and u the coordinate on the fibre.

One may consider what happens when one contracts the zero-section to a point: If k = 1, then it is a −1-curve and thus contracts to a smooth point, and indeed Z₁ is just C² blown up at the origin. On Z₂ the zero-section contracts to an ordinary double point on {x, y, w}/(xw = y²). For higher values of k, the resulting singularity is more complicated.

Bundles over Z_k

We now consider rank-2 holomorphic vector bundles over Z_k. It was shown in [1] that all such bundles are algebraic extensions of line bundles (which are themselves algebraic). As such they are given by a single transition function on the overlap of the two charts on Z_k that we described above, which is simply a 2×2-matrix whose entries are holomorphic, hence polynomials, in (z, z⁻¹, u). The standard form of this matrix is simply

where p is a polynomial in (z, z⁻¹, u) called the extension class. The integer −j₁− j₂ = c₁ is the first Chern class of the bundle.

For most purposes, it is possible to tensor the given bundle by a line bundle of arbitrary degree, so that one obtains a new bundle with first Chern class c₁ = 0 or 1. The first case, bundles with vanishing first Chern class, arises from SU(2)-instantons and are the object of our study, and we write j = j₁ = −j₂. The extension class p determines an element of Ext¹_Zk(O(j), O(−j)).

Instanton width and height

We will now define two integral invariants of the holomorphic rank-2 vector bundle that we defined above. By the Kobayashi-Hitchin correspondence (see e.g. [3]), holomorphic rank-2 bundles with c₂ = k correspond to SU(2)-instantons of charge k, and thus we are led to call the invariants the instanton width and height. They are precisely the two terms in the expression for the local Euler characteristic of the bundle, which in turn is defined in terms of the H⁰ and H¹ local cohomologies, i.e. spaces of holomorphic sections:

Consider a section (a(z, u), b(z, u)) given over the northern hemisphere, where a and b are polynomials in z and u. Then on the southern hemisphere the section is given by T(a, b) = (z^j a + p b, z^−j b), where it has to be holomorphic in z⁻¹ and z^k u. While this puts restrictions on the powers of z, all powers of u are allowed, and thus the space of sections is infinite-dimensional.

The solution is to use the theory of Formal Functions and consider the formal neighbourhoods l_n on which only powers of u up to n are allowed. Clearly l₀ is the zero-section, and restricted to l₀ the bundle splits as O(−j) ⊕ O(j). The result is that the inverse limit over n exists and can be computed by only computing the sections on the first few formal neighbourhoods. One obtains a C[[z, u]]-module of sections. The direct image of this module under the contraction of l₀ is called M, and the module Q defined by the double-dual exact sequence 0 → M → M^∨∨ → Q → 0 computes precisely the stalk at zero of the corresponding double-dual quotient of the direct image under the contraction of the sheaf of sections. The dimension of Q is called the instanton width.

[Need to define and explain instanton height, which for k = 1 is given by a simple formula, see [2].]

The sum of instanton width and height is the second Chern class of the bundle, written c₂. It is an open question whether all positive integers arise as second Chern classes of holomorphic rank-2 bundles on Z_k.

A complete write-up of the algorithm has been submitted [6].

References and papers

• Elizabeth Gasparim, Holomorphic bundles on O(−k) are algebraic, Comm. Algebra 25, no. 9, 3001–3009 (1997)
• Elizabeth Gasparim and Irena Swanson, Computing instanton numbers of curve singularities, J. Symbolic Computation 40, no. 2, 965–978 (2005)
• Elizabeth Gasparim, Thomas Köppe and Pushan Majumdar, Local holomorphic Euler characteristic and instanton decay. Pure Appl. Math. Q. 4 no. 2, Special Issue: In honor of Fedya Bogomolov, Part 1, 161–179 (2008) (Arxiv: 0709.2577).
• Edoardo Ballico, Elizabeth Gasparim and Thomas Köppe, Vector bundles near negative curves: moduli and local Euler characteristic. To appear in Communications in Algebra. (Arxiv: math/0404012)
• Edoardo Ballico, Elizabeth Gasparim and Thomas Köppe, Local moduli of holomorphic bundles. Journal of Pure and Applied Algebra 213, 397–408 (2009) (Arxiv: 0803.4216)
• Thomas Köppe, Computations of instanton invariants. Submitted for publication. (Arxiv: 0905.2745)

© 2007 by Thomas Köppe and Elizabeth Gasparim. This is version 2009-05-17.