ABELIAN MONOPOLE: POSITIVE DIMENSION CASE

Nik. Tyurin

Abstract. In the short paper we discuss some properties of the Seiberg — Witten invariants in the positive dimension case. In this case one can deduce the theory to the zero dimension case, using glueing procedure with \overline{ℂℙ}^{2}’s and in some special case one can extend the relationship between Seiberg — Witten and Gromov invariants to a priori non symplectic case.

1 Introduction

Four years ago new invariants of smooth structures on riemannian 4- manifolds were introduced by Witten in his seminal work [W].

These invariants themselfs are very close to the Donaldson invarinats, and the connection in a physics manner was established by Witten and Moore (the modern version — in [MW]).

Geometrical aspects of such relationship were discovered by A. Tyurin and V. Pidstrigach ([PT]) together with the introduction of a new version of Seiberg — Witten system of differential equations, known as B - monopoles.

It seems to exist a deep connection of Seiberg — Witten theory with the complex geometry, but in this short article we’ll extend the framework of the complex geometry to the case when underlying manifolds doesn’t admit a complex (or an almost complex) structure. As we’ll see this case corresponds to the positive dimensional case, when the moduli spaces have positive dimensions.

We are begining with the basic notions and facts for Seiberg — Witten theory.

First of all, let us recall that for a riemannian metric g and an orientation over X one has two projective bundles

ℙ^{±} → X,

which are P U(2)- bundles and called twistor bundles. Since w_{1}(X) — the first Stifel — Witney class — equals to zero, one can lift these two bundles to a pair of U(2) - bundles W^{±}. Such lifting depends on a choice of the first Chern class c_{1}(W^{±}) = c ∈ H^{2}(X, ℤ) and the last one has to satisfy the equality

c ≡ w_{1}(X)(mod2)

and is called a choice of Spin^{ℂ} - structure over X.

Two spinor bundles W^{±} satisfy the next condition

Typeset by AMS-TEX

1. T^{ℂ}X ≅ Hom(W^{−}, W^{+}), and moreover on the real level

T X ≅ Hom_{h}(W^{−}, W^{+}),

where the last one is the bundle on hermitian homomorphisms, which are compatible with both hermitian structures on W^{±}.

2. adW^{±} ≅ Λ^{±}, where the last are two eigenspaces of the Hodge star operator, induced by our metric g and our fixed orientation.

Any hermitian connection a ∈ A_{h}(detW^{+}) on the determinant line bundle detW^{+} together with the Levi - Civitae connection induces a hermitian connection on the whole W^{+} with the covariant derivative

d_{a} : Γ(W^{+}) → Γ(W^{+} ⊗ T^{∗}X).

The configuration space consists of pairs (a, ϕ), where a ∈ A_{h}(detW^{+}), and ϕ ∈ Γ(W^{+}). The first equation of the system reads as

D_{a}(ϕ) = 0,

where D_{a} is coupled Dirac operator, defined by the following composition

D_{a} : Γ(W^{+}) → Γ(W^{+}) ⊗ Γ((W^{+})^{∗} ⊗ W^{−}) → Γ(W^{−}),

where we are using the first property of our Spin^{ℂ} - structure.

The second equation is defined by the second property

ıF_{a}^{+} = −(ϕ ⊗ \overline{ϕ})_{0},

where zero denotes the traceless part of the corresponding homomorphism.

The next step — to introduce gauge group action on the configuration space. By itself it is infinite dimensional, and the space of solution of the system is, but after factorization by the action of the gauge group

G = Aut_{h}(detW^{+}),

we get a finite dimensional manifold M_{X}(g, c), which depends on our choice of Spin^{ℂ} - structure and our metric g. This manifold is smooth for generic metric if b_{2}^{+}(X) > 0, and moreover, the topological type of this manifold doesn’t depend on the choice of metric in general if b_{2}^{+}(X) > 1. Moreover, this manifold is compact and orientable, and one can fix an orientation by choosing orientation on the determinant line

detH^{0}(X, ℝ) ⊗ detH^{1}(X, ℝ) ⊗ H^{+}(X, ℝ).

Let us choose and fix an orientation.

The moduli space M_{X}(g, c) has virtual dimension

v.dimM_{X}(g, c) = \frac{1}{4}(c^{2} − 2χ − 3σ),

where χ, σ — are the euler characteristic and the signature of the base manifold X. For generic metric this dimension is achived, and the first version of the numerical invariant is as follows. Let the dimension be zero. Then for this Spin^{ℂ} - structure c ∈ H^{2}(X, ℝ) we have compact oriented zero dimensional moduli space. This M_{X}(g, c) should be a finite set of points, and we can calculate the number of these points with signs to get a numer, which is called the Seiberg — Witten invariant.

The point is that the zero dimensionality condition is absolutly known in the complex geometry. This condition is equivalent to the existence condition for an almost complex structure J over X with canonical class K_{J} ∈ H^{2}(X, ℝ).

For the positive dimensional case there exists procedure, defining the invariant, but only for even case.

Namely, it’s known that there exists the basic homological class h ∈ H^{2}(C^{∗}, ℤ), where the space C^{∗} is defined as the quotient space of irreducible pairs

(A_{h}(detW^{+}) × Γ_{∗}(W^{+}))/G.

Then if we fix a point x ∈ X we get the following fibration

1 → G_{0} → G → U(1)_{x} → 1,

so G_{0} is the subgroup of all gauge transformation, which act identically on the fiber (detW^{+})_{x}. Then one has fibration

C_{O}^{∗} → C^{∗},

where

C_{0}^{∗} = (A_{h}(detW^{+}) × Γ_{∗}(W^{+}))/G_{0},

and this fibration is a principal U(1) - bundle on C^{∗}. It’s natural to take the first Chern class of this principal bundle, thus we get our h.

Remark. If H^{1}(X, ℝ) ≠ 0 then there exist other cohomological classes, which one could try to exploit. But we are interested in the simply connected case.

If the moduli space M_{X}(g, c) has positive even dimension, one gets the number

N_{SW} (c) =< h^{d}, [M_{X}(g, c)] >,

where

d = \frac{1}{2}dimM_{X}(g, c).

Thus one has the map

N_{SW} (H^{2}(X, ℤ)) → ℤ,

but the last one is defined only for subset Spin_{0} ⊂ H^{2}(X, ℤ), which is

Spin_{0} = \{c ∈ H^{2}(X, ℤ)|c ≡ w_{1}(mod2), \frac{1}{8}(c^{2} − 2χ − 3σ) ∈ ℤ\}.

Remark. It’s obvious that if \frac{χ+σ}{4} ∈ ℤ, then Spin_{0} ≡ Spin, and in the other case Spin_{0} = ∅.

Our aim is to decribe in the geometrical framework the construction for the positive dimensional case.

2 The main theorem

The geometrical aspects are contained in the following

Theorem. Let in our setting for given simply connected X

dimM_{X}(g, c) = 2d.

Let us consider connected sum Y = X♯d\overline{ℂℙ}^{2} over d points x_{1}, ..., x_{d} ∈ X. Then the invariant, defined by M_{X}(c) equals to the invariant, defined by M_{Y} (c + ∑_{i=1}^{d} 3h_{i}).

Remark. It’s known, that the comological spaces of connected sum X_{1}♯X_{2} decompose as follows

H^{i}(X_{1}♯X_{2}) = H^{i}(X_{1}) ⊕ H^{i}(X_{2}), i ≠ 0, dimX_{i}.

So, in regorous manner we have to denote the images of all classes of the summands, but for brievity we’ll omitt all notations, introduced by the gluing procedure.

Thus there exists an alternative: to take a set P = {x_{1}, ..., x_{d}} of points x_{i} ∈ X, defining d representations of the cohomological class h ∈ H^{2}(C^{∗}, ℤ), and then to compute the invariant, or to take the same set P, form the connected sum with zero dimensional moduli space and compute the invariant.

Both approach’s are quite usefull, but as we’ll disscus in a future the last one gives some extension of the notion of almost complex structure.

Let us start with some arithmetical basis for our theorem. Namely, there is

Lemma. Consider in our setting a Spin^{ℂ} - structure c_{0} ∈ H^{2}(X, ℤ). Then the connected sum Y = X♯k\overline{ℂℙ}^{2} has an almost complex structure with canonical class of the form K_{Y} = c_{0} + ∑_{i=1}^{k} d_{i}h_{i} if and only if the number

d = \frac{1}{4}(c^{2} − 2χ(X) − 3σ(X))

is even.

To prove this let us recall that for the connected sum

\begin{matrix}χ(Y) = χ(X) + χ(k\overline{ℂℙ}^{2}) − 2 = χ(X) + k, \\ σ(Y) = σ(X) + σ(k\overline{ℂℙ}^{2}) = σ(X) − k. \end{matrix}

Then a class c_{0} + ∑_{i=1}^{k} d_{i}h_{i} to be canonical for an almost complex structure should satisfy the following two conditions:

1. to be a Spin^{ℂ} - structure d_{i} must be odd;

2. (c_{0} + ∑_{i=1}^{k} d_{i}h_{i})^{2} = 2(χ(X) + k) + 3(σ(X) − k),

thus

c_{0}^{2} − ∑_{i=1}^{k} d_{i}^{2} = 2χ(X) + 3σ(X) − k.

After rearrengment, one gets

c_{0}^{2} − 2χ(X) − 3σ(X) = ∑_{i=1}^{k}(d_{i}^{2} − 1).

So as d_{i} are odd number, the right side is divisible by 8, so the left side must do the same. So

\frac{1}{4}(c^{2} − 2χ(X) − 3σ(X))

must be even.

In this case we have the following solution:

d = k, and d_{i} = 3.

Remark. From the construction in [KMT] we know, that adding to X a copy of \overline{ℂℙ}^{2} doesn’t change the invariant for new Spin^{ℂ} - structure c_{0} +h, so essentially the solution above is unique.

Now the proof is obvious: from the description above one gets that the invariant correspondig to the positive dimensional case is defined as the number of solutions from M_{SW} (c_{0}) with trivial spinor parts in the framing set P = {x_{1}, ..., x_{d}}. But if we consider the connected sum, correspondig to this set,

Y = X♯d\overline{ℂℙ}^{2}

then an extendable solution of the Seiberg — Witten system over X could have zeros in the point set P. Moreover, from complexity of coupled Dirac operator, orders of the zeros must be even, so in general of the second order. And just these solutions survive in the glueing procedure.

Example. Let us consider in our setting the following natural case. After gluing with d copies of \overline{ℂℙ}^{2} in the framing set P one gets symplectic manifold Y with symplectic form ω and canonical class K_{ω} = c_{0} + ∑_{i=1}^{d} 3h_{i}.

Then by the first basic Taubes theorem ([T1]) we have

N_{SW} (K_{ω}) = ±1.

So we have to show, that the original moduli space M_{X}(c_{0}) could define the invariant equal to ±1. But in this situation one can use the second basic Taubes theorem ([T3]) and the construction from [KMT] to deduce that N_{SW} (c_{0}) equals to N_{SW} (c_{0}+ ∑_{i=1}^{d} h_{i}), and the last number is equal to the Gromov invariant over our symplectic manifold Y, corresponding to the following homological class

∑_{i=1}^{d} h_{i}^{∗} ∈ H_{2}(Y, ℤ),

(see [T3]). The invariant is defined by the counting of pseudoholomorphic curves, representing this homological class and passing through

k = \frac{1}{2} ((∑_{i=1}^{d} h_{i}^{∗})^{2}− < K_{ω}, ∑_{i=1}^{d} h_{i}^{∗} >)

points in general position. But

k = \frac{1}{2}(−d + 3d) = d.

Thus we can choose d points each on the corresponding \overline{ℂℙ}^{2} and then we have to count the number of −1 - sphere’s passing through a choosen point. Of course, there is unigue such (-1) - sphere, so the Gromov invariant is equal to ±1.

3 The complex and the spinor geometries

First of all let we recall that for a Spin^{ℂ} - structure c_{0} ∈ H^{2}(X, ℤ) the topology of the spinor bundles is defined by c_{0}, namely

\begin{matrix}c_{2}(W^{+}) = \frac{1}{4}(c_{0}^{2} − 2χ − 3σ), \\ c_{2}(W^{−}) = \frac{1}{4}(c_{0}^{2} − 2χ − 3σ) + χ, \end{matrix}

so for the case when c_{2}(W^{+}) = 0 we get an almost complex structure with canonical class c_{0}. For a section ϕ ∈ Γ(W^{+}), satisfing

(ϕ)_{0} = ∅

we have the corresponding almost complex structure J_{ϕ}, defined as follows. Let us consider a nondegenerated everywhere self dual two form

ω_{ϕ} = (ϕ ⊗ \overline{ϕ})_{0} ∈ Γ(adW^{+} = Λ^{+}).

Then our original metric g together with this form ω_{ϕ} defines the third element in hermitian triple (g, J_{ϕ}, ω_{ϕ}).

So any nonvanishing spinor field defines the corresponding almost complex structure with canonical class c_{0} = c_{1}(W^{+}), and this almost complex structure is compatible with the given orientation of X.

But now we have more general case, when c_{2}(W^{+}) = 2d > 0. Then we can propose a generalization of the notion of almost complex structures, namely, we’ll define an almost complex structure with degeneration in a point set P = {x_{1}, ..., x_{d}}. To define it let us consider a spinor field ϕ with double zeros in the set P. Then on the punctured manifold X\P we have well defined almost complex structure J_{ϕ} as above. For a homological class [Σ] ∈ H_{2}(X, ℤ) consider the following space of ”pseudoholomorphic” representations. Namely a curve Σ, representing [Σ], is called pseudoholomorphic with respect to the given almost complex structure with degeneration in P if it is pseudoholomorphic and smooth on X\P in the usual sense (so for a point x ∈ X\P, J_{ϕ} in the point preserves the subspace T_{x}Σ of the ambient space T_{x}X).

Of course, in a point x_{i} ∈ P our curve Σ cann’t be smooth. For if it were one could extend our almost complex structure J_{ϕ} through the point x_{i} using T_{x_{i}}Σ and the subspace of T_{x_{i}}X ortogonal with respect to g. The doubleness of zero x_{i} for ϕ ensures us that the compatibility of orientations holds. And in points x_{i} ∈ P our curve Σ must have self - intersections.

But we are interested in such smooth curves Σ that

Σ ∩ P = ∅.

Let us denote the space of such curves H_{X}([Σ], ϕ, P). Again, Σ ∈ H_{X}([Σ], ϕ, P) if

1. Σ is a smooth representation of [Σ] ∈ H_{2}(X, ℤ);

2. ∀x ∈ X\P we have J_{ϕ}(T_{x}Σ) = T_{x}Σ.

At the first sight this space is noncompact. But in our setting there exists the following compactification. Namely, let us consider the case (as above) when for given Spin^{ℂ} - structure c_{0} there is an extension c_{0} + ∑_{i=1}^{d} 3h_{i} on Y = X♯d\overline{ℂℙ}^{2} such that c_{0} +∑_{i=1}^{d} 3h_{i} is canonical class K_{ω} of a symplectic structure ω on Y. Thus our almost complex structure with degeneration J_{ϕ} can be extended to global almost complex structure J_{Y} on whole Y with canonical class K_{ω}.

Then we can distinguish two cases: when the given curve Σ doesn’t contain any point from P then it still represents the same homological class [Σ] ∈ H_{2}(X, ℤ) ⊂ H_{2}(Y, ℤ). In other case after normalization Σ represents new homological class [Σ] − ∑_{L⊂P} h_{i}^{∗}, where L is the subset of points in P, contained in Σ.

So we can use the Gromov — Taubes construction and get the space

H_{Y} ([Σ], J_{Y})

of pseudoholomorphic curves in usual sense, representing the given class [Σ] over our symplectic manifold Y. We know that this space is smooth real compact orientable manifold with dimension equal to

dimH_{Y} ([Σ], J_{Y}) = ([Σ])^{2}− < K_{ω}, [Σ] >= [Σ]^{2}− < c_{0}, [Σ] >∈ ℤ.

Essentially this good space is a compactification of the previous one

\overline{H_{X}([Σ], ϕ, P}) = H_{Y} ([Σ], J_{Y}).

So one can define the version of the Gromov invariant in the case, when our original base manifold X admits the extension to a symplectic Y of the shape as above.

With our main theorem in mind we have to conclude this section with the following

Conjecture. Let X be compact simply connected orientable riemannian 4 - manifold with b_{2}^{+} > 1, which admits a Spin^{ℂ} - structure c_{0} with the invariant equal to ±1. Then there exists a symplectic manifold Y of the form Y = X♯d\overline{ℂℙ}^{2}, where d = \frac{1}{8}(c_{0}^{2} − 2χ − 3σ), with canonical class K_{ω} = c_{0} + ∑_{i=1}^{d} 3h_{i}.

If this conjecture is true then one can define the Gromov invariant for all homological classes containing in H_{2}(X, ℤ) and generalize the Taubes result about the relationship between the Gromov and the Seiberg — Witten invariants of X.

But in any case one can hope that there exists such a generalization for pseudoholomorphic curves with respect to almost complex structures with degeneration.

4 The complex and the riemannian geometries.

Now we want to discuss a relationship, very close to the previous one, between riemannian and almost complex geometries.

This relationship just reflects some properties of the Spin(4) representation over riemannian 4- manifolds, but we’ll consider only geometrical meaning of this one.

First of all, let us recall the definition of canonical bundles for almost complex structures in the presence of riemannian metrics.

Let J be an almost complex structure over the riemannian manifold X. In the complex case one has an appropriate complex coordinates on X, definig corresponding complex line bundle Λ^{2,0}. But now there are not any complex coordinates because we are in the almost complex case. So to define the same bundle one needs an additional structure over X. Namely, real bundle on 2- forms Λ^{2}T^{∗}X has the extension of the adjoint to J operator J^{∗} : T^{∗}X → T^{∗}X, (J^{∗})^{2} = −id, and this exention

Λ^{2}J^{∗} : Λ^{2}T^{∗}X → Λ^{2}T^{∗}X

has two eigenspaces with egenvalues ±1. The −1 - eigenspace is two dimensional real bundle (Λ^{2,0} ⊕ Λ^{0,2})_{ℝ} and we have to describe a complex structure on this one, transforming this real 2- bundle to complex line bundle Λ^{2,0}. To define such a structure we need a riemannian metric g, compatible with J. Let us choose such a metric, then we get the corresponding hermitian triple (g, J, ω) and the corresponding Hodge decomposition

Λ^{2} = Λ^{+} ⊕ Λ^{−}

into the sum of the eigenspaces of the Hodge star operator ∗. It’s well known that (Λ^{2,0} ⊕ Λ^{0,2})_{ℝ} is a subbundle of Λ^{+}, and the complementary part is trivial real line bundle ℝω. Thus for a section v ∈ Γ(Λ^{2,0} ⊕ Λ^{0,2})_{ℝ} we have the following action of the complex structure J_{g} corresponding to g:

J_{g}(v) = v × ω,

where × denotes the vector product in the 3-dimensional space.

Thus we get complex line bundle {(Λ^{2,0} ⊕ Λ^{0,2})_{ℝ}, J_{g}} which is canonical bundle for J. Obviously, the first Chern class of the budnle doesn’t depend on the choice of riemannian metric.

As we’ve seen any nonvanishing spinor field for a riemannian metric g defines an almost complex structure. But now our aim is to describe how one can reconstruct a compatible riemannian metric from a given almost complex structure.

First of all, let us fix a smooth structure on X and let J be a smooth almost complex structure. Locally it means that over any point x in the projectivization of the complexified tangent space ℙ(T_{x}^{ℂ}) = ℂℙ^{3} there are

1. a standard real structure Θ;

2. two projective lines l_{1}, l_{2} ⊂ ℂℙ^{3}, satisfying

Θ(l_{1}) = l_{2}.

Of course, the first one is

l_{1} = ℙ(T_{x}^{1,0})

and the second one is

l_{2} = ℙ(T_{x}^{0,1}),

where

T_{x}^{ℂ} = T_{x}^{1,0} ⊕ T_{x}^{0,1}

is the usual decomposition.

But a conformal class of a reimannian metric g locally is just a nondegenerate real quadric Q_{x} without real points (see [jT]), so the reconstruction is made by finding of a real quadric Q_{x} with respect to Θ without real points such that

∀x l_{i} ⊂ Q_{x}.

To do this we choose a quarenionic real structure q_{x} on l_{1}

q_{x} : l_{1} → l_{1}, ∀p ∈ l_{1} q_{x}(p) ≠ p.

Then we get the following 1 - 1 - correspondence between l_{1} and l_{2}:

R_{x}(p) ↦ Θ_{x}(q_{x}(p)), p ∈ l_{1}.

And now we can construct set of projectrive lines

< p, R_{x}(p) >⊂ ℂℙ^{3},

which forms nondegenerate real quadric Q_{x} with respect to Θ without real points.

After globalization we get a conformal class of riemannian metrics, compatible with original J.

Remark. It’s just a reflection of the spinor represenation theory, and one can use the same procedure to reconstruct a metric with signature (2,2), but we cann’t reconstruct a Lorentz metric by the same procedure, so the Lorentz geometry is far from the complex geometry. To reconstruct a (2,2) - metric it’s sufficient to construct set of projective lines

< p, Θ_{x}(p) >⊂ ℂℙ^{3},

which forms a nondegenerated real quadric with real points.

With all above in mind we’ll consider the geometry of hermitian triple space over a riemannian 4- manifold X.

Let X be such manifold, which admits almost complex structures. Then let we denote as P_{X} the space of all hermitian triples over X, compatible, of course, with given smoth structure

P_{X} = {(g, J, ω)|g(Jv, Ju) = g(v, u), ω(v, u) = g(v, Ju)},

for every vector fields v, u ∈ Γ(T X).

There are two projections:

p : P_{X} → K_{X} = {J|J^{2} = −id}

to the space of all almost complex structures over X and

q : P_{X} → H_{X} = {g}

to the homogeneous space of all riemannian metrics, compatible with given smooth structure.

We’ll describe the fibers of these two projections.

The first projection. As we’ve seen above, for an almost complex structure J one can reconstruct a conformal class of compatible riemannian metric, choosing fiberwise quaternionic real structures on the corresponding projective lines l_{1}. Thus a fiber of p is isomorphic to the direct product of the space of all sections of the corresponing principal P U(2) - bundle P_{l_{1}} and the space of positive nonvanishing real functions. We need the second one to lift from the corresponding conformal class to a reimannian metric from this class.

The second projection. The next construction can be done in the following way. There is the set of canonical classes over X

C_{X} = {c ∈ H^{2}(X, ℤ)| c ≡ w_{2}(mod2), c^{2} = 2χ + 3σ},

defined topologically. Then a fiber of q is the direct product of the spaces of all nonvanishing spinor fields, corresponding to each class from C_{X}.

Of course, all the space P_{X} is infinite dimensional, so we need (as in the gauge theories) to impose the action of the gauge group of all C^{∞} - automorphisms of T X:

G = AutT X

(or, may be, another one). Then we can define the quotient space

T_{X} = P_{X}/G,

and the questions are:

1. the structure of this one;

2. the existence of a nontrivial invariant, defined by this space.

The point is that if X and X′ are diffeomorphic, then the spaces P_{X} and P_{X′} are isomorphic at least topologically (of course, in this case we have the corresponding isomorphism between C_{X} and C_{X′}, induced by the given diffeomorphism).

Now we’ll postpone these question, hoping that in the nearest future the answers would be found.

At the end I would like to express my gratitude to all staff in CIMAT for hospitality, but especially to Professors X. Gomez - Mont, S. Recillas, C. Gomez - Mont, R. Vila, L. Brambila, R. Carrion, A. Castorena and D. Dolgopyat for friendful attention, many non formal and valubale disscusions and remarks.

References

References

[D] S.K. Donaldson, ”The Seiberg — Witten equations and 4 - manifolds”, Bulletin of the AMS, vol. 33, 1, Jan. 1996;.

[DK] P.B. Kronheimer and S.K. Donaldson, ”The geometry of 4 - manifolds”, Oxford, Clarendon press, 1990..

[KT] D. Kotschik, J. Morgan, C.H. Taubes, ”Four manifolds without symplectic structures but with nontrivial Seiberg — Witten invariants”, Math. Res. Letters, vol 2, 1995, 797 — 808.

[MW] G. Moore, E. Witten, ”Integration over u - plane in Donaldson theory”, hep - th/ 9709193.

[PT] V. Pidstrigach and A. Tyurin, ”Localization of the Donaldson’s invariants along Seiberg — Witten classes”, Preprint Bielefeld, 075 (1995).

[T1] C.H. Taubes, ”The Seiberg — Witten invariants and symplectic forms”, Math. Res. Letters, vol 1, 1994, 809 — 822.

[T2] C.H. Taubes, ”More constraints on symplectic manifolds from Seiberg — Witten equations”, Math. Res. Letters, vol 2, 1995, 9 — 14.

[T3] C.H. Taubes, ”The Seiberg — Witten and Gromov invariants”, Math. Res. Letters, vol 2, 1995, 221 — 238.

[jT] N. Tyurin, ”Necessury and sufficient conditions for a deformation of a B - monopole into an instanton”, Izvestiya RAN, 1996, vol. 60:1, 211 — 224.