Guest blog by Monster from a U.K. research university founded in 1907
Let me try to explain in a maybe more physicistsfriendly way essentially the same things that dalpezzo already mentioned beneath the "blog post about 1729". It is an exercise for experts to notice the hidden assumptions and oversimplified explanations.
Let \(X\) be a compact manifold. Let us consider \(U(1)\) gauge theory on \(X\), i.e usual Maxwell's electromagnetism. Locally on \(X\), the gauge field is a 1form \(A\) and the field strength \(F=dA\) is a 2form. If \(X\) has nontrivial 2cycles, one can have nontrivial fluxes of \(F\) through these 2cycles. The number of topologically inequivalent 2cycles in \(X\) is \(B_2\), the second Betti number of \(X\) and, as the fluxes are quantized, a flux configuration is given by a collection \((n_1,\dots, n_{B_2})\) of \(B_2\) integers. It is easy to show that for any flux configuration, there exists a gauge field with the prescribed fluxes.
Assume that \(X\) has the extra structure of a complex manifold. It means that locally on \(X\), we have a notion of holomorphic coordinates \(z_i\) and antiholomorphic coordinates \(\bar{z}_i\). It is then possible to decompose the field strength as\[
F=F^{2,0}+F^{1,1}+F^{0,2}
\] where the \((2,0)\) part \(F^{2,0}\) only contains terms proportional to \(dz_i \wedge dz_j\), the \((1,1)\) part \(F^{1,1}\) only contains term proportional to \(dz_i \wedge d\bar{z}_j\), and the \((0,2)\) part \(F^{0,2}\) only contains terms proportional to \(d\bar{z}_i \wedge d\bar{z}_j\). In terms of indices, the \((p,q)\) part has \(p\) holomorphic indices and \(q\) antiholomorphic indices.
When the complex structures changes continuously, the holomorphic/antiholomorphic coordinates change continuously and so the above decomposition of the field strength changes continuously too. Now, given a complex structure on \(X\) and a flux configuration, one can ask the following question; is there a gauge field with the prescribed fluxes such that the associated field strength satisfies\[
F^{2,0}=F^{0,2}=0 \quad ?
\] This condition is a first order linear partial differential equation on the gauge field. It is always possible to solve it locally but an obstruction to glue these local solutions and to obtain a global solution can exist.
As the equation is linear, the set of flux configurations such that there exists a solution is a sublattice of the lattice of flux configurations. The Picard number of \(X\) is the rank of this sublattice, i.e. the number of independent flux configurations generating the space of flux configurations such that there exists a solution to the equation \(F^{2,0}=F^{0,2}=0\). The Picard number is an integer between \(0\) and \(B_2\) and in general depends on the complex structure of \(X\).
The field strength \(F\) of a gauge field configuration satisfies flux quantization but it is not in general the case of \(F^{2,0}\), \(F^{1,1}\) or \(F^{0,2}\). In general their fluxes are complex numbers. So it is useful to introduce the space of complexified flux configurations made of \(B_2\)uples \((a_1,...,a_{B_2})\) of complex numbers. The space of integral fluxes is a discrete subset of this complex vector space. One can show that the decomposition in \((2,0)\), \((1,1)\), and \((0,2)\) parts extend to the space of complexified flux configurations. The complex dimension of the space
of complexified flux configurations of type \((p,q)\) is called the Hodge number \(h^{p,q}\) of \(X\). One has\[
B_2=h^{2,0}+h^{1,1}+h^{0,2}
\] and \[
h^{2,0}=h^{0,2}.
\] One can show that the Hodge numbers do not change when the complex structure of \(X\) moves continuously but the corresponding subspaces of the space of complexified flux configurations in general moves continuously. The (integral) flux configurations such that there exists a gauge
field with these prescribed fluxes such that \(F^{2,0}=F^{0,2}=0\) are exactly the (integral) flux configurations leaving inside the \((1,1)\) subspace of the space of complexified flux configurations. In particular, the Picard number is always between \(0\) and \(h^{1,1}\).
So the picture to have is mind is the following: a big complex vector space, a discrete lattice of integral points and a specific subspace inside it. The Picard number measures the amount of integral points in the specific subspace. When the complex subspace moves, due to a change in the complex structure of \(X\), the Picard number in general changes. More precisely, when the complex subspace only moves a bit, an integral point which was not inside cannot become suddenly inside but an integral point inside can suddenly moves out. It means that the Picard number can get enhanced at special points of the moduli space of complex structures on \(X\). The whole subtlety of the Picard number comes from this interplay between the discrete set of (integral) flux configurations and the continuous subspace of complexified flux configurations of type \((1,1)\).
If \(h^{2,0}=0\), the above subtlety is not here: any complexified configuration is of type \((1,1)\), the equation \(F^{2,0}=F^{0,2}=0\) has always a solution and the Picard number is simply the second Betti number \(B_2\) and in particular does not depend on the complex structure. It is what happens for projective spaces, del Pezzo surfaces or CalabiYau manifolds (in the strict sense: holonomy equal to \(SU(n)\) and not just contained in \(SU(n)\)). In all these examples, the Picard number is not something interesting: it is something we already knew, i.e. \(B_2\).
To have the full subtle story of variations of Picard numbers, one needs to have \(h^{2,0}\) non zero. It is for example the case for complex tori or \(K3\) surfaces. For \(K3\) surfaces, we have\[
B_2=22, \quad h^{2,0}=h^{0,2}=1, \quad h^{1,1}=20.
\] The moduli space of complex structures on a \(K3\) surface is of complex dimension 20. A generic \(K3\) surface has Picard number \(0\). There is a special locus of complex codimension \(1\) at which the Picard number is enhanced to \(1\). There is a special locus of complex codimension \(2\) at which the Picard number is enhanced to \(2\) and so on until a special locus of complex codimension \(20\), i.e. dimension \(0\), at which the Picard number is enhanced to \(20\), its maximal possible value. Each of these special locus of complex dimension \(k\) is fairly complicated: it is a countable union of varieties of dimension \(k\).
For example, the space of \(K3\) surfaces of Picard number \(20\) is of dimension \(0\) but it is made of a countably infinite number of points and is in fact dense in the full moduli space of \(K3\) surfaces: it is as the rational numbers in the real numbers and so the full picture of the moduli space of \(K3\) surfaces with the various loci of given Picard numbers is extremely intricate.
Under nice hypothesis (\(X\) algebraic), there is a more geometric interpretation of the Picard number. A flux configuration is the data for each 2cycle of an integer. If \(X\) is of (real) dimension \(n\), one can interpret these integers as prescribed intersection numbers with the various 2cycles for a \((n2)\)cycle. If \(X\) is a (algebraic) complex manifold, one can show that the existence of a gauge field with \(F^{2,0}=F^{0,2}=0\) and prescribed flux configuration is equivalent to the existence of an holomorphic representative for the \((n2)\)cycle determined by the flux configuration. In other words, the Picard number measures the amount of holomorphic hypersurfaces (complex codimension \(1\), i.e. real codimension \(2\)) in \(X\). For a \(K3\) surface, of real dimension \(4\), i.e. complex dimension \(2\), an holomorphic hypersurface is the same thing that an holomorphic curve (complex dimension \(1\), i.e. real dimension \(2\)). For example, a generic \(K3\) surface has Picard number \(0\) and so has no holomorphic curves in it. In contrary, a \(K3\) surface with high Picard number has many holomorphic curves in it and so a rich complex geometry. Any nontrivial holomorphic geometry in a \(K3\) surface in general requires a high enough Picard number.
For instance, to compactify Ftheory on a \(K3\) surface, one needs an elliptic fibration with a section. The fiber of the elliptic fibration is a nontrivial holomorphic curve in the \(K3\) surface and similarly for the image of the section, and so such \(K3\) surface has at least Picard number \(2\). This kind on restriction on the allowed \(K3\) surfaces for a Ftheory compactification is not very surprising: IIB superstring compactified on a \(K3\) surface is dual to heterotic string on \(T^4\) and Ftheory compactified on a \(K3\) surface is dual to heterotic string on \(T^2\). As there are clearly less parameters in \(T^2\) that in \(T^4\), the range of allowed \(K3\) on the Ftheory side has to be somehow limited.
Similarly, I think that when one writes a \(K3\) surface in a relatively simple explicit form, something one wants to do for explicit computations and explicit checks of various dualities, one generally obtains a \(K3\) surface with relatively high Picard number precisely because the ability to write a simple description of an object is a sign of its deeper and richer structure.
But in general, it seems quite difficult to find a direct physical meaning to the Picard number or a jump in the Picard number. For example, moving in the moduli space of \(K3\) surfaces, when the Picard number jumps, nothing happens to the topology, nothing becomes singular, it is a really a subtle modification of the complex geometry and so does not correspond to something as brutal as a topology change transition or gauge symmetry enhancement. It is has a physical meaning, this one has to be relatively subtle. I can think of two examples of such physical meaning and both are about \(K3\) surfaces of maximal Picard number, i.e. Picard number \(20\).
The first one is due to Moore and is about the attractor mechanism. Let us have a look at type IIB superstring compactified on the CalabiYau 3fold obtained as a product of a \(K3\) surface by an elliptic curve. The vector multiplet moduli space is the complex moduli space of this CalabiYau
and so in particular contains the moduli space of \(K3\) surfaces. One can construct BPS black holes in four dimensions by wrapping D3branes over 3cycles of the CalabiYau. The possible charges for this BPS black holes form a infinite discrete set. The choice of a vacua of the theory is a choice of asymptotic value for the vectomultiplet moduli at infinity. But when we have a BPS black hole and when we move from infinity toward the black hole, the value of the vector multiplet get modified, as prescribed by the supergravity equation of motion. The attractor mechanism describes this evolution of the moduli and asserts that the value at the horizon of the black hole only depends on the charge of the black hole and not of the asymptotic value (as expected by general black hole entropy considerations: the entropy only depends on the charge and so the local geometry near the horizon should also only depends on the charge). So for each choice of charge, there should be a particular \(K3\) surface describing the compactified geometry at the horizon. The result is that these \(K3\) surfaces are exactly the \(K3\) surfaces of maximal Picard number.
Second, Gukov and Vafa have speculated about when a nonlinear sigma model, defining a two dimensional conformal field theory, is in fact a rational conformal field theory. A rational CFT is a CFT with the local fields organized in finitely many irreducible representations of a chiral algebra. The rationality of a CFT is a quite subtle property, not preserved in general under deformations. Applied to the case of \(K3\) surfaces, the proposal of Gukov and Vafa asserts that the supersymmetric sigma model of target a \(K3\) surface is a rational SCFT if and only the \(K3\) surface has maximal Picard number. I think that the validity of this proposal is an open question.
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