Underlying the Non-Perturbative Quantum Geometry of the Quartic Gauge Couplings in 8D.

A lot can be learned by simply focussing on the leading singularities in the moduli space of the effective theory. However, for the sake of performing really non-trivial quantitative tests of the heterotic/F-theory duality, we should try harder in order to reproduce the exact functional form of the couplings ∆_eff(T) from K3 geometry. The hope is, of course, to learn something new about how to do exact non-perturbative computations in D-brane physics.

More specifically, the issue is to eventually determine the extra contributions to the geometric Green’s functions. Having a priori no good clue from first principles how to do this, the results of the previous section, together with experience with four dimensional compactifications with N = 2 supersymmetry, suggest that somehow mirror symmetry should be a useful tool.

The starting point is the observation that threshold couplings of similar structure appear also in four dimensional, N = 2 supersymmetric compactifications of type II strings on Calabi-Yau threefolds. More precisely, these coupling functions multiply operators of the form TrF_G² (in contrast to quartic operators in d = 8), and can be written in the form

∆^(4d)_eff ∼ ln[λ^α₁(1-λ)^α₂(λ’)³] + γ(λ) —– (1)

which is similar to Green’s function

∆eff (λ) = ∆∑N-1prime form (λ) + δ(λ)

It is to be noted that a Green’s function is in general ambiguous up to the addition of a finite piece, and it is this ambiguous piece to which we can formally attribute those extra non-singular, non-perturbative corrections.

The term δ(λ) contributes to dilation flat coordinate. The dilation S is a period associated with the CY threefold moduli space, and like all period integrals, it satisfies a system of linear differential equations. This differential equation may then be translated back into geometry, and this then would hopefully give us a clue about what the relevant quantum geometry is that underlies those quartic gauge couplings in eight dimensions.

The starting point is the families of singular K3 surfaces associated with which are the period integrals that evaluate to the hypergeometric functions. Generally, period integrals satisfy the Picard-Fuchs linear differential equations.

The four-dimensional theories are obtained by compactifying the type II strings on CY threefolds of special type, namely they are fibrations of the K3 surfaces over P^l. The size of the P¹ yields then an additional modulus, whose associated fiat coordinate is precisely the dilaton S (in the dual, heterotic language; from the type II point of view, it is simply another geometric modulus). The K3-fibered threefolds are then associated with enlarged PF systems of the form:

L_N(z, y) = θ_z(θ_z – 2θ_y) – z(θ_z + 1/2N)(θ_z + 1/2 – 1/2N)

L₂(y) = θ_y² – 2y(2θ_y +1)θ_y —– (2)

For perturbative, one-loop contributions on the heterotic side (which capture the full story in d = 8, in contrast to d = 4), we need to consider only the weak coupling limit, which corresponds to the limit of large base space: y ∼ e^-S → 0. Though we might now be tempted to drop all the θ_y ≡ y∂y terms in the PF system, we better note that the θ_y term in L_N(z, y) can a non-vanishing contribution, namely in particular when it hits the logarithmic piece of the dilaton period, S = -In[y] + γ. As a result one finds that the piece , that we want to compute satisfies in the limit y → 0 the following inhomogenous differential equation

L_N . (γϖ₀)(z) = ϖ₀(z) —– (3)

We now apply the inverse of this strategy to our eight dimensional problem. Since we know from the perturbative heterotic calculation what the exact answer for δ must be, we can work backwards and see what inhomogenous differential equation the extra contribution δ(λ) obeys. It satisfies

L_N^⊗2 . (δϖ₀)(z) = ϖ₀²(z) —– (4)

whose homogenous operator

L_N^⊗2(z) = θ_z³ – z(θ_z + 1 – 1/N)(θ_z + 1/2)(θ_z + 1/N) —– (5)

is the symmetric square of the K3 Picard-Fuchs operator. This means that its solution space is given by the symmetric square of the solution space of L_N(z), i.e.,

L_N^⊗2 . (ϖ₀², ϖ₀ϖ₁, ϖ₁²) = 0 —– (6)

Even though the inhomogenous PF equation (4) concisely captures the extra corrections in the eight-dimensional threshold terms, the considerations leading to this equation have been rather formal and it would be clearly desirable to get a better understanding of what it mathematically and physically means.

Note that in the four dimensional situation, the PF operator L_N(z), which figures as homogenous piece in (3), is by construction associated with the K3 fiber of the threefold. By analogy, the homogenous piece of equation (4) should then tell us something about the geometry that is relevant in the eight dimensional situation. Observing that the solution space (6) is given by products of the K3 periods, it is clear what the natural geometrical object is: it must be the symmetric square Sym²(K3) = (K3 x K3)/Ζ₂. Being a hyperkähler manifold, its periods (not subject to world-sheet instanton corrections) indeed enjoy the factorization property exhibited by (6).

Formal similarity of the four and eight-dimensional string compactifications: the underlying quantum geometry that underlies the quadratic or quartic gauge couplings appears to be given by three- or five-folds, which are fibrations of K3 or its symmetric square, respectively. The perturbative computations on the heterotic side are supposdly reproduced by the mirror maps on these manifolds in the limit where the base P^l‘s are large.

The occurrence of such symmetric products is familiar in D-brane physics. The geometrical structure that is relevant to us is however not just the symmetric square of K3, but rather a fibration of it, in the limit of large base space – this is precisely what the content of the inhomogenous PF equation (4) is. It is however not at all obvious to us why this particular structure of a hyperkähler-fibered five-fold would underlie the non-perturbative quantum geometry of the quartic gauge couplings in eight dimensions.

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The Second Trichotomy. Thought of the Day 120.0

The second trichotomy (here is the first) is probably the most well-known piece of Peirce’s semiotics: it distinguishes three possible relations between the sign and its (dynamical) object. This relation may be motivated by similarity, by actual connection, or by general habit – giving rise to the sign classes icon, index, and symbol, respectively.

According to the second trichotomy, a Sign may be termed an Icon, an Index, or a Symbol.

An Icon is a sign which refers to the Object that it denotes merely by virtue of characters of its own, and which it possesses, just the same, whether any such Object actually exists or not. It is true that unless there really is such an Object, the Icon does not act as a sign; but this has nothing to do with its character as a sign. Anything whatever, be it quality, existent individual, or law, is an Icon of anything, in so far as it is like that thing and used as a sign of it.

An Index is a sign which refers to the Object that it denotes by virtue of being really affected by that Object. It cannot, therefore, be a Qualisign, because qualities are whatever they are independently of anything else. In so far as the Index is affected by the Object, it necessarily has some Quality in common with the Object, and it is in respect to these that it refers to the Object. It does, therefore, involve a sort of Icon, although an Icon of a peculiar kind; and it is not the mere resemblance of its Object, even in these respects which makes it a sign, but it is the actual modification of it by the Object. 

A Symbol is a sign which refers to the Object that it denotes by virtue of a law, usually an association of general ideas, which operates to cause the Symbol to be interpreted as referring to that Object. It is thus itself a general type or law, that is, a Legisign. As such it acts through a Replica. Not only is it general in itself, but the Object to which it refers is of general nature. Now that which is general has its being in the instances it will determine. There must, therefore, be existent instances of what the Symbol denotes, although we must here understand by ‘existent’, existent in the possibly imaginary universe to which the Symbol refers. The Symbol will indirectly, through the association or other law, be affected by those instances; and thus the Symbol will involve a sort of Index, although an Index of a peculiar kind. It will not, however, be by any means true that the slight effect upon the Symbol of those instances accounts for the significant character of the Symbol.

The icon refers to its object solely by means of its own properties. This implies that an icon potentially refers to an indefinite class of objects, namely all those objects which have, in some respect, a relation of similarity to it. In recent semiotics, it has often been remarked by someone like Nelson Goodman that any phenomenon can be said to be like any other phenomenon in some respect, if the criterion of similarity is chosen sufficiently general, just like the establishment of any convention immediately implies a similarity relation. If Nelson Goodman picks out two otherwise very different objects, then they are immediately similar to the extent that they now have the same relation to Nelson Goodman. Goodman and others have for this reason deemed the similarity relation insignificant – and consequently put the whole burden of semiotics on the shoulders of conventional signs only. But the counterargument against this rejection of the relevance of the icon lies close at hand. Given a tertium comparationis, a measuring stick, it is no longer possible to make anything be like anything else. This lies in Peirce’s observation that ‘It is true that unless there really is such an Object, the Icon does not act as a sign ’ The icon only functions as a sign to the extent that it is, in fact, used to refer to some object – and when it does that, some criterion for similarity, a measuring stick (or, at least, a delimited bundle of possible measuring sticks) are given in and with the comparison. In the quote just given, it is of course the immediate object Peirce refers to – it is no claim that there should in fact exist such an object as the icon refers to. Goodman and others are of course right in claiming that as ‘Anything whatever ( ) is an Icon of anything ’, then the universe is pervaded by a continuum of possible similarity relations back and forth, but as soon as some phenomenon is in fact used as an icon for an object, then a specific bundle of similarity relations are picked out: ‘ in so far as it is like that thing.’

Just like the qualisign, the icon is a limit category. ‘A possibility alone is an Icon purely by virtue of its quality; and its object can only be a Firstness.’ (Charles S. PeirceThe Essential Peirce_ Selected Philosophical Writings). Strictly speaking, a pure icon may only refer one possible Firstness to another. The pure icon would be an identity relation between possibilities. Consequently, the icon must, as soon as it functions as a sign, be more than iconic. The icon is typically an aspect of a more complicated sign, even if very often a most important aspect, because providing the predicative aspect of that sign. This Peirce records by his notion of ‘hypoicon’: ‘But a sign may be iconic, that is, may represent its object mainly by its similarity, no matter what its mode of being. If a substantive is wanted, an iconic representamen may be termed a hypoicon’. Hypoicons are signs which to a large extent makes use of iconical means as meaning-givers: images, paintings, photos, diagrams, etc. But the iconic meaning realized in hypoicons have an immensely fundamental role in Peirce’s semiotics. As icons are the only signs that look-like, then they are at the same time the only signs realizing meaning. Thus any higher sign, index and symbol alike, must contain, or, by association or inference terminate in, an icon. If a symbol can not give an iconic interpretant as a result, it is empty. In that respect, Peirce’s doctrine parallels that of Husserl where merely signitive acts require fulfillment by intuitive (‘anschauliche’) acts. This is actually Peirce’s continuation of Kant’s famous claim that intuitions without concepts are blind, while concepts without intuitions are empty. When Peirce observes that ‘With the exception of knowledge, in the present instant, of the contents of consciousness in that instant (the existence of which knowledge is open to doubt) all our thought and knowledge is by signs’ (Letters to Lady Welby), then these signs necessarily involve iconic components. Peirce has often been attacked for his tendency towards a pan-semiotism which lets all mental and physical processes take place via signs – in the quote just given, he, analogous to Husserl, claims there must be a basic evidence anterior to the sign – just like Husserl this evidence before the sign must be based on a ‘metaphysics of presence’ – the ‘present instant’ provides what is not yet mediated by signs. But icons provide the connection of signs, logic and science to this foundation for Peirce’s phenomenology: the icon is the only sign providing evidence (Charles S. Peirce The New Elements of Mathematics Vol. 4). The icon is, through its timeless similarity, apt to communicate aspects of an experience ‘in the present instant’. Thus, the typical index contains an icon (more or less elaborated, it is true): any symbol intends an iconic interpretant. Continuity is at stake in relation to the icon to the extent that the icon, while not in itself general, is the bearer of a potential generality. The infinitesimal generality is decisive for the higher sign types’ possibility to give rise to thought: the symbol thus contains a bundle of general icons defining its meaning. A special icon providing the condition of possibility for general and rigorous thought is, of course, the diagram.

The index connects the sign directly with its object via connection in space and time; as an actual sign connected to its object, the index is turned towards the past: the action which has left the index as a mark must be located in time earlier than the sign, so that the index presupposes, at least, the continuity of time and space without which an index might occur spontaneously and without any connection to a preceding action. Maybe surprisingly, in the Peircean doctrine, the index falls in two subtypes: designators vs. reagents. Reagents are the simplest – here the sign is caused by its object in one way or another. Designators, on the other hand, are more complex: the index finger as pointing to an object or the demonstrative pronoun as the subject of a proposition are prototypical examples. Here, the index presupposes an intention – the will to point out the object for some receiver. Designators, it must be argued, presuppose reagents: it is only possible to designate an object if you have already been in reagent contact (simulated or not) with it (this forming the rational kernel of causal reference theories of meaning). The closer determination of the object of an index, however, invariably involves selection on the background of continuities.

On the level of the symbol, continuity and generality play a main role – as always when approaching issues defined by Thirdness. The symbol is, in itself a legisign, that is, it is a general object which exists only due to its actual instantiations. The symbol itself is a real and general recipe for the production of similar instantiations in the future. But apart from thus being a legisign, it is connected to its object thanks to a habit, or regularity. Sometimes, this is taken to mean ‘due to a convention’ – in an attempt to distinguish conventional as opposed to motivated sign types. This, however, rests on a misunderstanding of Peirce’s doctrine in which the trichotomies record aspects of sign, not mutually exclusive, independent classes of signs: symbols and icons do not form opposed, autonomous sign classes; rather, the content of the symbol is constructed from indices and general icons. The habit realized by a symbol connects it, as a legisign, to an object which is also general – an object which just like the symbol itself exists in instantiations, be they real or imagined. The symbol is thus a connection between two general objects, each of them being actualized through replicas, tokens – a connection between two continua, that is:

Definition 1. Any Blank is a symbol which could not be vaguer than it is (although it may be so connected with a definite symbol as to form with it, a part of another partially definite symbol), yet which has a purpose.

Axiom 1. It is the nature of every symbol to blank in part. [ ]

Definition 2. Any Sheet would be that element of an entire symbol which is the subject of whatever definiteness it may have, and any such element of an entire symbol would be a Sheet. (‘Sketch of Dichotomic Mathematics’ (The New Elements of Mathematics Vol. 4 Mathematical Philosophy)

The symbol’s generality can be described as it having always blanks having the character of being indefinite parts of its continuous sheet. Thus, the continuity of its blank parts is what grants its generality. The symbol determines its object according to some rule, granting the object satisfies that rule – but leaving the object indeterminate in all other respects. It is tempting to take the typical symbol to be a word, but it should rather be taken as the argument – the predicate and the proposition being degenerate versions of arguments with further continuous blanks inserted by erasure, so to speak, forming the third trichotomy of term, proposition, argument.