Academic literature on the topic 'Holomorphicity'

String theory is so far the best candidate for quantization of gravity. Its very definition is however somewhat unsatisfactory, as a nonperturbative definition is still not completely clear. An important step in this direction has been to realize that the space of the states of this theory will finally include not only states coming from strings, but also from higher-dimensional extended objects, that were christened D-branes. Though in the perturbative formulation these latter objects can only be understood in terms of open strings, in the nonperturbative theory strings and branes will appear on the same footing. It is therefore compelling to understand as many properties of these objects as one can. This is not only out of the intellectual curiosity of fully understanding the structure and formulation of an interesting branch of mathematics, as string theory has happened to become; but for the very physical relevance of the theory itself. As an example, one of the requirements for any theory to supersede the current standard model of particle physics is to explain the observed parameters of the latter. This is currently the main drawback of string theory, given the huge number of possible compactifications to four dimensions, with little or no theoretical reason of preference among them. This is the physical ground of the huge mathematical problem of describing the moduli space of string vacua. D-branes are relevant to this problem because of their above mentioned role in nonperturbative string theory, and in particular because they give additional moduli. The best understood way to probe nonperturbative features of a supersymmetric theory is to study its BPS states; so one is lead to the study of those configurations of branes which do not break supersymmetry completely. We will study two separate applications of this method, both with the aim of understanding the nonperturbative structure of string theory. The first is the proposal known as Matrix String Theory (MST) [62, 21]. The idea that a secondquantized theory of strings, which describes multi-strings states, can be linked with matrices, is somewhat an old one, but it was revived more recently after the emergence of Matrix Theory [3]. After the initial proposal, the essential lines of a proof were shown [12]. The main tool in this proof were BPS solutions of the theory. These tum out to be given by Hitchin equations in 2 dimensions F- i[x,xt]w =cw, DX=O, (0.0.1) where Fis the curvature of the gauge field of the theory, D the (1, 0) part of its covariant derivative, and X is a matrix valued field. There is then a well-known technique to produce, out of these data, a covering of the base space, and a bundle on it. In our case the covering will be a Riemann surface with marked points; in the method of [12] these are used to reproduce the S-matrix of the Green-Schwarz string. The question arises, however, of whether this method can give the correct moduli space for string scatterings: this will be the theme of our first chapter. We will see that several interesting phenomena happen. First, there is a partial discretization of the moduli space, whose spacing goes however to zero in the limit in which MST is expected to reduce to perturbative string theory. Second, in general one has to allow these Riemann surfaces to be singular. Physically, this comes because we are embedding them in the four dimensions of spacetime, and they look singular; taking in account fluctuations in the extra dimensions, the singularities get resolved. Along the way, we will describe an interesting way of arriving to a known relationship between the Newton polygon description of a plane curve and its genus, involving toric geometry. Many of the mathematical concepts and methods introduced in this chapter will be used in the later part as well. Another interesting way to understand nonperturbative features of the theory, and at the same time to tackle an important part of the string theory moduli problem we alluded to above, is to consider branes in Calabi-Yau manifolds. The moduli space of Calabi-Yau compactifications without branes has been already thoroughly studied (see for example [40] for a review); inclusion of branes complicates enormously the problem. In perspective, the moduli space of branes should give an infinite covering of the closed string moduli space: for each closed string background there are several allowed brane configurations. As part of such a program, we will start with an already interesting part: fixing a Calabi-Yau and considering its closed string Kahler moduli space. In chapter 2 we will introduce the main ingredients for what follows. In particular, it will be noted a fact that will allow us to split the problem in two. This is already visible in Hitchin equations (0.0.1), which are again met in particular cases in this context: these equations consist of a real equation and of a complex holomorphic one. This splitting has a deeper meaning than one may think; it corresponds to the splitting of the equations which describe moduli spaces of vacua of supersymmetric gauge theories in D-terms and F-terms. Moreover, in the cases we will consider (varying Kahler moduli only) Fterms (holomorphic equations) are not modified [16]. It makes thus sense to consider solutions to the latter first, and to postpone analysis of the former (which, for reasons to become clear, will be referred to as a stability problem). This first step will be done in chapter 3. We will see that, in two particular points of the Kahler moduli space (called large volume limit and Gepner point), branes can be described by two different kind of objects: coherent sheaves and quiver representations. Since the moduli space is connected, a continuous interpolation between them should be possible: after having noted that this is nothing but a generalized McKay correspondence, we will find this interpolation in several examples, using Bondal theorem [10], a generalization to Fano varieties of Beilinson's one [6]. The most natural formulation of this theorem is in terms of derived category, and we will argue then that this is the most natural classifying object for D-branes as long as one neglects stability matters. These will be then the subject of chapter 4. Again, the scheme will be here to understand tractable points of the moduli space and try then to extend the analysis to the whole of it. In the large volume limit, as we have mentioned, we will find generalizations of Hitchin equations. They involve in an essential way connections on the gauge bundle E on the brane; it turns out that a property of this bundle is equivalent to the existence of solutions to the equations. This property is called stability and it always involves properties of subbundles E' of the given bundle E; the equivalence of stability with existence of solutions to the equations is sometimes called Hitchin-Kobayashi correspondence. It is interesting to note that the equations have also the meaning of physical stability of branes, giving a nice coincidence of words. Away from the large volume limit, on a generic point of Kahler moduli space, these D-term equations will be deformed. The complete form of these deformed equations is not known, though some features were explored [58]. However, from worldsheet arguments one can argue for a deformed stability, known as II-stability [28], which should again be equivalent to solutions of the deformed equations, whatever they are. Since this deformed stability condition reduces to usual stability in the limit, this gives in a sense a string theory rederivation of Hitchin-Kobayashi correspondence. However, we have seen that, away from the large volume limit, the appropriate description is derived category, one would expect some kind of derived category stability to come in. The final word on this has not yet come, and this part will be more of a sketch of a program than a conclusive one. The final goal should be to understand completely what is the derived category stability coming from worldsheet arguments, and then rederive from this all the known stabilities of known equations in the tractable limits. Depending on the cases, these equations can involve, besides the connection on the gauge bundle Eon the brane, transverse scalars X (as in the Hitchin example we have seen) and/or tachyons; each of these equations has its own Hitchin-Kobayashi correspondence with a stability condition, and string theory should allow to rederive all of them as different cases of a unique derived category stability.

The critical phases of two dimensional lattice models are widely believed to be described by conformal quantum field theories in the scaling limit. In the past few years, formal proofs of the conformal invariance in different formulations of the Ising model have emerged which make pivotal use of some complex lattice observables. Due to their distinctive property of discrete holomorphicity, they are considered to be the lattice counterpart of holomorphic currents in the field theories. In this thesis, we study a weakened form of discrete holomorphicity that is known to be obeyed by natural generalizations of these observables to three important families of solvable models. The main result of this thesis is that discrete holomorphicity can be seen as a requirement stronger than the conditions of integrability of the models. That is, these conditions, the inversion relations and the Yang-Baxter equations, can be derived from this form of lattice holomorphicity. This finding is proposed as an explanation for the remarkable observation that discrete holomorphicity holds only on the integrable critical manifold of the weights. For the self-dual models, the duality conditions can also be established similarly. A key role in this argument is played by the rhombic embeddings of Baxter lattices. It is noted that, by the requirement of holomorphicity on every rhombus, the conformal spins of the observables are restricted to a discrete spectrum that label the solutions in which the angles of the rhombuses are interpreted as re-scaled spectral parameters. This interpretation allows the relationships between the spectral parameters in the integrability conditions to be seen as the criteria for geometric consistency of the rhombic embedding. The crossing symmetry of the models is then related to the alignment of the rhombuses with respect to the rapidity lines. It has also been shown here that values of the observables on the boundary of a simply connected domain remain unchanged by local rearrangement of rhombuses due to the Z-invariance of the models. This provides an independent characterization of the observables besides discrete holomorphicity and enables us to define them on the equivalence class of Baxter lattices with any given braiding of the rapidity lines. For an equivalence class, the holomorphicity equations then provide linear relations among its partition functions with different boundary conditions. The results of this thesis thus point to the essential, albeit somewhat obscure, role of integrability in the rigorous proofs.

Consider an open subset O of the complex plane and a function f : O ÝÑ C. We understand the concept of holomorphicity of a complex-valued function. Suppose a function f defined on O takes values in a locally convex topological vector space (the complex plane is one particular example of a locally convex topological vector space). We want to study the notion of holomorphicity for f in this general setting. In the paper “Sur certains espaces de fonctions holomorphes I” [1], Alexandre Grothendieck deals with the concept of holomorphicity of vector-valued functions and duality. We read, understand, reproduce and at times fill in necessary details of the content of section 2 and a portion of section 4 of the paper. We understand the extension of Lebesgue integral to vector-valued functions on a measure space, named after I.M. Gelfand and B.J. Pettis as Gelfand-Pettis integral or Pettis integral (page 77 in [2]). Using the definition of Pettis integral and some results from complex analysis we introduce three notions of holomorphicity for the function f: holomorphicity of f on the open set O, weak derivability of f at a point zo P O and strong differentiability of f at a point zo P O. In Chapter 2 (section 2 in [1]) of this thesis, we study the conditions under which the first two notions of holomorphicity, i.e., f is holomorphic on the open set O and f is weakly derivable at every point of O coincide. Further the same conditions will imply the strong differentiability of f at each point of O. Also, we study Cauchy’s integral formula and Taylor’s expansion for vector valued holomorphic functions in Chapter 2. In section 4 of the paper “Sur certains espaces de fonctions holomorphes I” [1], for a subset of the Riemann sphere, Grothendieck introduces the space Pp ,Eq as the space of all locally holomorphic functions on vanishing at 8 if 8 P . He further considers two locally convex topological vector spaces E and F in separate duality and proves that the spaces Pp 1,Eq and Pp 2, Fq, where 1 and 2 are complementary subsets of the Riemann sphere, are in separate duality under some general conditions. In Chapter 3 of this thesis, which constitute the main portion of the thesis, we study the space Pp ,Eq and further present the argument of Grothendieck for separating duality with all details

This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.