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Finite-g Strings

Benoît Vicedo

Dissertation submitted for the Degree of

Doctor of Philosophy

at the University of Cambridge

Department of Applied Mathematics and Theoretical Physics

& Trinity College

University of Cambridge, UK

July 2008

Acknowledgements

First of all, I am deeply indebted to my supervisor Nicholas Dorey, for all his invaluable advice and insight throughout the whole of my Ph.D, for his guidance during our collaborative work as well as his thoughtful input into my independent work.

I am extremely grateful to Harry Braden, for taking interest in my work, for reading my papers in great detail, for raising many important points with regards to technical issues as well as for the many useful and stimulating discussions on various aspects of finite-gap integration. Without his “rigour” this thesis would not be complete. I am also very grateful to Harry for inviting me to give various talks in Edinburgh.

I would like to thank Marc Magro and Jean-Michel Maillet from École Normale Supérieure de Lyon for giving me the opportunity to present my work there and interact with the members of the theoretical physics group. I am especially grateful to Marc for his careful reading of various parts of my work as well as for bringing up certain issues that needed elaboration.

I would also like to thank Keisuke Okamura and Ryo Suzuki for the fruitful collaboration and the many interesting email correspondences.

This work was supported by both a Trinity College Internal Graduate Studentship and an Engineering and Physical Sciences Research Council Grant.

Last but not least, I would like to thank my parents and brother for all their moral support and constant encouragement throughout my studies.

Declaration

This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. The research described in this dissertation was carried out in the Department of Applied Mathematics and Theoretical Physics, Cambridge University, between September 2004 and March 2008. Except where reference is made to the work of others, all the results are original and based on the following works of mine:

  1. “On the Dynamics of Finite-Gap Solutions in Classical String Theory”
    N. Dorey and B. Vicedo
    JHEP 0607, 014 (2006) hep-th/0601194

  2. “A Symplectic Structure for String Theory on Integrable Backgrounds”
    N. Dorey and B. Vicedo
    JHEP 0703, 045 (2007) hep-th/0606287

  3. “Semiclassical Quantisation of Finite-Gap Strings”
    B. Vicedo
    JHEP 0806, 086 (2008) arXiv:0803.1605 [hep-th]

These papers are referred to as [1], [2] and [3] respectively in the bibliography. The content of Part II is taken mostly from [2]. Part III is based on all three papers [1, 2, 3] and Part IV is entirely based on [3]. None of the original works contained in this dissertation has been submitted by me for any other degree, diploma or similar qualification.

The following is a list of my other publications, referred to as [4] and [5] in the bibliography. The main purpose of these papers is not discussed in this thesis although certain minor results from them are used:

  1. “Giant Magnons and Singular Curves”
    B. Vicedo
    JHEP 0712, 078 (2007) hep-th/0703180

  2. “Large winding sector of AdS/CFT”
    H. Hayashi, K. Okamura, R. Suzuki and B. Vicedo
    JHEP 0711, 033 (2007) arXiv:0709.4033 [hep-th]

Benoît Vicedo

Cambridge, UK

20th July 2008

Abstract

In view of one day proving the AdS/CFT correspondence, a deeper understanding of string theory on certain curved backgrounds such as is required. In this dissertation we make a step in this direction by focusing on .

It was discovered in recent years that string theory on admits a Lax formulation. However, the complete statement of integrability requires not only the existence of a Lax formulation, but also that the resulting integrals of motion are in pairwise involution. This idea is central to the first part of this thesis.

Exploiting this integrability we apply algebro-geometric methods to string theory on and obtain the general finite-gap solution. The construction is based on an invariant algebraic curve previously found in the case. However, encoding the dynamics of the solution requires specification of additional marked points. By restricting the symplectic structure of the string to this algebro-geometric data we derive the action-angle variables of the system.

We then perform a first-principle semiclassical quantisation of string theory on as a toy model for strings on . The result is exactly what one expects from the dual gauge theory perspective, namely the underlying algebraic curve discretises in a natural way. We also derive a general formula for the fluctuation energies around the generic finite-gap solution. The ideas used can be generalised to .

Contents

Chapter 0 Introduction/Review

0.1 The AdS/CFT conjecture

Over the past thirty years there has been a fascinating rivalry between string theory on the one hand and gauge theories on the other in an attempt to describe the physics of the strong interaction. Indeed, string theory was originally invented as a way of describing some of the observed peculiarities of the strong force between quarks, the quarks being thought of in this theory as bound together by strings. But this theory of the strong force never had much success and with the advent of gauge theories it was soon discarded and replaced by the far more successful QCD which describes the interaction between quarks in terms of gauge fields. Later though string theory resurged as a possible candidate for unifying all the forces of nature. In this modern interpretation of string theory the strong force is now described by encapsulating QCD as a low energy part of its dynamics. The gauge fields however are now derived secondary objects of the theory, the fundamental objects being the strings themselves.

There is however yet another use of string theory discovered by ’t Hooft [6] who realised that perturbation expansions of gauge field theory in the large limit resemble string theory genus expansions (see [7] for a review). Loosely speaking, in the limit (with the ’t Hooft coupling held fixed, denoting the gauge theory coupling), each Feynman diagram of the gauge theory can be attributed a topology and the Feynman diagram expansion breaks up into a sum over topologies. Schematically we have for example for the free energy

where each picture in the equation represents the sum over Feynman diagrams of the given topology. This reorganised sum of Feynman diagrams resembles a string perturbation expansion over Riemann surfaces with playing the role of the string coupling and the ’t Hooft coupling related to Planck’s constant on the world sheet. More generally the limit of correlation functions of (single-trace) gauge invariant operators is schematically given by

which in the string theory analogy resembles a correlation function of vertex operator insertions on the world sheet. In particular, any given gauge invariant operator should correspond to a certain string theory state . Of course the Feynman diagrams in perturbative () gauge theory are not literally smooth Riemann surfaces but the Feynman propagators merely suggest simplicial decompositions of Riemann surfaces. One can nevertheless imagine how in the regime, which requires a nonperturbative formulation of the theory, the number of vertices in a typical diagram would become huge and the Feynman diagrams would more closely approximate smooth Riemann surfaces. This beautiful observation about the large limit of gauge theories is at the heart of the concept of string/gauge dualities. Indeed, although the above analogy is far from rigourous it strongly suggests that gauge theories are intimately related to string theories on certain backgrounds, in that some gauge theories may admit dual descriptions in terms of string theories.

The AdS/CFT correspondence due to Maldacena [8] is a conjectured realisation of such a duality for a supersymmetric cousin of QCD, namely it relates four-dimensional supersymmetric Yang-Mills theory (SYM) with gauge group to type IIB superstring theory on (see [9] for a review). Concretely, at large ’t Hooft coupling , SYM theory is believed to have a dual description in terms of type IIB superstring theory on with equal radii of curvature such that . The string coupling in the AdS/CFT correspondence is not simply as above, but instead is given by

The extra factor of however does not affect the interpretation of the gauge theory perturbation expansions as genus expansions.

An important part of the AdS/CFT correspondence is establishing a ‘dictionary’ for translating the language of one theory into the other. That is, given a gauge theory operator , we need a way of determining its dual string theory state and vice versa. For this it is helpful to classify the states of both theories according to the global symmetries present. Both theories share the global (bosonic) symmetry group : in gauge theory corresponds to the conformal symmetry group (in dimensions) and to the R-symmetry (acting for instance in the fundamental representation on the scalar fields of SYM), whereas on the string theory side is the target space symmetry. States on either side thus fall into representations of this global symmetry labelled by the eigenvalues of the six Casimirs, the first three being for and the last three for . For instance the complex combinations , and of the scalars have R-charges equal to , and respectively.

Note that one of the Casimirs of plays a distinguished role. In string theory this is the energy eigenvalue of the Hamiltonian which generates time translation in . And according to the AdS/CFT conjecture, it should be identified with the eigenvalue of the Dilation operator of SYM. Therefore if is a string energy eigenstate of energy and its dual gauge invariant conformal operator with anomalous dimension , namely

then the AdS/CFT conjecture states that

(0.1.1)

Checking (0.1.1) for arbitrary seems a hopeless task since determining the energy spectrum of the string to all orders in would be incredibly difficult. A more modest goal, at least initially, would be to check the correspondence in the ’t Hooft limit where all diagrams on the gauge theory side become planar, and the string theory becomes free, i.e. the worldsheet is topologically a sphere. Even with this simplification the duality is still of strong/weak coupling type and is therefore very hard to test since the weak coupling regions of both theories (in which perturbative methods apply) are non-overlapping. Specifically, a conformal operator in the strong coupling limit should admit an equivalent description in terms of a classical string (), i.e. a worldsheet soliton. Conversely, a string moving on a highly curved background should have an equivalent description as a weakly coupled () gauge field. This makes the conjecture very hard to prove since we only have access to perturbative methods on both sides of the correspondence.

0.2 The large Spin/R-charge limit

Despite the strong/weak coupling obstruction, it was realised in the work of Berenstein, Maldacena and Nastase [10] that explicit tests of the correspondence could be made (beyond sectors protected by supersymmetry) if one took the further limit where is a certain charge, say . This observation was later generalised in a series of papers by Frolov and Tseytlin [11, 12, 13] to larger sectors of the correspondence by taking multiple charges to infinity.

To first get an intuitive understanding of the significance of these large charge limits we go back to the picture of the Feynman graphs turning into Riemann surfaces. Focusing on the scalar sector of SYM, consider single-trace conformal operators

where in the pictorial representation the black dots each represent a single operator . They form a closed chain by virtue of the trace in . Now the 2-point correlation function of can be written symbolically as

\psfragx \psfragy
,

where the right hand side represents the sum of all possible Wick contractions, i.e. Feynman diagrams connecting the operators at and . As before the Feynman diagrams suggest a simplicial decomposition of a Riemann surface (with boundaries). This simplicial decomposition may be refined in two ways: either one increases the coupling as before to increase the number of vertices in these Feynman diagrams, or one can also increase the number of constituent operators in .

For example a BMN operator is made up of a large number of reference fields and a small number of other “impurity” fields and . Its string theory dual, the BMN string, is almost point-like and has angular momentum on . More generally an operator may contain a large amount of impurities such as with . Its string theory dual, the Frolov-Tseytlin string, is spatially extended and spins with the three different angular momenta , and on . As explained above one expects such ‘long’ () single-trace conformal operators to have a stringy behaviour even at weak coupling .

Concretely, suppose one can expand both sides of (0.1.1) in terms of and . On the string side this is achieved by doing a semiclassical expansion in with held fixed. On the gauge side one could first expand in and then further expand each coefficient in . When such expansions for the semiclassical energy and the perturbative anomalous dimension exist and take on the similar form

(0.2.1)

then their respective coefficients, for say, could be compared directly, even though they have been obtained differently from both sides of the duality.

With this procedure for making quantitative tests of the correspondence in place, the immediate goal from both sides of the duality is clear. From the gauge theory perspective one faces the problem of diagonalising the dilatation operator on long single-trace conformal operators perturbatively in . Since it commutes with the Casimirs of it does not mix operators of different weights. For instance, its action on the complete set of operators composed solely of the two scalars and is given by

(0.2.2a)
The problem is therefore reduced to diagonalising the matrix . However, since we are interested in the limit this simple diagonalisation task quickly becomes intractable without recourse to numerical methods.
The task on the string theory side is to obtain the semiclassical energy spectrum of strings on to leading order in . This in turn requires complete knowledge of the classical string motions on such a background. Restricting attention to the sector corresponding to the operators discussed above, the problem is reduced to finding the general solution to the equations of motion for a string moving on . However, the equations of motion for the fields describing the embedding of the string into ,
(0.2.2b)

are second order nonlinear partial differential equations subject to the constraint . Solving them exactly therefore seems quite intractable as well.

0.3 Classical/Quantum Integrability

Fortunately, something of a miracle happens in both cases. By computing the 1-loop planar dilatation operator on single-trace operators of all six scalar fields of SYM, Minahan and Zarembo [14] discovered it was proportional to the Hamiltonian of the integrable spin chain with nearest-neighbour interactions. Subsequently the complete one-loop planar dilatation operator of SYM was computed by Beisert [15, 16] and identified with an super spin chain by Beisert and Staudacher in [17]. Integrability also seems to persist at higher loops [18, 19]. For the purpose of this thesis we shall focus on the sector at one-loop where the planar dilatation operator reduces to the famous Heisenberg spin chain Hamiltonian which is quantum integrable. Specifically we have

(0.3.1)

where is the set of Pauli matrices acting on the site of the spin chain. The tree-level term in (0.3.1) is just the common engineering dimension of the operators , which is also just the length of the spin chain.

The fact that the one-loop planar dilatation operator (0.3.1) is integrable implies that it can be diagonalised analytically for any length . As usual, the definition of quantum integrability requires the existence of a maximal set of commuting operators which includes the Hamiltonian. The construction of such operators in the Heisenberg spin chain proceeds in the usual way (see [20, 21, 22] for a general discussion on quantum integrable systems) by defining the Lax operator where is called the spectral parameter. Here the subscript indicates that the matrix acts on the site of the spin chain and the subscript indicates that the matrix acts on an extra ‘auxiliary’ site. The main object of interest is the monodromy matrix (which acts on all sites as well as the auxiliary site). Writing out the action on the auxiliary site in matrix form it reads

Its trace over the auxiliary site , the transfer matrix, generates the desired family of commuting operators since one can show [21]

In particular the Hamiltonian can be extracted as .

The diagonalisation of can therefore be achieved by simultaneously diagonalising the whole family of operators . For this one defines a reference state on the spin chain by the condition and looks for eigenvectors of the form

(0.3.2)

This is akin to the Fock space construction where the operator creates a magnon excitation on the spin chain with rapidity . One can show that (0.3.2) is an eigenstate of the transfer matrix if and only if the parameters satisfy the famous Bethe equations which in this sector read [21, 22]

(0.3.3)

The solutions of these equations are called Bethe roots.

To study the limit of (0.3.3) one starts by taking its logarithm,

(0.3.4)

where the mode numbers specify the branch of the logarithm. A careful study of these equations determines the location of the Bethe roots in the limit . Since all Bethe roots are of order it is convenient to introduce the scaled spectral parameter by . If the number of mode numbers is finite, say , and the number of Bethe roots with the same mode number is of order then one finds that the Bethe roots of a given mode number all agglomerate into a vertical ‘cut’ in the complex plane, see Figure 1.

\psfrag

C1 \psfragC2 \psfragC3

Figure 1: Typical configuration of Bethe roots as .

To characterise the density of the Bethe roots along the various cuts one introduces a function on the complex plane called the quasi-momentum which can then be shown to have a simple pole at and the property that its value jumps by across (see [23] for details). Moreover, its integral around any cut gives exactly the proportion of Bethe root lying on called the filling fraction,

(0.3.5)

where is a contour around the cut . Now by construction, a distribution of Bethe roots like the one in Figure 1 characterises the limit of a single-trace eigen-operator of the one-loop planar dilatation operator (0.3.1). Therefore by the reasoning of section 0.2 we expect it to match the description of a classical string solution on . To see this we now turn to the string theory side.

Recall that the task there involves finding exact solutions to a set of non-linear second order partial differential equations (0.2.2b) subject to a constraint, which in general is impossible. Fortunately, it was discovered by Bena, Polchinski and Roiban [24] that the equations of motion for a superstring on can be formulated as a flatness condition for a 1-parameter family of currents depending on a complex parameter . This is a necessary condition for the theory to be classically integrable. In the sector the lightcone components of these currents are

(0.3.6)

This connection is built out of where depends on the fields and specifies the embedding of the string into . The flatness condition (0.3.6) is equivalent to the equations of motion (0.2.2b). As we will show in this thesis, when written in this form (0.3.6) the equations of motion can be solved exactly.

As we review in chapter 6, the zero-curvature representation (0.3.6) of the equations of motion directly leads to the construction of an algebraic curve equipped with a meromorphic differential , starting from a given solution to (0.2.2b). In other words (0.3.6) provides an assignment

(0.3.7)

Moreover, the pair is independent of the worldsheet -coordinates and therefore encodes the integrals of motion of the solution . Thus all solutions to (0.2.2b) on the string theory side are classified by their respective algebraic curves. In the sector these curves are all hyperelliptic and can be represented in terms of cuts in the complex plane. In chapter 9 we will give a proof of the usual assumption that these cuts are all vertical in the complex plane, see Figure 2 (note that the path taken by the cuts is arbitrary as long as they join up all the branch points in pairs).

\psfrag

C1 \psfragC2 \psfragC3

Figure 2: Cut representation of a genus two Riemann surface.

The remarkable similarity between Figures 1 and 2 was first discovered by Kazakov, Marshakov, Minahan and Zarembo in their seminal paper [23] (see [25, 26] for shorter reviews). The quasi-momentum on the gauge theory side is identified here with the Abelian integral since its value also jumps across cuts by , . It also has simple poles but this time they are at rather than . This is because to compare with the gauge theory one needs to scale the spectral parameter on the string theory side by setting so that now has poles at where . In the limit the string theory then exactly reproduces the one-loop gauge theory result [23].

As we discussed above, by virtue of quantum integrability the one-loop planar dilatation operator belongs to a whole family of commuting operators encoded in the transfer matrix . Likewise, as we will see in chapter 5, on the string theory side the energy is the first member of a whole hierarchy of conserved Poisson commuting charges encoded in a classical analogue of the transfer matrix. Now by construction, a distribution of Bethe roots characterises an eigen-operator of and an algebraic curve characterises a classical string solution. Therefore the matching of the classical string theory algebraic curve with the thermodynamic limit () of the one-loop Bethe root distribution provides a complete check in the sector of the equality between the coefficients in the expansion (0.2.1) for the spectrum of the quantum operator on the one hand and the range of the classical phase-space function on the other. The construction of the algebraic curve was later generalised to the sector [27], to the non-compact sector [28] and eventually to the full supersymmetric case [29]. This curve was then successfully compared in [30] against the full spectrum of SYM single-trace operators in the Frolov-Tseytlin limit.

To take the comparison to the next order in it was shown in [23] that a further change of spectral parameter was necessary on the gauge theory side. If one first renames the spectral parameter as , so that equations such as (0.3.5) now read the same with the relabelling ,

(0.3.8)

then the change of spectral parameter required to match the string theory results (expressed in terms ) is defined by the Zhukovsky map

(0.3.9)

This can also be written as in terms of the unscaled variables and . As we will show in chapter 8 the spectral parameter is in fact the natural choice on the string theory side since it brings the symplectic structure to the canonical Darboux form. Furthermore, the filling fractions are also naturally expressed in terms of it, as in (0.3.8). With this change of variables the two-loop gauge theory result was shown to exactly match the next order in of the classical string theory algebraic curve (see [23, p27] for details). This provides a test of the correspondence in the sector at the level of the coefficient in the expansion (0.2.1). Despite this perfect agreement at two-loop, the next coefficient in the expansion (0.2.1) on both sides of the correspondence were found to disagree, which has become known as the ‘three-loop discrepancy’ [31]. This mismatch however is not in conflict with the AdS/CFT correspondence and can be attributed to an order-of-limits effect [32, 33]. Indeed, on the string theory side one takes the classical limit before expanding in whereas on the gauge theory side the perturbation expansion in precedes the expansion in . In other words, the procedures described in section 0.2 for testing the AdS/CFT correspondence rely on the assumption that the following diagram [32]

is commutative. Yet, assuming the AdS/CFT correspondence holds, the mismatch at three-loop clearly shows otherwise and with hindsight the agreement for the coefficients and seems quite fortuitous.

One way to circumvent this difficulty would be to directly quantise string theory on . The main objective of the work presented in this thesis was to make a step towards obtaining the leading semiclassical corrections to the string spectrum and possibly gain some insight in view of one day performing an exact quantisation of string theory on . The more modest task of obtaining the semiclassical string spectrum would provide the set of coefficients in the expansion (0.2.1) from the string theory side. These could then be perturbatively tested against the corresponding coefficients obtained from the gauge theory side. In this short introduction we have mostly been concerned with the sector corresponding classically to bosonic strings moving in an submanifold of . This restriction is legitimate because at the classical level it is a consistent truncation of the full superstring theory on . At the quantum level however, even if we semiclassically quantise a solution in the subspace we know that quantum fluctuations will leave this subspace and so quantum mechanically one ought to consider the full target-space . Despite this, in this thesis we will continue focusing on the subspace as a toy model. The reason for doing this is that the subsector is the only one for which the complete set of solutions is explicitly know [1, 2], which is a necessary prerequisite for performing a semiclassical study of any system.

0.4 Outline of the thesis

Part I The first two chapters of this thesis contain all the necessary background material on the theory of Riemann surfaces [34, 35, 36, 37, 38, 39, 40, 41, 42] and semiclassical quantisation of finite-dimensional systems [43, 44, 45, 46, 47, 48, 49, 50, 51, 52] required for Parts III and IV respectively. Since the theory of Riemann surfaces plays such an important role in Part III, for completeness we cover the relevant aspects of it in some detail in chapter 1.

Part II In chapter 3 we give a review of bosonic strings theory on from the Lagrangian point of view and express it in terms of the principal chiral model subject to the Virasoro constraints. In chapter 4 we rephrase everything from the Hamiltonian perspective discussing the implementation of the Virasoro and static gauge constraints in the Dirac formalism. Finally, in chapter 5 we tackle the question of integrability of bosonic strings on . We start by reviewing the construction of the Lax connection and monodromy matrix in section 5.1 and the extraction of the local conserved charges in section 5.2. Section 5.3 is based on [2] in which we show that the integrals of motion previously obtained are also in involution. This is the complete statement of integrability of string theory on . We then exploit this in section 5.4 to construct the integrable hierarchy of the string as in [3].

Part III In this Part we put to full use the integrability unveiled in Part II to construct the general solution to the equations of motion for a string on following [1, 2, 3] as well as [4] for the last section. Section 6 is a review of the construction of the KMMZ curve [23] encoding the integrals of motion of a finite-gap solution. We show in section 7 that the reconstruction of the solution requires additional data, namely a finite set of points on the KMMZ curve. This completes the set of so called algebro-geometric data. We express the general finite-gap solution explicitly in terms of this data using Riemann -functions on the curve. In section 8 we derive the restriction of the symplectic structure of the string to the algebro-geometric data. The resulting finite-dimensional symplectic structure is canonical if the spectral parameter used is given by the Zhukovsky map. We then perform a standard change of variables to action-angle variables, obtaining explicit expressions for these in terms of the algebro-geometric data. In section 9 we discuss the necessary constraints on the data to obtain physical finite-gap solutions. In particular we derive the reality conditions on the KMMZ curve, showing that all the branch points must lie off the real axis in the sector.

Part IV In chapter 10 we use the knowledge of classical solutions acquired in Part III to perform a semiclassical analysis of bosonic string theory on from first principles. We derive a general and simple formula for extracting the fluctuation energies from the KMMZ curve in terms of a well defined meromorphic differential on the curve, namely the quasi-energy. We use these fluctuation energies to show formally (without regularising) that their sum leads to the discretisation of the KMMZ curve in the sense that all the fillings get half-integer quantised, including those of the singular points which are classically empty. The calculation therefore serves as a toy model for understanding from the finite-gap perspective the origin of the discretisation of the algebraic curve when leading order semiclassical corrections are included.

\psfrag

1\psfrag2\psfrag3\psfrag4\psfrag6\psfrag7\psfrag8\psfrag9\psfrag10\psfrag51\psfrag52\psfrag53\psfrag54

Figure 3: Chapter dependence guideline.

Part I Background

Chapter 1 Riemann surfaces

“Donuts. Is there anything they can’t do?”

Homer Simpson

This chapter is intended as a self contained review, based on [34, 35, 36, 37, 38, 39, 40, 41, 42], of those aspects from the theory of Riemann surfaces relevant to Part III of this thesis. The most important concepts and results required in the theory of finite-gap integration are found in section 1.5. Section 1.6 is a discussion of singular algebraic curves which are fundamental to chapters 6 and 10. Finally, section 1.7 discusses the relation of a curve to its Jacobian, an object of great importance in Parts III and IV.

1.1 Definition & Examples

Consider a real two-dimensional (connected) topological manifold , that is a second-countable Hausdorff space locally homeomorphic to , and let be an open cover of , i.e. . Then the fact that is locally homeomorphic to means we can find homeomorphisms called local charts from each to open subsets . We are interested in doing complex analysis on and so we use the homeomorphisms to locally equip with the analytic structure of . For instance, a function will be called holomorphic if is a holomorphic map in the usual sense.

\psfrag

z1 \psfragz2 \psfragU1 \psfragU2 \psfragt21 \psfragM

Figure 1.1: Compatibility conditions.

But for this analytic structure to have any meaning globally on we need a compatibility condition between charts on overlapping sets ensuring that is holomorphic iff is, for any . Thus we say that two charts and are (holomorphically) compatible if

called the transition function, is holomorphic as a function from to , c.f. for a differentiable manifold is required to be differentiable. If the charts are all compatible they are said to form a complex atlas and two complex atlases are compatible if is a complex atlas. Any atlas can be extended to a maximal atlas consisting of all charts compatible with . A maximal atlas is also called a complex structure.

Definition 1.1.1.

A Riemann surface is a real two-dimensional (complex one-dimensional) connected manifold equipped with a complex structure.

Remark One great advantage of working with a Riemann surface as opposed to simply dealing with the underlying two-dimensional differentiable manifold is that one can apply all the local concepts and powerful theorems of complex analysis using the local homeomorphisms with . However, just as with differentiable manifolds, these local homeomorphisms are not canonical because they depends on the choice of chart , and so the only objects one can consider on a Riemann surface are ones whose definitions are chart invariant.

Examples

The following are basic examples of Riemann surfaces that will be important later:

  • Any connected open domain equipped with a single chart .

  • The Riemann sphere (the one-point compactification of ) equipped with two charts

    with holomorphic transition functions .

  • Any non-singular algebraic curve defined by the zero-locus

    of a polynomial in and . The non-singular criteria means that and never both vanish on . By the implicit function theorem the variable (resp. ) can be taken as a local chart near points where (resp. ) and (resp. ) is analytic so this defines a complex structure on .

Remark In the neighbourhood of a singular point , the curve looks like an intersection of several complex-lines and so there is no neighbourhood of locally homeomorphic to . When encountering singular algebraic curves we will therefore have to desingularise them by a process to be explained later.

1.2 Holomorphic maps

Definition 1.2.1.

A continuous mapping

between Riemann surfaces is called holomorphic (or analytic) if for every local chart on and every local chart on with , the mapping

is holomorphic as a map from to .

Remark This definition is independent of the choice of charts and by holomorphicity of the transition functions to another set of charts and . Moreover, because holomorphicity is a local concept, all the usual local properties of holomorphic functions on will persist for holomorphic maps. For instance, any holomorphic map is open, i.e. sends open sets to open sets .

A holomorphic mapping into is called a holomorphic function. A holomorphic mapping into is called a meromorphic function. The ring of holomorphic functions on is denoted by and the field of meromorphic functions on by .

Local behaviour

A holomorphic function is locally injective around all but finitely many points of . That is, there exists a finite collection of points such that for all other points the restriction to a neighbourhood of is injective. The points around which fails to be locally injective are called branch points. These statements are made precise by the following Lemma:

Lemma 1.2.2.

Let be a holomorphic map and . Then there exists local charts near , such that is given by

Proof.

Choose local charts on vanishing at and on vanishing at . Now is holomorphic with so we can write it as for some holomorphic with . Since is non-vanishing on a disc around the origin it has a root and so . Defining a new coordinate the result follows. ∎

Thus a holomorphic map locally looks like the map . Hence in a small neighbourhood the number of solutions to the equation when approaches is . We see that the number appearing in Lemma 1.2.2 has an invariant geometrical meaning for the map and cannot depend of the choice of chart used to represent . It is called the valency or the ramification number of at . The number is called the branch number of at .

Definition 1.2.3.

A point for which is called a branch point of .

Lemma 1.2.4.

The branch points of a holomorphic map are isolated.

Proof.

Let be a branch point of . Then by Lemma 1.2.2, there exists a neighbourhood and coordinate with for which takes the local form . But the map is locally injective for so for any . ∎

Corollary 1.2.5.

If is compact, then has finitely many branch points.

Global behaviour

The local property that a holomorphic map is open (which follows from Lemma 1.2.2) implies a far reaching global property of holomorphic maps on compact Riemann surfaces:

Theorem 1.2.6.

Let be compact and a non-constant holomorphic map. Then is surjective () and N is compact.

Proof.

Since is not constant, is open (a holomorphic mapping is open). But is compact so is compact (the continuous image of a compact set is compact) and hence closed (a compact subset of a Hausdorff space is closed). So is a non-empty open and closed subset of , and since is connected we have . ∎

In fact one can be a lot more precise. Not only is any attained by , but every is assumed the same number of times, counting multiplicities.

Theorem 1.2.7.

Let be a non-constant holomorphic function with compact. Then there exists such that for any the equation has precisely solutions (counting multiplicities), i.e.

Proof.

Let . By Theorem 1.2.6 the equation has at least one solution. The number of solutions is finite because otherwise they would accumulate in and hence would be the constant map (since a non-zero holomorphic function has isolated zeroes). Now by Lemma 1.2.2 there exists neighbourhoods of and of with respect to which is of the local form in . Since has zeroes near it follows that is constant in . By compactness one can cover by finitely many and so remains constant over . ∎

We say that is an -sheeted ‘branched’ covering of , referring to the fact that branch points are the multiple solutions of , see Figure 1.2.

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Figure 1.2: Branched covering.
Definition 1.2.8.

The number is called the degree of and we write .

Applying Theorem 1.2.7 with implies that a non-constant meromorphic function on a compact Riemann surface assumes every value in the same number of times. In particular, has as many zeroes as poles, provided they are counted correctly with multiplicities.

Remark A single non-constant meromorphic function completely determines the complex structure of . Indeed, using Lemma 1.2.2 and the charts of , a local chart vanishing at is constructed as follows (with )

1.3 Topology

In this section we temporarily forget about the complex structure of Riemann surfaces and describe their topologies as real two-dimensional topological manifolds. Accordingly, all the charts on a surface in this section are homeomorphisms into , that is . As before we still assume the surface is connected and hence path connected.

Definition 1.3.1.

A manifold is orientable if there exists an atlas such that the transition functions preserve orientation.

Proposition 1.3.2.

Every Riemann surface is orientable.

Proof.

Holomorphic functions preserve orientation since by the Cauchy-Riemann equations the Jacobian of such a transformation is positive,

The following theorem and corollary give a complete classification of the possible topologies for a Riemann surface. The proof of Theorem 1.3.3, which we omit, usually relies on the fact that every compact surface is triangulable [34] and proceeds by cutting and gluing the triangulation to arrive at the final desired polygon form:

Theorem 1.3.3.

[34] Every compact orientable surface is homeomorphic either to the sphere or to a polygon with edges () identified pairwise in such a way that the orientations of these edges with respect to are opposite () and with all vertices identified.

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Remark The -gon described by Theorem 1.3.3 is a lift of to its universal covering space . We shall denote it since it can be obtained from by cutting along certain cycles. The identification process described in Theorem 1.3.3 corresponds to applying the covering map , in other words . The simply connected domain will come in handy later for defining branches of multi-valued functions on and so we give it a name:

Definition 1.3.4.

The -gon of Theorem 1.3.3 is called the normal form of .

In its normal form representation, the topology of is not very transparent since the edges and vertices still need to be identified following the prescription in Theorem 1.3.3. The next corollary describes the closed surface resulting from these identifications.

Corollary 1.3.5.

[34] Every compact orientable surface is homeomorphic to a sphere with handles, that is to when or to the -fold connected sum of torii when .

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Proof.

Using Theorem 1.3.3 we just have to show that the normal form is homeomorphic to a -fold connected sum of torii (a -fold torus). We proceed by induction on . We start by cutting the -gon into two polygons. The first has the 4 edges and a new edge . The second has the remaining edges and the edge .

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Next we make the identification of edges and vertices in each of these two polygons using the induction hypothesis. We end up on the one hand with a torus with a disc cut out, whose boundary is , and on the other hand a -fold torus with a disc cut out, whose boundary is .

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