Holomorphic Matrix Models and Special Geometry 

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Wilsonian effective actions of supersymmetric theories

For a given Lagrangian one can also introduce what is known as the Wilsonian effective action [141], [142]. Take λ to be some energy scale and define the Wilsonian effective Lagrangian Lλ as the local Lagrangian that, with λ imposed as an ultraviolet cut-off, reproduces precisely the same results for S-matrix elements of processes at momenta below λ as the original Lagrangian L. In general, masses and coupling constants in the Wilsonian action will depend on λ and usually there are infinitely many terms in the Lagrangian. Therefore, the Wilsonian action might not seem very attractive. However, it can be shown that its form is quite simple in the case of supersymmetric theories. Supersymmetric field theories are amazingly rich and beautiful. Independently on whether they turn out to be the correct description of nature, they certainly are useful to understand the structure of quantum field theory. This is the case since they often possess many properties and characteristic features of non-supersymmetric field theories, but the calculations are much more tractable, because of the higher symmetry. For an introduction to supersymmetry and some background material see [21], [134], [135]. Here we explain how one can calculate the Wilsonian effective superpotential in the case of N = 1 supersymmetric theories.

Period integrals on local Calabi-Yau manifolds and Riemann surfaces

We mentioned already in the introduction that one building block that is necessary to obtain the effective superpotential (1.14) of gauge theories is given by the integrals of the holomorphic (3, 0)-form ­, which exists on any (local) Calabi-Yau manifold, over all the (relative) three-cycles in the manifold. Here we will analyse these integrals on the space Xdef , and review how they map to integrals on a Riemann surface, which is closely related to the local Calabi-Yau manifold we are considering.
Let us first concentrate on the definition of ­. Xdef is given by a (non-singular) hypersurface in C4. Clearly, on C4 there is a preferred holomorphic (4, 0)-form, namely dx∧dv∧dw∧dz. Since Xdef is defined by F = 0, where F is a holomorphic function in the x, v,w, z, the (4, 0)-form on C4 induces a natural holomorphic (3, 0)-form on Xdef .
To see this note that dF is perpendicular to the hypersurface F = 0. Then there is a unique holomorphic (3, 0)-form on F = 0, such that dx ∧ dv ∧ dw ∧ dz = ­ ∧ dF. If z 6= 0 it can be written as ­ = dx ∧ dv ∧ dw ∧ dz dF = dx ∧ dv ∧ dw 2z .

The holomorphic matrix model

The fact that the holomorphic matric model is relevant in this context was first discovered by Dijkgraaf and Vafa in [43], who noticed that the open topological B-model on Xres is related to a holomorphic matrix model with W as its potential. Then, in [45] they explored how the matrix model can be used to evaluate the effective superpotential of a quantum field theory. A general reference for matrix models is [59], particularly important for us are the results of [25]. Although similar to the Hermitean matrix model, the holomorphic matrix model has been studied only recently. In [95] Lazaroiu described many of its intriguing features, see also [91]. The subtleties of the saddle point expansion in this model, as well as some aspects of the special geometry relations were first studied in our work [P5].

The partition function and convergence properties

We begin by defining the partition function of the holomorphic one-matrix model following [95]. In order to do so, one chooses a smooth path γ : R → C without selfintersection, such that γ˙ (u) 6= 0 ∀u ∈ R and |γ(u)| → ∞ for u → ±∞. Consider the ensemble ¡(γ) of1 ˆN × ˆN complex matrices M with spectrum spec(M) = {λ1, . . . λ ˆN } in2 γ and distinct eigenvalues, ¡(γ) := {M ∈ C ˆN × ˆN : spec(M) ⊂ γ, all λm distinct} .

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The saddle point approximation for the partition function

Recall that our goal is to calculate the integrals of ζ = ydx on the Riemann surface (4.53) using matrix model techniques. The tack will be to establish relations for these integrals which are similar to the special geometry relations (4.2). After we have obtained a clear cut understanding of the function F appearing in these relations, we can use the matrix model to calculate F and therefore the integrals. A natural candidate for this function F is the free energy of the matrix model, or rather, since we are working in the large ˆN limit, its planar component F0(t). However, F0(t) depends on t only and therefore it cannot appear in relations like (4.2). To remedy this we introduce a set of sources Ji and obtain a free energy that depends on more variables. In this subsection we evaluate this source dependent free energy and its Legendre transform F0(t, S) in the planar limit using a saddle point expansion [P5]. We start by coupling sources to the filling fractions,6 Z(gs, ˆN , J) := 1 ˆN ! Z γ dλ1 . . . Z.

Table of contents :

Setting the Stage
I Gauge Theories, Matrix Models and Geometric Transitions
1 Introduction and Overview
2 Effective Actions
2.1 The 1PI effective action and the background field method
2.2 Wilsonian effective actions of supersymmetric theories
2.3 Symmetries and effective potentials
3 Riemann Surfaces and Calabi-Yau Manifolds
3.1 Properties of Riemann surfaces
3.2 Properties of (local) Calabi-Yau manifolds
3.2.1 Aspects of compact Calabi-Yau manifolds
3.2.2 Local Calabi-Yau manifolds
3.2.3 Period integrals on local Calabi-Yau manifolds and Riemann surfaces
4 Holomorphic Matrix Models and Special Geometry
4.1 The holomorphic matrix model
4.1.1 The partition function and convergence properties
4.1.2 Perturbation theory and fatgraphs
4.1.3 Matrix model technology
4.1.4 The saddle point approximation for the partition function
4.2 Special geometry relations
4.2.1 Rigid special geometry
4.2.2 Integrals over relative cycles
4.2.3 Homogeneity of the prepotential
4.2.4 Duality transformations
4.2.5 Example and summary
5 Superstrings, the Geometric Transition and Matrix Models
5.1 Superpotentials from string theory with fluxes
5.1.1 Pairings on Riemann surfaces with marked points
5.1.2 The superpotential and matrix models
5.2 Example: Superstrings on the conifold
5.3 Example: Superstrings on local Calabi-Yau manifolds
6 B-Type Topological Strings and Matrix Models
7 Conclusions
II M-theory Compactifications, G2-Manifolds and Anomalies
8 Introduction
9 Anomaly Analysis of M-theory on Singular G2-Manifolds
9.1 Gauge and mixed anomalies
9.2 Non-Abelian gauge groups and anomalies
10 Compact Weak G2-Manifolds
10.1 Properties of weak G2-manifolds
10.2 Construction of weak G2-holonomy manifolds with singularities
11 The Hoˇrava-Witten Construction
12 Conclusions
III Appendices 
A Notation 
A.1 General notation
A.2 Spinors
A.2.1 Clifford algebras and their representation
A.2.2 Dirac, Weyl and Majorana spinors
A.3 Gauge theory
A.4 Curvature
B Some Mathematical Background 
B.1 Useful facts from complex geometry
B.2 The theory of divisors
B.3 Relative homology and relative cohomology
B.3.1 Relative homology
B.3.2 Relative cohomology
B.4 Index theorems
C Special Geometry and Picard-Fuchs Equations 
C.1 (Local) Special geometry
C.2 Rigid special geometry
D Topological String Theory 
D.1 Cohomological field theories
D.2 N = (2, 2) supersymmetry in 1+1 dimensions
D.3 The topological B-model
D.4 The B-type topological string
E Anomalies 
E.1 Elementary features of anomalies
E.2 Anomalies and index theory
IV Bibliography 

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