Statistics – Computation
Scientific paper
Oct 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.6083m&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 20, pp. 6083-6085 (2006).
Statistics
Computation
Scientific paper
This book is a find. Mariño meets the challenge of filling in less than 200 pages the need for an accessible review of topological gauge / gravity duality. He is one of the pioneers of the subject and a clear expositor. It is no surprise that reading this book is a great pleasure.
The existence of dualities between gauge theories and theories of gravity remains one of the most surprising recent discoveries in mathematical physics. While it is probably fair to say that we do not yet understand the full reach of such a relation, the impressive amount of evidence that has accumulated over the past years can be regarded as a substitute for a proof, and will certainly help to delineate the question of what is the most fundamental quantum mechanical theory.
As has been the case in the past, it is in the context of Witten's 'topological' quantum theories that the mathematical framework is well enough established to firmly ground, and fully benefit from, the development of the physical theories. This book makes an important contribution to this new chapter in the math / physics interaction.
There are two main instances of topological gauge/gravity duality. In the A-model, Chern Simons gauge theory on the 3-sphere is related to the closed topological string theory on the local Calabi Yau 3-fold {\mathcal O}_{{\mathbb P}^1}(-1) \oplus{\mathcal O}_{{\mathbb P}^1} (-1), also known as the resolved conifold (Gopakumar-Vafa duality). In the B-model, certain types of matrix models are related on the gravity side to topological strings on certain cousins of the deformed conifold (Dijkgraaf-Vafa duality).
In both cases, and similarly to the more physical AdS/CFT correspondence, the duality can be discovered by realizing the gauge theory as the target space theory of open strings ending on particular D-branes in a geometry closely related to the closed string background of the gravity theory. The A-branes supporting Chern Simons theory are wrapped on the Lagrangian three-sphere inside of T*S3, while the B-branes supporting the matrix models are wrapped on holomorphic curves in a certain class of toric Calabi Yau 3-folds. The gravity sides are reached via appropriate 'geometric transitions'.
It is worth remarking that while the embedding in string theory gives a credible justification of the duality as well as a heuristic derivation, it also touches on at least as many questions as it answers: Are we restricted to non-compact Calabi Yau manifolds? Does the Chern Simons theory have to live on the 3-sphere (or a Lens space) or could it be a more general three-manifold? Why are we restricted to B-branes wrapping 2-cycles? Can we derive the duality from worldsheet considerations? Can we see open strings on the gravity side? What is the relevance of four-dimensional topological gauge theory? Certainly fully answering these questions requires mastering the 'phenomenology' of topological gauge/gravity duality, and this is precisely what this book helps to achieve.
There are several important applications of these topological dualities. The A-model version is useful for the all-genus solution of the topological string on certain local Calabi Yau manifolds via the topological vertex. It also gives a new point of view on the theory of invariants of knots and three-manifolds via the incorporation of Wilson loops, which are dual to certain D-branes on the string theory side. On the other hand, the main application of the B-model topological gauge / gravity duality is to superpotential computations in four-dimensional N=1 gauge theories via the classical BCOV interpretation of topological amplitudes as computing F-terms in an effective space-time theory.
The presentation is extremely well-balanced with an emphasis on computational techniques. This aspect in particular, and despite the large amount of required background material will facilitate access to the rich and fascinating subjects that are explained in the book. While written from the perspective of a mathematical physicist, it will certainly also be useful for mathematicians willing to learn about the recent physical predictions for enumerative geometry.
Here is a brief summary of the book. The journey begins with matrix models and an introduction to various techniques for the computation of integrals of the form Z = \frac{1}{vol(U(N))}\int dM e^{-\bigl[\frac1{2g_s} TrM^2 + \frac{1}{g_s}\sum_p\frac{g_p}{p} Tr M^p\bigr]} \,, including perturbative expansion, large-N approximation, saddle point analysis, and the method of orthogonal polynomials. The second chapter, on Chern Simons theory, is the longest and probably the most complete one in the book. Starting from the action \frac{k}{4\pi}\int Tr\bigl(A\wedge d A + \frac23 A\wedge A\wedge A\bigr) we meet Wilson loop observables, the associated perturbative 3-manifold invariants, Witten's exact solution via the canonical duality to WZW models, the framing ambiguity, as well as a collection of results on knot invariants that can be derived from Chern Simons theory and the combinatorics of U (∞) representation theory. The chapter also contains a careful derivation of the large-N expansion of the Chern Simons partition function, which forms the cornerstone of its interpretation as a closed string theory. Finally, we learn that Chern Simons theory can sometimes also be represented as a matrix model.
The story then turns to the gravity side, with an introduction to topological sigma models (chapter 3) and topological string theory (chapter 4). While this presentation is necessarily rather condensed, (and the beginner may wish to consult as well some of the by now standard references on the subject), it serves its purpose as a review of the basic definitions and main objectives of that field. Chapter 5 delivers the tools for the construction of a class of Calabi Yau manifolds as topological string backgrounds, and introduces geometric transitions, which as mentioned above is the preferred way to access gauge / gravity dualities.
The third part of the book is the synthesis to the topological gauge/gravity dualities recalled above. Chapter 6 reviews the basic philosophies behind the correspondence between large-N gauge theories and string theory, following 't Hooft, Maldacena, and Gopakumar-Vafa. Chapters 7 and 8 give the heuristic derivation via open topological strings and geometric transitions, and collect the central results checking the dualities in the main examples in both A- and B-model. Applications to the topological vertex, knot invariants, and supersymmetric gauge theories, are covered in the final two chapters of the book.
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