Abstract

In this thesis we study AdS5/CFT4 correspondence and its applications on the half-BP; sector preserving sixteen supercharges. The AdS/CFT correspondence under consider ation is the most familiar one, namely, type IIB superstring theory on AdS5 x S5 am a maximally supersymmetric gauge theory (superconformal Yang-Mills theory) in fou: dimension. We focus on a special sector of gauge theory that contains gauge-invarian operators, i.e. half-BPS circular Wilson loops and half-BPS chiral primary operators which can be represented as D-brane excitations on the dual bulk side.

Two main spotlights in this thesis are holographical Wilson/Polyakov loop correlator; and giant graviton (half-BPS chiral primary operator) correlators in terms of non-critica. c = 1 string theory. That is, we consider their large 't Hooft coupling behaviors using classical gravity (or string) theory.

In the former case, the correlator preserves no supersymmetry and its strong coupling limit is predicted from an F-string/D-brane system in AdS5 x 85 geometry. On the other hand, the latter case has no dependence on gauge couplings, hence its rigorous result can be deduced exactly through a complex matrix model. In spite of this, we uncover a correspondence between this correlator and tachyon scatterings in non-critical c = 1 string theory. This relation can be thought of as a topological feature behind the highly supersymmetrically protected sector of super Yang-Mills.

An overview

Chapter 2

The starting point is devoted to an introduction of AdS/CFT correspondence. That is, gravity can be identified as gauge theory in certain parameter region and ultimately to all regions. We analyze AdS/CFT correspondence from two aspects. We first consider the dual geometry AdS5 x M5 when M5 = S5. This is the standard prototype of AdS/CFT, which has received most attention so far. Its finite temperature (or non-extremal) cousin, which will be adopted in chapter five, is also covered. Unlike the above extremal zero- temperature case, at finite temperature, the ratio of inverse temperature to the radius of S5 serves as a reference for the occurrence of Hawking-Page phase transition. This phenomenon is interpreted as confinement/deconfinement transition in the gauge theory context.

Secondly, a less supersymmetric M5 = T1,1 base manifold is chosen to demonstrate another example of AdS/CFT. We will use its dual gauge theory known as a quiver type one in chapter six. The reason is that the gauge theory moduli space is just the transverse Calabi-Yau cone幼onifold and shares many key features with topological strings on the conifold. In addition, since one can obtain smooth T1,1 by blowing up an orbifold S5/Z2, we show that this process amounts to break sixteen supercharges to eight ones through turning on a mass deformation on the field theory side.

Chapter 3

In this chapter, we include various materials to illustrate the intimate relationship between three-dimensional Chern-Simons gauge theory and Al = 4 SYM, concerning Wilson loops. They serve as basic building blocks of this thesis. First of all, we introduce Ooguri-Vafa operator, which best shows how holographical brane configurations can be, more or less, predicted from pure gauge theory considerations, namely, Wilson loop generating function.

Based on this, we go to review the construction of holographical duals of half-BPS circular Wilson loops. In the case of D5-branes embedded in AdS5 x S5, first proposed by Satoshi Yamaguchi, the construction itself bears a strong resemblance to baryon vertices found by Witten soon after Maldacena's AdS/CFT. By restricting ourselves to half-BPS Wilson loops, an exact gauge calculation is possible because supersymmetry protection controls all quantum corrections. This is carried out by the so-called "Gaussian matrix model" . We review this reduced model and, by taking large 't Hooft coupling limit, an interpolation can be made for strong coupling regime where the perfect match with gravity result is shown.

Finally, we discuss some features shared by Chern-Simons gauge theory and Al = 4 SYM. The main idea is that on both sides there are bubbling geometries due to a large number of probe branes, i.e. they backreact on the original AdS5 x S5 and the resulting modified geometry is one-to-one identified with a fermion droplet picture. This fascinat-ing description can be summarized as a "Maya-geometry correspondence". Here, Maya diagrams arise from Young tableaux which ultimately classify all free fermion droplets from matrix models. Moreover, the matrix model can be either the above Gaussian oneor a Kontsevich-type one appearing on the mirror topological B-model side.

Chapter 4

This chapter analyzes in detail two Wilson loop correlators, following the work of myself and Satoshi Yamaguchi. Their gravity duals are proposed as F-string/D-brane systems and their phase structures are studied thereof.

Chapter 5

This chapter extends the analysis of two temporal Wilson loop correlators to finite temper-ature case. In this case, the gravity dual becomes an F-string/D-brane system embedded in the AdS black hole geometry. As well, the phase structures is studied thereof.

Chapter 6

This chapter is devoted to discuss another kind of gauge/string correspondence. That is, a special (half-BPS) sector in .A/ = 4 SYM is mapped to non-critical c = 1 strings. The main result is the finding that both S-matrices are exactly the same up to a parameter map. Although the reason is still obscure, we can only say that it is again supersymmetry protection that makes only topological information survives, which is insensitive to fiuc-mations. This statement is then supported by an equivalence between c = 1 string theory at self-dual radius and topological strings on the simplest Calabi-Yau cone-conifold.

To say more on this point, we introduce some backgrounds of B-model topological strings and its dual (Dijkgraaf-Vafa or Kontsevich) matrix model description. Since DV nodel is known to be dual to the far infrared-red (IR) region of certain ./V = 1 gauge ;heory in confining phase, we find an interesting tri-ality between N = 1, c = 1 and ,opological strings. This observation in turn sheds light on a future work we will mention later.

Chapter 7

Ve give conclusions and address two relevant points as our concluding remarks of this hesis. We propose an observation concerning Nekrasov's instanton counting from c = 1 )erspective. This may be thought of as a by-product of the preceding tri-ality because ;olomorphic information of A = 2 Seiberg-Witten theory, i.e. the prepotential including astanton corrections, still survives in the descending N = 1 theory dual to c = 1 strings s stressed.

Another aspect is an application of the aforementioned Diikgraaf-Vafa-Kontsevichduality. We show. its physical meaning can be interpreted as "Seiberg duality" 1, which manifests an electric-magnetic duality for Al = 1 gauge theory with matters. This part is based on the discussion with Kazutoshi Ohta.

'Roughly speaking, a proposed weak-coupling theory at IR plays the complimentary role for another UV free but IR strongly-coupling V = 1 gauge theory.

本論文は7つの章及び2つのAppendixからなる。まず第1章で研究の背景と動機及び研究結果の概要が述べられ、さらに論文全体の構成が提示される。そして第2章において,本論文の研究の主題であるAdS/CFT対応についてのレビューが与えられている。ここではとりわけ、後の章の内容に関連する2つの場合、すなわちAdS5×M5においてM5=S5の場合(及びその有限温度の場合の議論)と、M5=T1,1の場合について詳しく述べられる。後者については、topological stringとの関係や超対称性の低減等についても言及されている。第3章では、Wilson loopに関連して、3次元Chern-Simons ゲージ理論とN=4の超対称性ヤンミルズ(SYM)理論との関係が説明され、Wilson loopの生成母関数とbraneの配位との関連の議論が与えられている。この関連に基づいて、half-BPSセクターにおける円周に沿ったWilson loopに対応するものとして、AdS5×S5におけるD5-branesの双対的な描像が述べられ、後の章での議論の基礎が与えられている。第4章では、第3章で与えた双対性の描像に基づき、具体的にWilson loopの2点相関関数を評価している。従来より知られた相関関数を拡張して、k階の反対称表現を含む相関関数を考察し、これに対する計算結果に基づいて、双対なF-string/D-brane系の相転移現象(Gross-Ooguri相転移)を議論している。なお、整合性の吟味として、本論文で求めたWilson loopの2点相関関数が、k=1のときにこれまでの知られた相関関数と完全に一致することが確かめられている。続く第5章は、第4章の分析を有限温度の場合に拡張したもので、AdS側がblack holeを許す幾何構造を持つことが示され、相転移の状況の変化も併せて議論されている。第6章においては、half-BPSセクターにおけるN=4SYM理論と、noncriticalなc=1 stringとの関係が議論される。この関係の観察は複素行列模型を通してなされ、それぞれの理論が一種の自由fermion系に置き換えられ、これを基礎に相関関数とタキオン散乱に関するS行列の等価性が示されている。最後の第7章では、結論として本論文の結果とその意義が述べられるとともに、今後の関連した研究トピックとして、Nekrasovのinstanton counting及びSeiberg双対性への応用が議論されている。また2つのAppendixでは、楕円積分の表式等の本文に対する技術的補足がなされている。

本論文はAdS/CFT対応を中心課題として、D-braneを伴う相関関数を用いた種々の計算を通して上の双対性の吟味や新しい側面を探ったものであり、その寄与は大きく2つにわけて考えることができる。まず1つは、上記のような双対性の概念に基づいて、Wilson loopの評価に際して従来より知られた相関関数をk階の反対称表現を含むものに拡張することに成功したことである。本論文で考察したのは平行した2つの円周に沿うWilson loopの2点相関関数であるが、SYM理論側では非常に困難な摂動的な計算に依らずとも、双対的な描像を用いて計算が可能であることを実際に示しており、かつその結果に対して、対応する双対なF-string/D-brane系でのloop間の距離に関する応答としての物理的解釈を行うことが可能であって、(有限温度の場合を含め)その相転移現象と密接に関連していることを示したことは、十分に意義のあることである。

もう1つは、half-BPSセクターにおけるN=4SYM理論のchiral primary場の相関関数が、すべてのgenusでc=1 string理論のタキオン散乱に関するS行列と(位相部分を除いて)一致することを示すことに成功したことである。これは一見繋がりの無い両者の理論が、複素行列模型を通して共通の数理物理的基盤を持つことを意味しており、非常に興味深い発見だと言える。また、これらの分析を行うにあたり、hemitian Gaussian 行列模型を用いてそのS行列を求める等、高度な操作と巧妙な手法を駆使しており、用いられた理論的分析は技術的にも高いものと判断する。

以上述べたように、本論文はD-braneを伴う相関関数の数理物理的な分析を通して、ゲージ理論と弦理論との関連の解明に寄与するものであり、 学位論文として十分な内容を含んでいると判断できる。

なお、本論文の4章の内容は、山口哲氏との共同研究であるが、論文提出者が主体となって分析を行ったものであり、論文提出者の寄与が十分であると判断する。なお、5章、6章は論文提出者個人の研究に基づくものである。

したがって、論文提出者に博士(理学)の学位を授与できると認める。