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Gauge Origami and BPS/CFT correspondence

Go Noshita

Abstract

Gauge origami is a generalized supersymmetric gauge theory defined on several intersecting space-time components. It provides a systematic way to consider generalizations of instantons. In this thesis, we explore the gauge origami system in $\mathbb{R}^{1,1}\times \mathbb{C}^{4}$ and its BPS/CFT correspondence. String theoretically, instantons of the gauge origami system arise from D0-branes bound to D$(2p)$-branes wrapping cycles in $\mathbb{C}^{4}$. The low energy theory of the D0-branes is understood as an $\mathcal{N}=2$ supersymmetric quiver quantum mechanical system and the Witten index of it produces the instanton partition function. We define a $q$-deformed quiver Cartan matrix associated to this quiver structure and introduce vertex operators associated with the D-branes and show that the contour integral formula for the Witten index has a nice free field realization. Such free field realization leads to the concept of BPS $qq$-characters or BPS quiver W-algebras, which are generalizations of the conventional deformed W-algebras. The $qq$-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the $qq$-characters of D6 and D8-branes represent novel types of $qq$-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these $qq$-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of $qq$-characters of D-branes in lower dimensions results in higher-dimensional D-brane $qq$-characters. We also investigate quadratic relations among these $qq$-characters. Furthermore, we explore the relationship with the representations of the quantum toroidal $\mathfrak{gl}_{1}$.

Gauge Origami and BPS/CFT correspondence

Abstract

Gauge origami is a generalized supersymmetric gauge theory defined on several intersecting space-time components. It provides a systematic way to consider generalizations of instantons. In this thesis, we explore the gauge origami system in and its BPS/CFT correspondence. String theoretically, instantons of the gauge origami system arise from D0-branes bound to D-branes wrapping cycles in . The low energy theory of the D0-branes is understood as an supersymmetric quiver quantum mechanical system and the Witten index of it produces the instanton partition function. We define a -deformed quiver Cartan matrix associated to this quiver structure and introduce vertex operators associated with the D-branes and show that the contour integral formula for the Witten index has a nice free field realization. Such free field realization leads to the concept of BPS -characters or BPS quiver W-algebras, which are generalizations of the conventional deformed W-algebras. The -characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the -characters of D6 and D8-branes represent novel types of -characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these -characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of -characters of D-branes in lower dimensions results in higher-dimensional D-brane -characters. We also investigate quadratic relations among these -characters. Furthermore, we explore the relationship with the representations of the quantum toroidal .

Paper Structure

This paper contains 178 sections, 66 theorems, 856 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rise of BPS/CFT correspondence
  3. Generalized gauge theory
  4. Gauge origami
  5. Quantum algebra of gauge origami
  6. QQuiver Cartan matrix
  7. Definition of D-brane vertex operators
  8. Operator formalism of $qq$-characters
  9. ADHM theory and supersymmetric localization
  10. Yang--Mills theory and instantons
  11. ADHM data and LMNS formula
  12. ADHM data
  13. LMNS formula
  14. Five-dimensional lift up
  15. Jefferey--Kirwan residue and Young diagrams
  16. ...and 163 more sections

Key Result

Theorem 1.0.2

For each D-brane (D0, D2, D4, D6, D8), we define the corresponding vertex operators as We have multiple copies of vertex operators if there are multiple ways that the D-branes can wrap. Then, the contour integral formula of the $k$-instanton partition function takes the form as where $\mathsf{V}_{i}(x)$ is an operator written from $\{\mathsf{X}_{A}(x),\mathsf{W}_{\bar{a}}(x),\mathsf{Z}(x)\}$.

Figures (3)

  • Figure 1: Left: The four vertices of the tetrahedron correspond to the $\mathbb{C}_{a}\,\,(a\in\underline{\textbf{4}})$, the six edges connecting two vertices of the tetrahedron correspond to the $\mathbb{C}^{2}_{A}\,\,(A\in\underline{\textbf{6}})$, the four faces surrounded by three vertices and the three edges connecting them correspond to the complex three-planes $\mathbb{C}^{3}_{\bar{a}}\,\,(a\in\underline{\textbf{4}})$, and the whole tetrahedron correspond to the $\mathbb{C}^{4}_{\underline{\textbf{4}}}$. Right: The toric diagram of $\mathbb{C}^{4}$. The vertices correspond to the co-dimension one subvariety $\mathbb{C}^{3}$, while the edges and faces correspond to co-dimension two and three subvarieties $\mathbb{C}^{2},\mathbb{C}$, respectively. Both of the description are obtained by taking the dual of each polytopes.
  • Figure 3: Coordinates of the plane partition
  • Figure :

Theorems & Definitions (113)

  • Definition 1.0.1: Def. \ref{['def:totalqquiverCartan']}
  • Theorem 1.0.2: Thm. \ref{['thm:freefieldconclusion']}
  • Theorem 1.0.3: Thm. \ref{['thm:spiked-qq-BPSCFT']}, \ref{['thm:tetra-origamiBPSCFT']}
  • Theorem 1.0.4: Thm. \ref{['thm:tetrascreening']}
  • Conjecture 1.0.5: Conj. \ref{['conj:D8commutativity']}
  • Corollary 1.0.6: Cor. \ref{['cor:D6commutativity']}
  • Lemma 2.3.1
  • proof
  • Lemma 2.3.2
  • Theorem 2.3.3
  • ...and 103 more