[Paper Review] N=1 Supersymmetry, Deconstruction, and Bosonic Gauge Theories

Robbert Dijkgraaf, Cumrun Vafa|ArXiv.org|Feb 3, 2003

Black Holes and Theoretical PhysicsPhysics and Astronomy58 references63 citations

TL;DR

This paper establishes a direct correspondence between N=1 supersymmetric gauge theories in four dimensions and matrix models via deconstruction, showing that F-terms in these theories are captured by c=1 non-critical bosonic string amplitudes at self-dual radius. The key result is that the full holomorphic geometry of Calabi-Yau threefold compactifications, including the deformed conifold, emerges from quiver matrix models, with Chern-Simons and unitary matrix models deconstructing higher-dimensional theories.

ABSTRACT

We show how the full holomorphic geometry of local Calabi-Yau threefold compactifications with N=1 supersymmetry can be obtained from matrix models. In particular for the conifold geometry we relate F-terms to the general amplitudes of c=1 non-critical bosonic string theory, and express them in a quiver or, equivalently, super matrix model. Moreover we relate, by deconstruction, the uncompactified c=1 theory to the six-dimensional conformal (2,0) theory. Furthermore, we show how we can use the idea of deconstruction to connect 4+k dimensional supersymmetric gauge theories to a k-dimensional internal bosonic gauge theory, generalizing the relation between 4d theories and matrix models. Examples of such bosonic systems include unitary matrix models and gauged matrix quantum mechanics, which deconstruct 5-dimensional supersymmetric gauge theories, and Chern-Simons gauge theories, which deconstruct gauge theories living on branes wrapped over cycles in Calabi-Yau threefolds.

Motivation & Objective

• To establish a direct connection between N=1 supersymmetric gauge theories in 4D and matrix models through deconstruction.
• To show that F-terms in N=1 Yang-Mills theories are equivalent to correlation functions in c=1 non-critical bosonic string theory at self-dual radius.
• To generalize the matrix model description to higher-dimensional gauge theories using internal bosonic gauge theories.
• To reconstruct the full Calabi-Yau threefold geometry, including holomorphic data, from the planar diagrams of matrix models.
• To demonstrate that deconstruction via matrix models encodes the full string theory, including holomorphic and potentially D-term data at superconformal fixed points.

Proposed method

• Use of quiver matrix models to describe the low-energy dynamics of N=1 supersymmetric gauge theories, particularly the affine Â1 quiver theory.
• Mapping of F-terms in N=1 theories to correlation functions in c=1 non-critical bosonic string theory at self-dual radius via the ring of holomorphic functions.
• Application of deconstruction to relate 4+k dimensional N=1 theories to k-dimensional internal bosonic gauge theories, such as unitary matrix models and Chern-Simons theories.
• Construction of a supermatrix model from a topological brane/antibrane system, realizing the c=1 string as a supermatrix model.
• Use of period matrices of spectral curves to compute gauge couplings in the 4D theory, with the curve derived from the matrix model's superpotential.
• Reduction of 5D N=1 SYM with hypermultiplets to a 1D matrix quantum mechanics model with action involving covariant derivatives and superpotential terms.

Experimental results

Research questions

• RQ1 How can the F-terms of N=1 supersymmetric Yang-Mills theory be captured by a matrix model description?
• RQ2 What is the precise correspondence between c=1 non-critical bosonic strings at self-dual radius and N=1 gauge theories?
• RQ3 How does deconstruction via matrix models allow for the reconstruction of Calabi-Yau threefold geometry from 4D gauge theories?
• RQ4 Can higher-dimensional N=1 gauge theories be deconstructed into lower-dimensional internal bosonic gauge theories using matrix models?
• RQ5 To what extent do matrix models encode the full string theory, including non-holomorphic D-term data, at superconformal fixed points?

Key findings

• The F-terms of the affine Â1 quiver theory are equivalent to correlation functions in the c=1 non-critical bosonic string at self-dual radius.
• The c=1 string at k times the self-dual radius corresponds to the F-terms of an Â_{2k-1} affine quiver theory, with the infinite radius limit corresponding to Â_∞.
• The matrix model description captures the full holomorphic geometry of the deformed conifold, including the period matrix of the spectral curve, which gives the 4D gauge coupling.
• Unitary matrix models and gauged matrix quantum mechanics deconstruct 5D N=1 supersymmetric Yang-Mills theories with hypermultiplets.
• Chern-Simons gauge theories in 3D (real or holomorphic) deconstruct gauge theories on D6-branes wrapped over 3-cycles and D9-branes over Calabi-Yau threefolds.
• At superconformal fixed points, the F-terms uniquely determine the theory, suggesting that the matrix model encodes the full string theory implicitly through holomorphic data.