In this review we describe our current understanding of the properties of open string tachyons on an unstable D-brane or brane–antibrane system in string theory. The various string theoretic methods used for this study include techniques of two-dimensional conformal field theory, open string field theory, boundary string field theory, noncommutative solitons, etc. We also describe various attempts to understand these results using field theoretic methods. These field theory models include toy models like singular potential models and p-adic string theory, as well as more realistic version of the tachyon effective action based on Dirac–Born–Infeld type action. Finally we study closed string background produced by the "decaying" unstable D-branes, both in the critical string theory and in the two-dimensional string theory, and describe the open string completeness conjecture that emerges out of this study. According to this conjecture the quantum dynamics of an unstable D-brane system is described by an int...

Tachyon Dynamics in Open String Theory

We study Euclidean 3D $ \mathcal{N} = 2 $ supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) × U(1) or U(1) × U(1) We show that, when a suitable background U(1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore $ \mathcal{N} = 2 $ supersymmetric gauge theories We present the Lagrangian and supersymmetry rules for general gauge theories The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch For the squashed sphere with U(1) × U(1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b ≠ 1

SUSY gauge theories on squashed three-spheres

We extend theformulaforpartitionfunctions of $$\mathcal{N}=2 $$
 superconformalgauge theories on S
 3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of $$\mathcal{N}=4 $$
 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to $$\mathcal{N}=2 $$
 .

Notes on SUSY gauge theories on three-sphere

For 3-dimensional field theories with $ \mathcal{N} = 2 $
 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number ofsuch large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the super potential. In all our $ \mathcal{N} = 2 $
 superconformal examples, the local maximization of F yields answers that scale as N
 3/2 and agree with the dual M-theory backgrounds AdS
 4 × Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the “F-theorem” that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N
 5/3 at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere

The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.

/pdf/abjm-theory-as-a-fermi-gas-56l32svhc5.pdf

ABJM theory as a Fermi gas

Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in $ \mathcal{N} = 2 $ superconformal (quiver) gauge theories in the large number limit of colors. Focusing at planar and large ‘t Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in $ \mathcal{N} = 4 $ super Yang-Mills theory with respect to ‘t Hooft coupling. For theory with gauge group SU(N) coupled to 2N fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth — at most, it can grow as a power of ‘t Hooft coupling. For theory with gauge group SU(N)×SU(N) and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups. We find Wilson loop in untwisted sector grows exponentially large as in $ \mathcal{N} = 4 $ super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of the two ‘t Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loops in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet. We suggest intuitive interpretation that in both classes of theory holographic dual background must involve string scale geometry even at planar and large ‘t Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicates that holographic dual of these gauge theories is provided by certain non-critical string theories.

Exact results and holography of Wilson loops in $ \mathcal{N} = 2 $ superconformal (quiver) gauge theories

Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in N=2 superconformal (quiver) gauge theories. Focusing at planar and large `t Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in N=4 super Yang-Mills theory. For theory with gauge group SU(N) coupled to 2N fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth -- at most, it can grow a power of `t Hooft coupling. For theory with gauge group SU(N) x SU(N) and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups. We find Wilson loop in untwisted sector grows exponentially large as in N=4 super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of two `t Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loop in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet. We suggest intuitive interpretation that in both type theories holographic dual background must involve string scale geometry even at planar and large `t Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicate that holographic dual of these gauge theories is provided by certain non-critical string theories.

Exact Results and Holography of Wilson Loops in N=2 Superconformal (Quiver) Gauge Theories

On Large N Solution of ABJM Theory

We investigate another supersymmetric Chern-Simons theory called the orientifold ABJM theory, which replaces the unitary supergroup structure of the ABJM theory with an orthosymplectic one. Its non-perturbative structure is completely clarified by considering the duplication of the quiver.

/pdf/instanton-effects-in-orientifold-abjm-theory-37iumrqyc5.pdf

Instanton Effects in Orientifold ABJM Theory

Recently it was found that the density matrix for a certain orthosymplectic Chern-Simons theory matches with that for the ABJM theory with the odd chiral projection. We prove this fact for a general case with the inclusion of fractional branes. We also identify the first few diagonal Gopakumar-Vafa invariants for the grand potential constructed from the chirally projected density matrix.

/pdf/orthosymplectic-chern-simons-matrix-model-and-chirality-3d6r3r3ln8.pdf

Takao Suyama

Papers

Exact results and holography of Wilson loops in $ \mathcal{N} = 2 $ superconformal (quiver) gauge theories

Exact Results and Holography of Wilson Loops in N=2 Superconformal (Quiver) Gauge Theories

On Large N Solution of ABJM Theory

Instanton Effects in Orientifold ABJM Theory

Orthosymplectic Chern-Simons matrix model and chirality projection