Review of localization in geometry
Review of localization in geometry: equivariant cohomology, characteristic classes, Atiyah-Bott formula, Atiyah-Singer equivariant index formula, Mathai-Quillen formalism
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arXiv subjects
Vasily Pestun
Publications and source records attributed to Vasily Pestun.
Localization techniques in quantum field theories
This is the foreword to the special volume on localization techniques in quantum field theory. We summarize the individual chapters and their relation.
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Quantum geometry and quiver gauge theories
We study macroscopically two dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product $\mathbb{T}^{d} \times \mathbb{R}^{2}_ε$ of a $d$-dimensional torus and a two dimensional cigar with $Ω$-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^{1+d}$, for $d=0,1,2$, and the Yangian $\mathbf{Y}_ε(\mathfrak{g}_Γ)$, quantum affine algebra $\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_Γ)$, or the quantum elliptic algebra $\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_Γ)$ associated to Kac-Moody algebra $\mathfrak{g}_Γ$ for quiver $Γ$.
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Localization of the four-dimensional N=4 SYM to a two-sphere and 1/8 BPS Wilson loops
We localize the four-dimensional N=4 super Yang-Mills theory on a four-sphere to the two-dimensional constrained Hitchin/Higgs-Yang-Mills (cHYM) theory on a two-sphere S^2. We show that expectation values of certain 1/8 BPS supersymmetric Wilson loops on S^2 in the 4d N=4 SYM is captured by the 2d cHYM theory. We further argue that expectation values of Wilson loops in the cHYM theory agree with the prescription "two-dimensional bosonic Yang-Mills excluding instanton contributions". Hence, we support the recent conjecture by Drukker, Giombi, Ricci and Trancanelli on the 1/8 BPS Wilson loops on S^2 in the 4d N=4 SYM.
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Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4 supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2 and the N=2* supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2 superconformal gauge theory is treated similarly.
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Correlators of Wilson loops and local operators from multi-matrix models and strings in AdS
We study correlation functions of Wilson loops and local operators in a subsector of N=4 SYM which preserves two supercharges. Localization arguments allow to map the problem to a calculation in bosonic two-dimensional Yang-Mills theory. In turn, this can be reduced to computing correlators in certain Gaussian multi-matrix models. We focus on the correlation function of a Wilson loop and two local operators, and solve the corresponding three-matrix model exactly in the planar limit. We compare the strong coupling behavior to string theory in AdS_5xS^5, finding precise agreement. We pay particular attention to the case in which the local operators have large R-charge J \sim sqrt{lambda} at strong coupling.
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Exact Results for 't Hooft Loops in Gauge Theories on S^4
The path integral of a general N=2 supersymmetric gauge theory on S^4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find - obtained using localization techniques - captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S^4. As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.
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On the instantons and the hypermultiplet mass of N=2* super Yang-Mills on S^4
We show that the physical N=4 super Yang-Mills theory on a four-sphere with an arbitrary gauge group receives no instanton contributions, by clarifying the relation between the hypermultiplet mass and the equivariant parameters of the mass-deformed theory preserving N=2 supersymmetry. The correct relation also implies that N=4 superconformal Yang-Mills theory with gauge group SU(2) corresponds to Liouville theory on a torus with the insertion of a non-trivial operator, rather than the identity as have been claimed in the literature.
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Supersymmetric Wilson loops in N=4 SYM and pure spinors
We study supersymmetric Wilson loop operators in four-dimensional N=4 super Yang-Mills theory. We show that the contour of a supersymmetric Wilson loop is either an orbit of some conformal transformation of the space-time (case I), or an arbitrary contour in the subspace where local superalgebra generator is a pure spinor (case II). In the more interesting case II we find and classify all pairs (Q,W) of the supercharges and the corresponding operators modulo the action of the global symmetry group.
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The 1/2 BPS 't Hooft loops in N=4 SYM as instantons in 2d Yang-Mills
We extend the recent conjecture on the relation between a certain 1/8 BPS subsector of 4d N=4 SYM on S^2 and 2d Yang-Mills theory by turning on circular 1/2 BPS 't Hooft operators linked with S^2. We show that localization predicts that these 't Hooft operators and their correlation functions with Wilson operators on S^2 are captured by instanton contributions to the partition function of the 2d Yang-Mills theory. Based on this prediction, we compute explicitly correlation functions involving the 't Hooft operator, and observe precise agreement with S-duality predictions.
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Notes on supersymmetric Wilson loops on a two-sphere
We study a recently discovered family of 1/8-BPS supersymmetric Wilson loops in N=4 super Yang-Mills theory and their string theory duals. The operators are defined for arbitrary contours on a two-sphere in space-time, and they were conjectured to be captured perturbatively by 2d bosonic Yang-Mills theory. In the AdS dual, they are described by pseudo-holomorphic string surfaces living on a certain submanifold of AdS_5 x S^5. We show that the regularized area of these string surfaces is invariant under area preserving diffeomorphisms of the boundary loop, in agreement with the conjecture. Further, we find a connection between the pseudo-holomorphicity equations and an auxiliary sigma-model on S^3, which may help to construct new 1/8-BPS string solutions. We also show that the conjectured relation to 2d Yang-Mills implies that a connected correlator of two Wilson loops is computed by a Hermitian Gaussian two-matrix model. On the AdS dual side, we argue that the connected correlator is described by two disconnected disks interacting through the exchange of supergravity modes, and we show that this agrees with the strong coupling planar limit of the two-matrix model.
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Correlators of local operators and 1/8 BPS Wilson loops on S^2 from 2d YM and matrix models
We propose that, in N=4 Super Yang-Mills theory, correlation functions of certain 1/8 BPS Wilson loops and local operators inserted on a S^2 in space-time may be computed in terms of analogous observables in the "zero-instanton" sector of 2d Yang-Mills theory. The Wilson loops are mapped to the standard Wilson loops of the 2d theory, as recently conjectured, while the local operators are mapped to powers of the 2d field strength. We give several perturbative checks of the correspondence, and derive from 2d Yang-Mills a two-matrix model for the correlator of a local operator and a Wilson loop of arbitrary shape. We show that the strong coupling planar limit of the two-matrix model precisely agrees with a string theory calculation in AdS_5 x S^5.
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Topological strings in generalized complex space
A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $ρ$ on $X$. In the present construction the algebra of $Q$-transformations automatically closes off-shell, the model transparently depends only on $J$, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector $β$ and recover holomorphic noncommutative Kontsevich $*$-product.
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Black hole entropy and topological strings on generalized CY manifolds
The H. Ooguri, A. Strominger and C. Vafa conjecture $Z_{BH}=|Z_{top}|^2$ is extended for the topological strings on generalized CY manifolds. It is argued that the classical black hole entropy is given by the generalized Hitchin functional, which defines by critical points a generalized complex structure on $X$. This geometry differs from an ordinary geometry if $b_1(X)$ does not vanish. In a critical point the generalized Hitchin functional equals to Legendre transform of the free energy of generalized topological string. The examples of $T^6$ and $T^2 \times K3$ are considered in details.
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The Hitchin functionals and the topological B-model at one loop
The quantization in quadratic order of the Hitchin functional, which defines by critical points a Calabi-Yau structure on a six-dimensional manifold, is performed. The conjectured relation between the topological B-model and the Hitchin functional is studied at one loop. It is found that the genus one free energy of the topological B-model disagrees with the one-loop free energy of the minimal Hitchin functional. However, the topological B-model does agree at one-loop order with the extended Hitchin functional, which also defines by critical points a generalized Calabi-Yau structure. The dependence of the one-loop result on a background metric is studied, and a gravitational anomaly is found for both the B-model and the extended Hitchin model. The anomaly reduces to a volume-dependent factor if one computes for only Ricci-flat Kahler metrics.
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N=4 super Yang-Mills matrix integrals for almost all simple gauge groups
In this paper the partition function of N=4 D=0 super Yang-Mills matrix theory with arbitrary simple gauge group is discussed. We explicitly computed its value for all classical groups of rank up to 11 and for the exceptional groups G_2, F_4 and E_6. In the case of classical groups of arbitrary rank we conjecture general formulas for the B_r, C_r and D_r series in addition to the known result for the A_r series. Also, the relevant boundary term contributing to the Witten index of the corresponding supersymmetric quantum mechanics has been explicitly computed as a simple function of rank for the orthogonal and symplectic groups SO(2N+1), Sp(2N), SO(2N).
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On non-abelian low energy effective action for D-branes
Connection between the partition function for the 2D sigma model with boundary pertubations and the low energy effective action for massless fields from in the open string theory is discussed. In the non-abelian case with a stack of $N$ D-branes, the terms up to the order of $α'^3$ are found
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