Motivated by M-theory compactification on elliptic Calabi-Yau threefolds, we present a correspondence between networks of small resolutions for singular elliptic fibrations and Coulomb branches of five-dimensional N=1 gauge theories. While resolutions correspond to subchambers of the Coulomb branch, partial resolutions correspond to higher codimension loci at which the Coulomb branch intersects the Coulomb-Higgs branches. Flops between different resolutions are identified with reflections on the Coulomb branch. Physics aside, this correspondence provides an interesting link between elliptic fibrations and representation theory.
These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.
We present a simple algebraic construction of all the small resolutions for the SU(5) Weierstrass model. Each resolution corresponds to a subchamber on the Coulomb branch of the five-dimensional N=1 SU(5) gauge theory with matter fields in the fundamental and two-index antisymmetric representations. This construction unifies all previous resolutions found in the literature in a single framework.
Many N=1 heterotic string compactifications exhibit physically mysterious singularities at codimension one in the moduli space of vacua. At these singularities, Yukawa couplings of charged fields develop poles as a function of the moduli. We explain these conformal field theory singularities, in a class of examples, as arising from non-perturbative gauge dynamics of non-perturbative gauge bosons (whose gauge coupling is the sigma model coupling) in the string theory.
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singular…
We study the behavior of the dynamical fermion mass when infrared divergences and mass shell singularities are present in a gauge theory. In particular, in the massive Schwinger model in covariant gauges we find that the pole of the fermion propagator is divergent and gauge dependent at one loop, but the leading singularities cancel in the quenched rainbow approximation. On the other hand, in physical gauges, we find that the dynamical fermion mass is finite and gauge independent at least up to one loop.
IR divergences of a non-commutative U(1) Maxwell theory are discussed at the one-loop level using an interpolating gauge to show that quadratic IR divergences are independent not only from a covariant gauge fixing but also independent from an axial gauge fixing.
We consider D3 branes at orbifolded conifold singularities which are not quotient singularities. We use toric geometry and gauged linear sigma model to study the moduli space of the gauge theories on the D3 branes. We find that topologically distinct phases are related by a flop transition. It is also shown that an orbifold singularity can occur in some phases if we give expectation values to some of the chiral fields.
It is shown, by explicit calculation in the axial gauge, that the renormalization group flow for the Wilson loop in perturbation theory does exhibit singularities and consequently it can not eventually reproduce the gauge invariant result, when the infrared cut off is removed
We conjecture that the leading two-derivative tree-level amplitudes for gluons and gravitons can be derived from gauge invariance together with mild assumptions on their singularity structure. Assuming locality (that the singularities are associated with the poles of cubic graphs), we prove that gauge-invariance in just (n-1) particles together with minimal power-counting uniquely fixes the amplitude. Unitarity in the form of factorization then follows from locality and gauge invariance. We also give evidence for a stronger conjecture: assuming only that singularities occur when the sum of a s…
In this work, we first review the issues on the singularities and the boundary conditions in light cone gauge and how to regularize them properly. Then we will further review how these singularities and the boundary conditions can result in the gauge link at the infinity in the light cone direction in the Drell-Yan process. Except for reviewing, we also have verified that the gauge link at the light cone infinity has no dependence on the path not only for the Abelian field but also for non-Abelian gauge field.
We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifolds ${\bf C^3/Z_2\times Z_2}$,${\bf C^3/Z_2\times Z_3}$ and ${\bf C^4/Z_2\times Z_2 \times Z_2}$.
The axial-gauge boson propagator contains 1/(n.k)^p-type singularities. These singularities have generally been treated by inventing prescriptions for them. We propose an alternative procedere for treating these singularities in the path-integral formalism using the known way of treating the 1/k^{2n}-type singularities in Lorentz-type gauges. For this purpose we use a finite field-dependent BRS transformation that inerpolates between the Lorentz and axial-type gauges. We arrive at the ε-dependent tree propagator in axial-type gauges.
We point out that the recent conjecture relating large N gauge theories to string theory in anti-de Sitter spaces offers a resolution in principle of many problems in black hole physics. This is because the gauge theory also describes spacetimes which are not anti-de Sitter, and include black hole horizons and curvature singularities.
Some gauge theories with Coulomb branches exhibit singularities in perturbation theory, which are usually resolved by nonperturbative physics. In string theory this corresponds to the resolution of timelike singularities near the core of orientifold planes by effects from F or M theory. We propose a new mechanism for resolving Coulomb branch singularities in three dimensional gauge theories, based on Chern-Simons interactions. This is illustrated in a supersymmetric SU(2) Yang-Mills-Chern-Simons theory. We calculate the one loop corrections to the Coulomb branch of this theory and find a resul…
We study a class of extremal transitions between topological distinct Calabi--Yau manifolds which have an interpretation in terms of the special massless states of a type II string compactification. In those cases where a dual heterotic description exists the exceptional massless states are due to genuine strong (string-) coupling effects. A new feature is the appearance of enhanced non-abelian gauge symmetries in the exact nonperturbative theory.
We study the operator product expansion of two non-interacting chiral currents in the presence of external gauge fields in four dimensional euclidean space. We obtain the operator singularity in terms of the beta function of the free energy density and a connection on the space of external gauge fields, by imposing the consistency between the variational formula, introduced previously by the author, and both the renormalization group equations and the equation of motion for the currents. As a byproduct, we derive a euclidean version of the anomalous commutator of two currents in an appendix.
In N=1 supersymmetric U(N) gauge theory with adjoint matter $Φ$ and polynomial tree-level superpotential $W(Φ)$, the massless fluctuations about each quantum vacuum are generically described by $U(1)^n$ gauge theory for some n. However, by tuning the parameters of $W(Φ)$ to non-generic values, we can reach singular vacua where additional fields become massless. Using both the matrix model prescription and the strong-coupling approach, we study in detail three examples of such singularities: the singularities of the n=1 branch, intersections of n=1 and n=2 branches, and a class of N=1 Argyres-D…
We study brane configurations that give rise to large-N gauge theories with eight supersymmetries and no hypermultiplets. These configurations include a variety of wrapped, fractional, and stretched branes or strings. The corresponding spacetime geometries which we study have a distinct kind of singularity known as a repulson. We find that this singularity is removed by a distinctive mechanism, leaving a smooth geometry with a core having an enhanced gauge symmetry. The spacetime geometry can be related to large-N Seiberg-Witten theory.
The propagators in axial-type, light-cone and planar gauges contain 1/(η\cdot k)^p-type singularities. These singularities have generally been treated by inventing prescriptions for them. In this work, we propose an alternative procedure for treating these singularities in the path integral formalism using the known way of treating the singularities in Lorentz gauges. To this end, we use a finite field-dependent BRS transformation that interpolates between Lorentz-type and the axial-type gauges. We arrive at the $ε$-dependent tree propagator in the axial-type gauges. We examine the singularity…
Using ``Tate's algorithm,'' we identify loci in the moduli of F-theory compactifications corresponding to enhanced gauge symmetry. We apply this to test the proposed F-theory/heterotic dualities in six dimensions. We recover the perturbative gauge symmetry enhancements of the heterotic side and the physics of small $SO(32)$ instantons, and discover new mixed perturbative/non-perturbative gauge symmetry enhancements. Upon further toroidal compactification to 4 dimensions, we derive the chain of Calabi-Yau threefolds dual to various Coulomb branches of heterotic strings.
The structure of singularities in perturbative massless gauge theories is investigated in coordinate space. The pinch singularities in coordinate-space integrals occur at configurations of vertices which have a direct interpretation in terms of physical scattering of particles in real space-time in the same way as for the loop momenta in the case of momentum-space singularities. In the analysis of vertex functions in coordinate space, the well-known factorization into hard, soft, and jet functions is found. By power-counting arguments, it is found that coordinate-space integrals of vertex func…
We consider semi-direct gauge mediation models of supersymmetry breaking where the messengers are composite fields and their supersymmetric mass is naturally generated through quartic superpotential couplings. We show that such composite messenger models can be easily embedded in quiver gauge theories arising from D-branes at Calabi-Yau singularities, and argue that semi-direct gauge mediation is in fact a very natural option for supersymmetry breaking in D-brane models. We provide several explicit examples and discuss their salient phenomenological properties.
We discuss spacetime singularity resolution in the context of the gravity/gauge correspondence, for brane systems which give rise to gauge theories with eight supercharges and no hypermultiplets. The discussion is aimed at non-experts. Writeup of talk on hep-th/9911161 with C.V.Johnson and J.Polchinski, given in various forms at: PASCOS-99, Aspen winter conference `Way Beyond the Standard Models', Banff CIAR Gravity+Cosmology Programme Meeting.
We use the earlier results on the correlations of axial gauge Green's functions and the Lorentz gauge Green's functions obtained via finite field-dependent BRS transformations to study the question of the correct treatment of 1/(eta\cdot k)^p- type singularities in the axial gauge boson propagator. We show how the known treatment of the 1/(k^2)^n-type singularity in the Lorentz-type gauges can be used to write down the axial propagator via field transformation. We examine the singularity structure of the latter and find that the axial propagator so constructed has $no$ spurious poles, but a co…
The gauge theory on a set of D3-branes at a toric Calabi-Yau singularity can be encoded in a tiling of the 2-torus denoted dimer diagram (or brane tiling). We use these techniques to describe the effect on the gauge theory of geometric operations partially smoothing the singularity at which D3-branes sit, namely partial resolutions and complex deformations. More specifically, we describe the effect of arbitrary partial resolutions, including those which split the original singularity into two separated. The gauge theory correspondingly splits into two sectors (associated to branes in either si…
Via partial resolution of Abelian orbifolds we present an algorithm for extracting a consistent set of gauge theory data for an arbitrary toric variety whose singularity a D-brane probes. As illustrative examples, we tabulate the matter content and superpotential for a D-brane living on the toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. Moreover, we discuss the non-uniqueness of the general problem and present examples of vastly different theories whose moduli spaces are described by the same toric data. Our methods provide new tools for calculating gauge theories which flo…
We consider a gauge/gravity dual model of thermalization which consists of a collapsing thin matter shell in asymptotically Anti-de Sitter space. A central aspect of our model is to consider a shell moving at finite velocity as determined by its equation of motion, rather than a quasi-static approximation as considered previously in the literature. By applying a divergence matching method, we obtain the evolution of singularities in the retarded unequal time correlator $G^R(t,t')$, which probes different stages of the thermalization. We find that the number of singularities decreases from a fi…
Perturbative QCD calculations in the light-cone gauge have long suffered from the ambiguity associated with the regularization of the poles in the gluon propagator. In this work we study sub-gauge conditions within the light-cone gauge corresponding to several known ways of regulating the gluon propagator. Using the functional integral calculation of the gluon propagator, we rederive the known sub-gauge conditions for the theta-function gauges and identify the sub-gauge condition for the principal value (PV) regularization of the gluon propagator's light-cone poles. The obtained sub-gauge cond…
We investigate backgrounds of Type IIB string theory with null singularities and their duals proposed in hep-th/0602107. The dual theory is a deformed N=4 Yang-Mills theory in 3+1 dimensions with couplings dependent on a light-like direction. We concentrate on backgrounds which become AdS_5 x S^5 at early and late times and where the string coupling is bounded, vanishing at the singularity. Our main conclusion is that in these cases the dual gauge theory is nonsingular. We show this by arguing that there exists a complete set of gauge invariant observables in the dual gauge theory whose correl…
In this paper we discuss the singularities in the Yukawa and gauge couplings of N=1 compactifications of the SO(32) heterotic string in four space-time dimensions. Such singularities can arise from the strong coupling dynamics of a confined non-perturbative gauge group.
We point out that the metrics recently proposed by K. Balasubramanian and J. McGreevy \cite{BalaMcGreevyLifshitz} as gravitational duals to Lifshitz Chern-Simons gauge theories contain both a hidden null singularity and a region of closed timelike curves accessible to asymptotic observers. Like the singularity in the original Liftshitz spacetime given by Kachru, Liu, and Mulligan, this singularity does not include large $α'$ or $g_s$ corrections and hence appears to be singular in string theory as well as classically.
We consider Type 0B D3-branes placed at conical singularities and analyze in detail the conifold singularity. We study the non supersymmetric gauge theories on their worldvolume and their conjectured dual gravity descriptions. In the ultraviolet the solutions exhibit a logarithmic running of the gauge coupling. In the infrared we find confining solutions and IR fixed points.
The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, Stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities.
We point out that the leading infrared singular terms in the effective actions of noncommutative gauge theories arising from nonplanar loop diagrams have a natural interpretation in terms of the matrix model (operator) formulation of these theories. In this formulation (for maximal spatial noncommutativity), noncommutative space arises as a configuration of an infinite number of D-particles. We show that the IR singular terms correspond to instantaneous linear potentials between these D-particles resulting from the zero point energies of fluctuations about this background. For theories with fe…
The boundary chiral ring of a 2d gauged linear sigma model on a Kähler manifold $X$ classifies the topological D-brane sectors and the massless open strings between them. While it is determined at small volume by simple group theory, its continuation to generic volume provides highly non-trivial information about the $D$-branes on $X$, related to the derived category $D^\flat(X)$. We use this correspondence to elaborate on an extended notion of McKay correspondence that captures more general than orbifold singularities. As an illustration, we work out this new notion of McKay correspondence fo…
In contrast to the familiar (2,2) case, the singularities which arise in the (0,2) setting can be associated with degeneration of the base Calabi-Yau manifold {\it and/or}\/ with degenerations of the gauge bundle. We study a variety of such singularities and give a procedure for resolving those which can be cured perturbatively. Among the novel features which emerge are models in which smoothing singularities in the base yields a gauge {\it sheaf}\/ as opposed to a gauge bundle as the structure to which left moving fermions couple. Supersymmetric $σ$-models with target data being an appropriat…
Building on author's previous results in singular semi-Riemannian geometry and singular general relativity, the behavior of gauge theory at singularities is analyzed. The usual formulations of the field equations at singularities are accompanied by infinities which block the evolution equations, mainly because the metric is singular, hence the usual differential operators, constructed from the metric, blow up. However, it is possible to give otherwise equivalent formulations of the Einstein, Maxwell and Yang-Mills equations, which in addition admit solutions which can be extended beyond the si…
We improve on a recently constructed graphical representation of the supergravity 7-brane solution and apply this refined representation to re-study the open string description of the A-D-E-singularities in F-theory on K3. A noteworthy feature of the graphical representation is that it provides the complete global branch cut structure of the 7-brane solution which plays an important role in our analysis. We first identify those groups of branes which when made to coincide lead to the A-D-E-gauge groups. We next show that there is always a sufficient number of open BPS strings to account for al…
Using a geometric realization of the $SU(2)_R$ symmetry and a procedure of factorisation of the gauge and $SU(2)_R$ charges, we study the small instanton singularities of the Higgs branch of supersymmetric $U(1)^r$ gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges. In particular, we obtain an extension of the ordinary ADE singularities for hyperKahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces.
Recent scenarios of phenomenologically realistic string compactifications involve the existence of gauge sectors localized on D-branes at singular points of Calabi-Yau threefolds. The spectrum and interactions in these gauge sectors are determined by the local geometry of the singularity, and can be encoded in quiver diagrams. We discuss the physical models arising for the simplest case of orbifold singularities, and generalize to non-orbifold singularities and orientifold singularities. Finally we show that relatively simple singularities lead to gauge sectors surprisingly close to the standa…
Discontinuity of gauge theory in the gauge condition $n\cdot\partial n\cdot A=0$, which emerges at $n\cdot k=0$, is studied here. Such discontinuity is different from that one confronts in axial gauge and can not be regularized by conventional analytical continuation method. The Faddeev-Popov determinate of the gauge $n\cdot\partial n\cdot A=0$, which is solved explicitly in the manuscript, behaves like a $δ$-functional of gauge potentials once singularities in the functional integral is neglected and the length along $n^μ$ direction of the space tends to infinity. As a sequence, perturbation …
Recent work has provided a direct string calculation of the internal coordinate dependence of gauge field tadpoles on the orbifold C^3/Z_3. We investigate the structure of these profiles in momentum and coordinate space representations using both analytical and numerical methods. The twisted states are determined to be localized within a few string lengths of the singularity. This provides an example of how delta-like singularities, which are typical for field theories on orbifolds, are avoided in string theory. A systematic expansion of the full string result allows us to speculate on how the…
We investigate the singularity structure of the $(-1)^F$ graded partition function in QCD with $n_f \geq 1$ massive adjoint fermions in the large-$N$ limit. Here, $F$ is fermion number and $N$ is the number of colors. The large $N$ partition function is made reliably calculable by taking space to be a small three-sphere $S^3$. Singularites in the graded partition function are related to phase transitions and to Hagedorn behavior in the $(-1)^F$-graded density of states. We study the flow of the singularities in the complex "inverse temperature" $β$ plane as a function of the quark mass. This a…
We compute the Nekrasov partition function of gauge theories on the (resolved) toric singularities C^2/Γin terms of blow-up formulae. We discuss the expansion of the partition function in the ε_1,ε_2 \to 0 limit along with its modular properties and how to derive them from the M-theory perspective. On the two-dimensional conformal field theory side, our results can be interpreted in terms of representations of the direct sum of Heisenberg plus W_N-algebrae with suitable central charges, which can be computed from the fan of the resolved toric variety.We provide a check of this correspondence b…
We propose a definition of irregular vertex operators in the H3+ WZW model. Our definition is compatible with the duality [1] between the H3+ WZW model and Liouville theory, and we provide the explicit map between correlation functions of irregular vertex operators in the two conformal field theories. Our definition of irregular vertex operators is motivated by relations to partition functions of N=2 gauge theory and scattering amplitudes in N=4 gauge theory
We describe a class of 4d N=1 compactifications of the $SO(32)$ heterotic/type I string theory which are destabilized by nonperturbatively generated superpotentials. In the type I description, the destabilizing superpotential is generated by a one instanton effect or gaugino condensation in a nonperturbative $SU(2)$ gauge group. The dual, heterotic description involves destabilization due to worldsheet instanton or $\it half$ worldsheet instanton effects in the two cases. A genericity argument suggests that a (global) supersymmetry-breaking model of Intriligator and Thomas might be typical in …
We calculate the three- and four-loop corrections to the massless fermion propagator in three-dimensional quenched Quantum Electrodynamics with four-component fermions. The three-loop correction is finite and gauge invariant but the four-loop one has singularities except in the Feynman gauge where it is also finite. Our results explicitly show that, up to four loops, gauge-dependent terms are completely determined by lower order ones in agreement with the Landau-Khalatnikov-Fradkin transformation.
Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional…
M theory compactifications on G_2 holonomy manifolds, whilst supersymmetric, require singularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) holonomy manifolds which correspond to the properties of three dimensional fi…
We find the explicit operatorial form of renormalon-type singularities in abelian gauge theory. Local operators of dimension six take care of the first U.V. renormalon, non local operators are needed for I.R. singularities. In the effective lagrangian constructed with these operators non local imaginary parts appearing in the usual perturbative expansion at large orders are cancelled.
We present a first step towards generalizing the work of Seiberg and Witten on N=2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus $n-1$ Riemann surfaces to underly the quantum moduli space of $SU(n)$ N=2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible re…
We study the effects produced by D-brane instantons on the holomorphic quantities of a D-brane gauge theory at an orbifold singularity. These effects are not limited to reproducing the well known contributions of the gauge theory instantons but also generate extra terms in the superpotential or the prepotential. On these brane instantons there are some neutral fermionic zero-modes in addition to the ones expected from broken supertranslations. They are crucial in correctly reproducing effects which are dual to gauge theory instantons, but they may make some other interesting contributions vani…
AdS/CFT predicts a precise relation between the central charge a, the scaling dimensions of some operators in the CFT on D3-branes at conical singularities and the volumes of the horizon and of certain cycles in the supergravity dual. We review how a quantitative check of this relation can be performed for all toric singularities. In addition to the results presented in hep-th/0506232, we also discuss the relation with the recently discovered map between toric singularities and tilings; in particular, we discuss how to find the precise distribution of R-charges in the quiver gauge theory using…
Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profile…
The singularity structure of the Coulomb and Higgs branches of good $3d$ $\mathcal{N}=4$ circular quiver gauge theories (CQGTs) with unitary gauge groups is studied. The central method employed is the Kraft--Procesi transition. CQGTs are described as a generalisation of a class of linear quivers. This class degenerates into the familiar class $T_ρ^σ(SU(N))$ in the linear case, however the circular case does not have the degeneracy and so the class of CQGTs contains many more theories and much more structure. We describe a collection of good, unitary, CQGTs from which the entire class can be fo…
Type II string theory compactified on a Calabi-Yau manifold, with a singularity modeled by a hypersurface in an orbifold, is considered. In the limit of vanishing string coupling, one expects a non gravitational theory concentrated at the singularity. It is proposed that this theory is holographicly dual to a family of ``non-critical'' superstring vacua, generalizing a previous proposal for hypersurfaces in flat space. It is argued that a class of such singularities is relevant for the study of non-trivial IR fixed points that appear in the moduli space of four-dimensional N=2 SQCD: SU(N_c) ga…
