Coulomb Branch Research Articles

Effective action of 6D F-theory with U(1) factors: rational sections make Chern-Simons terms jump

We derive the six-dimensional (1, 0) effective action arising from F-theory on an elliptically fibered Calabi-Yau threefold with multiple sections. The considered theories admit both non-Abelian and Abelian gauge symmetries. Our derivation employs the M-theory to F-theory duality in five-dimensions after circle reduction. Five-dimensional gauge and gravitational Chern-Simons terms are shown to arise at one-loop by integrating out massive Coulomb branch and Kaluza-Klein modes. In the presence of a non-holomorphic zero section, we find an improved systematic for performing the F-theory limit by using the concept of the extended relative Mori cone. In this situation Kaluza-Klein modes can become lighter than Coulomb branch modes and a jump in the Chern-Simons levels occurs. By determining Chern-Simons terms for various threefold examples we are able to compute the complete six-dimensional charged matter spectrum and show consistency with six-dimensional anomalies.

Aharony dualities for 3d theories with adjoint matter

We study Aharony dualities for 3d N = 2 gauge theories of classical gauge group with one adjoint and fundamental matters. We work out the 3d superconformal index for the dual pairs to find the perfect matchings. Along with it, we enumerate the independent monopole operators parametrizing the Coulomb branches and confirm the nonperturbative truncation of the chiral rings, consistent with the proposed dualities.

On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

Abstract In previous work we have shown that the equivariant index of multi-centered $ \mathcal{N}=2 $ black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke’s result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package “CoulombHiggs.m” implementing the Coulomb and Higgs branch formulae.

Exact results in D = 2 supersymmetric gauge theories

Abstract We compute exactly the partition function of two dimensional $ \mathcal{N} $ = (2, 2) gauge theories on S 2 and show that it admits two dual descriptions: either as an integral over the Coulomb branch or as a sum over vortex and anti-vortex excitations on the Higgs branches of the theory. We further demonstrate that correlation functions in two dimensional Liouville/Toda CFT compute the S 2 partition function for a class of $ \mathcal{N} $ = (2, 2) gauge theories, thereby uncovering novel modular properties in two dimensional gauge theories. Some of these gauge theories flow in the infrared to Calabi-Yau sigma models — such as the conifold — and the topology changing flop transition is realized as crossing symmetry in Liouville/Toda CFT. Evidence for Seiberg duality in two dimensions is exhibited by demonstrating that the partition function of conjectured Seiberg dual pairs are the same.

Examples of emergent type IIB backgrounds from matrices

We study models of emergent space associated with the Coulomb branch, non-commutative and β deformations of the N=4 super-Yang–Mills theory, extending a previous work on the undeformed conformal case. The idea is to compute the effective action for D-instantons from the microscopic four-dimensional open-string description and to compare with the non-abelian D-instanton action in the dual ten-dimensional supergravity background. To linear order in the deformation parameter, the D-instantons can probe the full space–time geometry and we can derive all the supergravity fields in this way. We find a perfect match with the known supergravity solutions, including for the Neveu–Schwarz and Ramond–Ramond forms.

Rigorous test of non-conformal holography: Wilson loops in $ \mathcal{N} $ = 2∗ theory

N=2* gauge theory in four space-time dimensions arises as a deformation of the parent N=4 supersymmetric Yang-Mill theory under which its hypermultiplet acquires a mass. This theory, on the one hand, has a known supergravity dual and on the other hand is amenable to localization. We explain, using localization, a dynamical selection of the supergravity Coulomb branch vacuum at large N. We also demonstrate that large Wilson loops obey perimeter law, again finding exact match between direct field-theory calculations and string theory. Additionally, we compute free energy at large 't Hooft coupling.

$ {{\mathcal{W}}_3} $ irregular states and isolated $ \mathcal{N}=2 $ superconformal field theories

We explore the proposal that the six-dimensional (2, 0) theory on the Riemann surface with irregular punctures leads to a four-dimensional gauge theory coupled to the isolated $ \mathcal{N}=2 $ superconformal theories of Argyres-Douglas type, and to two-dimensional conformal field theory with irregular states. Following the approach of Gaiotto-Teschner for the Virasoro case, we construct $ {{\mathcal{W}}_3} $ irregular states by colliding a single SU(3) puncture with several regular punctures of simple type. If n simple punctures are colliding with the SU(3) puncture, the resulting irregular state is a simultaneous eigenvector of the positive modes L n , . . . , L 2n and W 2n , . . . , W 3n of the $ {{\mathcal{W}}_3} $ algebra. We find the corresponding isolated SCFT with an SU(3) flavor symmetry as a nontrivial IR fixed point on the Coulomb branch of the SU(3) linear quiver gauge theories, by confirming that its Seiberg-Witten curve correctly predicts the conditions for the $ {{\mathcal{W}}_3} $ irregular states. We also compare these SCFT’s with the ones obtained from the BPS quiver method.

NILPOTENT ORBITS AND CODIMENSION-2 DEFECTS OF 6d${\mathcal N} = (2, 0)$ THEORIES

We study the local properties of a class of codimension-2 defects of the 6d [Formula: see text] theories of type J = A, D, E labeled by nilpotent orbits of a Lie algebra [Formula: see text], where [Formula: see text] is determined by J and the outer-automorphism twist around the defect. This class is a natural generalization of the defects of the six-dimensional (6d) theory of type SU (N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the four-dimensional (4d) central charges a and c and to the flavor central charge k.

Anomaly cancellation and abelian gauge symmetries in F-theory

We study 4D F-theory compactifications on singular Calabi-Yau fourfolds with fluxes. The resulting $\mathcal{N}=1$ effective theories can admit non-Abelian and U(1) gauge groups as well as charged chiral matter. In these setups we analyze anomaly cancellation and the generalized Green-Schwarz mechanism. This requires the study of 3D $\mathcal{N}=2$ theories obtained by a circle compactification and their M-theory duals. Reducing M- theory on resolved Calabi-Yau fourfolds corresponds to considering the effective theory on the 3D Coulomb branch in which certain massive states are integrated out. Both 4D gaugings and 3D one-loop corrections of these massive states induce Chern-Simons terms. All 4D anomalies are captured by the one-loop terms. The ones corresponding to the mixed gauge-gravitational anomalies depend on the Kaluza-Klein vector and are induced by integrating out Kaluza-Klein modes of the U(1) charged matter. In M-theory all Chern-Simons terms classically arise from G 4-flux. We find that F-theory fluxes implement automatically the 4D Green-Schwarz mechanism if non-trivial geometric relations for the resolved Calabi-Yau fourfold are satisfied. We confirm these relations in various explicit examples and elucidate the general construction of U(1) symmetries in F-theory. We also compare anomaly cancellation in F-theory with its analog in Type IIB orientifold setups.

M5-BRANES, TORIC DIAGRAMS AND GAUGE THEORY DUALITY

We explore a duality between five-dimensional supersymmetric linear quiver gauge theories compactified on a circle, which are the five-dimensional uplifts of four-dimensional superconformal linear quiver theories. We find a correspondence between the gauge theory parameters of two dual theories, under which identical infrared effective coupling constants are obtained on the Coulomb branch. Two independent approaches using M-theory and the topological string theory give a consistent result.

Classification of complete $N = 2$ supersymmetric theories in 4 dimensions

We define the notion of a complete N = 2 supersymmetric theory in 4 dimensions as a UV complete theory for which all the BPS central charges can be arbitrarily varied as we vary its Coulomb branch parameters, masses, and coupling constants. We classify all such theories whose BPS spectrum can be obtained via a quiver diagram. This is done using the 4d/2d correspondence and by showing that such complete N = 2 theories map to quivers of finite mutation type. The list of such theories is given by the (generalized) Gaiotto theories consisting of two 5-branes wrapping Riemann surfaces with punctures, as well as 11 additional exceptional cases, which we identify.

From black holes to quivers

Abstract Middle cohomology states on the Higgs branch of supersymmetric quiver quantum mechanics — also known as pure Higgs states — have recently emerged as possible microscopic candidates for single-centered black hole micro-states, as they carry zero angular momentum and appear to be robust under wall-crossing. Using the connection between quiver quantum mechanics on the Coulomb branch and the quantum mechanics of multi-centered black holes, we propose a general algorithm for reconstructing the full moduli-dependent cohomology of the moduli space of an arbitrary quiver, in terms of the BPS invariants of the pure Higgs states. We analyze many examples of quivers with loops, including all cyclic Abelian quivers and several examples with two loops or non-Abelian gauge groups, and provide supporting evidence for this proposal. We also develop methods to count pure Higgs states directly.

Scaling BPS solutions and pure-Higgs states

Abstract Depending on the value of the coupling, BPS states of type II string theory compactified on a Calabi-Yau manifold can be described as multicenter supergravity solutions or as BPS states in a quiver gauge theory. While states that spread into the Coulomb-branch states can be mapped one-to-one to supergravity states, this is not automatically so for the majority of Higgs-branch states. In this paper we explicitly compute the BPS spectrum of the Higgs branch of a three-center quiver with a closed loop, and identify the subset of states that are in one-to-one correspondence with Coulomb/supergravity multicenter states. We also show that there exist additional “pure-Higgs” states, that exist if and only if the charges of the centers can form a scaling solution. Using generating function techniques we compute the large charge degeneracy of the “pure-Higgs” sector and show that it is always exponential. We also construct the map between Higgs- and Coulomb-branch states, discuss its relation to the Higgs-Coulomb map of one of the authors and Verlinde, and argue that the pure Higgs states live in the kernel of this map. Given that these states have no obvious description on the Coulomb branch or in supergravity, we discuss whether they can correspond to a single-center black hole or can be related to more complicated horizonless configurations.

On renormalization group flows and the a-theorem in 6d

We study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d=6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow a_UV - a_IR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p^6 in the low energy limit of n-point scattering amplitudes of the dilaton for n > 3. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p^6. The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS_7 x S^4. Given the confirmation in two distinct models, we attempt to use dispersion relations to prove that the anomaly flow is positive in general. Unfortunately the 4-point matrix element of the Euler anomaly is proportional to stu and vanishes for forward scattering. Thus the optical theorem cannot be applied to show positivity. Instead the anomaly flow is given by a dispersion sum rule in which the integrand does not have definite sign. It may be possible to base a proof of the a-theorem on the analyticity and unitarity properties of the 6-point function, but our preliminary study reveals some difficulties.

Darboux coordinates and instanton corrections in projective superspace

By demanding consistency of the Legendre transform construction of hyperk\"ahler metrics in projective superspace, we derive the expression for the Darboux coordinates on the hyperk\"ahler manifold. We apply these results to study the Coulomb branch moduli space of 4D, N=2 super-Yang-Mills theory (SYM) on R^3 x S^1, recovering the results by GMN. We also apply this method to study the electric corrections to the moduli space of 5D, N=1 SYM on R^3 x T^2 and give the Darboux coordinates explicitly.

Instanton Counting and Wall-Crossing for Orbifold Quivers

Noncommutative Donaldson–Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative \({{\mathcal N}=2}\) gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated with the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson–Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.

S-duality in twistor space

Abstract In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space $ {\mathcal{M}_H} $ must carry an isometric action of the modular group SL(2, $ \mathbb{Z} $ ), inherited from the S-duality symmetry of type IIB string theory in ten dimensions. We investigate how this modular symmetry is realized at the level of the twistor space of $ {\mathcal{M}_H} $ , and construct a general class of SL(2, $ \mathbb{Z} $ )-invariant quaternion-Kähler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions. This family should include $ {\mathcal{M}_H} $ corrected by D3-D1-D(-1)-instantons (with five-brane corrections ignored) and, after taking a suitable rigid limit, the Coulomb branch of five-dimensional $ \mathcal{N} = {2} $ gauge theories compactified on a torus, including monopole string instantons. These results allow us to considerably simplify the derivation of the mirror map between type IIA and IIB fields in the sector where only D1-D(-1)-instantons are retained.

Large N limit of β-ensembles and deformed Seiberg-Witten relations

We study the beta-ensemble that represents conformal blocks of Liouville theory on the sphere. This quantity is related through AGT conjecture to the Nekrasov instanton partition function of 4d $\mathcal{N}=2$ SU(2) gauge theory with four flavors. We focus on the large N limit, equivalent to the Nekrasov-Shatashvili limit where one of the Omega-background deformation parameters is vanishing. A quantized Seiberg-Witten differential form is defined perturbatively in h-bar as the singular part of the beta-ensemble resolvent. Using the Dyson collective field action, we show that the free energy obeys the Seiberg-Witten relations. As suggested by Mironov and Morozov, the quantized differential form can be obtained from the classical one by the action of a differential operator in the hypermultiplet masses and the Coulomb branch modulus.

The conformal anomaly of M5-branes

We show that the conformal anomaly for N M5-branes grows like $N^3$. The method we employ relates Coulomb branch interactions in six dimensions to interactions in four dimensions using supersymmetry. This leads to a relation between the six-dimensional conformal anomaly and the conformal anomaly of N=4 Yang-Mills. Along the way, we determine the structure of the four derivative interactions for the toroidally compactified (2,0) theory, while encountering interesting novelties in the structure of the six derivative interactions.

Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds

The 6d effective action of F-theory compactified on an elliptically fibred CalabiYau threefold is determined using an M-theory lift. The non-Abelian gauge theory arises on a stack of seven-branes located on the singularities of the threefold. The gauge theory is analysed in the 5d Coulomb branch by resolving the Calabi-Yau threefold. The reduction includes an arbitrary number of 6d tensor multiplets for which the self- and anti-self duality constraints can be imposed in the 5d M-theory set-up. Higher curvature terms are crucial for anomaly cancellation in six dimensions, and are obtained from corrections to the 11d supergravity action. The chiral spectrum in a 6d supergravity theory is linked to the form of the Green-Schwarz couplings via anomalies. We find that these data are encoded in the 5d effective action via one loop corrections.