5d Gauge Theories Research Articles

Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on T 2 × T 4, via their partition functions

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T 4 with flat metric using Dirac quantization. In addition to an $$ \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry, it possesses $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry that is electromagnetic S-duality. We show explicitly how this $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2, 0) tensor theory, by computing the partition function of a single fivebrane compactified on T 2 times T 4, which has $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right)\times \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry. If we identify the couplings of the abelian gauge theory $$ \tau =\frac{\theta }{2\pi }+i\frac{4\pi }{e^2} $$ with the complex modulus of the T 2 torus $$ \tau ={\beta}^2+i\frac{R_1}{R_2} $$ , then in the small T 2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.

Topological invariants for gauge theories and symmetry-protected topological phases

We study the braiding statistics of particle-like and loop-like excitations in 2D and 3D gauge theories with finite, Abelian gauge group. The gauge theories that we consider are obtained by gauging the symmetry of gapped, short-range entangled, lattice boson models. We define a set of quantities --- called {\it topological invariants} --- that summarize some of the most important parts of the braiding statistics data for these systems. Conveniently, these invariants are always Abelian phases, even if the gauge theory supports excitations with non-Abelian statistics. We compute these invariants for gauge theories obtained from the exactly soluble group cohomology models of Chen, Gu, Liu and Wen, and we derive two results. First, we find that the invariants take different values for every group cohomology model with finite, Abelian symmetry group. Second, we find that these models exhaust all possible values for the invariants in the 2D case, and we give some evidence for this in the 3D case. The first result implies that every one of these models belongs to a distinct SPT phase, while the second result suggests that these models may realize all SPT phases. These results support the group cohomology classification conjecture for SPT phases in the case where the symmetry group is finite, Abelian, and unitary.

Lifting 4d dualities to 5d

In this paper we set out to further explore the connection between isolated $$ \mathcal{N}=2 $$ SCFT’s in four dimensions and $$ \mathcal{N}=1 $$ SCFT’s in five dimensions. Using 5-brane webs we are able to provide IR Lagrangian descriptions in terms of 5d gauge theories for several classes of theories including the so-called T N theories. In many of these we find multiple dual gauge theory descriptions. The connection to 4d theories is then used to lift 4d $$ \mathcal{N}=2 $$ S-dualities that involve weakly-gauging isolated theories to 5d gauge theory dualities. The 5d description allows one to study the spectrum of BPS operators directly, using for example the superconformal index. This provides additional non-trivial checks of enhanced global symmetries and 4d dualities.

Non-Abelian string and particle braiding in topological order: ModularSL(3,Z)representation and(3+1)-dimensional twisted gauge theory

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $\omega_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $\omega_{3(b)}$, dimensionally reduced from the 4-cocycle $\omega_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

Realization of the lepton flavor structure from point interactions

We investigate a 5d gauge theory on S1 with point interactions. The point interactions describe extra boundary conditions and provide three generations, the charged lepton mass hierarchy, the lepton flavor mixing and tiny degenerated neutrino masses after choosing suitable boundary conditions and parameters. The existence of the restriction in the flavor mixing, which appears from the configuration of the extra dimension, is one of the features of this model. Tiny Yukawa couplings for the neutrinos also appears without the see-saw mechanism nor symmetries in our model. The magnitude of CP violation in the leptons can be a prediction and is consistent with the current experimental data.

Realization of the lepton flavor structure from point interactions

We investigate a 5d gauge theory on $S^1$ with point interactions. The point interactions describe extra boundary conditions and provide three generations, the charged lepton mass hierarchy, the lepton flavor mixing and tiny degenerated neutrino masses after choosing suitable boundary conditions and parameters. The existence of the restriction in the flavor mixing, which appears from the configuration of the extra dimension, is one of the features of this model. Tiny Yukawa couplings for the neutrinos also appears without the see-saw mechanism nor symmetries in our model. The magnitude of CP violation in the leptons can be a prediction and is consistent with the current experimental data.

Duality and enhancement of symmetry in 5d gauge theories

We study various cases of dualities between $\cal{N}$$=1$ 5d supersymmetric gauge theories. We motivate the dualities using brane webs, and provide evidence for them by comparing the superconformal index. In many cases we find that the classical global symmetry is enhanced by instantons to a larger group including one where the enhancement is to the exceptional group $G_2$.

3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators

3d N=2 partition functions on the squashed three-sphere and on the twisted product S2xS1 have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes squashed three-sphere and S2xS1 partition functions but the two cases involve different inner products, the S-pairing and the id-pairing respectively. We define a class of q-deformed CFT correlators where conformal blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing respectively. Applying the bootstrap approach to a class of degenerate correlators we are able to derive three-point functions. We show that degenerate correlators can be mapped to 3d partition functions while the crossing symmetry of CFT correlators corresponds to the flop symmetry of 3d gauge theories. We explore how non-degenerate q-deformed correlators are related to 5d partition functions. We argue that id-pairing correlators are associated to the superconformal index on S4xS1 while S-pairing three-point function factors capture the one-loop part of S5 partition functions. This is consistent with the interpretation of S2xS1 and squashed three-sphere gauge theories as codimension two defect theories inside S4xS1 and S5 respectively.

Features of a 2d gauge theory with vanishing chiral condensate

The Schwinger model with Nf ≥ 2 flavors is a simple example for a fermionic model with zero chiral condensate Σ (in the chiral limit). We consider numerical data for two light flavors, based on simulations with dynamical chiral lattice fermions. We test properties and predictions that were put forward in the recent literature for models with Σ = 0, which include IR conformal theories. In particular, we probe the decorrelation of low lying Dirac eigenvalues, and we discuss the mass anomalous dimension and its IR extrapolation. Here, we encounter subtleties, which may urge caution with analogous efforts in other models, such as multi-flavor QCD.

2d gauge theories and generalized geometry

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M \equiv TM \oplus T^*M$ by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure $D \subset \mathbb{T}M$ (or, more generally, the choide of a "small Dirac-Rinehart sheaf" $\cal{D}$), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: A gauging of $\mathfrak{g}$ of a standard sigma model with Wess-Zumino term exists, \emph{iff} there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid $M \times \mathfrak{g}\to M$ into $D\to M$ (or the algebraic analogue of the morphism in the case of $\cal{D}$). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.

Intersecting branes, domain walls and superpotentials in 3d gauge theories

We revisit the Hanany-Witten brane construction of 3d gauge theories with N=2 supersymmetry. Instantons are known to generate a superpotential on the Coulomb branch of the theory. We show that this superpotential can be viewed as arising from the classical scattering of domain wall solitons. The domain walls live on the worldvolume of the fivebranes and their existence relies on the recent observation that the charged hypermultiplet at the intersection of perpendicular D-branes has non-canonical kinetic terms. We further show how Dp branes may be absorbed at the intersection of perpendicular D(p+4)-branes where they appear as BPS sigma-model lumps.

Dynamics of 3D SUSY gauge theories with antisymmetric matter

We investigate the IR dynamics of N=2 SUSY gauge theories in 3D with antisymmetric matter. The presence of the antisymmetric fields leads to further splitting of the Coulomb branch. Counting zero modes in the instanton background suggests that more than a single direction along the Coulomb branch may remain unlifted. We examine the case of SU(4) with one or two antisymmetric fields and various flavors in detail. Using the results for the corresponding 4D theories, we find the IR dynamics of the 3D cases via compactification and a real mass deformation. We find that for the s-confining case with two antisymmetric fields, a second unlifted Coulomb branch direction indeed appears in the low-energy dynamics. We present several non-trivial consistency checks to establish the validity of these results. We also comment on the expected structure of general s-confining theories in 3D, which might involve several unlifted Coulomb branch directions.

Global Properties of Supersymmetric Theories and the Lens Space

We compute the supersymmetric partition function on $${L(r,1)\times \mathbb{S}^1}$$ , the lens space index, for 4d gauge theories related by supersymmetric dualities and involving non simply-connected groups. This computation is sensitive to the global properties of the underlying gauge group and to discrete theta angle parameters and thus distinguishes versions of dualities differing by such. We explicitly discuss $${\mathcal{N}=1}$$ so(N c ) Seiberg dualities and $${\mathcal{N}=4\,su(N_c)}$$ S-dualities.

Topological strings and 5d T N partition functions

We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau threefolds. The theories include certain non-Lagrangian theories such as the T_N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T_3 theory as well as Sp(1) gauge theories with N_f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T_N theory. We compute the partition function of the E_7 theory via this prescription, and find the E_7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams.

Gauge/gravity duality and RG flows in 5d gauge theories

We discuss RG flows in 5d gauge theories triggered by VEVs of either mesonic or baryonic operators. As a warm-up, we explicitly discuss the counterpart of these flows in 4d gauge theories with N=2 supersymmetry by focusing on the A1 theory. As opposed to the N=1 case, in cases with 8 supercharges we need to solve a more involved PDE. In the 5d case the boundary conditions for such equation play a crucial role in order to reproduce the expected spectrum.

Walls, lines, and spectral dualities in 3d gauge theories

In this paper we analyze various half-BPS defects in a general three dimensional $ \mathcal{N} $ = 2 supersymmetric gauge theory $ \mathcal{T} $ . They correspond to closed paths in SUSY parameter space and their tension is computed by evaluating period integrals along these paths. In addition to such defects, we also study wall defects that interpolate between $ \mathcal{T} $ and its SL(2, $ \mathbb{Z} $ ) transform by coupling the 3d theory to a 4d theory with S-duality wall. We propose a novel spectral duality between 3d gauge theories and integrable systems. This duality complements a similar duality discovered by Nekrasov and Shatashvili. As another application, for 3d $ \mathcal{N} $ = 2 theories associated with knots and 3-manifolds we compute periods of (super) A-polynomial curves and relate the results with the spectrum of domain walls and line operators.

On the 6d origin of discrete additional data of 4d gauge theories

Starting with a choice of gauge algebras, specification of a 4d gauge theory involves additional data, namely the gauge groups and the discrete theta angles. Equivalently, one needs to specify the set of charges of allowed line operators. In this note, we study how these additional data are represented in 6d, when the 4d theory in question is an N=4 super Yang-Mills theory or an N=2 class S theory. We will see that the Z_N symmetry of the so-called T_N theory plays an important role. As a byproduct, we will find that the superconformal index of class S theories can be refined so that it can give 2d q-deformed Yang-Mills theory with different gauge groups associated to the same gauge algebra.

Vacuum structure in 3D supersymmetric gauge theories

Based on a talk given at the Pomeranchuk memorial conference at ITEP in June 2013, we review the vacuum dynamics in 3d supersymmetric Yang–Mills–Chern–Simons theories with and without extra matter multiplets. By analyzing the effective Born–Oppenheimer Hamiltonian in a small spatial box, we calculate the number of vacuum states (Witten index) and examine their structure for these theories. The results are identical to those obtained by other methods.

Notes on Mayer expansions and matrix models

Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N=2 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them with similar relations on the grand canonical side. It leads to relate the multi-point densities, fundamental objects of the matrix model, to the generating functions of multi-rooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model.

On novel string theories from 4d gauge theories

We investigate strings theories as defined from four dimensional gauge theories. It is argued that novel (super)string theories exist up to 26 dimensions. Some of them may support weakly curved geometries.