Tests of Seiberg-like dualities in three dimensions

We provide non-trivial checks of $ \mathcal{N} $ = 4, D = 3 mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as (A m−1 , D n ), where m and n are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n ≥ 4, arbitrary ranks of the gauge groups and varying m, we test the conjectured duality by proving the precise equality of the S 3 partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur’s Pfaffian identity, which are discussed in the paper.

We investigate features of duality in three dimensional N=2 Chern-Simons matter theories conjectured to describe M2 branes at toric Calabi Yau four-fold singularities. For 3D theories with non-chiral 4D parents we propone a Seiberg-like duality which turns out to be a toric duality. For theories with chiral 4D parents we discuss the conditions under which that Seiberg-like duality leads to toric duality. We comment on such duality in 3D theories without 4D parents.

We explore 1d vortex dynamics of 3d supersymmetric Yang-Mills theories, as inferred from factorization of exact partition functions. Under Seiberg-like dualities, the 3d partition function must remain invariant, yet it is not a priori clear what should happen to the vortex dynamics. We observe that the 1d quivers for the vortices remain the same, and the net effect of the 3d duality map manifests as 1d Wall-Crossing phenomenon; although the vortex number can shift along such duality maps, the ranks of the 1d quiver theory are unaffected, leading to a notion of fundamental vortices as basic building blocks for topological sectors. For Aharony-type duality, in particular, where one must supply extra chiral fields to couple with monopole operators on the dual side, 1d wall-crossings of an infinite number of vortex quiver theories are neatly and collectively encoded by 3d determinant of such extra chiral fields. As such, 1d wall-crossing of the vortex theory encodes the particle-vortex duality embedded in the 3d Seiberg-like duality. For mathcal{N} = 4, the D-brane picture is used to motivate this 3d/1d connection, while, for mathcal{N} = 2, this 3d/1d connection is used to fine-tune otherwise ambiguous vortex dynamics. We also prove some identities of 3d supersymmetric partition functions for the Aharony duality using this vortex wall-crossing interpretation.

We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to\infty$, with $\lambda = N/k$ and $\frac{T^2 V_{2}}{N}$ held fixed where $T$ is the temperature and $V_{2}$ the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of $\lambda$. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of ${2\pi \over k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{2 \pi \lambda}$. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of $\lambda$. At low temperatures our partition function smoothly matches onto the $N$ and $\lambda$ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large $N$ Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.

This thesis presents a number of results on partition functions for four-dimensional supersymmetric black holes. These partition functions are important tools to explain the entropy of black holes from a microscopic point of view. Such a microscopic explanation was desired after the association of a macroscopic entropy to black holes in the 70's, based on the analogies between black hole physics and thermodynamics. The correct microscopic account of black hole entropy was achieved in string theory and M-theory during the 90's, and a crucial role is played by D-branes and M-branes. The black holes, which are studied in this thesis, are supersymmetric solutions of four-dimensional $\mathcal{N}=2$ supergravity, which carry both electric and magnetic charges. An important feature of the global geometry is the the near-horizon geometry, which is AdS$_2\times S^2$ and where the K\ahler moduli are fixed at their attractor values. The horizon area of this class of black holes is given by $S_\mathrm{BH}=\pi|Z|^2=\pi \sqrt{\frac{2}{3}p^3 (q_{\bar 0}+\frac{1}{2}q^2)}$, where $p^a$ and $(q_{\bar 0},q_a)$ are respectively the electric and magnetic charges, and $Z$ is the central charge. The combination $\hat q_{\bar 0}=q_{\bar 0}+\frac{1}{2}q^2$ is required to be positive for black holes. The first motivation to introduce a black hole partition function $\mathcal{Z}_\mathrm{BH}$, is to explain $S_\mathrm{BH}$ microscopically. An analysis of the attractor equations suggest that $\mathcal{Z}_\mathrm{BH}$ is naturally expanded in $q_{\bar 0}$ and $q_a$, while the $p^a$ are kept fixed. In other more physical words, the electric charges are in a macrocanonical ensemble and the magnetic charges in a microcanonical ensemble. If higher derivative contributions are included in the supergravity action, the entropy receives corrections. The form of these corrections suggests that $\mathcal{Z}_\mathrm{BH}$ is well approximated by the square of the topological string partition function $|\mathcal{Z}_\mathrm{top}|^2$. Topological string theory is a simplified version of string theory, which allows the computation of many quantities using elaborate mathematical techniques, for example $\mathcal{Z}_\mathrm{top}$ basically enumerates the holomorphic maps of a Riemann surface into a Calabi-Yau threefold. The conjecture that $\mathcal{Z}_\mathrm{BH}=|\mathcal{Z}_\mathrm{top}|^2$, is the second motivation for this thesis. The third motivation is the correspondence between a theory including gravity in Anti-de Sitter (AdS) space and a conformal field theory (CFT) on the boundary of the AdS-space. Part of the near-horizon geometry of the black holes in the eleven dimensions is AdS$_3$, whose boundary is a two-dimensional torus. The correspondence suggests that the CFT$_2$ partition function equals the one of the theory in the bulk of AdS$_3$. Therefore, the CFT$_2$ partition function should admit an expansion which is natural for an AdS$_3$-(super)gravity partition function. Dijkgraaf {\it et al.} proposed in 2000 that an SCFT partition function can be rewritten as a Poincar\'e series, which is a sum over the coset $\Gamma_\infty\backslash\Gamma$. Every element in the coset corresponds to a semi-classical saddle point geometry, providing therefore evidence for the AdS/CFT correspondence. Chapter \ref{chap:bheinstein} explains these notions rather heuristically in a bosonic setting, subsequent chapters are more precise. Chapter \ref{chap:bhmtheory} explains how the black holes arise as a solution of 11-dimensional M-theory, and how this can account for the entropy microscopically. Four-dimensional supergravity appears in this context as the reduction of M-theory on a six-dimensional Calabi-Yau $X$ times a circle $S^1_\mathrm{M}$. The heavy objects, which source the black holes, are M5-branes. These correspond to the magnetic charges whereas the electric charges are generated by fluxes on the worldvolume of the M5-brane and momentum around $S^1_\mathrm{M}$. The six-dimensional M5-branes wrap a four-dimensional divisor $P$ of $X$ together with $S^1_\mathrm{M}$ and the Euclidean time circle $S^1_\mathrm{t}$. The parameters of the theory can be chosen such that the low energy approximation to M-theory is valid. The M5-brane low energy degrees of freedom can be reduced to the $T^2$, which is formed by the two circles. There, the degrees of freedom combine to an $\CN=(4,0)$ superconformal field theory. The central charges of the holomorphic and anti-holomorphic sector can be determined using index formulas. The relevant partition function for this SCFT is a (modified) elliptic genus. The symmetries of the theory determine that the elliptic genus transforms covariantly under modular transformations, which makes it possible to derive the Cardy formula for the entropy. This gives the correct leading behavior of the entropy. For a specific identification of the parameters, the CFT partition function is equal to the (divergent) black hole partition function. An important property of the elliptic genera is the decomposition into theta functions and a vector-valued modular form. The principal part of the Laurent expansion of the vector-valued modular form gives rise to the definition of the ``polar spectrum'' of the SCFT. Chapter \ref{chap:vectforms} is devoted to an analysis of meromorphic vector-valued modular forms. It is shown how the Fourier coefficients can be expressed as an infinite sum over the coset $\Gamma_\infty\backslash \Gamma$. If the polar degeneracies are known, the non-polar degeneracies can be determined with an arbitrary accuracy, improving on the leading order estimate by the Cardy formula. In addition is shown how the vector-valued modular form can be written as a sum over $\Gamma_\infty\backslash \Gamma$. The sum is a regularized Poincar\'e series, and an improvement of the proposal by Dijkgraaf {\it et al.}. The regularization leads in general to an anomaly, which is canceled if the polar degeneracies satisfy a number of constraints. This number can be determined using the Selberg trace formula. The dimension of the space of vector-valued modular forms is simply given by the number of polar terms minus the number of constraints. With the results of Chapter \ref{chap:vectforms}, Chapter \ref{chap:interpretation} revisits the motivations of Chapter \ref{chap:bheinstein}. The regularized Poincar\'e series confirms the AdS$_3$/CFT$_2$ correspondence, since the sum over $\Gamma_\infty\backslash \Gamma$ is suggestive of a semi-classical sum over AdS$_3$-geometries. If the complex structure $\tau$ is varied, the most contributing geometry to $\mathcal{Z}_{\mathrm{BH}}$ might jump, which is a nice manifestation of Hawking-Page phase transitions. The Poincar\'e series are essentially a sum over $\Gamma_\infty\backslash \Gamma$ of the polar spectrum, which lies classically below the cosmic censorship bound. Therefore, one can view the series heuristically as a sum over all geometries (including black holes) of the states which do not collapse into a black hole. This is also how the connection between black holes and topological strings can be understood. The degeneracies of charged BPS-particles (M2-branes) in the near-horizon geometry are enumerated by $|\mathcal{Z}_\mathrm{top}|^2$. Therefore, $|\mathcal{Z}_\mathrm{top}|^2$ appears for every saddle point geometry in $\mathcal{Z}_\mathrm{BH}$. The conjecture is now elucidated for strong topological string coupling constant ($\hat q_{\bar 0}\gg p^3$), since then a single AdS$_3$-geometry dominates the partition function. Also the case of weak topological string couple is discussed. It is shown that for this part of the spectrum, the elliptic genus confirms the leading behavior of two-centered solutions. The approximation $\mathcal{Z}_\mathrm{BH}\sim |\mathcal{Z}_\mathrm{top}|^2$ is however no longer valid. The constraints on the polar degeneracies by modularity are strongest if the number of polar terms is small. This is generically not the case for the black hole and AdS$_3$ applications, which are discussed before. The vector-valued modular forms appear however at many places in theoretical physics, for example rational conformal field theory and $\CN=4$ supersymmetric gauge theory on a four-manifolds $M$. The partition functions of such gauge theories with gauge group $U(N)$ are closely related to M5-brane elliptic genera. If certain conditions are satisfied, the partition function of this theory is the generating function of the Euler characteristic of instanton moduli spaces. Section \ref{sec:n4ym} performs an analysis of the partition functions of the $U(N)$ theories on $\mathbb{CP}^2$. This confirms the older results in the literature for $N=1$ and $2$. A new generating function is proposed for the Euler numbers of $SU(3)$ moduli spaces.

The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold XN,M (for given (N, M)) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold XN,M is dual to XN ′,M ′ if NM = N′M′ and gcd(N, M) = gcd(N′, M′). Combining this result with the triality structure, we currently argue for a large number of dual quiver gauge theories, whose instanton partition functions can be computed explicitly as specific expansions of the topological partition function {mathcal{Z}}_{N,M} of XN,M. We illustrate this web of dual theories by studying explicit examples in detail. We also undertake first steps in further analysing the extended moduli space of XN,M with the goal of finding other dual gauge theories.

We study a method to compute a topological phase factor of partition function for pure Chern-Simons theory incorporating the supersymmetric localization. We develop a regularization preserving supersymmetry and the topological phase appears as a result of the supersymmetric regularization. Applying this method to pure Chern-Simons theory on lens space we compute the background dependent phase factor coming from the Chern-Simons term. We confirm that the partition function computed in this method enjoys a couple of level rank dualities including the one recently proposed in arXiv:1607.07457 for all ranks and levels within our numerical calculation. We also present a phase factor with which the lens space partition function exhibits the perfect match between any level rank dual pair including the total phase.

We investigate backgrounds of Type IIB string theory with null singularities and their duals proposed in S. R. Das, J. Michelson, K. Narayan, S. P. Trivedi, hep-th/0602107. The dual theory is a deformed $\mathcal{N}=4$ Yang-Mills theory in $3+1$ dimensions with couplings dependent on a lightlike direction. We concentrate on backgrounds which become ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ 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 correlation functions are nonsingular at all times. The two-point correlator for some operators calculated in the gauge theory does not agree with the result from the bulk supergravity solution. However, the bulk calculation is invalid near the singularity where corrections to the supergravity approximation become important. We also obtain pp-waves which are suitable Penrose limits of this general class of solutions, and construct the matrix membrane theory which describes these pp-wave backgrounds.

We study type IIB brane configurations engineering 3d flavoured ABJ(M) theories with Yang-Mills kinetic terms, which flow to IR fixed points describing M2 branes at a class of toric Calabi-Yau fourfold singularities. The type IIB construction provides a bridge between M-theory geometry and field theory, and allows to identify the superconformal field theories with fixed quiver diagram, Chern-Simons levels and superpotential, differing by the ranks of the gauge groups, which we associate to dual AdS_4\timesY_7 backgrounds of M-theory without or with torsion G-fluxes sourced by fractional M2 branes in Y_7, when Y_7 is smooth. The analysis includes the Q^{1,1,1} and Y^{1,2}(CP^2) geometries. We also comment on duality cascades and on the interplay between torsion G-fluxes in M-theory and partial resolutions.

We consider supersymmetric gauge theories on S5 with a negative Yang-Mills coupling in their large N limits. Using localization we compute the partition functions and show that the pure SU(N) gauge theory descends to an SU(N/2)+N/2× SU(N/2)−N/2× SU(2) Chern-Simons gauge theory as the inverse ’t Hooft coupling is taken to negative infinity for N even. The Yang-Mills coupling of the SU(N/2)±N/2 is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd N case has somewhat different behavior. We then study the SU(N/2)N/2 pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider USp(2N) theories with an antisymmetric hypermultiplet and Nf< 8 fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the SU(N) theories in the large N limit.

In this thesis we apply the technique of localization to three-dimensional N=2 superconformal field theories. We consider both theories which are exactly superconformal, and those which are believed to flow to nontrivial superconformal fixed points, for which we consider implicitly these fixed points. We find that in such theories, the partition function and certain supersymmetric observables, such as Wilson loops, can be computed exactly by a matrix model. This matrix model consists of an integral over g, the Lie algebra of the gauge group of the theory, of a certain product of 1-loop factors and classical contributions. One can also consider a space of supersymmetric deformations of the partition function corresponding to the set of abelian global symmetries. In the second part of the thesis we apply these results to test dualities. We start with the case of ABJM theory, which is dual to M-theory on an asymptotically AdS_4xS^7 background. We extract strong coupling results in the field theory, which can be compared to semiclassical, weak coupling results in the gravity theory, and a nontrivial agreement is found. We also consider several classes of dualities between two three-dimensional field theories, namely, 3D mirror symmetry, Aharony duality, and Giveon-Kutasov duality. Here the dualities are typically between the IR limits of two Yang-Mills theories, which are strongly coupled in three dimensions since Yang-Mills theory is asymptotically free here. Thus the comparison is again very nontrivial, and relies on the exactness of the localization computation. We also compare the deformed partition functions, which tests the mapping of global symmetries of the dual theories. Finally, we discuss some recent progress in the understanding of general three-dimensional theories in the form of the F-theorem, a conjectured analogy to the a-theorem in four dimensions and c-theorem in two dimensions, which is closely related to the localization computation.

Following [1] we study a QCD-like gauge theory using a non-supersymmetric setup in type IIB string theory. The setup includes an O3 plane and N D3 anti-branes and it realises a USp(2N) ‘electric’ gauge theory with four “quarks” in the two-index antisymmetric representation and six heavy scalars in the adjoint representation. Using S-duality we obtain a dual ‘magnetic’ theory that includes SO(2N−1) gauge theory with six scalars in the adjoint representation and four heavy “quarks” in the two-index symmetric representation. The dual theory provides a description of confinement and dynamical symmetry breaking of the form SU(4)→SO(4). We extend the results of [1] by adding masses to the quarks in the electric side and deriving parts of the chiral Lagrangian using the dual magnetic theory. In particular, we derive the Gell-Mann-Oakes-Renner (GMOR) relation for Nambu-Goldstone bosons (pions) using the magnetic theory.

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on S3 and the reduced quantum field theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding a topological theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function. Published by the American Physical Society 2024

Many gauge theories in three dimensions flow to interacting conformal field theories in the infrared. We define a new class of local operators in these conformal field theories that are not polynomial in the fundamental fields and create topological disorder. They can be regarded as higher-dimensional analogs of twist and winding-state operators in free 2-D CFTs. We call them monopole operators for reasons explained in the text. The importance of monopole operators is that in the Higgs phase, they create Abrikosov-Nielsen-Olesen vortices. We study properties of these operators in three-dimensional gauge theories using large N_f expansion. For non-supersymmetric gauge theories we show that monopole operators belong to representations of the conformal group whose primaries have dimension of order N_f. We demonstrate that these monopole operators transform non-trivially under the flavor symmetry group. We also consider topology-changing operators in the infrared limits of N=2 and N=4 supersymmetric QED as well as N=4 SU(2) gauge theory in three dimensions. Using large N_f expansion and operator-state isomorphism of the resulting superconformal field theories, we construct monopole operators that are primaries of short representation of the superconformal algebra and compute their charges under the global symmetries. Predictions of three-dimensional mirror symmetry for the quantum numbers of these monopole operators are verified. Furthermore, we argue that some of our large-N_f results are exact. This implies, in particular, that certain monopole operators in N=4 3-D SQED with N_f=1 are free fields. This amounts to a proof of 3-D mirror symmetry in these special cases.

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.