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A bstract The mass spectrum of 1 + 1dimensional SU( N ) gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large N limit using LightCone Quantization. Here we extend this approach to theories with small values of N , exhibiting explicit results for N = 2 , 3, and 4. In the context of Discretized LightCone Quantization, we develop a procedure based on the CayleyHamilton theorem for determining which states of the large N theory become null at finite N . For the lowlying bound states, we find that the squared masses divided by g 2 N , where g is the gauge coupling, have very weak dependence on N . The coefficients of the 1 /N 2 corrections to their large N values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a KacMoody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to g 2 N/π , we find evidence that the spectrum exhibits bosonfermion degeneracies, in agreement with the supersymmetry of the model at any value of N .more » « lessFree, publiclyaccessible full text available April 1, 2024

A bstract Threedimensional $$ \mathcal{N} $$ N = 4 superconformal field theories contain 1d topological sectors consisting of twisted linear combinations of halfBPS local operators that can be inserted anywhere along a line. After a conformal mapping to a round threesphere, the 1d sectors are now defined on a great circle of S 3 . We show that the 1d topological sectors are preserved under the squashing of the sphere. For gauge theories with matter hypermultiplets, we use supersymmetric localization to derive an explicit description of the topological sector associated with the Higgs branch. Furthermore, we find that the dependence of the 1d correlation functions on the squashing parameter b can be removed after appropriate rescalings. One can introduce real mass and FayetIliopolous parameters that, after appropriate rescalings, modify the 1d theory on the squashed sphere precisely as they do on the round sphere. In addition, we also show that when a generic 3d $$ \mathcal{N} $$ N = 4 theory is deformed by real mass parameters, this deformation translates into a universal deformation of the corresponding 1d theory.more » « less

A bstract We study the massdeformed sphere free energy of threedimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals. Building on previous observations, we conjecture a proportionality relation between the sphere free energy on the boundary and the prepotential of the fourdimensional $$ \mathcal{N} $$ N = 2 supergravity theory in the bulk. We verify this formula by explicit computation in several examples of supergravity theories with vector multiplets and hypermultiplets.more » « less

A bstract We study which bulk couplings contribute to the S 3 free energy F ( $$ \mathfrak{m} $$ m ) of threedimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals, potentially deformed by boundary realmass parameters m. In particular, we show that F ( $$ \mathfrak{m} $$ m ) is independent of a large class of bulk couplings that include nonchiral Fterms and all Dterms. On the other hand, in general, F ( $$ \mathfrak{m} $$ m ) does depend nontrivially on bulk chiral Fterms, such as prepotential interactions, and on bulk realmass terms. These conclusions can be reached solely from properties of the AdS superalgebra, $$ \mathfrak{osp} $$ osp (24). We also consider massive vector multiplets in AdS, which in the dual field theory correspond to long singletrace superconformal multiplets of spin zero. We provide evidence that F ( $$ \mathfrak{m} $$ m ) is insensitive to the vector multiplet mass and to the interaction couplings between the massive vector multiplet and massless ones. In particular, this implies that F ( $$ \mathfrak{m} $$ m ) does not contain information about scaling dimensions or OPE coefficients of singletrace long scalar $$ \mathcal{N} $$ N = 2 superconformal multiplets.more » « less

null (Ed.)A bstract Twodimensional SU( N ) gauge theory coupled to a Majorana fermion in the adjoint representation is a nice toy model for higherdimensional gauge dynamics. It possesses a multitude of “gluinoball” bound states whose spectrum has been studied using numerical diagonalizations of the lightcone Hamiltonian. We extend this model by coupling it to N f flavors of fundamental Dirac fermions (quarks). The extended model also contains mesonlike bound states, both bosonic and fermionic, which in the large N limit decouple from the gluinoballs. We study the large N meson spectrum using the Discretized LightCone Quantization (DLCQ). When all the fermions are massless, we exhibit an exact $$ \mathfrak{osp} $$ osp (14) symmetry algebra that leads to an infinite number of degeneracies in the DLCQ approach. More generally, we show that many singletrace states in the theory are threshold bound states that are degenerate with multitrace states. These exact degeneracies can be explained using the KacMoody algebra of the SU( N ) current. We also present strong numerical evidence that additional threshold states appear in the continuum limit. Finally, we make the quarks massive while keeping the adjoint fermion massless. In this case too, we observe some exact degeneracies that show that the spectrum of mesons becomes continuous above a certain threshold. This demonstrates quantitatively that the fundamental string tension vanishes in the massless adjoint QCD 2 without explicit fourfermion operators.more » « less

A bstract We study the space of 3d $$ \mathcal{N} $$ N = 6 SCFTs by combining numerical bootstrap techniques with exact results derived using supersymmetric localization. First we derive the superconformal block decomposition of the fourpoint function of the stress tensor multiplet superconformal primary. We then use supersymmetric localization results for the $$ \mathcal{N} $$ N = 6 U( N ) k × U( N + M ) −k ChernSimonsmatter theories to determine two protected OPE coefficients for many values of N, M, k . These two exact inputs are combined with the numerical bootstrap to compute precise rigorous islands for a wide range of N, k at M = 0, so that we can nonperturbatively interpolate between SCFTs with Mtheory duals at small k and string theory duals at large k . We also present evidence that the localization results for the U(1) 2 M × U (1 + M ) − 2 M theory, which has a vectorlike large M limit dual to higher spin theory, saturates the bootstrap bounds for certain protected CFT data. The extremal functional allows us to then conjecturally reconstruct lowlying CFT data for this theory.more » « less

A bstract When the SU( N ) $$ \mathcal{N} $$ N = 4 superYangMills (SYM) theory with complexified gauge coupling τ is placed on a round foursphere and deformed by an $$ \mathcal{N} $$ N = 2preserving mass parameter m , its free energy F ( m, τ, $$ \overline{\tau} $$ τ ¯ ) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative $$ {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left{{}_m}_{=0}\right. $$ ∂ m 4 F m τ τ ¯ m = 0 of the sphere free energy and the integrated stresstensor multiplet fourpoint function in the $$ \mathcal{N} $$ N = 4 SYM theory. We then apply this exact relation, along with various other constraints derived in previous work (coming from analytic bootstrap, the mixed derivative $$ {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left{{}_m}_{=0}\right. $$ ∂ τ ∂ τ ¯ ∂ m 2 F m τ τ ¯ m = 0 , and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large N and large ’t Hooft coupling λ expansion of the $$ \mathcal{N} $$ N = 4 SYM correlator at separated points. In particular, we determine the leading large λ term in the $$ \mathcal{N} $$ N = 4 SYM correlation function at order 1 /N 8 . This is three orders beyond the planar limit.more » « less

A bstract We study modular invariants arising in the fourpoint functions of the stress tensor multiplet operators of the $$ \mathcal{N} $$ N = 4 SU( N ) superYangMills theory, in the limit where N is taken to be large while the complexified YangMills coupling τ is held fixed. The specific fourpoint functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the $$ \mathcal{N} $$ N = 2 ∗ theory with respect to the squashing parameter b and mass parameter m , evaluated at the values b = 1 and m = 0 that correspond to the $$ \mathcal{N} $$ N = 4 theory on a round sphere. At each order in the 1 /N expansion, these fourth derivatives are modular invariant functions of ( τ, $$ \overline{\tau} $$ τ ¯ ). We present evidence that at halfinteger orders in 1 /N , these modular invariants are linear combinations of nonholomorphic Eisenstein series, while at integer orders in 1 /N , they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the lowenergy expansion of the fourgraviton amplitude in type IIB superstring theory in tendimensional flat space and have interesting implications for the structure of the analogous expansion in AdS 5 × S 5 .more » « less

A bstract We study the fourpoint function of the lowestlying halfBPS operators in the $$ \mathcal{N} $$ N = 4 SU( N ) superYangMills theory and its relation to the flatspace fourgraviton amplitude in type IIB superstring theory. We work in a large N expansion in which the complexified YangMills coupling τ is fixed. In this expansion, nonperturbative instanton contributions are present, and the SL(2 , ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the massdeformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed $$ \mathcal{N} $$ N = 4 SYM theory, which in turn constrains the fourpoint correlator at separated points. In a normalization where the twopoint functions are proportional to N 2 − 1 and are independent of τ and $$ \overline{\tau} $$ τ ¯ , we find that the terms of order $$ \sqrt{N} $$ N and $$ 1/\sqrt{N} $$ 1 / N in the large N expansion of the fourpoint correlator are proportional to the nonholomorphic Eisenstein series $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$ E 3 2 τ τ ¯ and $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$ E 5 2 τ τ ¯ , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB Smatrix arising from R 4 and D 4 R 4 contact interactions, which, for the R 4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $$ {N}^{\frac{1}{2}m} $$ N 1 2 − m with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of nonholomorphic Eisenstein series with halfinteger index, which are manifestly SL(2 , ℤ) invariant.more » « less