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The 4D Superconformal Index from q-deformed 2D Yang-Mills
Abstract and Figures
We identify the 2d topological theory underlying the N=2 4d superconformal
index with an explicit model: q-deformed 2d Yang-Mills. By this route we are
able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's
T_N theories.
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... Coulomb branch operator Integrated current [19] Orbifold C 2 /Z M Change CFT to coset [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] 1d Change CFT by Drinfeld-Sokolov reduction [72][73][74][75][76][77][78][79][80][81][82][83][84][85] Symmetry-breaking wall Verlinde loop [43] 3d S-duality domain wall Modular kernel [86][87][88][89][90] Boundary Boundary CFT [91,92] 4d Coupling to a tinkertoy Vertex operator [5,[93][94][95][96][97][98][99][100][101][102][103][104][105][106][107][108][109] We discuss some offshoots of the AGT correspondence in section 9. Placing the 6d theory onto other product spaces M × C (with some twist) leads to interesting relations between theories on M and on C: the index/qYang-Mills (YM) correspondence [110], the 3d/3d correspondence [111], the 2d/4d correspondence [112]. In another direction, some class S theories (especially linear quiver gauge theories) can be realized as reductions of 5d N = 1 theories. ...
... The wave functions can be computed order by order in p, q, t, but are not known in closed form. In the Schur limit q = t correlators are functions of q only (p-dependent terms are Q-exact), wavefunctions are proportional to Schur polynomials, and the corresponding TQFT is q-deformed 2d Yang-Mills theory [110]. In the more general Macdonald limit p = 0, wavefunctions are essentially Macdonald polynomials in q, t and the TQFT must be deformed by changing the measure in the path integral of q-YM theory [667]. ...
Starting with a gentle approach to the Alday–Gaiotto–Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School ‘YRISW 2020’ to appear in a special issue of J. Phys. A. Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD (quantum chromodynamics) to non-Lagrangian theories) obtained by twisted compactification of 6d N = ( 2 , 0 ) superconformal theories on a Riemann surface C . This 6d construction yields the Coulomb branch and Seiberg–Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of a 4d N = 2 S U ( 2 ) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres–Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilson–’t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.
A bstract
We study the unflavored Schur indices in the $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory for the B n , C n , D n , G 2 gauge groups. We explore two methods, namely the character expansion method and the Fermi gas method, to efficiently compute the q -series expansion of the Schur indices to some high orders. Using the available data and the modular properties, we are able to fix the exact formulas for the general gauge groups up to some high ranks and discover some interesting new features. We also identify some empirical modular anomaly equations, but unlike the case of A n groups, they are quite complicated and not sufficiently useful to fix exact formulas for gauge groups of arbitrary rank.
A bstract
We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for 𝒩 = 4 U( N ) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q -analogues of multiple zeta values ( q -MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.
We study superconformal indices of 4d compactifications of the 6d minimal (DN+3,DN+3) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with AN,CN and A1N global symmetries. We mainly concentrate on C2 case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our C2 operator and previously derived A2 generalization of van Diejen model. Finally, we also consider compactifications with AN-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.
A bstract
We use consistent truncations in supergravity to show that the backreaction of N rotating M5-branes wrapped on a Riemann surface leads to asymptotically AdS 5 black hole solutions of 11d supergravity. We discuss the thermodynamic properties of these black holes focusing on their supersymmetric limit. The Bekenstein-Hawking entropy of the supersymmetric black holes scales as N ³ and can be reproduced by the superconformal index of the holographically dual 4d $$ \mathcal{N} $$ N = 1 SCFTs of class $$ \mathcal{S} $$ S .
A bstract
We study the Schur line defect correlation functions in $$ \mathcal{N} $$ = 4 and $$ \mathcal{N} $$ = 2 ∗ U( N ) super Yang-Mills (SYM) theory. We find exact closed-form formulae of the correlation functions of the Wilson line operators in the fundamental, antisymmetric and symmetric representations via the Fermi-gas method in the canonical and grand canonical ensembles. All the Schur line defect correlators are shown to be expressible in terms of multiple series that generalizes the Kronecker theta function. From the large N correlators we obtain generating functions for the spectra of the D5-brane giant and the D3-brane dual giant and find a correspondence between the fluctuation modes and the plane partition diamonds.
A bstract
We study the compactification of 4D $$ \mathcal{N} $$ N = 3 superconformal field theories (SCFTs) on S ¹ , focusing on the relation between the 4D superconformal index and 3D partition function on the squashed sphere $$ {S}_b^3 $$ S b 3 . Since the center $$ \mathfrak{u} $$ u (1) of the $$ \mathfrak{u} $$ u (3) R-symmetry of the 4D theory can mix with an $$ \mathcal{N} $$ N = 6 abelian flavor symmetry in three dimensions, the precise 4D/3D relation for the global symmetry is not obvious. Focusing on the case in which the 3D theory is the ABJM theory, we demonstrate that the above R-symmetry mixing can be precisely identified by considering the Schur limit (and/or its $$ \mathcal{N} $$ N = 3 cousin) of the 4D index. As a result, we generalize to the ABJM theories recent discussions on the connection between supersymmetry enhancement of the 4D index and squashing independence of the $$ {S}_b^3 $$ S b 3 partition function.
A bstract
Mixed anomalies, higher form symmetries, two-group symmetries and non-invertible symmetries have proved to be useful in providing non-trivial constraints on the dynamics of quantum field theories. We study mixed anomalies involving discrete zero-form global symmetries, and possibly a one-form symmetry, in 3d $$ \mathcal{N} $$ N ≥ 3 gauge theories using the superconformal index. The effectiveness of this method is demonstrated via several classes of theories, including Chern-Simons-matter theories, such as the U(1) k gauge theory with hypermultiplets of diverse charges, the T (SU( N )) theory of Gaiotto-Witten, the theories with 𝔰𝔬(2 N ) 2 k gauge algebra and hypermultiplets in the vector representation, and variants of the Aharony-Bergman-Jafferis (ABJ) theory with the orthosymplectic gauge algebra. Gauging appropriate global symmetries of some of these models, we obtain various interesting theories with non-invertible symmetries or two-group structures.
A bstract
We find closed-form expressions for the Schur indices of 4d $$ \mathcal{N} $$ N = 2 * super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.
We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult. Comment: 25 pages, 11 figures
General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the “classical ” special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on author’s habilitation thesis [Spi7] containing a more detailed account of the subject. Contents 1. General definition of univariate elliptic hypergeometric series and integrals 2 2. An overview of classical hypergeometric functions 3 3. Elliptic functions versus balanced, well poised and very well
A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns
out to be e.g. the ordinary Yang–Mills theory, the G/G gauged WZNW model or the Poisson σ-model that underlies the Kontsevich
quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the q-deformation of the ordinary Yang–Mills partition function in the sense that the series over the weights is replaced by the
same series over the q-weights. For q equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the
Verlinde formula.
We count the number of bound states of BPS black holes on local Calabi–Yau three-folds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a q-deformed Yang–Mills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of O(e−N), ZBH is given as a sum of a square of chiral blocks, each of which corresponds to a specific D-brane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The subleading chiral blocks involve topological string amplitudes with D-brane insertions at (2g−2) points on the Riemann surface analogous to the Ω points in the large N 2d Yang–Mills theory. The finite N amplitude provides a non-perturbative definition of topological strings in these backgrounds. This also leads to a novel non-perturbative formulation of c=1 non-critical string at the self-dual radius.
We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1. Comment: 32 pages, 8 figures; v2: minor corrections, published version
We study the gauge/gravity duality for theories with four dimensional ${\cal N}=2$ supersymmetries. We consider the large class of generalized quiver field theories constructed recently by one of us (D.G.). These field theories can also be viewed as the IR limit of M5 branes wrapping a Riemann surface with punctures. We give a prescription for constructing the corresponding geometries and we discuss a few special cases in detail. There is a precise match for various quantities between the field theory and the M-theory description. Comment: 46 pages, 18 figures, V4: Figure fixed
We demonstrate the agreement between the Higgs branches of two \({\mathcal{N}=2}\) theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E
6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkähler quotient of a linear space and another hyperkähler quotient involving the minimal nilpotent orbit of E
6, modulo the identification of the twistor lines.
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