Starting from 6D superconformal field theories (SCFTs) realized via F-theory, we show how reduction on a circle leads to a uniform perspective on the phase structure of the resulting 5D theories, and their possible conformal fixed points. Using the correspondence between F-theory reduced on a circle and M-theory on the corresponding elliptically fibered Calabi--Yau threefold, we show that each 6D SCFT with minimal supersymmetry directly reduces to a collection of between one and four 5D SCFTs. Additionally, we find that in most cases, reduction of the tensor branch of a 6D SCFT yields a 5D generalization of a quiver gauge theory. These two reductions of the theory often correspond to different phases in the 5D theory which are in general connected by a sequence of flop transitions in the extended Kahler cone of the Calabi--Yau threefold. We also elaborate on the structure of the resulting conformal fixed points, and emergent flavor symmetries, as realized by M-theory on a canonical singularity.

6D SCFTs and Phases of 5D Theories

Starting from 6D superconformal field theories (SCFTs) realized via F-theory, we show how reduction on a circle leads to a uniform perspective on the phase structure of the resulting 5D theories, and their possible conformal fixed points. Using the correspon-dence between F-theory reduced on a circle and M-theory on the corresponding elliptically fibered Calabi-Yau threefold, we show that each 6D SCFT with minimal supersymmetry directly reduces to a collection of between one and four 5D SCFTs. Additionally, we find that in most cases, reduction of the tensor branch of a 6D SCFT yields a 5D generalization of a quiver gauge theory. These two reductions of the theory often correspond to different phases in the 5D theory which are in general connected by a sequence of flop transitions in the extended Kahler cone of the Calabi-Yau threefold. We also elaborate on the structure of the resulting conformal fixed points, and emergent flavor symmetries, as realized by M-theory on a canonical singularity.

/pdf/6d-scfts-and-phases-of-5d-theories-4gg2l5cnz4.pdf

6D SCFTs and phases of 5D theories

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

/pdf/infrared-computations-of-defect-schur-indices-ps3o2x2x5q.pdf

Infrared Computations of Defect Schur Indices

We explore the non-perturbative Dyson-Schwinger equations obeyed by the partition functions of the $\Omega$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $\Omega$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case.

Opers, surface defects, and Yang-Yang functional

We investigate the relation between 3d
 $$ \mathcal{N} $$
 = 2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S2× S1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recently proposed duality relating the U(N) theory with one flavor and one adjoint to a WZ model can be regarded as the 3d ancestor of the evaluation formula for the DF integral representation of the 3-point correlator. We are also able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N) theory to the 5d T2 theory.

/pdf/from-3d-dualities-to-2d-free-field-correlators-and-back-1uqe09hkpu.pdf

From 3d dualities to 2d free field correlators and back

Non-perturbative aspects of $$ \mathcal{N}=2 $$
 supersymmetric gauge theories of class $$ \mathcal{S} $$
 are deeply encoded in the algebra of functions on the moduli space $$ {\mathrm{\mathcal{M}}}_{\mathrm{flat}} $$
 of flat SL(N )- connections on Riemann surfaces. Expectation values of Wilson and ’t Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on $$ {\mathrm{\mathcal{M}}}_{\mathrm{flat}} $$
 . Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda field theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank class $$ \mathcal{S} $$
 theories.

/pdf/line-operators-in-theories-of-class-mathcal-s-quantized-40vn2448p8.pdf

Line operators in theories of class $\mathcal{S}$, quantized moduli space of flat connections, and Toda field theory

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $\mathcal{W}$-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat $\mathrm{SL}(3)$-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel-Nielsen coordinates from Teichmuller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.

Toda conformal blocks, quantum groups, and flat connections

In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory $T_2$. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the $T_2$ theories representing trifundamental half hypermultiplets in $N=2$, $d=4$ field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two $T_2$ vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the $T_2$ vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.

/pdf/trinion-conformal-blocks-from-topological-strings-1elnxino35.pdf

Trinion Conformal Blocks from Topological strings

https://link.springer.com/content/pdf/10.1007/JHEP07(2023)120.pdf

Keeping matter in the loop in dS3 quantum gravity

We study the quantum modular properties of $\hat{ Z}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have $n$ junction nodes with definite signature and for rank $r$ gauge group $G$, that $\hat{ Z}^G$ is related to a quantum modular form of depth $nr$. We prove this for $G=SU(3)$ and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of $\hat{ Z}^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\hat{ Z}^G$ with $G=SU(2)$ and $SU(3)$, for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for $SU(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.

pdf/quantum-modular-hat-z-g-invariants-2toolkcf.pdf

Quantum Modular $\hat Z^G$-Invariants

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