The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S
 3 ×  time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the $${\mathcal N}=4$$
 theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS
 5 × S
 5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS
 5, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the $${\mathcal N}=4$$
 theory. We note that entropy for some small supersymmetric AdS
 5 black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the $${\mathcal N}=4$$
 Yang Mills theory.

An Index for 4 dimensional super conformal theories

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0001/CNTP-2007-0001-0001-a001.pdf

Electric-Magnetic Duality And The Geometric Langlands Program

http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q = 0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a sub-group). They can also have ’t Hooft anomalies, which prevent us from gauging them, but lead to ’t Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

/pdf/generalized-global-symmetries-mg00hfmchr.pdf

Generalized Global Symmetries

https://www.math.uni-hamburg.de/home/runkel/PDF/tft1.pdf

TFT construction of RCFT correlators I: partition functions

We explain how D-branes on group manifolds are stabilized against shrinking by quantized worldvolume U(1) fluxes. Starting from the Born-Infeld action in the case of the SU(2) group manifold we derive the masses, multiplicities and spectrum of small fluctuations of these branes, and show that they agree exactly with the predictions of Conformal Field Theory, to all orders in the α' expansion. We discuss the generalization to other groups and comment on an apparent paradox: why are the `RR charges' of these branes not quantized?

http://cds.cern.ch/record/429982/files/0003037.pdf

Flux stabilization of D-branes

We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore-Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A-A-bimodules. 
The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties. 
We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore-Seiberg data.

TFT construction of RCFT correlators I: Partition functions

https://www.math.uni-hamburg.de/home/runkel/PDF/def.pdf

Christoph Schweigert

Papers

TFT construction of RCFT correlators I: partition functions

Flux stabilization of D-branes

TFT construction of RCFT correlators I: Partition functions

Duality and defects in rational conformal field theory

TFT construction of RCFT correlators: III: simple currents