Membranes and sheaves.

Nekrasov, Nikita; Okounkov, Andrei

doi:10.14231/AG-2016-015

Membranes and sheaves.

Apr 8, 2014

77 pages

Published in:

Algebr. Geom. 3 (2016) 3, 320-369

Published: 2016

e-Print:

1404.2323 [math.AG]

DOI:

10.14231/AG-2016-015

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Abstract: (FIZ Karlsruhe)

Having a glimpse at the title and then authors of this paper, mathematicians would immediately realize the difficulty to follow the paper as well as its importance in mathematical physics. Membranes are basic objects in M-theory which is believed by string physicists to be The Theory of Everything, while (quasi-)coherent sheaves are fundamentally important in algebraic geometry, representation theory, etc. which are much more familiar to mathematicians.Given a Calabi-Yau (CY) 5-fold Z, the partition function of M-theory on a S1-bundle Z˜×S1 of Z should be the trace of certain operator on the Hilbert space of the theory on Z. By supersymmetry (SUSY), the operator is the square of a Dirac type operator D. The index of D is a mathematically interesting object and serves as the virtual representation of all symmetries of the physical theory. For M2-branes in Z, the configuration space is roughly speaking the space of all Riemann surfaces in it. The moduli space M of SUSY M2-branes is hard to describe in general and expected to be some compactification of the moduli space of immersed holomorphic curves.This paper aims to guess what the space M as well as the corresponding index sheaf might be. The proposal is via a conjectural relation with the so-called Donaldson-Thomas (DT) theory on 3-folds in the following geometric setting. Let X be a smooth 3-fold and L1, L2 be two line bundles on it with L1⊗L2=KX. Define Z=Tot(L1⊕L2) to be the total space, which is a non-compact CY 5-fold with a C∗-action on fibers preserving the CY 5-form. Using the cycle maps to the Chow variety of X, the authors conjecture a relation (see Conjecture 2.1) between the virtual structure sheaf of M and of the DT moduli space (in the Pandharipande-Thomas chamber). Concrete examples are studied to support the conjecture.As defining M2-brane index and (K-theoretical) DT theory involves a choice of square roots of certain determinant line bundles over the moduli spaces, the authors prove an existence result for them. The result is later heavily used by many other authors like C. Brav et al. [J. Singul. 11, 85–151 (2015; Zbl 1325.14057)] and Y.-H. Kiem and J. Li [“Categorification of Donaldson-Thomas invariants via perverse sheaves”, Preprint, arXiv:1212.6444 (opens in new tab)] in the definition of cohomological DT theory on CY 3-folds.When the CY 5-fold is simply Z=X×C2, an rigidity result for DT theory is used in the simplification of calculations. Moreover, when X is toric, the authors discuss how the K-theoretical DT invariants may be expressed in terms of the K-theoretical topological vertex of A. Iqbal et al. [“The refined topological vertex”, J. High Energy Phys. 2009, No. 10, Paper No. 069 (2009; doi:10.1088/1126-6708/2009/10/069 (opens in new tab))].What we review here already shows the great influence of this paper on the mathematical understanding of M-theory and DT invariants. In fact, there are also many other interesting insights, ideas and results which we have not been able to touch and they deserve more careful reading. We leave them to the readers to explore.

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