Matrix models, geometric engineering and elliptic genera

By T. Hollowood, Amer Iqbal, C. Vafa. 2003

324 citations recorded by the source corpus.

We compute the prepotential of = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R4. We study the compactifications of = 2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T2 combines the Kahler and complex moduli of T2 and the mass parameter into the period matrix of a genus 2 curve.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fda86b558e22ee4bf8bb5977cc8516d491d10938

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Sergei Gukov
2. C. Vafa
3. Tudor Dimofte
4. Marcos Mariño
5. D. Gaiotto
6. Mina Aganagic
7. E. Witten
8. A. Klemm

In Nuclear and High Energy Physics

1. C. Vafa
2. Sergei Gukov
3. Marcos Mariño
4. Tudor Dimofte
5. D. Gaiotto
6. E. Witten
7. A. Klemm
8. Mina Aganagic

In Statistical and Nonlinear Physics

1. Marcos Mariño
2. C. Vafa
3. A. Klemm
4. Sergei Gukov
5. Mina Aganagic
6. Yuji Tachikawa
7. D. Gaiotto
8. Tudor Dimofte

Citations in the corpus

• Refined, Motivic, and Quantum (2009, background): “Thus, in certain examples the refined BPS invariants can be related to equivariant instanton counting [3,4] or to categorification of quantum group invariants [5].”
• REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS (2008, background): “ing possible application. It has been argued the Nekrasov’s partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces [11, 14, 13]. As far as we know (see appendix C for examples), the refined BPS state counting always ”
• Quantum Geometry and Quiver Gauge Theories (2013, methodology): “N = 2 ∗ theory), the phase space of classical integrable system is the moduli space of non-commutative U(1) instantons on T 4 was studied in [123, 145, 146].”
• Quantum geometry of refined topological strings (2011, methodology): “Equivalently, we could have obtained FBPS = logZBPS from a one-loop, Schwinger type computation which involves integrating out M2 brane particles in a gravi-photon background parameterized by ǫ1 and ǫ2 that is no longer self-dual [20].”
• Orbifolds of M-strings (2013, methodology): “As a main tool we will use the topological vertex and its refinement [5–7, 15, 16], to compute the degeneracy of BPS states.”
• Vortex Counting and Lagrangian 3-Manifolds (2010, methodology): “ion for four-dimensional gauge theories, in the context of BPS/instanton counting. We summarize a set of conventions that leads to a consistent identication of ZBPS with Zinst K theory . (See, e.g., [29] for further information.) At the origin of the Coulomb branch, an N= 2 gauge theory with gauge g”
• Topological Strings and Integrable Hierarchies (2003, background): “Example 4: Integrable structure of the topological vertex The topological vertex was introduced in [9] as the building block for topological string amplitudes on local, toric Calabi-Yau threefolds (for further properties and applications of the topological vertex, see for example [44,45,46,47,48,49]”
• Gauge theory loop operators and Liouville theory (2009, methodology): “Progress in realizing the Nekrasov instanton partition function [11, 12] for arbitrary ǫ1 and ǫ2 have been made in topological string theory, where amplitudes sewn from a “refined” topological vertex [60, 61] were shown to reproduce Nekrasov’s partition function.”
• The refined topological vertex (2007, methodology): “amplitudes suggests such a possibility which was explored in [ 8 ].”
• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2 (2015, background): “Such toric geometries were considered originally in [19].”
• Liouville Correlation Functions from Four-Dimensional Gauge Theories (2009, background): “This just reflects the difference in the conventions in [3] and in [10, 11] of the zero of the hypermultiplet mass.”
• M-Strings (2013, background): “The geometry of the blow-up is a local Calabi-Yau and is given by the periodic toric diagram [10, 15] in Fig.”
• Knot Homology from Refined Chern-Simons Theory (2011, unknown): “This defines a refinement of the partition function of the closed topological string [20,21,22], depending on the extra parameter t and agreeing with the partition function of the ordinary topological string at q = t.”