2015

A topologically twisted index for three-dimensional supersymmetric theories

Benini, Francesco, Zaffaroni, Alberto

Understand

We provide a general formula for the partition function of three-dimensional $\mathcal{N}=2$ gauge theories placed on $S^2 \times S^1$ with a topological twist along $S^2$, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields.

The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes.
The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries.
We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities.

Built on

T. T. Wu and C. N. Yang, Dirac Monopole Without Strings: Monopole Harmonics
1976
Earlier work this paper cites.
P. Goddard, J. Nuyts, and D. I. Olive, Gauge Theories and Magnetic Charge
1977
Earlier work this paper cites.
A. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions
1984
Earlier work this paper cites.
A. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions
1984
Earlier work this paper cites.
C. Beem, T. Dimofte, and S. Pasquetti, Holomorphic Blocks in Three Dimensions
Original
1986
Earlier work this paper cites.
E. P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory
1988
Earlier work this paper cites.
E. Witten, Quantum Field Theory and the Jones Polynomial
1989
Earlier work this paper cites.
M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model
1993
Earlier work this paper cites.
E. Witten, Phases of 𝒩 = 2 \mathcal{N}{=}2 theories in two-dimensions
1993
Earlier work this paper cites.
L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions
1995
Earlier work this paper cites.
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories
1995
Earlier work this paper cites.
D. R. Morrison and M. R. Plesser, Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties
1995
Earlier work this paper cites.
K. A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric S P ( N c ) SP(N_{c}) gauge theories
1995
Earlier work this paper cites.
H.-C. Kao, K.-M. Lee, and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories
1996
Earlier work this paper cites.
O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. Strassler, Aspects of 𝒩 = 2 \mathcal{N}{=}2 supersymmetric gauge theories in three-dimensions
1997
Earlier work this paper cites.
Springer, New York, 1997
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory · 1997
Earlier work this paper cites.
M. Brion and M. Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue
1999
Earlier work this paper cites.
A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov
2004
Earlier work this paper cites.
K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional 𝒩 = ( 2 , 2 ) \mathcal{N}{=}(2,2) Theories
2007
Earlier work this paper cites.
S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, and M. Ando, Cluster decomposition, T-duality, and gerby CFT’s
2007
Earlier work this paper cites.
A. Giveon and D. Kutasov, Seiberg Duality in Chern-Simons Theory
Original
2009
Earlier work this paper cites.
S. Kim, The Complete superconformal index for 𝒩 = 6 \mathcal{N}{=}6 Chern-Simons theory
Original
2009
Earlier work this paper cites.

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