We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt* geometry. In the case of 3 dimensions, the parameter space is (T2 x R)N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to S2 x S1 or S3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT's. In the case of 4 dimensions the parameter space is of the form XM,N = (T3 * R) M * T3N, and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries. © The Authors.

tt* geometry in 3 and 4 dimensions

Cecotti, Sergio;
2014-01-01

Abstract

We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt* geometry. In the case of 3 dimensions, the parameter space is (T2 x R)N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to S2 x S1 or S3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT's. In the case of 4 dimensions the parameter space is of the form XM,N = (T3 * R) M * T3N, and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries. © The Authors.
2014
2014
5
1
110
055
10.1007/JHEP05(2014)055
Cecotti, Sergio; Gaiotto, D.; Vafa, C.
File in questo prodotto:
File Dimensione Formato  
art2014.pdf

accesso aperto

Tipologia: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 1.4 MB
Formato Adobe PDF
1.4 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/11824
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 42
  • ???jsp.display-item.citation.isi??? 40
social impact