The planar limit of N $$ \mathcal{N} $$ = 2 superconformal field theories
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Received: March 13, 2020 Accepted: May 1, 2020 Published: May 27, 2020
Bartomeu Fiol, Jairo Mart´ınez-Montoya and Alan Rios Fukelman Departament de F´ısica Qu` antica i Astrof´ısica i, Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ı i Franqu`es 1, Barcelona 08028, Catalonia, Spain
E-mail: [email protected], [email protected], [email protected] Abstract: We obtain the perturbative expansion of the free energy on S 4 for four dimensional Lagrangian N = 2 superconformal field theories, to all orders in the ’t Hooft coupling, in the planar limit. We do so by using supersymmetric localization, after rewriting the 1-loop factor as an effective action involving an infinite number of single and double trace terms. The answer we obtain is purely combinatorial, and involves a sum over tree graphs. We also apply these methods to the perturbative expansion of the free energy at finite N , and to the computation of the vacuum expectation value of the 1/2 BPS circular Wilson loop, which in the planar limit involves a sum over rooted tree graphs. Keywords: Matrix Models, Supersymmetric Gauge Theory, Wilson, ’t Hooft and Polyakov loops ArXiv ePrint: 2003.02879
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)136
JHEP05(2020)136
The planar limit of N = 2 superconformal field theories
Contents 1 Introduction partition function of N = 2 superconformal Yang-Mills theories The 1-loop factor as an effective action Partition function and color invariants at finite N Free energy at large N
4 5 7 9
3 The 1/2 BPS circular Wilson loop 3.1 Wilson loops at finite N 3.2 Wilson loops at large N
16 18 19
G for the classical groups A Z1-loop and Sint
22
B Explicit planar free energy up to 13th order
23
1
Introduction
Part of the theoretical appeal of supersymmetric gauge theories is that, for certain questions, they allow more analytical control than their non-supersymmetric counterparts. An outstanding example is supersymmetric localization, which allows to reduce the evaluation of certain quantities of 4d N = 2 super Yang-Mills (SYM) theories to matrix integrals [1]. For instance, the partition function on S 4 is reduced to Z ZS 4 =
da e
−
8π 2 Tr(a2 ) g2 YM
Z1-loop |Zinst |2
(1.1)
where Z1-loop is a factor that arises from a 1-loop computation, while Zinst is the instanton contribution. Similarly, the expectation value of a 1/2 BPS circular Wilson loop hWR i is also reduced to a matrix integral [1]. The fact that four dimensional questions admit zero dimensional answers constitutes a dramatic simplification, but still leaves the formidable task of evaluating these matrix integrals. A first approach consists of restricting the integrals to a Cartan subalgebra of the Lie algebra. In a second approach [2–4], the integrals are over the full Lie algebra, and the 1-loop factor in (1.1) is rewritten as an effective action. For N = 4 super Yang Mills theories, both Z1-loop = 1 and |Zinst |2 = 1 in (1.1) [1]. The free energy can be easily computed [5], bu
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