We study the two-dimensional Yang-Mills theory with four supercharges in the large-N limit. By using thermal boundary conditions, we analyze the internal energy and the distribution of scalars. We compare their behavior to the maximally supersymmetric case with sixteen supercharges, which is known to admit a holographic interpretation. Our lattice results for the scalar distribution show no visible dependence on N and the energy at strong coupling appears independent of temperature

Dhindsa, NS

Jha, RG

Joseph, A

Schaich, D

University of Liverpool Repository

Large-N limit of two-dimensional Yang–Mills theory withfour superchargesNavdeep Singh Dhindsa,a,∗ Raghav G. Jha,b Anosh Josepha and David SchaichcaDepartment of Physical Sciences, Indian Institute of Science Education and Research - Mohali,Knowledge City, Sector 81, SAS Nagar, Punjab 140306, IndiabPerimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, CanadacDepartment of Mathematical Sciences, University of Liverpool,Liverpool L69 7ZL, United KingdomE-mail: navdeep.s.dhindsa@gmail.com, raghav.govind.jha@gmail.com,anoshjoseph@iisermohali.ac.in, david.schaich@liverpool.ac.ukWe study the two-dimensional Yang–Mills theory with four supercharges in the large-N limit. Byusing thermal boundary conditions, we analyze the internal energy and the distribution of scalars.We compare their behavior to the maximally supersymmetric case with sixteen supercharges,which is known to admit a holographic interpretation. Our lattice results for the scalar distributionshow no visible dependence on N and the energy at strong coupling appears independent oftemperature.The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021Zoom/Gather@Massachusetts Institute of Technology∗Speaker© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh Dhindsa1. Introduction and motivationLarge-N studies of various supersymmetric Yang–Mills (SYM) theories can help us understandthe non-perturbative nature of their holographic dual counterparts [1–3]. Regularizing these theorieson a latticewould open up a first-principles approach to study these quantumfield theories. However,it is not so straightforward to regularize these theories on a lattice (see Ref. [4] for a recent review).Supersymmetric theories with a relatively larger number of supersymmetries can be discretized ona lattice with the help of twisting or orbifolding [5–7].In this work we consider a two-dimensional SYM theory with four supercharges. A twistedversion of this theory can be regularized on a lattice, with the help of a geometrical discretizationscheme. The resultant lattice theory preserves one supercharge at finite lattice spacing [8]. Themotivations for studying two-dimensional four-supercharge SYM theory on a lattice are the follow-ing. This theory is one of the simplest two-dimensional supersymmetric gauge theories to studyon the lattice with a well-defined continuum limit. It does not exhibit the numerical sign problemwhen the continuum limit is properly taken [8–10]. The maximally supersymmetric counterpartof this theory, i.e., the two-dimensional theory with sixteen supercharges, has a holographic dual.Although the theory under consideration does not have a gravity dual, we study this to understandwhether it has features that belong to the same universality class as that of its sixteen-superchargecounterpart. Our goal is to look for the existence of bound states in this theory at large N , and tostudy the thermodynamics of the theory. For computing these observables, we impose anti-periodicboundary conditions for fermions along the temporal direction. It has been previously observedthat with periodic boundary conditions the scalars clump around origin even in the presence of flatdirections [11].We present preliminary results on the energy density and distribution of scalars for this theory.These results are obtained for a range of temperatures with several numbers of colors N ≤ 12. InSec. 2 we briefly discuss two-dimensionalN = (2,2) SYM and its lattice construction. The resultsare discussed in Sec. 3. Future directions are discussed in Sec. 4.2. Two-dimensional N = (2,2) SYMTo study our target theory in a flat Euclidean spacetime, we will use the method of topologicaltwisting. The two-dimensional N = (2,2) SYM is constructed from the dimensional reductionof N = 1 SYM in four dimensions. The parent theory has a global symmetry group G =SO(4)E ×U(1), where SO(4)E is the Euclidean rotation symmetry and U(1) is the chiral symmetry.After dimensionally reducing this to two dimensions, the global symmetry group becomes G =SO(2)E × SO(2)R1 × U(1)R2 , where SO(2)E is the Euclidean rotation symmetry, SO(2)R1 is therotation symmetry along the reduced dimensions and U(1)R2 is the chiral symmetry of the theory.To maximally twist the theory in d dimensions,1 the R symmetry group should contain SO(d) as asubgroup. The R symmetry group in our target theory has two SO(2) factors as U(1) is also locallyequivalent to SO(2). Hence we can twist this theory by combining SO(2)E with either U(1)R2or SO(2)R1 , which are called the A-model and B-model twists, respectively. We work with the1See Ref. [12] for a review of maximally twisted SYM theories on a lattice.2Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh DhindsaB-model twist for which the twisted rotation group takes the formSO(2)′ = diag(SO(2)E × SO(2)R1). (1)The theory has four bosonic and four fermionic degrees of freedom. It also has four real super-charges. After twisting, the fermions and supercharges decompose into integer-spin representationsof SO(2)′. In the twisted formalism, the gauge field (Aa) and scalars (Xa) transform identically, andhence we can combine them to form a complexified gauge field Aa = Aa + iXa. Thus the actionin the twisted form is composed of a complexified gauge field Aa and the twisted fermions η, ψa,and χab = −χba. The action is given byS =N4λQ∫d2x Tr(χabFab + η[Da,Da]−12ηd), (2)where λ is the ’t Hooft coupling. The complexified field strengths are Fab = [Da,Db] andF ab =[Da,Db], with Da = ∂a + Aa and Da = ∂a + Aa denoting the complexified covariantderivatives. The nilpotent supercharge Q acts on the twisted fields in the following way:QAa = ψa QAa = 0 Qψa = 0Q χab = −F ab Qη = d Qd = 0, (3)where d is a bosonic auxiliary field with equation of motion d =[Da,Da]. After performing theQ variation on the action and integrating out the auxiliary field d, we getS =N4λ∫d2x Tr(−F abFab +12[Da,Da]2− χabD[a ψ b] − ηDaψa). (4)Now, upon using the geometrical discretization we arrive at the arrangement in which the two-dimensional N = (2,2) SYM lives on a two-dimensional square lattice spanned by two orthogonalunit vectors. The complexified gauge field is mapped to a complexified Wilson link, Aa(x) →Ua(n) on the lattice. Its fermionic superpartner ψa(n) also transforms as a link variable. The fieldη(n) lives on a site. The 2-form field χab lives on the diagonal of the unit cell. The orientations ofthe fields on the lattice ensure gauge invariance.The continuum derivatives are replaced by covariant difference operators according to thegeometrical discretization scheme prescribed in Ref. [13]. The curl-like covariant derivativesand divergence-like covariant derivatives become forward- and backward-difference operators,respectively. We haveD(−)a fa(n) = fa(n)Ua(n) − Ua(n − µ̂a) fa(n − µ̂a),D(+)a fb(n) = Ua(n) fb(n + µ̂a) − fb(n)Ua(n + µ̂b). (5)The lattice action then becomesS =N4λlat∑nTr(−F ab(n)Fab(n) +12(D(−)a Ua(n))2− χab(n)D(+)[aψ b](n) − η(n)D(−)a ψa(n)).(6)3Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh DhindsaIn addition, to control the flat directions in the theory, we add a scalar potential term to the actionwith a tunable parameter µ. Hence the complete action is given byStotal = S +N4λlatµ2∑n,aTr(Ua(n)Ua(n) − IN)2. (7)3. Preliminary resultsThe theory is analyzed on symmetric lattices with aNt = β = L = aNx , where β and L aredimensionful temporal and spatial extents, respectively. The lattice spacing is denoted by a, and Ntand Nx are the number of lattice sites along the temporal and spatial directions, respectively. Anti-periodic boundary conditions are imposed along the temporal direction for fermions. The fields aremade dimensionless in terms of the ’t Hooft coupling λ. Hence the dimensionless temporal (andspatial) extent becomes rt =√λβ = 1/t, which also serves as an effective coupling and inversedimensionless temperature. We set the scalar potential parameter to be µ = ζ rtNt = ζ√λa = ζ√λlat,making ζ the parameter we tune. The flat directions of the continuum theory are recovered in thelimit ζ2 → 0.The calculations are performed using the publicly available software2 described in Ref. [14].For the preliminary analysis, we consider several numbers of colors up to N = 12 and lattice sizesup to 322 over a range of temperatures. 0 0.2 0.4 0.6 0.8 11/7 1/5 1/3PRELIMINARYN = 12, Nt = 24Etζ = 0.50ζ = 0.40ζ = 0.30ζ → 0.00 0 2 4 6 8 10 12 1  2  3PRELIMINARYN = 12, Nt = 24Etζ = 0.301.01926(8508)t2, χ2/d.o.f = 0.8941Figure 1: Energy density (Eq. 8) for t < 1 with different ζ values (left) and t ≥ 1 with ζ = 0.3 (right). Bothplots are for N = 12 and lattice size 242.We first report on the energy densityE =3λlat(1 −23N2SB), (8)where SB is the bosonic action averaged over the lattice volume and λlat is the dimensionless ’t Hooftcoupling. The results for temperatures t < 1 and t ≥ 1 are shown in Fig. 1. At high temperatures(weak effective couplings rt ), the E ∝ t2 behavior of the energy density is the same as that ofthe maximally supersymmetric theory as reported in Refs. [15, 16], with a different coefficient.2github.com/daschaich/susy4Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh DhindsaHowever, at low temperatures (strong effective couplings rt ) the ζ2 → 0 energy density appears tobe independent of the temperature, in contrast to the maximally supersymmetric case where a cubictemperature dependence is known [17].The other observable we study is the extent of the scalars, which we define as〈Scalar2〉=〈1NNtNxNtNx∑i=1Tr(X2i )〉. (9)Our preliminary analyses of〈Scalar2〉at rt = 5 for different N and Nt values are shown in Fig. 2. 0.028 0.032 0.036 0.04 0.044 0  0.1  0.2  0.3  0.4  0.5  0.6rt = 5.0, Nt = 24PRELIMINARY<Scalar2>1/Nζ = 0.3ζ = 0.4ζ = 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0  0.1  0.2  0.3  0.4  0.5  0.6rt = 5.0, N = 12PRELIMINARY<Scalar2>ζNt = 16Nt = 24Nt = 32Figure 2: The dependence of the scalar extent (Eq. 9) on N ∈ {2,4,8,12} for different ζ at fixed rt = 5 on242 lattices (left), and its dependence on ζ for different lattice sizes at fixed rt = 5 and N = 12 (right).Considering the large-N limit in the left-panel plot of Fig. 2, we observe no visible dependenceon N . These results suggest that the scalars seem to clump around the origin even for small U(N)gauge groups, provided we are working with sufficiently large lattices. For smaller lattices, sayNt = 16, the right-panel plot in Fig. 2 suggests that the scalars may not converge to a definite valuein the ζ2 → 0 limit.For N = 12 and Nt = 24 the behavior of the scalars in the low-temperature region of strongeffective couplings rt is shown in Fig. 3. The distribution of scalars for a fixed rt is estimated byconsidering several ζ values and taking the ζ2 → 0 limit. As the rt → 0 limit is taken〈Scalar2〉approaches a nonzero value, which indicates that the scalars do not show any run-away behaviorin the presence of flat directions and hence the existence of bound state at finite temperature isobserved. The ∼r3t dependence we observe is very different from the ∼1/rt behavior predicted forthe maximally supersymmetric theory.4. Conclusions and way forwardThe preliminary analysis reported in this work indicates the existence of the bound states ofscalars at finite temperatures for two-dimensional N = (2,2) SYM. Even though we need largelattices to construct the bound state (and to control the sign problem [8–10]), we observe thatthere is no significant dependence on the gauge group U(N). Even modest values of N & 4 appearto provide access to the large-N limit of the theory.5Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh Dhindsa 0 0.04 0.08 0.12 0.16 0  1  2  3  4  5  6  7PRELIMINARYN = 12, Nt = 24<Scalar2>rt0.00789(19) + 0.000316(3).rt3, χ2/d.o.f = 2.5992 0 0.05 0.1 0.15 0.2 0  0.1  0.2  0.3  0.4  0.5  0.6<Scalar2>ζrt = 3.0rt = 5.0rt = 7.0Figure 3: The observable〈Scalar2〉given by Eq. (9) is shown in the ζ2 → 0 limit for various couplings. Inthe inset, we show how the ζ2 → 0 limit is taken. The results are for N = 12 on 242 lattices.By looking at the temperature dependence of the energy density and the distribution of thescalars we can say that the two-dimensionalN = (2,2) SYM theory seems to behave very differentlyat low temperatures compared to its maximal N = (8,8) counterpart.We also want to investigate whether there is a deconfinement transition in theN = (2,2) theoryas we move from small to large lattice sizes. In the sixteen-supercharge theory, there is a phasetransition which is dual to a transition between different black hole solutions [15, 16]. We have seensigns of theN = (2,2)SYMcenter symmetry being broken for small lattices (high temperatures) andrestored for large lattices, but more careful analysis is needed. We are continuing our investigationsby considering larger lattice sizes to do systematic studies of continuum extrapolations, as well asobtaining more robust results for the dependence on the number of colors over a wider range oftemperatures.AcknowledgmentsNSD thanks the Council of Scientific and Industrial Research (CSIR), Government of India, forthe financial support through a research fellowship (Award No. 09/947(0119)/2019-EMR-I). RGJwould like to thank Masanori Hanada for discussions. Research at Perimeter Institute is supportedin part by the Government of Canada through the Department of Innovation, Science and EconomicDevelopment and by the Province of Ontario through the Ministry of Colleges and Universities.The work of AJ was supported in part by the Start-up Research Grant (No. SRG/2019/002035)from the Science and Engineering Research Board (SERB), Government of India, and in part by aSeed Grant from the Indian Institute of Science Education and Research (IISER) Mohali. DS wassupported by UK Research and Innovation Future Leader Fellowship MR/S015418/1 and STFCgrant ST/T000988/1. Numerical calculations were carried out at the University of Liverpool.6Large-N limit of two-dimensional Yang–Mills theory with four supercharges Navdeep Singh DhindsaReferences[1] J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv.Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200].[2] L. Motl, Proposals on nonperturbative superstring interactions, hep-th/9701025.[3] N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, Supergravity and thelarge-N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004[hep-th/9802042].[4] D. Schaich, Progress and prospects of lattice supersymmetry, Proc. Sci. LATTICE2018(2019) 005 [1810.09282].[5] A.G. Cohen, D.B. Kaplan, E. Katz and M. Unsal, Supersymmetry on a Euclidean space-timelattice. 1. A Target theory with four supercharges, JHEP 0308 (2003) 024[hep-lat/0302017].[6] F. 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Large-N limit of two-dimensional Yang-Mills theory with four supercharges

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Large-N limit of two-dimensional Yang-Mills theory with four supercharges

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