Mean string field theoryLandau-Ginzburg theory for 1-form symmetries

Nabil IqbalIn collaboration with J. McGreevy

(to appear)

Global symmetriesGlobal symmetries are important. Let’s review ordinary U(1) 1-index currents:

An ordinary current counts particles. Local operators create particles, and are charged under ordinary symmetries.

O(x) ! ei↵O(x)<latexit sha1_base64="wYalRBUeO6nXsj55hTCVfLSGNk4=">AAACFnicbVDLSgMxFM3UV62vqks3wSLUhWWmCrosunFnBfuATi130kwbmskMSUYsQ7/Cjb/ixoUibsWdf2PazsK2HggczrmX3HO8iDOlbfvHyiwtr6yuZddzG5tb2zv53b26CmNJaI2EPJRNDxTlTNCaZprTZiQpBB6nDW9wNfYbD1QqFoo7PYxoO4CeYD4joI3UyZ+4Aeg+AZ7cjIqPx9jVIab3CXOBR30Y4Vm7ky/YJXsCvEiclBRQimon/+12QxIHVGjCQamWY0e6nYDUjHA6yrmxohGQAfRoy1ABAVXtZBJrhI+M0sV+KM0TGk/UvxsJBEoNA89Mjq9U895Y/M9rxdq/aCdMRLGmgkw/8mOOTfZxR7jLJCWaDw0BIpm5FZM+SCDaNJkzJTjzkRdJvVxyTkvl27NC5TKtI4sO0CEqIgedowq6RlVUQwQ9oRf0ht6tZ+vV+rA+p6MZK93ZRzOwvn4By3efIg==</latexit>

What are global symmetries good for? Landau told us: use them to classify phases of matter.

hO(x)i = 0<latexit sha1_base64="hUUF6uU18i8N1POeS6COlAm88n8=">AAACCXicbVA9SwNBEN2LXzF+RS1tFoMQm3AXBW2EoI2dEcwH5I4wt9kkS/b2jt09MRxpbfwrNhaK2PoP7Pw37l1SaOKDgcd7M8zM8yPOlLbtbyu3tLyyupZfL2xsbm3vFHf3miqMJaENEvJQtn1QlDNBG5ppTtuRpBD4nLb80VXqt+6pVCwUd3ocUS+AgWB9RkAbqVvELgcx4NQNQA8J8ORmUn44dmUm4gtsd4slu2JnwIvEmZESmqHeLX65vZDEARWacFCq49iR9hKQmhFOJwU3VjQCMoIB7RgqIKDKS7JPJvjIKD3cD6UpoXGm/p5IIFBqHPimMz1YzXup+J/XiXX/3EuYiGJNBZku6scc6xCnseAek5RoPjYEiGTmVkyGIIFoE17BhODMv7xImtWKc1Kp3p6WapezOPLoAB2iMnLQGaqha1RHDUTQI3pGr+jNerJerHfrY9qas2Yz++gPrM8femeZhg==</latexit>

Unbroken

hO(x)i 6= 0<latexit sha1_base64="mxGg1CdGpCl3MC+9LpgmdSp46Kk=">AAACDHicbVDLSgMxFM3UV62vqks3wSLUTZmpgi6LbtxZwT6gU0omvW1DM5kxyYhl6Ae48VfcuFDErR/gzr8xM52Fth4IHM45l9x7vJAzpW3728otLa+sruXXCxubW9s7xd29pgoiSaFBAx7ItkcUcCagoZnm0A4lEN/j0PLGl4nfugepWCBu9SSErk+Ggg0YJdpIvWLJ5UQMObg+0SNKeHw9LT8cuzIVsSvgDtsmZVfsFHiROBkpoQz1XvHL7Qc08kFoyolSHccOdTcmUjPKYVpwIwUhoWMyhI6hgvigunF6zBQfGaWPB4E0T2icqr8nYuIrNfE9k0x2VvNeIv7ndSI9OO/GTISRBkFnHw0ijnWAk2Zwn0mgmk8MIVQysyumIyIJ1aa/ginBmT95kTSrFeekUr05LdUusjry6AAdojJy0BmqoStURw1E0SN6Rq/ozXqyXqx362MWzVnZzD76A+vzByqimwc=</latexit>

Spontaneously broken

Critical point

Landau-Ginzburg theory

Unbroken Spontaneously broken

Describes the phase structure and low-lying modes in SSB phase. Sometimes (d > 4) usefully describes the phase transition too.

Close to the transition (…or to understand the phase structure…), we can often use a universal “Landau-Ginzburg” field theory:

S[�] =

Zddx

�@µ�@

µ�† �m2|�|2 � �|�|4�

<latexit sha1_base64="GovgrtTEH8Z9K0HrRK8O9HwNM3I=">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</latexit>

�(x)<latexit sha1_base64="G066fGpOyQTG9fUeSaVYqs0sClM=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxgv2AdinZNNuGZrMhyYpl6Y/w4kERr/4eb/4b0+0etPXBwOO9GWbmBZIzbVz32ymsrW9sbhW3Szu7e/sH5cOjto4TRWiLxDxW3QBrypmgLcMMp12pKI4CTjvB5Hbudx6p0iwWD2YqqR/hkWAhI9hYqdOXY1Z9Oh+UK27NzYBWiZeTCuRoDspf/WFMkogKQzjWuue50vgpVoYRTmelfqKpxGSCR7RnqcAR1X6anTtDZ1YZojBWtoRBmfp7IsWR1tMosJ0RNmO97M3F/7xeYsJrP2VCJoYKslgUJhyZGM1/R0OmKDF8agkmitlbERljhYmxCZVsCN7yy6ukXa95F7X6/WWlcZPHUYQTOIUqeHAFDbiDJrSAwASe4RXeHOm8OO/Ox6K14OQzx/AHzucPueyPKg==</latexit>

Order parameter: �(x) ! �(x)ei↵<latexit sha1_base64="1A2NXK/SO/nr5kL4LNnYOVkZ8so=">AAACCHicbVDLSgMxFM34rPU16tKFwSLUTZmpgi6LblxWsA/ojCWTZjqhmUxIMmIZunTjr7hxoYhbP8Gdf2PajqCtBwKHc+7l5pxAMKq043xZC4tLyyurhbXi+sbm1ra9s9tUSSoxaeCEJbIdIEUY5aShqWakLSRBccBIKxhcjv3WHZGKJvxGDwXxY9TnNKQYaSN17QNPRLR8fww9ncAfTm4z6iEmIjTq2iWn4kwA54mbkxLIUe/an14vwWlMuMYMKdVxHaH9DElNMSOjopcqIhAeoD7pGMpRTJSfTYKM4JFRejBMpHlcw4n6eyNDsVLDODCTMdKRmvXG4n9eJ9XhuZ9RLlJNOJ4eClMGTehxK7BHJcGaDQ1BWFLzV4gjJBHWpruiKcGdjTxPmtWKe1KpXp+Wahd5HQWwDw5BGbjgDNTAFaiDBsDgATyBF/BqPVrP1pv1Ph1dsPKdPfAH1sc3NTOY1g==</latexit>

The old Landau paradigm

These ideas define the Landau paradigm:

1. Phases of matter are classified by their patterns of broken and unbroken symmetries.

2. Critical points between phases can be studied by universal theories of the “order parameter”.

This works spectacularly well for many phases of matter, and is the foundation of textbook CMT.

But fails for many interesting things: e.g. topological order: fractional quantum Hall states, deconfined phases of gauge theory.

Can we do better?

‘Beyond-Landau’ critical points?

Landau paradigm part 2:At a critical point, the critical dofs are the fluctuations of theorder parameter.Apparent exceptions:• Direct transitions between states

which break di↵erent symmetries

(deconfined quantum critical points),

e.g. Neel to VBS in D = 2 + 1.

[Image: Alan Stonebraker]

• Transitions out of topologically-

ordered states (no local order

parameter).

[Image: Fradkin-Shenker]

Higher form global symmetriesIn the rest of this talk, we study higher-form global symmetries. (Gaiotto, Kapustin, Seiberg, Willett) , e.g. for a 1-form symmetry:

This current counts strings. Line operators create strings, and are charged under 1-form symmetries.

W [C] ! eiRC �W [C]

<latexit sha1_base64="l0tBNg+hUYiukXsP1OEa+yHbZ6Q=">AAACDHicbVC7TsMwFHXKq5RXgZHFokJiqpKCBGNFBxiLRB9SEirHdVqrthPZDlIV5QNY+BUWBhBi5QPY+BucNgO0HMnS0Tn36vqcIGZUadv+tkorq2vrG+XNytb2zu5edf+gq6JEYtLBEYtkP0CKMCpIR1PNSD+WBPGAkV4waeV+74FIRSNxp6cx8TkaCRpSjLSRBtVaz2350NMRJPcphR4VepC2MuhdI85RBnPbTNl1ewa4TJyC1ECB9qD65Q0jnHAiNGZIKdexY+2nSGqKGckqXqJIjPAEjYhrqECcKD+dhcngiVGGMIykeULDmfp7I0VcqSkPzCRHeqwWvVz8z3MTHV76KRVxoonA80NhwqDJnjcDh1QSrNnUEIQlNX+FeIwkwtr0VzElOIuRl0m3UXfO6o3b81rzqqijDI7AMTgFDrgATXAD2qADMHgEz+AVvFlP1ov1bn3MR0tWsXMI/sD6/AHP4Zo0</latexit>

d� = 0<latexit sha1_base64="LpeX5CeKdfNVf0rpkGyPFrPLIF4=">AAAB8nicbVBNS8NAEN34WetX1aOXxSJ4KkkV9CIUPeixgv2ANJTJZtMu3c2G3Y1QQn+GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6acaeO6387K6tr6xmZpq7y9s7u3Xzk4bGuZKUJbRHKpuiFoyllCW4YZTrupoiBCTjvh6Hbqd56o0kwmj2ac0kDAIGExI2Cs5Ee9OxAC8DV2+5WqW3NnwMvEK0gVFWj2K1+9SJJM0MQQDlr7npuaIAdlGOF0Uu5lmqZARjCgvqUJCKqDfHbyBJ9aJcKxVLYSg2fq74kchNZjEdpOAWaoF72p+J/nZya+CnKWpJmhCZkvijOOjcTT/3HEFCWGjy0Bopi9FZMhKCDGplS2IXiLLy+Tdr3mndfqDxfVxk0RRwkdoxN0hjx0iRroHjVRCxEk0TN6RW+OcV6cd+dj3rriFDNH6A+czx+7s5A8</latexit>

C<latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit>

Examples1. Maxwell electrodynamics with matter has a single U(1) 1-form symmetry, associated with conservation of magnetic flux.

Charged line operator is the t’Hooft line.

2. Pure SU(N) gauge theory has a ZN 1-form symmetry (“center symmetry”). Charged line operator is the fundamental Wilson line.

2. 3d Ising model has a Z2 1-form symmetry associated with the integrity of domain walls. (Makes the most sense in the gauge

formulation of the model). Charged line operator is the 3d Ising defect.

Fermionic strings from 3d Ising model.

[Polyakov, 80s]

Disorder operator:

µ(C) ⌘Q

hiji?SC ,@SC=C e�2��i�j .

µ is independent of local changes in SC by �i ! ��i

symmetry. SC is a branch cut for �i.

a1···aL(C) ⌘ µ(C)QL

s=1�(xs + eas)

(xs = center of link s) satisfies

a1···aL(C) = cosh(2�) a1···as+1,as+1,···aL(C)

� sinh(2�) a1···as�1,a0s,as+2,a0

s+2,as+1,···aL(C +⇧as)

Links like free Dirac particles, connected by

unbreakability of domain wall.

This description is shared by the RNS superstring.

µ�xµ � x0

µ

�|physi = 0.

Higher form global symmetries

These symmetries do everything that normal symmetries do! (anomalies, hydrodynamics, Goldstone bosons, etc.)

Follow Landau: we should use them to classify phases of matter. (From examples: this has to do with confinement).

In this talk: I will fill in the missing space in this box:

Normal symmetries

Higher-form symmetries

Landau-Ginzburg theory

?

Framework to describe condensation of strings? Notoriously hard problem, smells like string field theory…

IngredientsWhat is the analogue of Landau-Ginzburg theory? We will construct everything by analogy.

S[�] =

Zddx

�@µ�@

µ�† �m2|�|2 � �|�|4�

<latexit sha1_base64="GovgrtTEH8Z9K0HrRK8O9HwNM3I=">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</latexit>

�(x)<latexit sha1_base64="G066fGpOyQTG9fUeSaVYqs0sClM=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxgv2AdinZNNuGZrMhyYpl6Y/w4kERr/4eb/4b0+0etPXBwOO9GWbmBZIzbVz32ymsrW9sbhW3Szu7e/sH5cOjto4TRWiLxDxW3QBrypmgLcMMp12pKI4CTjvB5Hbudx6p0iwWD2YqqR/hkWAhI9hYqdOXY1Z9Oh+UK27NzYBWiZeTCuRoDspf/WFMkogKQzjWuue50vgpVoYRTmelfqKpxGSCR7RnqcAR1X6anTtDZ1YZojBWtoRBmfp7IsWR1tMosJ0RNmO97M3F/7xeYsJrP2VCJoYKslgUJhyZGM1/R0OmKDF8agkmitlbERljhYmxCZVsCN7yy6ukXa95F7X6/WWlcZPHUYQTOIUqeHAFDbiDJrSAwASe4RXeHOm8OO/Ox6K14OQzx/AHzucPueyPKg==</latexit>

Ordinary symmetry:

Function: Map from points to C

�(x) ! �(x)ei↵<latexit sha1_base64="1A2NXK/SO/nr5kL4LNnYOVkZ8so=">AAACCHicbVDLSgMxFM34rPU16tKFwSLUTZmpgi6LblxWsA/ojCWTZjqhmUxIMmIZunTjr7hxoYhbP8Gdf2PajqCtBwKHc+7l5pxAMKq043xZC4tLyyurhbXi+sbm1ra9s9tUSSoxaeCEJbIdIEUY5aShqWakLSRBccBIKxhcjv3WHZGKJvxGDwXxY9TnNKQYaSN17QNPRLR8fww9ncAfTm4z6iEmIjTq2iWn4kwA54mbkxLIUe/an14vwWlMuMYMKdVxHaH9DElNMSOjopcqIhAeoD7pGMpRTJSfTYKM4JFRejBMpHlcw4n6eyNDsVLDODCTMdKRmvXG4n9eJ9XhuZ9RLlJNOJ4eClMGTehxK7BHJcGaDQ1BWFLzV4gjJBHWpruiKcGdjTxPmtWKe1KpXp+Wahd5HQWwDw5BGbjgDNTAFaiDBsDgATyBF/BqPVrP1pv1Ph1dsPKdPfAH1sc3NTOY1g==</latexit>

d↵ = 0<latexit sha1_base64="c6DDEDemI/RaFpFdhwpnxvuGo3k=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoBeh6MVjBfsBaSibzaZdutkNuxuhhP4MLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0w508Z1v53S2vrG5lZ5u7Kzu7d/UD086miZKULbRHKpeiHWlDNB24YZTnupojgJOe2G47uZ332iSjMpHs0kpUGCh4LFjGBjJT/qY56OMLpB7qBac+vuHGiVeAWpQYHWoPrVjyTJEioM4Vhr33NTE+RYGUY4nVb6maYpJmM8pL6lAidUB/n85Ck6s0qEYqlsCYPm6u+JHCdaT5LQdibYjPSyNxP/8/zMxNdBzkSaGSrIYlGccWQkmv2PIqYoMXxiCSaK2VsRGWGFibEpVWwI3vLLq6TTqHsX9cbDZa15W8RRhhM4hXPw4AqacA8taAMBCc/wCm+OcV6cd+dj0Vpyiplj+APn8wfx1ZBf</latexit>

Higher form symmetry:

[C]<latexit sha1_base64="loC6greMWb+b2F54/p3pOVFhyIA=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMdiLx4r2A9IQ9lsN+3SzWbZ3Qgl9Ed48aCIV3+PN/+N2zQHbX0w8Hhvhpl5oeRMG9f9dkobm1vbO+Xdyt7+weFR9fikq5NUEdohCU9UP8SaciZoxzDDaV8qiuOQ0144bS383hNVmiXi0cwkDWI8FixiBBsr9QZSM78VDKs1t+7mQOvEK0gNCrSH1a/BKCFpTIUhHGvte640QYaVYYTTeWWQaioxmeIx9S0VOKY6yPJz5+jCKiMUJcqWMChXf09kONZ6Foe2M8Zmole9hfif56cmug0yJmRqqCDLRVHKkUnQ4nc0YooSw2eWYKKYvRWRCVaYGJtQxYbgrb68TrqNundVbzxc15p3RRxlOINzuAQPbqAJ99CGDhCYwjO8wpsjnRfn3flYtpacYuYU/sD5/AEWzI9n</latexit>

Functional: map from the space of curves to C

[C] ! [C]eiRC �

<latexit sha1_base64="n6YFBF89wcbWwaxOFOs+cdxIA3Y=">AAACEXicbVDLSgMxFM3UV62vqks3wSJ0VWaqoMtiF7qsYB/QGYdMmmlDk8yQZIQyzC+48VfcuFDErTt3/o1pO4K2HggczrmXm3OCmFGlbfvLKqysrq1vFDdLW9s7u3vl/YOOihKJSRtHLJK9ACnCqCBtTTUjvVgSxANGusG4OfW790QqGolbPYmJx9FQ0JBipI3kl6turGi/6UFXR/CHk7uUulRoP21m0L1CnKPML1fsmj0DXCZOTiogR8svf7qDCCecCI0ZUqrv2LH2UiQ1xYxkJTdRJEZ4jIakb6hAnCgvnSXK4IlRBjCMpHlCw5n6eyNFXKkJD8wkR3qkFr2p+J/XT3R44aVUxIkmAs8PhQmDJv20HjigkmDNJoYgLKn5K8QjJBHWpsSSKcFZjLxMOvWac1qr35xVGpd5HUVwBI5BFTjgHDTANWiBNsDgATyBF/BqPVrP1pv1Ph8tWPnOIfgD6+Mbo7ac6A==</latexit>

d� = 0<latexit sha1_base64="LpeX5CeKdfNVf0rpkGyPFrPLIF4=">AAAB8nicbVBNS8NAEN34WetX1aOXxSJ4KkkV9CIUPeixgv2ANJTJZtMu3c2G3Y1QQn+GFw+KePXXePPfuG1z0NYHA4/3ZpiZF6acaeO6387K6tr6xmZpq7y9s7u3Xzk4bGuZKUJbRHKpuiFoyllCW4YZTrupoiBCTjvh6Hbqd56o0kwmj2ac0kDAIGExI2Cs5Ee9OxAC8DV2+5WqW3NnwMvEK0gVFWj2K1+9SJJM0MQQDlr7npuaIAdlGOF0Uu5lmqZARjCgvqUJCKqDfHbyBJ9aJcKxVLYSg2fq74kchNZjEdpOAWaoF72p+J/nZya+CnKWpJmhCZkvijOOjcTT/3HEFCWGjy0Bopi9FZMhKCDGplS2IXiLLy+Tdr3mndfqDxfVxk0RRwkdoxN0hjx0iRroHjVRCxEk0TN6RW+OcV6cd+dj3rriFDNH6A+czx+7s5A8</latexit>

Γ is a closed 1-form

More ingredientsOrdinary symmetry: Higher form symmetry:

Ordinary derivative compares field at two nearby points.

Area derivative (Polyakov, Migdal…). Beautiful geometric construction that compares field between two nearby curves!

�

��µ⌫(�) [C]

<latexit sha1_base64="fcGpYS3zs5Z4htSpca8cnqrIKZM=">AAACH3icbVDLSgMxFM34tr6qLt0Ei6CbMqOiLkU3LhWsFpqx3Mlk2tAkMyQZoQzzJ278FTcuFBF3/o1pnYVWD4Qczrn3JvdEmeDG+v6nNzU9Mzs3v7BYW1peWV2rr2/cmDTXlLVoKlLdjsAwwRVrWW4Fa2eagYwEu40G5yP/9p5pw1N1bYcZCyX0FE84Beukbv2IJBpoQWImLJTVjYnhPQl3BZE5UXm5S4SbGMNeiUlmeOc87NYbftMfA/8lQUUaqMJlt/5B4pTmkilLBRjTCfzMhgVoy6lgZY3khmVAB9BjHUcVSGbCYrxfiXecEuMk1e4oi8fqz44CpDFDGblKCbZvJr2R+J/XyW1yEhZcZbllin4/lOQC2xSPwsIx14xaMXQEqObur5j2wQVmXaQ1F0IwufJfcrPfDA6a+1eHjdOzKo4FtIW20S4K0DE6RRfoErUQRQ/oCb2gV+/Re/bevPfv0imv6tlEv+B9fgFCXaOw</latexit>

@

@xµ�(x)

<latexit sha1_base64="l/u0X9IDkL2+4HASvOW4OrCI7HI=">AAACEnicbZBNS8MwGMfT+TbnW9Wjl+AQtstop6DHoRePE9wLrHWkWbqFpWlJUtko/Qxe/CpePCji1ZM3v43pVkQ3/xD48X+eJ8nz9yJGpbKsL6Owsrq2vlHcLG1t7+zumfsHbRnGApMWDlkouh6ShFFOWooqRrqRICjwGOl446us3rknQtKQ36ppRNwADTn1KUZKW32z6vgC4cSJkFAUsfSH4OQucYI4TaETjWhlUu2bZatmzQSXwc6hDHI1++anMwhxHBCuMENS9mwrUm6SXY8ZSUtOLEmE8BgNSU8jRwGRbjJbKYUn2hlAPxT6cAVn7u+JBAVSTgNPdwZIjeRiLTP/q/Vi5V+4CeVRrAjH84f8mEEVwiwfOKCCYMWmGhAWVP8V4hHSGSmdYkmHYC+uvAztes0+rdVvzsqNyzyOIjgCx6ACbHAOGuAaNEELYPAAnsALeDUejWfjzXiftxaMfOYQ/JHx8Q1Jh551</latexit>

�xµ<latexit sha1_base64="XGuo9xGXygVJa6sxqKitIWyvJ8s=">AAAB9XicbVBNSwMxEM36WetX1aOXYBE8ld0q6LHoxWMF+wHttmSzs21okl2SrFqW/g8vHhTx6n/x5r8xbfegrQ8GHu/NMDMvSDjTxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKDRozGPVDogGziQ0DDMc2okCIgIOrWB0M/VbD6A0i+W9GSfgCzKQLGKUGCv1uiFwQ/BTL+uKdNIvld2KOwNeJl5OyihHvV/66oYxTQVIQznRuuO5ifEzogyjHCbFbqohIXREBtCxVBIB2s9mV0/wqVVCHMXKljR4pv6eyIjQeiwC2ymIGepFbyr+53VSE135GZNJakDS+aIo5djEeBoBDpkCavjYEkIVs7diOiSKUGODKtoQvMWXl0mzWvHOK9W7i3LtOo+jgI7RCTpDHrpENXSL6qiBKFLoGb2iN+fReXHenY9564qTzxyhP3A+fwCmkZKc</latexit>

��µ⌫<latexit sha1_base64="6KBSgrPxcxp2Yb1tDxpp6fMojOI=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy4r2Ac0sUwmk3bozCTMQyixC3/FjQtF3Pob7vwbp20W2nrgwuGce7n3nihjVGnP+3aWlldW19ZLG+XNre2dXXdvv6VSIzFp4pSlshMhRRgVpKmpZqSTSYJ4xEg7Gl5P/PYDkYqm4k6PMhJy1Bc0oRhpK/XcwyAmTCMYKNrn6D4PuAmEGffcilf1poCLxC9IBRRo9NyvIE6x4URozJBSXd/LdJgjqSlmZFwOjCIZwkPUJ11LBeJEhfn0/jE8sUoMk1TaEhpO1d8TOeJKjXhkOznSAzXvTcT/vK7RyWWYU5EZTQSeLUoMgzqFkzBgTCXBmo0sQVhSeyvEAyQR1jaysg3Bn395kbRqVf+sWrs9r9SvijhK4Agcg1PggwtQBzegAZoAg0fwDF7Bm/PkvDjvzsesdckpZg7AHzifP1htllE=</latexit>

C<latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit>

Zddx

<latexit sha1_base64="araWcu9VHkCtDVsZKnQ4cbOIyto=">AAAB83icbVDLSgMxFL3js9ZX1aWbYBFclZkq6LLoxmUF+4DOWDKZTBuaSYYkI5ahv+HGhSJu/Rl3/o1pOwttPRA4nHMv9+SEKWfauO63s7K6tr6xWdoqb+/s7u1XDg7bWmaK0BaRXKpuiDXlTNCWYYbTbqooTkJOO+HoZup3HqnSTIp7M05pkOCBYDEj2FjJ95kwKHrIowl66leqbs2dAS0TryBVKNDsV778SJIsocIQjrXueW5qghwrwwink7KfaZpiMsID2rNU4ITqIJ9lnqBTq0Qolso+m2Gm/t7IcaL1OAntZILNUC96U/E/r5eZ+CrImUgzQwWZH4ozjoxE0wJQxBQlho8twUQxmxWRIVaYGFtT2ZbgLX55mbTrNe+8Vr+7qDauizpKcAwncAYeXEIDbqEJLSCQwjO8wpuTOS/Ou/MxH11xip0j+APn8weumZFz</latexit>

In action, we integrate over all points.

Z[dC]

<latexit sha1_base64="ZOxB2CBS85uUh92sGafgrB5fYYo=">AAAB8HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LLYjcsK9iHToWQymTY0yQxJRihDv8KNC0Xc+jnu/BvTdhbaeiBwOOdecu4JU860cd1vZ219Y3Nru7RT3t3bPzisHB13dJIpQtsk4YnqhVhTziRtG2Y47aWKYhFy2g3HzZnffaJKs0Q+mElKA4GHksWMYGOlxz6TBvlRMxhUqm7NnQOtEq8gVSjQGlS++lFCMkGlIRxr7XtuaoIcK8MIp9NyP9M0xWSMh9S3VGJBdZDPA0/RuVUiFCfKPhtgrv7eyLHQeiJCOymwGellbyb+5/mZiW+CnMk0M1SSxUdxxpFJ0Ox6FDFFieETSzBRzGZFZIQVJsZ2VLYleMsnr5JOveZd1ur3V9XGbVFHCU7hDC7Ag2towB20oA0EBDzDK7w5ynlx3p2PxeiaU+ycwB84nz8sJI/+</latexit>

In action, perform functional integral over the space of all curves.

Mean string field theory actionPresent the action of mean string field theory:

Idea: like normal Landau-Ginzburg, write down most general* dynamics for string field that respects the symmetries…

Basic degree of freedom Ψ[C] is a functional; even evaluating the action requires a functional integral over the space of closed curves.

Difficult but well-posed problem: what does this theory describe?

Easier than “real” string field theory; not UV complete (see Rey 1989) * “most general” is hard; in this talk I neglect some things.

S =

Z[dC]

✓1

L[C]

Ids

� †[C]

��µ⌫(s)

� [C]

��µ⌫(s)+M2 †[C] [C] + �| [C]|4

◆

<latexit sha1_base64="qRt83i+1x3itNp0KsRlLt2Cr4ys=">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</latexit>

Unbroken phase of mean string field theory

V ( ) =<latexit sha1_base64="z7ThVfTVrQisT9Ui/CTlRarCgeU=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBDiJexGQS9C0IvHCOaByRJmJ73JkNnZZWZWCCF/4cWDIl79G2/+jZNkD5pY0FBUddPdFSSCa+O6387K6tr6xmZuK7+9s7u3Xzg4bOg4VQzrLBaxagVUo+AS64Ybga1EIY0Cgc1geDv1m0+oNI/lgxkl6Ee0L3nIGTVWemyUOonmZ+SadAtFt+zOQJaJl5EiZKh1C1+dXszSCKVhgmrd9tzE+GOqDGcCJ/lOqjGhbEj72LZU0gi1P55dPCGnVumRMFa2pCEz9ffEmEZaj6LAdkbUDPSiNxX/89qpCa/8MZdJalCy+aIwFcTEZPo+6XGFzIiRJZQpbm8lbEAVZcaGlLcheIsvL5NGpeydlyv3F8XqTRZHDo7hBErgwSVU4Q5qUAcGEp7hFd4c7bw4787HvHXFyWaO4A+czx/Js4+u</latexit>

Hard to solve string field equations of motion. In limit curve very large, find area law solution to quadratic action!

[C] ⇠ e�MArea[C]<latexit sha1_base64="lrOdOWTLYTlNRXQkzQm7Ag1cO6Q=">AAACDXicbVC7TsMwFHV4lvIKMLJYFCQWqqQgwVjowoJUJPqQklA5rtNatZPIdpCqKD/Awq+wMIAQKzsbf4PTZoCWI1k6Oufe63uPHzMqlWV9GwuLS8srq6W18vrG5ta2ubPbllEiMGnhiEWi6yNJGA1JS1HFSDcWBHGfkY4/auR+54EISaPwTo1j4nE0CGlAMVJa6pmHbiyp0/CgKymH5D49uYEuR2ooeHqpB2kry3pmxapaE8B5YhekAgo0e+aX249wwkmoMENSOrYVKy9FQlHMSFZ2E0lihEdoQBxNQ8SJ9NLJNRk80kofBpHQL1Rwov7uSBGXcsx9XZkvKme9XPzPcxIVXHgpDeNEkRBPPwoSBlUE82hgnwqCFRtrgrCgeleIh0ggrHSAZR2CPXvyPGnXqvZptXZ7VqlfFXGUwD44AMfABuegDq5BE7QABo/gGbyCN+PJeDHejY9p6YJR9OyBPzA+fwCXZ5tF</latexit>

Expected behavior in the unbroken phase! “Confinement” without gauge bosons (relation to loop formulation of QCD by Migdal, Makeenko?)

S =

Z[dC]

✓1

L[C]

Ids

� †[C]

��µ⌫(s)

� [C]

��µ⌫(s)+M2 †[C] [C] + �| [C]|4

◆

<latexit sha1_base64="qRt83i+1x3itNp0KsRlLt2Cr4ys=">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</latexit>

Unbroken phase:

Broken phase of mean string field theory

Broken phase:

V ( ) =<latexit sha1_base64="z7ThVfTVrQisT9Ui/CTlRarCgeU=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBDiJexGQS9C0IvHCOaByRJmJ73JkNnZZWZWCCF/4cWDIl79G2/+jZNkD5pY0FBUddPdFSSCa+O6387K6tr6xmZuK7+9s7u3Xzg4bOg4VQzrLBaxagVUo+AS64Ybga1EIY0Cgc1geDv1m0+oNI/lgxkl6Ee0L3nIGTVWemyUOonmZ+SadAtFt+zOQJaJl5EiZKh1C1+dXszSCKVhgmrd9tzE+GOqDGcCJ/lOqjGhbEj72LZU0gi1P55dPCGnVumRMFa2pCEz9ffEmEZaj6LAdkbUDPSiNxX/89qpCa/8MZdJalCy+aIwFcTEZPo+6XGFzIiRJZQpbm8lbEAVZcaGlLcheIsvL5NGpeydlyv3F8XqTRZHDo7hBErgwSVU4Q5qUAcGEp7hFd4c7bw4787HvHXFyWaO4A+czx/Js4+u</latexit>

S =

Z[dC]

✓1

L[C]

Ids

� †[C]

��µ⌫(s)

� [C]

��µ⌫(s)+M2 †[C] [C] + �| [C]|4

◆

<latexit sha1_base64="qRt83i+1x3itNp0KsRlLt2Cr4ys=">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</latexit>

[C] = v<latexit sha1_base64="3zFmpVk5AwPoHEMsPJU6Ux+hRMI=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgl6EYi8eK9gP2F1KNs22odlsSLKFUvo3vHhQxKt/xpv/xrTdg7Y+GHi8N8PMvEhypo3rfjuFjc2t7Z3ibmlv/+DwqHx80tZppghtkZSnqhthTTkTtGWY4bQrFcVJxGknGjXmfmdMlWapeDITScMEDwSLGcHGSkEgNfMbIbpDY9QrV9yquwBaJ15OKpCj2St/Bf2UZAkVhnCste+50oRTrAwjnM5KQaapxGSEB9S3VOCE6nC6uHmGLqzSR3GqbAmDFurviSlOtJ4kke1MsBnqVW8u/uf5mYlvwykTMjNUkOWiOOPIpGgeAOozRYnhE0swUczeisgQK0yMjalkQ/BWX14n7VrVu6rWHq8r9fs8jiKcwTlcggc3UIcHaEILCEh4hld4czLnxXl3PpatBSefOYU/cD5/AH+WkKw=</latexit>

(This is like a perimeter law, i.e. local dependence on C).

[C] = veiRC A(x)

<latexit sha1_base64="kRfgctp4/9GAKGQ38pzoYGBDz2E=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhahbkpSBd0I1W5cVrAPSGKZTCft0MkkzEyKJWTpxl9x40IRt36CO//G6WOhrQcuHM65l3vv8WNGpbKsbyO3tLyyupZfL2xsbm3vmLt7TRklApMGjlgk2j6ShFFOGooqRtqxICj0GWn5g9rYbw2JkDTid2oUEy9EPU4DipHSUsc8dGNJnZoHL+EQkvuUQpdy1UlrGbwqPZxkHbNola0J4CKxZ6QIZqh3zC+3G+EkJFxhhqR0bCtWXoqEopiRrOAmksQID1CPOJpyFBLppZNHMnislS4MIqGLKzhRf0+kKJRyFPq6M0SqL+e9sfif5yQquPBSyuNEEY6ni4KEQRXBcSqwSwXBio00QVhQfSvEfSQQVjq7gg7Bnn95kTQrZfu0XLk9K1avZ3HkwQE4AiVgg3NQBTegDhoAg0fwDF7Bm/FkvBjvxse0NWfMZvbBHxifP2CfmEo=</latexit>

Integrate out curves, find gapless gauge-boson Goldstone mode, as expected on general grounds (Hofman, NI; Lake). Can show other modes are gapped.

Structure of phases works out! (In 4d EM, this is the photon).

Low-lying fluctuations:

S ⇠ #v2Z

ddx(dA)2<latexit sha1_base64="rV0BpEaP5K/8z2028nsndQsljoU=">AAACCnicbVC7TsMwFHXKq5RXgJHFECGVpUoKEowFFsYi6ENq0spxnNaq40S2U1FFnVn4FRYGEGLlC9j4G9zHAC1HutLROffq3nv8hFGpbPvbyC0tr6yu5dcLG5tb2zvm7l5dxqnApIZjFoumjyRhlJOaooqRZiIIinxGGn7/euw3BkRIGvN7NUyIF6EupyHFSGmpYx7eQVfSCLoWHLTL0KVcwaCdBaMHWAwuT7TUMS27ZE8AF4kzIxaYodoxv9wgxmlEuMIMSdly7ER5GRKKYkZGBTeVJEG4j7qkpSlHEZFeNnllBI+1EsAwFrr0KRP190SGIimHka87I6R6ct4bi/95rVSFF15GeZIqwvF0UZgyqGI4zgUGVBCs2FAThAXVt0LcQwJhpdMr6BCc+ZcXSb1cck5L5dszq3I1iyMPDsARKAIHnIMKuAFVUAMYPIJn8ArejCfjxXg3PqatOWM2sw/+wPj8AUsVmBk=</latexit>

Transition?

There is a ”mean-field” transition at Mc = 0.

S =

Z[dC]

✓1

L[C]

Ids

� †[C]

��µ⌫(s)

� [C]

��µ⌫(s)+M2 †[C] [C] + �| [C]|4

◆

<latexit sha1_base64="qRt83i+1x3itNp0KsRlLt2Cr4ys=">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</latexit>

[C] ⇠ exp⇣�pM2 �M2

cA⌘

<latexit sha1_base64="tb3Ieeh1Ndt4Ie9ZQcW7LGUna+g=">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</latexit>

Mean-field scaling of string tension at critical point! Compare with data?

3d Ising numerics (‘80s): same exponent is 1.26 ≠ 0.5.

Upper critical dimension of MSFT is (at least) 8 > 3. Also possible issues w/ universality, but still fun to compare.

Future directions

We do not yet fully understand this theory; (ask me about things I swept

under the rug!) but I think we can.

If we use higher form symmetries (and anomalies), what is the new Landau paradigm? Does it include all phases of matter?

Can string theory – in this putative “mean string field theory” form -- help us understand critical phenomena and new fixed points? Or vice-versa?

Fermionic strings from 3d Ising model.

[Polyakov, 80s]

Disorder operator:

µ(C) ⌘Q

hiji?SC ,@SC=C e�2��i�j .

µ is independent of local changes in SC by �i ! ��i

symmetry. SC is a branch cut for �i.

a1···aL(C) ⌘ µ(C)QL

s=1�(xs + eas)

(xs = center of link s) satisfies

a1···aL(C) = cosh(2�) a1···as+1,as+1,···aL(C)

� sinh(2�) a1···as�1,a0s,as+2,a0

s+2,as+1,···aL(C +⇧as)

Links like free Dirac particles, connected by

unbreakability of domain wall.

This description is shared by the RNS superstring.

µ�xµ � x0

µ

�|physi = 0.

Summary

1. Higher form symmetries are a new kind of global symmetry that may let us build a new Landau paradigm.

2. Possible to build a Landau-Ginzburg “mean string field theory” that non-perturbatively describes the condensation of strings.

3. Line operators has an area law in the unbroken phase, and Goldstone modes in the broken phase.

4. Theory is not yet well-understood; much to be explored.

��µ⌫<latexit sha1_base64="6KBSgrPxcxp2Yb1tDxpp6fMojOI=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy4r2Ac0sUwmk3bozCTMQyixC3/FjQtF3Pob7vwbp20W2nrgwuGce7n3nihjVGnP+3aWlldW19ZLG+XNre2dXXdvv6VSIzFp4pSlshMhRRgVpKmpZqSTSYJ4xEg7Gl5P/PYDkYqm4k6PMhJy1Bc0oRhpK/XcwyAmTCMYKNrn6D4PuAmEGffcilf1poCLxC9IBRRo9NyvIE6x4URozJBSXd/LdJgjqSlmZFwOjCIZwkPUJ11LBeJEhfn0/jE8sUoMk1TaEhpO1d8TOeJKjXhkOznSAzXvTcT/vK7RyWWYU5EZTQSeLUoMgzqFkzBgTCXBmo0sQVhSeyvEAyQR1jaysg3Bn395kbRqVf+sWrs9r9SvijhK4Agcg1PggwtQBzegAZoAg0fwDF7Bm/PkvDjvzsesdckpZg7AHzifP1htllE=</latexit>

C<latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> The End.

A little philosophy

Grand existing tradition since ‘70s to reformulate Yang-Mills theory in terms of gauge-invariant loop equations (Migdal, Makeenko, Polyakov…)

But what is Yang-Mills theory, really? Gauge symmetry is not a symmetry but a redundancy.

Candidate answer: pure Yang-Mills theory is a theory that has a ZN 1-form global symmetry. It is gratifying that the simplest way to build dynamics around that organizing principle gives us an area law in the unbroken phase.

Zd4xF a

µ⌫Fµ⌫a

<latexit sha1_base64="Wl3/s3obDVQCOv/t91uBvZR1xzQ=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6XgqiS1oMuiUFxWsA9o0jCZTNuhk0mYmYgl5Avc+CtuXCji1rU7/8Zpm4W2Hhg4c8693HuPHzMqlWV9G4W19Y3NreJ2aWd3b//APDzqyCgRmLRxxCLR85EkjHLSVlQx0osFQaHPSNefXM/87j0Rkkb8Tk1j4oZoxOmQYqS05JkVh3IFg0H9ATa91AkThyfZIEUZbA7yL0SZZ5atqjUHXCV2TsogR8szv5wgwklIuMIMSdm3rVi5KRKKYkaykpNIEiM8QSPS15SjkEg3nZ+TwYpWAjiMhH56ubn6uyNFoZTT0NeVIVJjuezNxP+8fqKGl25KeZwowvFi0DBhUEVwlg0MqCBYsakmCAuqd4V4jATCSidY0iHYyyevkk6tap9Xa7f1cuMqj6MITsApOAM2uAANcANaoA0weATP4BW8GU/Gi/FufCxKC0becwz+wPj8Ad7XnAE=</latexit>

Issues with universality

The action discussed is not the most general:1. Intrinsically stringy (topology-changing) terms exist:

C1<latexit sha1_base64="b342B6lxS9u4MYepTTuvsFWGG+U=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9FjsxWNF+wFtKJvtpF262YTdjVBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WBcn/ntJ1Sax/LRTBL0IzqUPOSMGis91Ptev1R2K+4cZJV4OSlDjka/9NUbxCyNUBomqNZdz02Mn1FlOBM4LfZSjQllYzrErqWSRqj9bH7qlJxbZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMT3vgZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTtCF4yy+vkla14l1WqvdX5dptHkcBTuEMLsCDa6jBHTSgCQyG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gC8MY1v</latexit>

C2<latexit sha1_base64="1qkECvOB+R2I9+RxbCR24LYk+kg=">AAAB6nicdVDLSsNAFL3xWeur6tLNYBFchaQPWnfFblxWtA9oQ5lMJ+3QySTMTIQS+gluXCji1i9y5984bSOo6IELh3Pu5d57/JgzpR3nw1pb39jc2s7t5Hf39g8OC0fHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+bC795TqVgk7vQspl6Ix4IFjGBtpNvmsDQsFB27WqpfVitoRcr1jFRqyLWdJYqQoTUsvA9GEUlCKjThWKm+68TaS7HUjHA6zw8SRWNMpnhM+4YKHFLlpctT5+jcKCMURNKU0Gipfp9IcajULPRNZ4j1RP32FuJfXj/RQd1LmYgTTQVZLQoSjnSEFn+jEZOUaD4zBBPJzK2ITLDERJt08iaEr0/R/6RTst2yXbqpFBtXWRw5OIUzuAAXatCAa2hBGwiM4QGe4Nni1qP1Yr2uWtesbOYEfsB6+wRSr43X</latexit>

C3<latexit sha1_base64="3iDpH6MRg0ngr4gVSKGCmFgZBDA=">AAAB6nicdVDLSsNAFL3xWeur6tLNYBFchaQPWnfFblxWtA9oQ5lMJ+3QySTMTIQS+gluXCji1i9y5984bSOo6IELh3Pu5d57/JgzpR3nw1pb39jc2s7t5Hf39g8OC0fHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+bC795TqVgk7vQspl6Ix4IFjGBtpNvmsDwsFB27WqpfVitoRcr1jFRqyLWdJYqQoTUsvA9GEUlCKjThWKm+68TaS7HUjHA6zw8SRWNMpnhM+4YKHFLlpctT5+jcKCMURNKU0Gipfp9IcajULPRNZ4j1RP32FuJfXj/RQd1LmYgTTQVZLQoSjnSEFn+jEZOUaD4zBBPJzK2ITLDERJt08iaEr0/R/6RTst2yXbqpFBtXWRw5OIUzuAAXatCAa2hBGwiM4QGe4Nni1qP1Yr2uWtesbOYEfsB6+wRUM43Y</latexit>

S = g

Z[dC1,2,3]�[C1 � (C2 + C3)]

†[C1] [C2] [C3]<latexit sha1_base64="pIu382+aN2OkeTpWmFQxT0CpW08=">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</latexit>

Likely important, but hard to treat! (Generated along RG? Confusion about meaning of locality in loop space)

2. Couplings could depend on proper length of curve, e.g.

M2(L[C]) = M20 +M2

11

L[C]⇤+ · · ·

<latexit sha1_base64="diCHnLrDehLKicaFdEnWIJOZvlk=">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</latexit>

Such issues are likely important near phase transition (especially if you want

to compare with lattice…). Need to build a framework for RG in loop space…