# Dual Description of Gauge Theories from an Iterative Noether Approach

###### Abstract

An iterative Noether scheme, advocated by Deser, is used to introduce gauge
invariant couplings to nonrelativistic matter with global symmetries related
to usual charge conservation and dipole conservation recently discussed in
fractonic theories. No reference to any gauge principle, fractonic or
otherwise, is required. A dual description is found where the theory is
defined either in terms of the usual vector gauge field or, alternatively, in
terms of higher derivatives of a symmetric tensor field given in fractonic
theories. A connection between these two descriptions is obtained by providing
an explicit map between the vector and tensor fields. This method yields a
novel ‘minimal’ prescription involving tensor fields which is identified with
the usual minimal prescription that involves vector fields, by using the
mapping. It also spells out the structure of the pure gauge field action in
both formulations. Extension of the abelian $U(1)$ invariance to the
nonabelian $SU(N)$ invariance is done for the standard formulation.

Rabin<EMAIL_ADDRESS>
S. N. Bose National Centre for Basic Sciences , JD Block, Sector III, Salt
Lake City,

Kolkata -700 106, India

##  1 Introduction

The recent study of a new class of nonrelativistic symmetric higher rank
tensor gauge theories [1, 2, 3, 4] has led to the intriguing possibility of
the existence of novel topological excitations, called fractons. Their
analysis has found applications in various condensed matter contexts [5, 6, 7,
8, 9] and even in gravitational physics [10, 11, 12]. They have the unusual
property of restricted motion which may vary from complete immobility to
mobility in subdimensional spaces. This is a consequence of nonstandard
conservation laws since the matter sector admits general global symmetries
which, apart from the usual charge symmetry, also has dipole symmetries. This
makes even the pure matter action (‘free’ fractonic theory) non-gaussian,
leading to nonlinear equations of motion. The matter action is constructed up
to leading orders in the derivatives by using the specific structure of the
charge and dipole moment symmetries. Interactions are introduced by exploiting
the fractonic gauge principle [13] which demands that the original invariance
of the pure matter sector under global symmetries is retained under the local
transformations. This requires the introduction of a new symmetric higher rank
tensor (fractonic) field with a specific transformation property.

Simultaneously, in a parallel but related development, it was shown [4] that
interactions may also be introduced using the usual vector field where the
gauge principle now ensures that all ordinary derivatives are replaced by
their covariant expressions. Thus there are two competing approaches, based on
usual vector fields (standard formulation) or on symmetric tensor fields
(nonstandard formulation).

Long back Deser [14] developed an approach where interactions are inducted by
an iterative Noether prescription without any need for a gauge principle. This
method, which emphasises the role of interactions, proceeds by adding to the
original theory a source that is obtained from the free part itself, followed
by the addition of a further source due to this one and so on till the
iteration stops due to the lack of any new source. It was originally applied
to obtain the $SU(2)$ Yang-Mills theory (a nonlinear theory) from the linear
Maxwell theory where self interactions were suitably inducted. Several
examples of this approach can be found including an extension to matter fields
whose interactions with gauge fields naturally lead to the minimal coupling
[15, 16, 17, 18, 19, 20]. All these applications consider relativistic
theories where the basic starting lagrangian is quadratic with linear
equations of motion.

In this paper we use this formalism in a completely different context, namely,
nonrelativistic theories where the pure matter sector is nonlinear. It is
therefore a new application of the method which has to be generalised to
incorporate the fact that the starting lagrangian is not quadratic and also
includes higher derivatives. The purpose of the paper, however, is not merely
to provide a new (and somewhat nontrivial) application. More importantly, it
yields fresh insights in the construction of interacting gauge theories
beginning from a matter sector that admits generalised global symmetry
yielding conservation of both charge and dipole moment.

Starting from a pure matter sector that has both these symmetries, the
implementation of the formalism leads to a dual description of interacting
gauge theories. In one scenario the vector current is gauged with the usual
vector field and the interacting theory turns out to be the one where all
ordinary derivatives are replaced by their covariant derivatives. Thus the
standard minimal coupling prescription obtained by the gauge principle is
reproduced. However there is another possibility. The consequence of two
global symmetries necessarily leads to a construction where the vector current
is expressed as the gradient of a second rank symmetric tensor current. This
comes out naturally in the Noether prescription. Instead of considering the
vector current as the source, the tensor current is taken as the source in the
iterative scheme. This leads to the fractonic gauge theory recently considered
in the literature [1, 2, 3, 4, 6, 12]. As already mentioned the two currents
are related by a spatial derivative. This is used to find an exact map between
the vector field and the tensor gauge field, thereby relating the two
descriptions. We further extend the analysis to include more global
symmetries. The dual picture still holds, except that the relation connecting
the gauge fields becomes different.

The iterative Noether scheme is introduced in section 2 where, starting from
the free Schroedinger theory, the minimally coupled interacting theory is
obtained without using any gauge principle. Dual description of gauge
theories, the central point of the paper, is treated in section 3. There is an
overview of conservation laws related to charge and dipole moment symmetries.
Next, the matter sector is given and interactions are found by the method of
iteration. This gives a dual description expressed either in terms of a
standard vector field or, alternatively, in terms of a symmetric second rank
fractonic tensor field. A connection between the two formulations is found.
The interactions involving the tensor field can also be obtained by a novel
‘minimal prescription’. Its connection with the conventional minimal
prescription is elucidated. The explicit structures for the pure gauge field
lagrangians are given. Finally, an extension to $SU(N)$ invariance is done.
Section 4 discusses similar findings if other conservation laws besides charge
and dipole moment exist. Conclusions are given in section 5.

##  2  Schroedinger Theory from Noether Approach

Here we show how interactions in the usual Schroedinger theory are introduced
by adopting an iterative Noether prescription [14] instead of resorting to any
gauge principle. Consider the free Schroedinger theory for a complex scalar
field,

${\cal
L}_{0}=\frac{i}{2}\phi^{*}({\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}_{0})\phi-\frac{1}{2m}\partial_{i}\phi^{*}\partial_{i}\phi$
(1)

It has a global $U(1)$ symmetry,

$\phi\rightarrow e^{i\alpha}\phi\,\,;\,\,\phi^{*}\rightarrow
e^{-i\alpha}\phi^{*}$ (2)

which leads to the conservation law,

$\partial_{0}j_{0}-\partial_{i}j_{i}=0$ (3)

from Noether’s first theorem. This theorem yields the explicit structures for
the currents,

$\alpha j^{0}=\frac{\partial{\cal
L}_{0}}{\partial(\partial_{0}\phi)}\delta\phi+c.c.\,\,;\,\,\alpha
j^{i}=\frac{\partial{\cal L}_{0}}{\partial(\partial_{i}\phi)}\delta\phi+c.c.$
(4)

where $c.c.$ stands for complex conjugate. Using $\delta\phi=i\alpha\phi$ we
obtain,

$j^{0}=-j_{0}=-\phi^{*}\phi\,\,\,;\,\,\,j^{i}=j_{i}=\frac{i}{2m}\Big{(}(\partial_{i}\phi)\phi^{*}-(\partial_{i}\phi^{*})\phi\Big{)}$
(5)

The standard way to introduce interactions is to make the transformation
parameter $\alpha$ local and adopt the gauge principle which ensures
invariance under the local symmetry by replacing the usual derivatives by
covariant derivatives. Here we follow a different path.

We obtain the final lagrangian by an iterative process. The analysis is done
separately for the time and space components. For the time component, the
first correction to the lagrangian is obtained by adding the term,

${\cal L}_{1}=j^{0}A_{0}=-\phi^{*}\phi A_{0}$ (6)

Since there are no derivatives, the Noether prescription (4) does not yield
anything and the iteration concludes. Thus the final lagrangian, for the time
part, is given by,

${\cal L}_{time}={\cal L}_{0}+{\cal
L}_{1}=\frac{i}{2}\phi^{*}({\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}_{0})\phi-\phi^{*}\phi
A_{0}$ (7)

which may also be expressed in the standard minimal coupling form,

${\cal
L}_{time}=\frac{i}{2}\Big{(}\phi^{*}D_{0}\phi-\phi(D_{0}\phi)^{*}\Big{)};\hskip
28.45274ptD_{0}\phi=(\partial_{0}+iA_{0})\phi$ (8)

We next repeat the analysis for the space part. The first correction is
obvious,

${\cal
L}_{1}=j^{i}A_{i}=\frac{i}{2m}\Big{(}(\partial_{i}\phi)\phi^{*}-(\partial_{i}\phi^{*})\phi\Big{)}A_{i}$
(9)

Since derivatives occur, there is a next correction to be evaluated by the
Noether prescription, replacing ${\cal L}_{0}$ by ${\cal L}_{1}$ in (4). We
find,

$\alpha j_{1}^{i}=\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\phi)}\delta\phi+c.c.$ (10)

which yields,

$j_{1}^{i}=-\frac{1}{m}\phi^{*}\phi A_{i}$ (11)

The contribution to the lagrangian is given by,

${\cal L}_{2}=\frac{1}{2}j_{1}^{i}A_{i}=-\frac{1}{2m}\phi^{*}\phi A_{i}^{2}$
(12)

The half factor is necessary to reproduce the expression for the current from
a variation of the action,

$j_{1}^{i}(x)=\frac{\delta}{\delta A_{i}(x)}\int\,{\cal L}_{2}$ (13)

Since there are no derivatives, the iteration stops and the final lagrangian
for the spatial part is given by adding the contributions,

${\cal L}_{space}={\cal L}_{0}+{\cal L}_{1}+{\cal
L}_{2}=-\frac{1}{2m}\partial_{i}\phi^{*}\partial_{i}\phi+\frac{i}{2m}\Big{(}(\partial_{i}\phi)\phi^{*}-(\partial_{i}\phi^{*})\phi\Big{)}A_{i}-\frac{1}{2m}\phi^{*}\phi
A_{i}^{2}$ (14)

which may be rearranged to display the minimal coupling form,

${\cal L}_{space}=-\frac{1}{2m}(D_{i}\phi)^{*}(D_{i}\phi)\hskip
28.45274ptD_{i}\phi=(\partial_{i}+iA_{i})\phi$ (15)

The complete interacting lagrangian, which is a sum of (8) and (15), is given
by,

${\cal L}_{I}={\cal L}_{time}+{\cal
L}_{space}=\frac{i}{2}\Big{(}\phi^{*}D_{0}\phi-\phi(D_{0}\phi)^{*}\Big{)}-\frac{1}{2m}(D_{i}\phi)^{*}(D_{i}\phi)$
(16)

Expectedly, the above lagrangian is invariant under the local gauge
transformations,

$\phi\rightarrow e^{i\alpha(x)\phi}\,\,;\,\,A_{\mu}\rightarrow
A_{\mu}-\partial_{\mu}\alpha$ (17)

The dynamics for the gauge field is given by the usual Maxwell type
term,222Contrary to the relativistic case, the coefficients $C_{1},C_{2}$,
multiplying the electric and magnetic parts can be different.

${\cal L}_{gauge}=C_{1}F_{0i}^{2}-C_{2}F_{ij}^{2}\hskip
28.45274ptF_{0i}=\partial_{0}A_{i}-\partial_{i}A_{0}\hskip
14.22636ptF_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$ (18)

The extension to the nonabelian case may be done along similar lines, This
analysis is deferred to section 3.2.3 when the higher derivative example is
treated.

##  3 Dual Description of Gauge Theories

Recently a new class of higher derivative gauge theories which exhibit, apart
from standard charge conservation, other conservation laws related to the
dipole symmetry have come to the fore. Their study has led to the discovery
and understanding of a new type of topological matter called fractons.

After a general overview that discusses the various global symmetries in
fractonic theories, we show that the Noether prescription correctly reproduces
the various conservation laws associated with them in a concrete example
involving complex scalars. The iterative Noether approach is then applied to
obtain the explicit couplings of these scalars to gauge fields. The higher
derivative nature of the theory allows for either minimal or nonminimal
couplings, leading to a dual description.

###  3.1 Overview of Symmetries and Conservation Laws

Let us consider some matter theory that has, as sources, $J_{0}$ and a
symmetric tensor $J_{ij}$ satisfying [4, 12],

$\partial_{o}J_{0}-\partial_{i}\partial_{j}J_{ij}=0$ (19)

This can be achieved, for instance, by coupling these sources to $A^{0}$ and
$A_{ij}$, respectively. Here $A_{ij}$ is a symmetric tensor gauge field that
can, among others, characterise a fractonic gauge field that transforms as,

$\delta A_{ij}=\partial_{i}\partial_{j}\alpha$ (20)

while $A_{0}$ satisfies the standard rule (17). Gauge invariance under these
transformations ensures the conservation law (19).

We now analyse the global symmetries. There is the obvious ordinary global
symmetry with currents,

$j_{0}=J_{0}\hskip 42.67912ptj_{i}=\partial_{j}J_{ij}$ (21)

satisfying,

$\partial_{0}j_{0}-\partial_{i}j_{i}=0$ (22)

with the usual conserved charge,

$Q=\int_{space}j_{0}=\int_{space}J_{0}$ (23)

Further, it has a vector global symmetry with currents,

$j_{0}^{i}=x^{i}J_{0}\hskip 28.45274ptj_{ij}=x^{j}\partial_{k}J^{ki}-J_{ij}$
(24)

having the conservation law,

$\partial_{0}j_{0}^{i}-\partial_{j}j_{ji}=0$ (25)

The corresponding conserved charge is given by,

$Q^{i}=\int_{space}j_{0}^{i}=\int_{space}x^{i}J_{0}$ (26)

The global symmetry (25), (26) is interpreted as a dipole symmetry just as the
usual one (23) is considered as a charge symmetry. It implies that a single
charge cannot move although restricted mobility is allowed for a system of
charges. These features are characteristic of fractons.

###  3.2 Matter Sector and Interactions

An explicit construction of the matter sector in terms of complex scalars,
satisfying the conservation laws (22) and (25), has been given in the
literature. We briefly review this construction and then show how gauge
couplings are introduced using the iterative Noether prescription. The
couplings can be interpreted either in terms of a standard gauge field or,
alternatively, in terms of a higher rank tensor field that occurs in fractonic
theories.

Since the theory obeys both charge and dipole conservation, the appropriate
symmetry transformation on the complex scalar takes the form,

$\phi(x)\rightarrow e^{i\alpha(x)}\phi(x)\hskip
28.45274pt\alpha(x)=\alpha_{0}+\alpha_{i}x^{i}$ (27)

where $\alpha_{0},\alpha_{i}$ are global (constant) parameters. The matter
lagrangian invariant under the above transformations modifies the spatial
derivative terms of (1) and is given by [13, 4],

${\cal
L}_{0}=\frac{i}{2}\phi^{*}({\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}_{0})\phi-s\,\partial_{i}(\phi^{*}\phi)\partial_{i}(\phi^{*}\phi)+t\,(i\phi^{*2}\partial_{i}\phi\partial^{i}\phi-i\phi^{2}\partial_{i}\phi^{*}\partial^{i}\phi^{*})+u\,|{\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi}|^{2}-\lambda|\phi|^{4}$
(28)

where the leading order terms containing fourth powers of the field as well as
the derivatives have only been retained.

The conserved currents are found by the Noether procedure. Noting that,

$\delta\phi=i\alpha\phi\hskip 28.45274pt\delta\phi^{*}=-i\alpha\phi^{*}$ (29)

the time component of the current is given by,

$\alpha j^{0}=\frac{\partial{\cal
L}_{0}}{\partial(\partial_{0}\phi)}\delta\phi+\frac{\partial{\cal
L}_{0}}{\partial(\partial_{0}\phi^{*})}\delta\phi^{*}=\alpha\phi^{*}\phi$ (30)

so that,

$j_{0}=\phi^{*}\phi$ (31)

This has the same form as conventional theories because there is no time
dependence in the transformation parameter $\alpha$.

For the computation of the space component, we consider the various terms one
by one using the same definition as (4). There is no contribution from the
term involving $s$. For the $t$ term the result is,

$\alpha_{0}j^{i}=\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\phi)}\delta\phi+\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\phi^{*})}\delta\phi^{*}=-\alpha_{0}\,\partial_{i}|\phi|^{4}$
(32)

so that,

$j_{i}=-\,\partial_{i}|\phi|^{4}$ (33)

This can be put in the form $j_{i}=\partial_{j}J_{ij}$ (see 21), where,

$J_{ij}=-\delta_{ij}|\phi|^{4}$ (34)

This relation was earlier given in [4] by using a polar decomposition of the
fractonic field that simplified the algebra. Thus the general analysis in
section 3.1 goes through and both conservation laws (22) and (25) hold.
Observe that we have taken only the constant parameter $\alpha_{0}$ in (32) to
keep parity with the standard Noether definition.

The analysis for the $u$ term in (28) is more involved since it involves
double derivatives on the fields. In this case the derivation of the Noether
current is based on the Euler-Lagrange equation valid up to second order
derivatives which is given by,

$\frac{{\partial}{\cal L}}{\partial\phi}-\partial_{\mu}\frac{{\partial}{\cal
L}}{\partial(\partial_{\mu}\phi)}+\partial_{\mu}\partial_{\nu}\frac{{\partial}{\cal
L}}{\partial(\partial_{\mu}\partial_{\nu}\phi)}=0$ (35)

This leads to the Noether current, which is a generalization of (4),

$\displaystyle\alpha_{0}j^{i}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\phi)}\delta\phi+\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\partial_{j}\phi)}\partial_{j}(\delta\phi)-\Big{(}\partial_{j}\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\partial_{j}\phi)}\Big{)}(\delta\phi)$ (36)
$\displaystyle+$ $\displaystyle\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\phi^{*})}\delta\phi^{*}+\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\partial_{j}\phi^{*})}\partial_{j}(\delta\phi^{*})-\Big{(}\partial_{j}\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\partial_{j}\phi^{*})}\Big{)}(\delta\phi^{*})$
(37)

Substituting the $u$-term from (28), keeping terms proportional to
$\alpha_{0}$, yields the following result,

$\displaystyle\alpha_{0}j^{i}=-i\alpha_{0}\,\partial_{j}\Big{(}\phi^{2}(\phi^{*}\partial_{i}\partial_{j}\phi^{*}-\partial_{i}\phi^{*}\partial_{j}\phi^{*})-\phi^{*2}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi)\Big{)}$
(38)

Equating terms proportional to $\alpha_{0}$, we find,

$j^{i}=\,\partial_{j}J_{ij}$ (39)

where,

$J_{ij}=-i\Big{(}\phi^{2}(\phi^{*}\partial_{i}\partial_{j}\phi^{*}-\partial_{i}\phi^{*}\partial_{j}\phi^{*})-\phi^{*2}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi)\Big{)}$
(40)

which reproduces the structure (21) from the general discussion. Incidentally,
this relation appeared earlier in [4] by using equations of motion in polar
form.

####  3.2.1 Gauge Couplings

In this section we discuss the introduction of gauge couplings to the matter
fields by adopting the iterative Noether prescription [14]. This method,
successful in conventional theories, also holds here, leading to new insights.
Especially, it provides a minimal coupling prescription for tensor gauge
fields encountered here. Also, this analysis is used to give a map that
connects the tensor field with the usual vector field.

We begin from the lagrangian (28) and first consider the temporal part. This
has the same form as standard theories, eventually leading to the minimally
coupled form (8).

The remaining terms involving spatial derivatives are now considered. For the
$s$ term, as already stated there is no contribution to the Noether current.
Thus there is no coupling associated with this term. To see that the same
conclusion also follows by adopting the minimal prescription, the ordinary
derivatives are replaced by covariant derivatives. The result is,

$\Big{(}(\partial_{i}-iA_{i})\phi^{*}\phi+\phi^{*}(\partial_{i}+iA_{i})\phi\Big{)}\Big{(}(\partial_{i}-iA_{i})\phi^{*}\phi+\phi^{*}(\partial_{i}+iA_{i})\phi\Big{)}$
(41)

The $A_{i}$ terms in each of the big brackets cancel out leaving just the
original terms. Thus there is no effect of the minimal coupling. This came out
naturally in the Noether scheme.

We next consider the $t$ term in (28). The current for this term was already
computed in (33). Thus the new addition in the lagrangian is given by,

${\cal L}_{1}=j_{i}A_{i}=-\,\partial_{i}|\phi|^{4}A_{i}$ (42)

The iterative Noether process continues since the lagrangian involves
derivatives. The corresponding current is obtained, as usual, by replacing
${\cal L}_{0}$ in (32) by ${\cal L}_{1}$,

$j^{i}=\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\phi)}\delta\phi+\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\phi^{*})}\delta\phi^{*}=0$ (43)

There is no contribution to the current and hence there is no correction to
the lagrangian. Also, the iteration terminates and the final expression for
the $t$ term is given by,

${\cal L}={\cal L}_{0}+{\cal
L}_{1}=(i\phi^{*2}\partial_{i}\phi\partial^{i}\phi-i\phi^{2}\partial_{i}\phi^{*}\partial^{i}\phi^{*})-\partial_{i}|\phi|^{4}A_{i}$
(44)

In this form invariance is realised by the fact that the change in the pure
matter sector under the local transformations ( where $\alpha(x)$ is arbitrary
and not confined to the special form given in (27) is precisely cancelled by
the change in the coupling term.

Although not obvious, the above lagrangian can be expressed in the standard
minimally coupled form,

${\cal
L}=i\phi^{*2}(\partial_{i}+iA_{i})\phi\,(\partial^{i}+iA^{i})\phi-i\phi^{2}(\partial_{i}-iA_{i})\phi^{*}\,(\partial^{i}-iA^{i})\phi^{*}$
(45)

Now gauge invariance under the local transformations becomes manifest, exactly
as happens for the usual case.

It is possible to rephrase the entire discussion in terms of nonminimal
couplings and nonstandard gauge fields, recently introduced in the literature
as fractonic gauge fields. The origin of this lies in the fact that the vector
current $j_{i}$ can be expressed as the gradient of a symmetric tensor current
$J_{ij}$ (see (33, 34). Thus, instead of writing the coupling term as in (42),
it is written as,

${\cal L}_{1}=J_{ij}A_{ij}=-\delta_{ij}|\phi|^{4}A_{ij}$ (46)

The complete lagrangian (44) is now expressed as,

${\cal L}={\cal L}_{0}+{\cal
L}_{1}=(i\phi^{*2}\partial_{i}\phi\partial^{i}\phi-i\phi^{2}\partial_{i}\phi^{*}\partial^{i}\phi^{*})-\delta_{ij}|\phi|^{4}A_{ij}$
(47)

Local gauge invariance333It is actually quasi invariance since the lagrangian
is invariant modulo a boundary term. is retained provided the tensor field
transforms as,

$A_{ij}\rightarrow A_{ij}+\partial_{i}\partial_{j}\alpha$ (48)

which reproduces the structure (20).

It is also possible to establish a connection between the standard and
nonstandard formulations by expressing the coupling in a dual form,

${\cal
L}_{1}=j_{i}A_{i}=\partial_{j}J_{ij}A_{i}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})J_{ij}$
(49)

by appropriately shifting derivatives.

Comparison with (46) yields 444This relation appeared earlier in footnote 10
of [4]. ,

$A_{ij}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})$ (50)

The consistency of this identification is proved by recalling the
transformation of the usual vector field (17) that immediately yields (48).
This gives a dual description of the theory where standard gauge fields with
minimal coupling are used or fractonic fields with nonminimal coupling555The
presence of derivatives in (50) indicates that the coupling $J_{ij}A_{ij}$ is
nonminimal.. Later on we discuss the complete theory, expressed either in
terms of a conventional gauge field or a fractonic field.

Let us now concentrate on the last $u$-term of the theory (28). The current is
defined in (39, 40). Hence the first correction to the lagrangian is given by,

${\cal
L}_{1}=-i\partial_{j}\Big{(}\phi^{2}(\phi^{*}\partial_{i}\partial_{j}\phi^{*}-\partial_{i}\phi^{*}\partial_{j}\phi^{*})-\phi^{*2}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi)\Big{)}A_{i}$
(51)

The ensuing analysis is simplified by rephrasing the theory in terms of a
fractonic field involving nonminimal coupling, following previously described
steps,

${\cal
L}_{1}=-i\Big{(}\phi^{2}(\phi^{*}\partial_{i}\partial_{j}\phi^{*}-\partial_{i}\phi^{*}\partial_{j}\phi^{*})-\phi^{*2}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi)\Big{)}A_{ij}$
(52)

where the field $A_{ij}$ is defined in (50). Since derivatives are involved,
the iterative process continues. The new current is obtained by concentrating
on the $\alpha_{0}$ part of (37), so that,

$\displaystyle j^{i}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\phi)}(i\phi)+\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\partial_{j}\phi)}\partial_{j}(i\phi)-\Big{(}\partial_{j}\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\partial_{j}\phi)}\Big{)}(i\phi)$ (53)
$\displaystyle+$ $\displaystyle\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\phi^{*})}(-i\phi^{*})+\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\partial_{j}\phi^{*})}\partial_{j}(-i\phi^{*})-\Big{(}\partial_{j}\frac{\partial{\cal
L}_{1}}{\partial(\partial_{i}\partial_{j}\phi^{*})}\Big{)}(-i\phi^{*})$ (54)
$\displaystyle=$ $\displaystyle 2\partial_{j}(|\phi|^{4}A_{ij})$ (55)

where ${\cal L}_{1}$ has been used from (52). In terms of the symmetric tensor
current, the result is,

$j^{i}=\partial_{j}J_{ij}\,,\hskip 28.45274ptJ_{ij}=2|\phi|^{4}A_{ij}$ (56)

Thus the contribution to the lagrangian is,

${\cal L}_{2}=|\phi|^{4}A_{ij}^{2}$ (57)

so that the functional derivative of the lagrangian (57) reproduces the
current (56),

$J_{ij}=\frac{\delta}{\delta A_{ij}}\int{\cal L}_{2}$ (58)

Since no further derivatives occur, the iteration is over and the complete
interacting lagrangian for the $u$\- term is given by adding the contributions
from (28), (52) and (57),

${\cal
L}=|{\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi}|^{2}-i\Big{(}\phi^{2}(\phi^{*}\partial_{i}\partial_{j}\phi^{*}-\partial_{i}\phi^{*}\partial_{j}\phi^{*})-\phi^{*2}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi)\Big{)}A_{ij}+|\phi|^{4}A_{ij}^{2}$
(59)

The iterative method clearly spells out the ‘minimal prescription’ that takes
one from the original ‘free’ theory, which here is the $u$-term in (28), to
the final form (59). The appropriate prescription is,

$\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi\rightarrow\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi-
iA_{ij}\phi^{2}$ (60)

Then the pure matter sector goes to the full interacting theory (59),

$|{\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi}|^{2}\rightarrow|\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi-
iA_{ij}\phi^{2}|^{2}$ (61)

reproducing (59). The currents in the interacting theory are likewise obtained
by the same prescription (60). From (40) we obtain,

$J_{ij}\rightarrow J_{ij}+2|\phi|^{4}A_{ij}$ (62)

The extra piece, expectedly, turns out to be the one found in (56).

Gauge invariance of the lagrangian (59) (or (60) is easily seen by noting the
covariant transformation property,

$(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi-
iA_{ij}\phi^{2})\rightarrow
e^{2i\alpha(x)}(\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi-
iA_{ij}\phi^{2})$ (63)

The nonstandard minimal prescription (60) can be related to the standard one
where ordinary derivatives are replaced by covariant derivatives. To see this
the double derivative term there is first symmetrised and then the minimal
prescription is imposed,

$\displaystyle\phi\partial_{i}\partial_{j}\phi$ $\displaystyle-$
$\displaystyle\partial_{i}\phi\partial_{j}\phi=\frac{1}{2}(\phi\partial_{i}\partial_{j}\phi+\phi\partial_{j}\partial_{i}\phi)-\partial_{i}\phi\partial_{j}\phi\rightarrow$
(64)
$\displaystyle\frac{1}{2}[\phi(\partial_{i}+iA_{i})(\partial_{j}+iA_{j})\phi$
$\displaystyle+$
$\displaystyle\phi(\partial_{j}+iA_{j})(\partial_{i}+iA_{i})\phi]-(\partial_{i}+iA_{i})\phi(\partial_{j}+iA_{j})\phi$
(65)

On simplification this expression yields,

$\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi\rightarrow{\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi}+\frac{i}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})\phi^{2}$
(66)

which, on using (50), is identical to (60).

In terms of the vector field, therefore, the interacting lagrangian, in the
standard formulation, has the structure,

${\cal
L}=|{\phi\partial_{i}\partial_{j}\phi-\partial_{i}\phi\partial_{j}\phi}+\frac{i}{2}(\partial_{i}A_{j}+\partial_{j}A_{i})\phi^{2}|^{2}$
(67)

One can also reproduce this form from the iterative Noether process by
directly working with the vector gauge field.

####  3.2.2 Gauge Field Lagrangian

To obtain the complete theory one has to include the gauge field lagrangian.
For the standard case, this is simply the conventional Maxwell type term given
in (18). Thus the complete lagrangian is given by,

${\cal
L}=C_{1}F_{0i}F_{0i}-C_{2}F_{ij}F_{ij}+\frac{i}{2}\phi^{*}({\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{0}})\phi+....$
(68)

and the ellipses denote terms that were calculated in the previous section and
involve both matter and gauge fields. Specifically these do not involve the
$A_{0}$ field666Note that in either description, the $A_{0}$ field appears. It
is the multiplier that enforces the relevant Gauss constraint, as shall be
shown shortly.. Since our purpose is to calculate the gauge transformation
properties, the irrelevant terms are dropped. The equation of motion for the
$A_{0}$ field is given by,

$2C_{1}\partial_{i}F_{0i}-\phi^{*}\phi=0$ (69)

The time derivative is eliminated in favour of the the canonical momenta which
is defined as,

$\pi_{i}=\frac{\partial{\cal L}}{\partial\dot{A}_{i}}=2C_{1}F_{0i}$ (70)

so that (69) takes the form,

$\partial_{i}\pi_{i}-\phi^{*}\phi=0$ (71)

The left side is just the Gauss operator that generates the standard gauge
transformation (17),

$\delta
A_{i}(x)=\\{A_{i}(x),\int_{space}\alpha\,(\partial_{j}\pi_{j}-\phi^{*}\phi)\\}=-\partial_{i}\alpha(x)$
(72)

For the dual description in terms of the nonstandard fractonic field $A_{ij}$,
the pure gauge field lagrangian is given by,777 Here also the coefficients of
the electric and magnetic terms can be different. These are taken equal for
simplifying the algebra.

${\cal L}_{gauge}=\frac{1}{2}(E_{ij}^{2}-B_{ij}^{2})$ (73)

where the electric and magnetic fields are defined as,

$E_{ij}=\partial_{i}\partial_{j}A_{0}+\partial_{0}A_{ij}\,,\hskip
28.45274ptB_{ij}=-\partial_{i}\partial_{k}A_{jk}+\partial_{j}\partial_{k}A_{ik}$
(74)

The above lagrangian is dictated by the fact that it is the simplest
generalisation of the Maxwell lagrangian that is gauge invariant under (48)
and (17). The complete lagrangian is therefore given by,

${\cal
L}=\frac{1}{2}(E_{ij}^{2}-B_{ij}^{2})+\frac{i}{2}\phi^{*}({\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{0}})\phi+....$
(75)

where the ellipses denote terms containing matter and gauge fields but do not
involve any $A_{0}$, similar to (68). The basic difference from (68) is in the
pure gauge terms, which is expressed in terms of the vector field $A_{i}$.
Note that while the gauge field lagrangian (73, 74) agrees with the form given
in the literature, the iterative Noether procedure is discussed in terms of
the vector field and conforms to the lagrangian (68).

The equation of motion for the $A_{0}$ field is given by,

$\partial_{i}\partial_{j}E_{ij}-\phi^{*}\phi=0$ (76)

The time derivative on $A_{ij}$ is eliminated in favour of the canonical
momenta,

$\pi_{ij}=\frac{\partial{\cal L}}{\partial(\partial_{0}A_{ij})}=E_{ij}$ (77)

so that (76) is expressed as,

$\partial_{i}\partial_{j}\pi_{ij}-\phi^{*}\phi=0$ (78)

the left side of which is the Gauss operator. Using the usual
$A_{ij}-\pi_{kl}$ Poisson algebra it is seen that this Gauss constraint
generates the desired transformation (48),

$\delta
A_{ij}(x)=\\{A_{ij}(x),\,\int_{space}\alpha\,(\partial_{k}\partial_{l}\pi_{kl}-\phi^{*}\phi)\\}=\partial_{i}\partial_{j}\alpha(x)$
(79)

It is observed that the coupling of the $A_{0}$ and matter fields in either
description remains unchanged. In both cases it acts as a lagrange multiplier
that enforces the Gauss constraint. The structural similarity of the Gauss
constraints (71) and (78) is noted. It ensures that, in both descriptions, the
scalar field transforms identically as,

$\delta\phi(x)=\\{\phi(x),\int_{space}\alpha\,(\partial_{k}\partial_{l}\pi_{kl}-\phi^{*}\phi)\\}=\\{\phi(x),\int_{space}\alpha\,(\partial_{k}\pi_{k}-\phi^{*}\phi)\\}=i\alpha\phi(x)$
(80)

This demonstrates the validity of the choice (75) as the lagrangan since the
correct gauge transformation properties are reproduced.

####  3.2.3 Extension to $SU(N)$ Invariance

The extension to the nonabelian invariance is carried following similar steps
as for conventional systems. In this case, however, the dual formulation
cannot be done. Only that in terms of the standard vector field is possible.
This is explicitly shown for the $t$-term in (28). Consider therefore,

${\cal
L}_{0}=(i\phi^{*2}\partial_{i}\phi_{k}\partial^{i}\phi_{k}-i\phi^{2}\partial_{i}\phi^{*}_{k}\partial^{i}\phi^{*}_{k})\,\,,\,\,\phi^{2}=\phi_{k}\phi_{k}$
(81)

where $k$ runs from $1$ to the dimension of the representation of $SU(N)$
under which the scalar fields transform. The lagrangian is invariant under the
global $SU(N)$ transformations (2). The corresponding Noether currents are
given by,

$j^{ia}\alpha^{a}=j^{i}=\frac{\partial{\cal
L}_{0}}{\partial(\partial_{i}\phi^{k})}\delta\phi^{k}+c.c.=2i\phi^{*2}\partial_{i}\phi_{k}(i\alpha^{a}(T^{a})_{kj}\phi_{j})+c.c$
(82)

so that,

$j^{ia}=-2(\phi^{*2}\partial_{i}\phi(T^{a})\phi+\phi^{2}\partial_{i}\phi^{*}(T^{*a})\phi^{*})$
(83)

where $T^{a}$ denotes the $SU(N)$ generators. This current, unlike its abelian
counterpart, cannot be written as the gradient of a tensor current like (39).
Hence the dual formulation in terms of a fractonic field cannot be done.
Nevertheless, it is feasible to continue with the Noether process and find the
gauge couplings. From (83), the contribution to the lagrangian is written as,

${\cal L}_{1}=-2\Big{(}\phi^{*2}(\partial_{i}\phi
T^{a}\phi)+\phi^{2}(\partial_{i}\phi^{*}T^{*a}\phi^{*})\Big{)}A_{i}^{a}$ (84)

Since there are derivative terms the iterative process continues. The next
stage Noether current is obtained by replacing ${\cal L}_{0}$ by ${\cal
L}_{1}$ in (82). The result is,

$j^{ia}\alpha^{a}=j^{i}=-2\phi^{*2}(T)^{a}_{kl}\phi_{l}i\alpha^{b}(T^{b})_{kj}\phi_{j}A_{i}^{a}+c.c.$
(85)

so that the current is given by,

$j^{ib}=-2i\phi^{*2}(T)^{a}_{kl}\phi_{l}(T^{b})_{kj}\phi_{j}A_{i}^{a}+c.c.$
(86)

Observing that the factor multiplying the gauge field is symmetric in $(a,b)$,
the contribution to the lagrangian comes out as,

${\cal
L}_{2}=-i\Big{(}\phi^{*2}(T)^{a}_{kl}\phi_{l}(T^{b})_{kj}\phi_{j}-\phi^{2}(T^{*a})_{kl}\phi^{*}_{l}(T^{*b})_{kj}\phi^{*}_{j}\Big{)}A_{i}^{a}A_{i}^{b}$
(87)

The iteration terminates. The final lagrangian is given by the sum of three
contributions (81), (84) and (87). The result is,

${\cal
L}=i\phi^{*2}(D_{i}\phi_{k})(D^{i}\phi_{k})-i\phi^{2}(D_{i}\phi_{k})^{*}(D^{i}\phi_{k})^{*}$
(88)

where the covariant derivatives are defined as,

$D_{i}\phi_{k}=\partial_{i}\phi_{k}+iA_{i}^{a}(T^{a})_{kl}\phi_{l}$ (89)

Thus the minimally coupled form is reproduced without referring to any gauge
principle. The other terms in the matter action (28) may be similarly treated.

It is worthwhile to stress that, in the non-abelian example, only the
‘ordinary’ $SU(N)$ transformation is a symmetry as there is no analogue of the
dipolar symmetry. This is further seen from (88) which is obtained by gauging
an ordinary $SU(N)$ symmetry.

##  4 Inclusion of Other Conservation Laws

So far our analysis was confined to the conservation laws generically
expressed in (22) and (25). Physically, these correspond to the charge (23)
and dipole (26) symmetries, respectively. However there is a possibility that
other conservation laws along these lines exist [1, 2, 11, 12]. Thus there
could be a situation where a dipole can move only normally to the dipole
moment imposing further restrictions on the mobility of particles. In analogy
with (26) this is expressed by the conservation of the quantity,

$Q=\int_{space}x^{2}J_{0}(x)$ (90)

It is possible to verify that conservation of all three charges (23, 26, 90)
holds if the global symmetry indicated by (19) is modified to,

$\partial_{o}J_{0}-(\partial_{i}\partial_{j}-\frac{1}{d}\delta_{ij}\partial^{2})J_{ij}=0$
(91)

where $d$ is the spatial dimension. Now the corresponding symmetry
transformation on the complex scalars takes the form (27) where,

$\alpha(x)=\alpha_{0}+\alpha_{i}x^{i}+\beta x^{2}$ (92)

with a constant $\beta$. A specific realisation of the matter action
satisfying this property may be done but this is not necessary for the ensuing
analysis.

We show that once again a dual description is possible, either in terms of a
standard gauge field or in terms of a symmetric traceless tensor field. A
connection between these two descriptions is derived which is obviously
different from the earlier example.

The conservation law (91) may be expressed in the form (22) where [12],

$j_{0}=J_{0}\,,\hskip
28.45274ptj_{i}=\partial_{j}J_{ji}-\frac{1}{d}\partial_{i}J_{kk}$ (93)

The temporal component remains identical to the earlier model. So we
concentrate on the spatial current. Coupling with the standard $A_{i}$ field
may be rephrased in favour of a symmetric tensor field as,

$j_{i}A_{i}=J_{ij}A_{ij}$ (94)

where,

$A_{ij}=-\frac{1}{2}(\partial_{i}A_{j}+\partial_{j}A_{i}-\frac{2}{d}\delta_{ij}\partial_{l}A_{l})$
(95)

The condition (94) is general and is a manifestation of the dual nature of the
fields. Any change in the relation between the vector and tensor currents is
precisely compensated by a corresponding change in the relation between the
vector and tensor gauge fields, such that the condition (94) is preserved. For
instance, in the present case, (93) and (95), respectively, give the relations
between the currents and the gauge fields. Similarly, for the earlier case
involving the charge and dipole symmetry only, the relation among the currents
is given by (21) while that between the gauge fields is (50). It is easy to
verify that, given any one relation among (93) and (95), the other can be
derived by exploiting (94). As an example, contracting (95) by $J_{ij}$,
appropriately shifting the derivatives, using (94), leads to (93).

The tensor field in (95) is traceless $(A_{ii}=0)$ and is a consequence of the
modified symmetry. Using (17) the transformation of the above tensor field is
given by,

$\delta
A_{ij}=\partial_{i}\partial_{j}\alpha-\frac{1}{d}\delta_{ij}\partial^{2}\alpha$
(96)

While the magnetic field (74) remains gauge invariant under this new
transformation, the electric field is not. To make it gauge invariant, the
definition of the electric is modified to,

$E_{ij}=(\partial_{i}\partial_{j}-\frac{1}{d}\delta_{ij}\partial^{2})A_{0}+\partial_{0}A_{ij}\,,\hskip
28.45274ptB_{ij}=-\partial_{i}\partial_{k}A_{jk}+\partial_{j}\partial_{k}A_{ik}$
(97)

The lagrangian of the theory is defined by (75) with the electric and magnetic
fields given above. The pure matter sector naturally is different but its
explicit form is not required here. We want to compute the Gauss constraint
from this lagrangian and deduce the transformation law (96). The equation of
motion for the $A_{0}$ field is altered from (76) to,

$(\partial_{i}\partial_{j}-\frac{1}{d}\delta_{ij}\partial^{2})E_{ij}-\phi^{*}\phi=0$
(98)

Since the momenta has the same functional form as (77), the Gauss constraint
follows from (98),

$(\partial_{i}\partial_{j}-\frac{1}{d}\delta_{ij}\partial^{2})\pi_{ij}-\phi^{*}\phi=0$
(99)

which reproduces the transformation (96),

$\delta
A_{ij}=\\{A_{ij}(x),\,\int_{space}\alpha[\,(\partial_{k}\partial_{l}-\frac{1}{d}\delta_{kl}\partial^{2})\pi_{kl}-\phi^{*}\phi]\\}=(\partial_{i}\partial_{j}-\frac{1}{d}\delta_{ij}\partial^{2})\alpha$
(100)

while the transformation of the scalar field remains same as before. This
shows that the suggested lagrangian yields an appropriate generator (Gauss
constraint) that generates the desired transformation laws. The formulation in
terms of the standard gauge field is carried out as usual with a Maxwell type
term (see (68)).

##  5 Conclusions

The gauge principle, which requires the invariance of a physical theory under
local gauge transformations, is a cornerstone in the development of modern
field theory and gravity, both relativistic and nonrelativistic [21]. It has
also formed the basis for introducing interactions in a free theory of matter
invariant under global symmetry transformations. Ordinary derivatives are
replaced by covariant derivatives such that the additional connection term
compensates for the difference in the transformation of the field at two
neighbouring points. Field strengths and/or curvatures are now defined using
the Ricci identity and the action can be written. Thus the origins of the
gauge principle are more geometrical than physical.

An alternative and more physical approach was presented by Deser [14] where
interactions, instead of the gauge principle, play the pivotal role. It is
based on an iterative Noether prescription where interactions are introduced
in a step by step process. The term found from the original source is used to
calculate the source for the next step and so on till the iteration terminates
when no new source is obtained. This method is therefore a bridge between the
two theorems of Noether. Starting from a theory that has a global symmetry
(Noether’s first theorem), the iteration eventually yields an interacting
theory invariant under local symmetry transformations (Noether’s second
theorem).

So far the iterative Noether scheme was applied to usual (non higher
derivative) systems [15, 16, 17, 18, 19, 20]. Here we presented an application
to higher derivative theories invariant under both charge and dipole
symmetries. Such theories have been discussed recently, both on general terms
[4] as well as by concrete examples [4, 13, 12]. They have generated
considerable interest since they predict a new type of topological matter,
called fractons, that have restricted mobility. The iterative prescription,
which begins from the free (quadratic) theory, has to be suitably generalised
to account for the fact that the new ‘free’ matter theory is no longer
quadratic. Also, in the present case, the presence of higher derivatives is an
additional complication. The application of the present scheme yields a dual
picture which is traced to the presence of multiple global symmetries.

The iterative Noether prescription allows the introduction of interactions in
two distinct ways, the origin of which is in the dual picture stated above. In
one case they are inducted by a vector field and the additional terms can be
written so that the complete theory is the standard minimally coupled theory,
where ordinary derivatives are replaced by their covariant expressions.
Alternatively, interactions may also be expressed in terms of a symmetric
tensor field recently encountered in fractonic theories. In fact this method
of introducing interactions naturally led to a nonstandard ‘minimal
prescription’ where the complete theory is expressed by replacing the ordinary
double derivatives by appropriate ‘covariant’ expressions involving the tensor
fields (see (60). Furthermore, an equivalence of these two minimal coupling
prescriptions is shown by exploiting a map that relates the vector and tensor
gauge fields.

The analysis also dictates the structure of the pure gauge term in the
lagrangian. Self consistency is established from the constraint structure of
the total lagrangian including gauge, matter and interacting sectors.
Extension to other fractonic theories with more conservation laws following
from additional global symmetries is shown.

It would be desirable to apply the methods formulated here to an analysis of
the current algebra in these theories. As we mentioned, the presence of
multiple global symmetries yields a dual description leading to distinct
currents. Their algebra, with or without the inclusion of gauge fields, could
possibly provide new sights into the structure of these theories.

Acknowledgements: This work was supported by a (DAE) Raja Ramanna Fellowship.

## References

  * [1] M. Pretko, Subdimensional particle structure of higher rank U(1) spin liquids, Phys. Rev. B 95, 115139 (2017), arXiv:1604.05329.
  * [2] M. Pretko, Generalized electromagnetism of subdimensional particles: A spin liquid story, Phys. Rev. B 96, 035119 (2017), arXiv:1606.08857
  * [3] O. Dubinkin, A. Rasmussen, and T. L. Hughes, Higher-form gauge symmetries in multipole topological phases, Ann. Phys. (N.Y.) 422, 168297 (2020), arXiv:2007.05539.
  * [4] N. Seiberg, Field theories with a vector global symmetry, SciPost Phys. 8, 050 (2020), arXiv:1909.10544.
  * [5] C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005), cond-mat/0404182.
  * [6] R. M. Nandkishore and M. Hermele, Fractons, Ann. Rev. Condensed Matter Phys. 10 (2019) 295–313, [arXiv:1803.11196], and references therein.
  * [7] M. Pretko, X. Chen, and Y. You, Fracton Phases of Matter, Int. J. Mod. Phys. A 35 (2020), no. 06 2030003, [arXiv:2001.01722].
  * [8] M. Qi, L. Radzihovsky, and M. Hermele, Fracton phases via exotic higher-form symmetry breaking, Ann. Phys. (N.Y.) 424, 168360 (2021), arXiv:2010.02254.
  * [9] N. Seiberg and S.-H. Shao, Exotic symmetries, duality, and fractons in 2+1-dimensional quantum field theory, SciPost Phys. 10, 027 (2021), arXiv:2003.10466.
  * [10] Kevin Slagle, Abhinav Prem, Michael Pretko Symmetric Tensor Gauge Theories on Curved Spaces, Annals of Physics 410 (2019) 167910, arXiv:1807.00827.
  * [11] Yi-Hsien Du, Umang Mehta, Dung Xuan Nguyen, and Dam Thanh Son Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry, arXiv:2103.09826.
  * [12] R. Banerjee, Hamiltonian formulation of higher rank symmetric fractonic gauge theories, arXiv:2105.04152; Eur. Phys. J. C (2022) 82:22.
  * [13] M. Pretko, The fracton gauge principle, Phys. Rev. B98, 115134 (2018).
  * [14] S. Deser, Self-interaction and gauge invariance, Gen. Rel. Grav. 1, 9, (1970); Since this paper is not readily available, Deser subsequently posted an arXiv: gr-qc/0411023 (V3) which is basically the original version.
  * [15] S. Deser, J.H. Kay and D.G. Boulware, Supergravity from selfinteraction, Physica (Amsterdam) 96A, 141 (1979).
  * [16] C. Aragone and J. Stephany, Nonabelian Chern-Simons topological coupling from selfinteraction, Rev. Bras. Fis. 16, 287, (1986).
  * [17] S. Deser and M. Henneaux, Gauge properties of conserved currents in abelian versus nonabelian theories, Mod. Phys. Letts. A10, 991 (1995).
  * [18] A. Khoudeir, Nonabelian antisymmetric vector coupling from selfinteraction, Mod. Phys. Letts. A11, 2489 (1996).
  * [19] A. Khoudeir, Nonabelian self duality from selfinteraction, Mod. Phys. Letts. A16, 2123 (2001).
  * [20] A. Basu, P. Majumdar and I. Mitra, Gauge invariant matter field actions from an iterative Noether coupling, Phys. Rev. D98, 105018 (2018).
  * [21] G. ’t Hooft, Under the Spell of the Gauge Principle, World Scientific Publishing Company, (1994).


any minor of a forest is a forest. Forests can be described by one forbidden
minor — a pseudograph formed by one loop. Indeed if a pseudograph has a cycle,
then by a sequence of deletions and contractions, one can get a single loop
from it.

9.12. Exercise. Describe the following classes of graphs by a single forbidden
minor.

  1. (a)

Graphs that do not contain trees with 5 end vertexes.

  2. (b)

Graphs in which any two cycles have at most one common vertex.

A more complicated example if given by the Pontryagin-–Kuratowski theorem
which states that _planar graphs are characterized by two forbidden minors:
$K_{5}$ and $K_{3,3}$._

Since toroidality is inherited to its minors, it can be described by a set of
forbidden minors. The complete list of the forbidden minors for this problem
has to be huge and it is not yet known.

9.13. Advanced exercise. Show that the graph on the diagram is not toroidal,
but every proper minor of this graph is toroidal.

### Other surfaces

One may draw graphs on other surfaces, for example, on the so called _surfaces
of genus $g$_. These surfaces can be obtained by attaching $g$ toruses to each
other;

a surface of genus $g=4$ is shown on the picture. Namely, to construct a
surface $S^{\prime}$ of genus $g+1$, start with a surface $S$ of genus $g$ and
a torus $T$, drill a hole in each and reconnect them to each other as
shown.333The described construction is called _connected sum_ ; so we can say
that _connected sum of a surface of genus $g$ and a torus is a surface of
genus $g+1$_.

It is natural to assume that the sphere has genus $0$ and torus has genus $1$.
(In general genus tells how many disjoint closed curves one could draw on the
surface so that they do not cut the surface into pieces.)

The simple regions of a drawing on a surface of genus $g$ can be defined the
same way as on the torus. Euler’s formula given in Theorem 9 admits the
following straightforward generalization

9.14. Theorem. For any drawing $G$ of a pseudograph on the surface of genus
$g$ we have

$p-q+r\geqslant 1-2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}g,$

where $p$, $q$, and $r$ denote the number of vertexes, edges and regions of
$G$.

Moreover, equality holds if all regions of $G$ are simple.

The seven color theorem also admits the following straightforward
generalization; it was proved by Percy John Heawood [19].

9.15. Theorem. If a graph $G$ admits a drawing on a surface of genus
$g\geqslant 1$, then its chromatic number cannot exceed

$\frac{7+\sqrt{1+48{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}g}}{2}.$

Note that for a sphere (that is, for $g=0$) the formula gives $4$, which is
the right bound for the chromatic number for planar graphs. However this is
just a coincidence, the proof of Heawood works only for $g\geqslant 1$.

The estimate in the last theorem is sharp. The latter was proved by
constructing a drawing on the surface of genus $g$ of the complete graphs
$K_{n}$ for any $n\leqslant\frac{7+\sqrt{1+48{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}g}}{2}$. The final step in this construction was made by Gerhhard Ringel
and Ted Youngs [32]. The solution uses the so called _rotations of graphs_ ;
which is a combinatiric way to encode a drawing of a graph on a surface. This
is the subject of [18, Chapter 10].

One may consider nonoriented surfaces; for example, the so called Möbius strip
shown on the diagram. It turns out that it is possible to generalize Euler’s
inequality and get an exact upper bound for the chromatic numbers for graphs
that can be drawn on such surfaces. (The problem is slightly harder for the so
called Klein bottle, but still, its difficulty is not comparable with the four
color theorem.)

9.16. Exercise. Draw the complete graph $K_{6}$ on a Möbius strip (assume it
is made from a transparent material).

##  Chapter 10 Rado graph

In this chapter, we consider one graph with many surprising properties. Unlike
most of the graphs we considered so far, this graph has an infinite set of
vertexes.

### Definition

Recall that a set is _countable_ if it can be enumerated by natural numbers
$1,2,\dots$; it might be infinite or finite.

A _countable graph_ is a graph with a countable set of vertexes; the set of
vertexes can be infinite or finite, but it can not be empty since we always
assume that a graph has a nonempty set of vertexes.

10.1. Definition. A _Rado graph_ is a countable graph satisfying the following
property:

Given two finite disjoint sets of vertexes $V$ and $W$, there exists a vertex
$v\notin V\cup W$ that is adjacent to any vertex in $V$ and nonadjacent to any
vertex in $W$.

The property in the definition will be called the _Rado property_ ; so we can
say that _for the sets of vertexes $V$ and $W$ in a graph, the Rado property
holds_ or _does not hold_.

10.2. Exercise. Show that any Rado graph has an infinite number of vertexes.

10.3. Exercise. Show that any Rado graph has diameter 2.

### Stability

The following exercises show that the Rado property is very stable — small
changes can not destroy it.

10.4. Exercise. Let $R$ be a countable graph.

  1. (a)

Assume $e$ is an edge in $R$. Show that $R-e$ is a Rado graph if and only if
so is $R$.

  2. (b)

Assume $v$ is a vertex in $R$. Show that $R-v$ is a Rado graph if so is $R$.

  3. (c)

Assume $v$ is a vertex in $R$. Consider the graph $R^{\prime}$ obtained from
$R$ by replacing each edge from $v$ by a non-edge, and each non-edge from $v$
by an edge (leaving the rest unchanged). Show that $R^{\prime}$ is a Rado
graph if and only if so is $R$.

10.5. Exercise. Assume the set of vertexes of a Rado graph is partitioned into
two subsets. Show that the subgraph induced by one of these subsets is Rado.

Hint: Let $P$ and $Q$ be the induced subgraphs in the Rado graph $R$. Assume
$P$ is not Rado; that is, there is a pair of finite vertex sets $V$ and $W$ in
$P$, such that any vertex $v$ in $R$ that meet the Rado property for $V$ and
$W$ does not lie in $P$ (and therefore it lies in $Q$). Use the pair of sets
$V$ and $W$ to show that $Q$ is Rado.

10.6. Exercise. Let $R$ be a Rado graph. Assume that $Z$ is the set of all
vertexes in $R$ adjacent to a given vertex $z$. Show that the subgraph induced
by $Z$ is Rado.

### Existence

10.7. Theorem. There is a Rado graph.

Proof. Let $G$ be a finite graph. Denote by $G^{\prime}$ the graph obtained
from $G$ according to the following rule: for each subset $V$ of vertexes in
$G$ add a vertex $v$ and connect it to all the vertexes in $V$.

Note that if $G$ has $p$ vertexes, then $G^{\prime}$ has $p+2^{p}$ vertexes —
it has $p$ vertexes of $G$ and $2^{p}$ additional vertexes — one for each of
$2^{p}$ subsets of the $p$-element set (including the empty set).

The original graph $G$ is an induced subgraph in $G^{\prime}$. Note also that
$G^{\prime}$ is finite — it has $p+2^{p}$ vertexes.

By construction, the Rado property holds in $G^{\prime}$ for any two sets $V$
and $W$ of vertexes in $G$ — the required vertex $v$ is the vertex in
$G^{\prime}$ that corresponds to the subset $V$.

Let $G_{1}$ be a graph with one vertex.

By repeating the construction, we get a sequence of graphs
$G_{1},G_{2},G_{3},\dots$, such that $G_{n+1}=G_{n}^{\prime}$ for any $n$. The
graphs $G_{1},G_{2},G_{3}$ are shown on the diagram.111It would be hard to
draw $G_{4}$ since it contains $1+2^{1}+2^{3}+2^{11}=2059$ vertexes, and it is
impossible to draw $G_{5}$ — it has $1+2^{1}+\nobreak 2^{3}+\nobreak
2^{11}+2^{2059}$ vertexes which exceeds by many orders the number of particles
in the observable universe.

Since $G_{n}$ is a subgraph of $G_{n+1}$ for any $n$, we can consider the
union of the graphs in the sequence $(G_{n})$; denote it by $R$. By
construction, each graph $G_{n}$ is a subgraph $R$ induced by finitely many
vertexes. Moreover, any vertex or edge of $R$ belongs to any $G_{n}$ with a
sufficiently large $n$.

Note that $R$ is Rado. Indeed, any two finite sets of vertexes $V$ and $W$
belong to $G_{n}$ for some $n$. From above, the Rado property holds for $V$
and $W$ in $G_{n+1}$, and therefore in $R$. ∎

Another construction. One could also construct a Rado graph by directly
specifying which vertexes are adjacent. Namely, consider the graph $R$ as on
the diagram with vertexes $r_{0},r_{1},\dots$

$r_{0}$$r_{1}$$r_{2}$$r_{3}$$r_{4}$$r_{5}$$r_{6}$$r_{7}$$r_{8}$

such that $r_{i}$ is adjacent to $r_{j}$ for some $i<j$ if the $i$-th bit of
the binary representation of $j$ is 1.

For instance, vertex $r_{0}$ is adjacent to all $r_{n}$ with odd $n$, because
the numbers whose 0-th bit is nonzero are exactly the odd numbers. Vertex
$r_{1}$ is adjacent to $r_{0}$ (since 1 is odd) and to all $r_{n}$ with
$n\equiv 2$ or $3\pmod{4}$; and so on.

10.8. Exercise. Show that the described graph is Rado.

### Uniqueness

In this section we will prove that any two Rado graphs are isomorphic, so
essentially there is only one Rado graph. First, let us prove a simpler
statement.

10.9. Theorem. Let $R$ be a Rado graph. Then any countable graph $G$ (finite
or infinite) is isomorphic to an induced subgraph of $R$.

Proof. Enumerate the vertexes of $G$ as $v_{1},v_{2},\dots$ (the sequence
might be finite or infinite).

It is sufficient to construct a sequence $r_{1},r_{2},\dots$ of vertexes in
$R$ such that $r_{i}$ is adjacent to $r_{j}$ if and only if $v_{i}$ is
adjacent to $v_{j}$. In this case, the graph $G$ is isomorphic to the subgraph
of $R$ induced by $\\{r_{1},r_{2}\dots\\}$.

We may choose any vertex of $R$ as $r_{1}$. Suppose that the sequence
$r_{1},\dots,r_{n}$ is constructed. If $G$ has $n$ vertexes, then the required
sequence is already constructed. Otherwise note that the Rado property implies
that there is a vertex $r_{n+1}$ in $R$ that is adjacent to $r_{i}$ for
$i\leqslant n$ if and only if $v_{n+1}$ is adjacent to $v_{i}$.

Clearly, the new vertex $r_{n+1}$ meets all the required properties. Repeating
this procedure infinitely many times, or until the sequence $(v_{n})$
terminates, produces the required sequence $(r_{n})$. ∎

10.10. Exercise. Show that any two vertexes in a Rado graph can be connected
by a path of length 10.

10.11. Theorem. Any two Rado graphs $R$ and $S$ are isomorphic.

Moreover any isomorphism $f_{0}\colon S_{0}\to R_{0}$ between finite induced
subgraphs in $R$ and $S$ can be extended to an isomorphism $f\colon
S\to\nobreak R$.

Note that Theorem 10 implies that $R$ is isomorphic to an induced subgraph in
$S$ and the other way around — $S$ is isomorphic to an induced subgraph in
$R$. For finite graphs these two properties would imply that the graphs are
isomorphic; see Exercise 10. As the following example shows, it does not hold
for infinite graphs. It is instructive to understand this example before going
into the proof.

The first graph $T$ on the diagram has an infinite number of vertexes, non of
which has degree 3. The second graph $T^{\prime}$ has exactly one vertex of
degree 3. Therefore these two graphs are not isomorphic.

Deleting the marked vertexes from one graph produces the other one. Therefore
$T$ is isomorphic to a subgraph of $T^{\prime}$ and the other way around.

The proof below uses the same construction as in the proof of Theorem 10, but
it is applied back and forth to ensure that the constructed subgraphs contain
all the vertexes of the original graph.

Proof. Once we have proved the second statement, the first statement will
follow if you apply it to single-vertex subgraphs $R_{0}$ and $S_{0}$.

Since the graphs are countable, we can enumerate the vertexes of $R$ and $S$,
as $r_{1},r_{2},\dots$ and $s_{1},s_{2},\dots$ respectively. We will construct
a sequence of induced subgraphs $R_{n}$ in $R$ and $S_{n}$ in $S$ with a
sequence isomorphisms $f_{n}\colon R_{n}\to S_{n}$.

Suppose that an isomorphism $f_{n}\colon R_{n}\to S_{n}$ is constructed.

If $n$ is even, set $m$ to be the smallest index such that $r_{m}$ not in
$R_{n}$. The Rado property guarantees that there is a vertex $s_{k}$ such that
for any vertex $r_{i}$ in $R_{n}$, $s_{k}$ is adjacent to $f_{n}(r_{i})$ if
and only if $r_{m}$ is adjacent to $r_{i}$. Set $R_{n+1}$ to be the graph
induced by vertexes of $R_{n}$ and $r_{m}$; further set $S_{n+1}$ to be the
graph induced by vertexes of $S_{n}$ and $s_{k}$. The isomorphism $f_{n}$ can
be extended to the isomorphism $f_{n+1}\colon R_{n+1}\to S_{n+1}$ by setting
$f_{n+1}(r_{m})=s_{k}$.

If $n$ is odd, we do the same, but backwards. Let $m$ be the smallest index
such that $s_{m}$ not in $S_{n}$. The Rado property guarantees that there is a
vertex $r_{k}$ which is adjacent to a vertex $r_{i}$ in $R_{n}$ if and only if
$f_{n}(r_{i})$ is adjacent to $s_{m}$. Set $R_{n+1}$ to be the graph induced
by vertexes of $R_{n}$ and $r_{k}$; further set $S_{n+1}$ to be the graph
induced by vertexes of $S_{n}$ and $s_{m}$. The isomorphism $f_{n}$ can be
extended to the isomorphism $f_{n+1}\colon R_{n+1}\to S_{n+1}$ by setting
$f_{n+1}(r_{k})=s_{m}$.

Note that if $f_{n}(r_{i})=s_{j}$, then $f_{m}(r_{i})=s_{j}$ for all
$m\geqslant n$. Therefore we can define $f(r_{i})=s_{j}$ if
$f_{n}(r_{i})=s_{j}$ for some $n$.

By construction we get that

  * $\diamond$

$f_{n}(r_{i})$ is defined for any $n>2{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}i$. Therefore $f$ is defined at any vertex of $R$.

  * $\diamond$

$s_{j}$ lies in the range of $f_{n}$ for any $n>2{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}j$. Therefore the range of $f$ contains all the
vertexes of $S$.

  * $\diamond$

$r_{i}$ is adjacent to $r_{j}$ if and only if $f(r_{i})$ is adjacent to
$f(r_{j})$.

Therefore $f\colon R\to S$ is an isomorphism. ∎

10.12. Exercise. Explain how to modify the proof of theorem above to prove the
following theorem.

10.13. Theorem. Let $R$ be a Rado graph. A countable graph $G$ is isomorphic
to a spanning subgraph of $R$ if and only if, given any finite set $V$ of
vertexes of $G$, there is a vertex $w$ that is not adjacent to any vertex in
$V$.

10.14. Exercise. Let $v$ and $w$ be two vertexes in a Rado graph $R$. Show
that there is an isomorphism from $R$ to itself that sends $v$ to $w$.

10.15. Exercise. Let $G$ and $H$ be two finite graphs. Assume $G$ is
isomorphic to a subgraph of $H$ and the other way around — $H$ is isomorphic
to a subgraph of $G$. Show that $G$ is isomorphic to $H$.

### The random graph

The following theorem explains why a Rado graph is also named _random graph_.

10.16. Theorem. Assume an infinite countable graph is chosen at random by
choosing an edge, independently and with probability $\tfrac{1}{2}$ for each
pair of vertexes. Then, with probability 1, the resulting graph Rado.

Proof. It is sufficient to show that for two given finite sets of vertexes $V$
and $W$, the Rado property fails with probability 0.

Assume $n=|V|+|W|$; that is, $n$ is the total number of vertexes in $V$ and
$W$. The probability that a given vertex $v$ outside of $V$ and $W$ satisfies
the Rado property for $V$ and $W$ is $\tfrac{1}{2^{n}}$. Therefore probability
that a given vertex $v$ does _not_ satisfy this property is
$1-\tfrac{1}{2^{n}}$.

Note that events that a given vertex does not satisfy the property are
independent. Therefore the probability that $N$ different vertexes
$v_{1},\dots,v_{N}$ outside of $V$ and $W$ do not satisfy the Rado property
for $V$ and $W$ is

$(1-\tfrac{1}{2^{n}})^{N}.$

This value tends to 0 as $N\to\infty$; therefore the event that no vertex is
correctly joined has probability 0. ∎

10.17. Exercise. Let $0<\alpha<1$. Assume an infinite countable graph is
chosen at random by selecting edges independently with probability $\alpha$
from the set of 2-element subsets of the vertex set. Show that with
probability 1, the resulting graph is a Rado graph.

### Remarks

The _Rado graph_ is also called the _Erdős–Rényi graph_ or _random graph_ ; it
was first discovered by Wilhelm Ackermann, rediscovered later by Paul Erdős
and Alfréd Rényi and yet by Richard Rado. Theorem 10 was discovered by Paul
Erdős and Alfréd Rényi. A good survey on the subject is written by Peter
Cameron [3].

##  Appendix A Corrections and additions

Here we include corrections and additions to [18].

### Correction to 3.2.2

The proof of Theorem 3.2.2 about decomposition of a cubic graph with a bridge
into 1-factors does not explain why ‘‘each bank has an odd number of
vertexes’’.

This is true since the 1-factor containing the bridge breaks all the vertexes
of each bank into pairs except the end vertex of the bridge.

### Correction to 8.4.1

There is an inaccuracy in the proof of Theorem 8.4.1 about stretchable planar
graphs. Namely, in the planar drawing of $G-h$, the region $R$ might be
unbounded.

To fix this inaccuracy, one needs to prove a slightly stronger statement.
Namely that any planar drawing of the maximal planar graph $G$ can be
stretched. That is, given a planar drawing of $G$, there is a stretched
drawing of $G$ and a bijection between the bounded (necessarily triangular)
regions such that corresponding triangles have the same edges of $G$ as the
sides.

The remaining part of the proof works with no other changes.

## Index

  * $2$-factor, 50
  * adjacency matrix, 25
  * adjacent, 5
  * alternating path, 60
  * assignment problem, 55
  * augmenting path, 60
  * bigraph, 60
  * Binet’s formula, 46
  * binomial coefficient, 8
  * bridge, 19
  * chromatic polynomial, 36
  * complete graph, 31
  * connected sum, 75
  * constant-recursive sequences, 22
  * countable graph, 77
  * countable set, 77
  * degree, 4
  * deletion-minus-contraction formula, 34
  * deletion-plus-contraction formula, 20
  * diameter, 14
  * digraph, 65
  * directed graph, 65
  * distance, 14
  * double factorial, 48
  * drawing, 68
  * edge, 4
  * edge cover, 64
  * end vertex, 4
  * exponential generating function, 43
  * fan, 21
  * Fibonacci numbers, 21
  * forbidden minior, 74
  * forest, 51
  * free term, 35
  * graph, 4
  * greedy algorithm, 55
  * incident, 5
  * isolated vertex, 4
  * Kirchhoff matrix, 26
  * Kirchhoff minor, 26
  * labeled graph, 25
  * ladder, 23
  * leading coefficient, 35
  * loop, 4
  * Lucas numbers, 24
  * Markov’s inequality, 15
  * matching, 37, 60
  * matching number, 38
  * matching polynomial, 38
  * minimum spanning tree proble, 55
  * minior, 74
  * monic polynomial, 35
  * multigraph, 4
  * nearest neighbor algorithm, 54
  * neighbor, 55
  * P=NP problem, 54
  * parallel edges, 4
  * perfect matching, 48
  * planar graph, 68
  * pseudograph, 4
  * Rado graph, 77
  * Rado property, 77
  * rooted tree, 52
  * rotation, 76
  * set of neighbors, 61
  * shortest path problem, 55
  * simple region, 69
  * spanning-tree polynomial, 39
  * square diagram, 68
  * star, 65
  * surface of genus $g$, 74
  * symmetric matrix, 25
  * toroidal graph, 68
  * traveling salesman problem, 54
  * typical property, 17
  * vertex, 4
  * vertex cover, 63
  * weight, 54

## References

  * [1] M. Aigner and G. Ziegler  ‘‘Proofs from The Book’’, 2014  DOI: 10.1007/978-3-662-44205-0
  * [2] N. Alon  ‘‘A simple algorithm for edge-coloring bipartite multigraphs’’  In _Inform. Process. Lett._ 85.6, 2003, pp. 301–302  DOI: 10.1016/S0020-0190(02)00446-5
  * [3] P. Cameron  ‘‘The random graph’’  In _The mathematics of Paul Erdős, II_ 14, Algorithms Combin., 1997, pp. 333–351  DOI: 10.1007/978-3-642-60406-5_32
  * [4] A. Cayley  ‘‘A theorem on trees’’  In _Quart. J. Math._ 23, 1889, pp. 376–378
  * [5] T.. Cormen, C.. Leiserson, R.. Rivest and C. Stein  ‘‘Introduction to algorithms’’, 2009
  * [6] P. Doyle and J.. Snell  ‘‘Random walks and electric networks’’ (arXiv:math/0001057 [math.PR]) 22, Carus Mathematical Monographs, 1984
  * [7] E. Egerváry  ‘‘Matrixok kombinatorius tulajdonságairól’’  In _Matematikai és Fizikai Lapok_ 38.1931, 1931, pp. 16–28
  * [8] P. Elias, A. Feinstein and C. Shannon  ‘‘A note on the maximum flow through a network’’  In _IRE Transactions on Information Theory_ 2.4 IEEE, 1956, pp. 117–119
  * [9] А.. Эвнин  ‘‘Вокруг теоремы Холла’’  In _Математическое образование_ Фонд математического образования и просвещения, 2005, pp. 2–23
  * [10] А.. Эвнин  ‘‘Хроматический многочлен графа в задачах’’  In _Математическое образование_ Фонд математического образования и просвещения, 2014, pp. 9–15
  * [11] L.. Ford and D.. Fulkerson  ‘‘Maximal flow through a network’’  In _Canad. J. Math._ 8, 1956, pp. 399–404  DOI: 10.4153/CJM-1956-045-5
  * [12] M. Gardiner  ‘‘The 2nd Scientific American book of mathematical puzzles and diversions’’, 1961
  * [13] C.. Godsil and I. Gutman  ‘‘On the theory of the matching polynomial’’  In _J. Graph Theory_ 5.2, 1981, pp. 137–144  DOI: 10.1002/jgt.3190050203
  * [14] W.. Gowers  ‘‘The two cultures of mathematics’’  In _Mathematics: frontiers and perspectives_ , 2000, pp. 65–78
  * [15] R. Graham, D. Knuth and O. Patashnik  ‘‘Concrete mathematics’’, 1994
  * [16] P. Hall  ‘‘On representatives of subsets’’  In _Journal of the London Mathematical Society_ 1.1 Wiley Online Library, 1935, pp. 26–30
  * [17] F. Harary and E. Palmer  ‘‘Graphical enumeration’’, 1973
  * [18] N. Hartsfield and G. Ringel  ‘‘Pearls in graph theory’’, 2003
  * [19] P.. Heawood  ‘‘Map color theorems’’  In _Quant. J. Math._ 24, 1890, pp. 332–338
  * [20] C. Jordan  ‘‘Calculus of Finite Differences’’, 1939
  * [21] D. Jungnickel  ‘‘Graphs, networks and algorithms’’ 5, Algorithms and Computation in Mathematics, 2008  DOI: 10.1007/978-3-540-72780-4
  * [22] G. Kalai  ‘‘The seventeen camels riddle, and Noga Alon’s camel proof and algorithms’’,  URL: https://gilkalai.wordpress.com/
  * [23] R.. Karp  ‘‘Reducibility among combinatorial problems’’  In _Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972)_ , 1972, pp. 85–103
  * [24] M. Katz and J. Reimann  ‘‘An introduction to Ramsey theory’’ Fast functions, infinity, and metamathematics 87, Student Mathematical Library, 2018
  * [25] D. Kőnig  ‘‘Gráfok és mátrixok’’  In _Matematikai és Fizikai Lapok_ 38, 1931, pp. 116–119
  * [26] J.. Kruskal  ‘‘On the shortest spanning subtree of a graph and the traveling salesman problem’’  In _Proc. Amer. Math. Soc._ 7, 1956, pp. 48–50  DOI: 10.2307/2033241
  * [27] M. Levi  ‘‘An Electrician’s (or a plumber’s) proof of Euler’s polyhedral formula’’  In _SIAM News_ 50.4, 2017
  * [28] O. Ollivier  ‘‘Jacobi’s bound’’,  URL: https://www.lix.polytechnique.fr/~ollivier/JACOBI/jacobiEngl.htm
  * [29] F. Petrov  ‘‘Generating function in graph theory’’, MathOverflow, 2017  URL: https://mathoverflow.net/q/287767
  * [30] S. Radziszowski  ‘‘Small Ramsey numbers’’  In _Electron. J. Combin._ 1, 1994, pp. Dynamic Survey 130  URL: https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1
  * [31] R. Read  ‘‘An introduction to chromatic polynomials’’  In _J. Combinatorial Theory_ 4, 1968, pp. 52–71
  * [32] G. Ringel and J… Youngs  ‘‘Solution of the Heawood map-coloring problem’’  In _Proc. Nat. Acad. Sci. U.S.A._ 60, 1968, pp. 438–445  DOI: 10.1073/pnas.60.2.438
  * [33] N. Robertson and P.. Seymour  ‘‘Graph minors. XX. Wagner’s conjecture’’  In _J. Combin. Theory Ser. B_ 92.2, 2004, pp. 325–357  DOI: 10.1016/j.jctb.2004.08.001
  * [34] M.. Shirdareh Haghighi and Kh. Bibak  ‘‘Recursive relations for the number of spanning trees’’  In _Appl. Math. Sci. (Ruse)_ 3.45-48, 2009, pp. 2263–2269