Topological Field Theory and Phase Transition

Jing Zhou; Jialun Ping

arXiv:1904.09892·hep-th·October 27, 2021

Topological Field Theory and Phase Transition

Jing Zhou, Jialun Ping

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TL;DR

This paper explores the relationship between topological field theories, Jones polynomials, and phase transitions, revealing Lee-Yang type zeros and phase transitions in these mathematical and physical frameworks.

Contribution

It demonstrates that zeros of the Jones polynomial correspond to Lee-Yang type phase transitions within topological twisted super Yang-Mills theory.

Findings

01

Jones polynomial zeros are Lee-Yang type.

02

Lee-Yang phase transition analyzed in Jones polynomial of torus knots.

03

Partition function expansion relates to Wilson loop expectation values.

Abstract

The partition function of the topological twisted super Yang-Mills field theory on the boundary can be expanded as Jones polynomial, which can be computed as expectation values of Wilson loop operators. We show that the zero of the Jones polynomial is Lee-Yang type. Moreover, Lee-Yang phase transition is also discussed in the Jones polynomial of torus knot and the topological twisted super Yang-Mills field theory.

Equations46

C S (A) = \frac{k}{4 π} \int_{M} T r (d A \land A + \frac{2}{3} A \land A \land A) .

C S (A) = \frac{k}{4 π} \int_{M} T r (d A \land A + \frac{2}{3} A \land A \land A) .

\int D A exp (i C S) .

\int D A exp (i C S) .

W (K, R) = T r P exp \int_{K} A .

W (K, R) = T r P exp \int_{K} A .

Z (M; K) = \int D A exp (i C S) i = 1 \prod r W (K_{i}, R) .

Z (M; K) = \int D A exp (i C S) i = 1 \prod r W (K_{i}, R) .

⟨ W (K, R)⟩ = \frac{\int D A exp ( i C S ) \prod _{i = 1}^{r} W ( K _{i} , R )}{\int D A exp ( i C S )},

⟨ W (K, R)⟩ = \frac{\int D A exp ( i C S ) \prod _{i = 1}^{r} W ( K _{i} , R )}{\int D A exp ( i C S )},

⟨ W (K, R) ⟩ = J_{K} (q) .

⟨ W (K, R) ⟩ = J_{K} (q) .

S_{mn} = \frac{2}{k + 2} sin (\frac{( m + 1 ) ( n + 1 ) π}{k + 2}) .

S_{mn} = \frac{2}{k + 2} sin (\frac{( m + 1 ) ( n + 1 ) π}{k + 2}) .

Z (S^{3}) = S_{00} = \frac{2}{k + 2} sin (\frac{π}{k + 2}) .

Z (S^{3}) = S_{00} = \frac{2}{k + 2} sin (\frac{π}{k + 2}) .

Z = i = 1 \sum n e^{\frac{μ N - E _{n}}{k T}},

Z = i = 1 \sum n e^{\frac{μ N - E _{n}}{k T}},

Z = 1 + a_{1} z + a_{2} z^{2} + \dots + a_{M} z^{M},

Z = 1 + a_{1} z + a_{2} z^{2} + \dots + a_{M} z^{M},

Z (T, V, z) = 0.

Z (T, V, z) = 0.

⟨ W (K, R)⟩ = \frac{Z _{M; K}}{Z _{S^{3}}} = \frac{S _{10}}{S _{00}} = q^{\frac{1}{2}} + q^{- \frac{1}{2}},

⟨ W (K, R)⟩ = \frac{Z _{M; K}}{Z _{S^{3}}} = \frac{S _{10}}{S _{00}} = q^{\frac{1}{2}} + q^{- \frac{1}{2}},

q = exp \frac{2 iπ}{k + 2} .

q = exp \frac{2 iπ}{k + 2} .

q^{\frac{1}{2}} + q^{- \frac{1}{2}} = 0,

q^{\frac{1}{2}} + q^{- \frac{1}{2}} = 0,

⟨ W_{h o p f} (K, R)⟩ = \frac{Z _{M, K}}{Z _{S^{3}}} = \frac{S _{11}}{S _{00}} = (q^{\frac{1}{2}} + q^{- \frac{1}{2}}) (q + q^{- 1}) .

⟨ W_{h o p f} (K, R)⟩ = \frac{Z _{M, K}}{Z _{S^{3}}} = \frac{S _{11}}{S _{00}} = (q^{\frac{1}{2}} + q^{- \frac{1}{2}}) (q + q^{- 1}) .

\frac{\int D A exp ( i C S ) \prod _{i = 1}^{r} W ( K _{i} , R )}{\int D A exp ( i C S )} = 0.

\frac{\int D A exp ( i C S ) \prod _{i = 1}^{r} W ( K _{i} , R )}{\int D A exp ( i C S )} = 0.

\int D A exp (i C S) i = 1 \prod r W (K_{i}, R) = 0.

\int D A exp (i C S) i = 1 \prod r W (K_{i}, R) = 0.

J (m, n) = 1 - q^{m + 1} - q^{n + 1} + q^{m + n},

J (m, n) = 1 - q^{m + 1} - q^{n + 1} + q^{m + n},

τ = \frac{θ}{2 π} + \frac{4 π i}{g ^{2}}

τ = \frac{θ}{2 π} + \frac{4 π i}{g ^{2}}

Z\left(\frac{a\tau+b}{c\tau+d}\right)\sim Z(\tau),\quad\left(\begin{array}[]{ll}{a}&{b}\\ {c}&{d}\end{array}\right)\in SL(2,\mathbb{Z}).

Z\left(\frac{a\tau+b}{c\tau+d}\right)\sim Z(\tau),\quad\left(\begin{array}[]{ll}{a}&{b}\\ {c}&{d}\end{array}\right)\in SL(2,\mathbb{Z}).

S_{twisted} = {Q, I} - 2 π i τ (n - Δ),

S_{twisted} = {Q, I} - 2 π i τ (n - Δ),

n = \frac{1}{8 π ^{2}} \int_{M} Tr F \land F .

n = \frac{1}{8 π ^{2}} \int_{M} Tr F \land F .

\begin{array}[]{c}{Z_{SU(N)}(-1/\tau)=\pm N^{-1+b_{1}-\frac{b_{2}}{2}}\left(\frac{\tau}{i}\right)^{\frac{w}{2}}Z_{SU(N)/\mathbf{Z}_{N}}(\tau)}\\ {=\pm N^{-\chi/2}\left(\frac{\tau}{i}\right)^{\frac{w}{2}}Z_{SU(N)/\mathbf{Z}_{N}}(\tau)}\end{array}.

\begin{array}[]{c}{Z_{SU(N)}(-1/\tau)=\pm N^{-1+b_{1}-\frac{b_{2}}{2}}\left(\frac{\tau}{i}\right)^{\frac{w}{2}}Z_{SU(N)/\mathbf{Z}_{N}}(\tau)}\\ {=\pm N^{-\chi/2}\left(\frac{\tau}{i}\right)^{\frac{w}{2}}Z_{SU(N)/\mathbf{Z}_{N}}(\tau)}\end{array}.

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Taxonomy

TopicsBlack Holes and Theoretical Physics · Theoretical and Computational Physics · Geometric and Algebraic Topology

Full text

Topological Field Theory and Phase Transition††thanks: Supported in part by the

National Natural Science Foundation of China (11775118 and 11535005).

Jing Zhou Jia-Lun Ping1

[email protected]

Department of Physics, Nanjing Normal University, Nanjing, Jiangsu 210097, China

The partition function of the topological twisted super Yang-Mills field theory on the boundary can be expanded as Jones polynomial, which can be computed as expectation values of Wilson loop operators. We show that the zero of the Jones polynomial is Lee-Yang type. Moreover, Lee-Yang phase transition is also discussed in the Jones polynomial of torus knot and the topological twisted super Yang-Mills field theory.*

00footnotetext: $\scriptstyle\copyright$ 2019 Chinese Physical Society and IOP Publishing Ltd

The Jones polynomial [2, 3] is a celebrated invariant of a knot in the three-dimensional space, it is discovered by Jones as an shoot of his work on von Neumann algebras. Then many descriptions and generalizations of the Jones polynomial were discovered immediately after Jones’s work. In fact, the polynomial has multiple relations to many aspects of mathematical physics, which including statistical mechanics, two dimensional conformal field theory, representations of braid groups and three-dimensional Chern-Simons gauge theory [4, 5, 6, 7, 8].

One of the most remarkable known quantum field theories in four dimensions is the N = 4 supersymmetric Yang-Mills theory. This theory has the largest supersymmetry for a four-dimensional theory without gravity. A long-standing conjecture asserts that this theory has a symmetry exchanging strong and weak coupling and exchanging electric and magnetic fields. It was realized by Vafa and Witten [9]through topological twisted the super Yang-Mills theory. Khovanov homology [9] is considered to be a topological theory in four dimensions super Yang-Mills field theory. The relation between the Khovanov homology and the knot is that the four-dimensional theory associated to Khovanov homology, when compactified on a circle, reduces to the three-dimensional theory that yields the Jones polynomial.

In 1952, Lee-Yang [11, 12] established a rigorous relation between the analytic properties of free energies and thermodynamics through continuation of the partition function to the complex plane of physical parameters. And they considered a general Ising model with the ferromagnetic interaction $J_{ij}>0$ under a magnetic field $h$ with the Hamiltonian. Then they proved that all the zeros of this partition function lie on the unit circle in the complex plane of $z$ . And one can find that the Lee-Yang zeros characterize the analytic properties of partition function and the thermodynamics systems. So, determining the Lee-Yang zeros is not only useful for a complete picture of thermodynamics and statistical physics but also important for studying phase transition of physics system. Surprisingly, Lee-Yang zeros are observed by measuring quantum coherence of a probe spin coupled to an Ising-type spin bath [13]. In other word, the zeros really exist.

Actually, Lee-Yang zero has go beyond the statistical physics. Maloney and Witten [14] shown that the Hawking-Page transition [15] in $AdS_{3}$ space is Lee-Yang type, while the original Lee-Yang phase transition is only for two dimensional Ising model. This is an important reason that we argue that the phase transition in topological field theory is Lee-Yang type. Next, let us interpret the Lee-Yang phase transition in the $AdS_{3}$ space in more detail.

The Hawking-Page transition can be seen from Lee-Yang condensation of zeros in the partition function for $k\rightarrow\infty$ . Actually, the partition function $Z(\tau)$ of three dimensional gravity is a modular function which computes at fixed temperature Im $\tau$ and angular potential Re $\tau$ . In the limit of infinite volume, these zeroes condense along the phase boundaries. So, it gives rise to phase transition. The analog of the infinite volume limit for the partition function $Z(\tau)$ is $k\rightarrow\infty$ because as we know $k=\ell/16G$ implies that k directly proportional to the $AdS_{3}$ radius. In this case, the partition function $Z(\tau)$ is non analytic which corresponding to the occurrence of phase transition.

Our aim here, however, is not to understand the phase transition in $AdS_{3}$ space but the zeros in Jones polynomial or topological field theory. Then, there is a natural question to ask: What do these zeros mean? Or, in other word, can phase transition happen in topological field theory? In the following work, we show that it may shed light on this issue.

The Chern-Simons action for a gauge theory with gauge group $G$ and gauge field $A$ on an oriented three-manifold $M$ can be write as [16],

[TABLE]

Here $k$ is an integer for topological reasons. All we really need to know for now about $CS(A)$ is that it is gauge-invariant. The Feynman path integral now is formally an integral over the infinite-dimensional space of connections $A$ . This is a basic construction in the quantum field theory,

[TABLE]

To including a knot which it is an embedded oriented loop $K\subset M$ , we make use of the loop of the connection $A$ around $K$ . Picking an irreducible representation $R$ of $K$ , then one can associate an observable, the trace of Wilson loop operator,

[TABLE]

Then we define a natural invariant of the pair ( $M$ , $K$ ), So the partition function can write as,

[TABLE]

This is a topological invariant of the knot $K$ in the three manifold $M$ , which depends only on $G$ , $R$ and $K$ . Then, Witten made use of the topological invariance of the theory to solve Chern-Simons topological theory on three manifold $M$ with collection of knots, which is a suitably normalized Chern-Simons partition function,

[TABLE]

equals to the Jones polynomial,

[TABLE]

For modular transformation and $G=SU(2)$ we have,

[TABLE]

And we obtain the partition function on $S^{3}$ [8, 17],

[TABLE]

we have shown that the three-dimensional quantum theory gives a definition of the Jones polynomial of a knot. An important fact is that we can expand the partition function to a polynomial. Then it is natural to study the zeros of this polynomial. However this is not the whole story, we need to know the Lee-Yang zero first. Let $Z$ be the canonical partition function which is written as,

[TABLE]

where $E_{n}$ is the energy, $\mu$ is the chemical potential and $N$ is the number of particles. If we expand partition function $Z$ to a polynomial of $z$ [18],

[TABLE]

where $z=e^{\frac{\mu}{kT}}$ . Here one should note that $a_{j}$ $\left(j=1,2,\cdots,M\right)$ are all positive real number. Now Let us consider the root of the function,

[TABLE]

Then from the above two equations, one can find that there is no positive real root for the equation. If analytical extension to the complex plane, we find that all the zeros are on the unit circle which is the celebrated Lee-Yang unit circle theorem. But it is not the whole story. In physics, phase transition is a very interesting phenomenon. It is argued that phase transition can only happen when the zeros of the partition function has real part. In this case, the partition function is not a analytic function now. This comment is very important for the zeros of the of Jones polynomial.

Since both of the partition function of Lee-Yang and Jones polynomial can expand to polynomial. Then there may be natural link between the two structures. Now Let us consider the zeros of the Jones polynomial. From the previous section, one may find that the expectation value of a Wilson loop in representation with spin $\frac{1}{2}$ is,

[TABLE]

where

[TABLE]

If $\langle\mathcal{W}(K,R)\rangle=0$ , then it is means,

[TABLE]

So one can find that there is only one root $-1$ which is on the unit circle. The next case is the expectation value of Hopf link with 2 spin $\frac{1}{2}$ Wilson loop which can write as

[TABLE]

Then one can find there are three roots which are $-1$ , $-i$ , $i$ . In fact, according to the Lee-Yang theorem, then there is no phase transition. Actually, the expectation value of the Wilson loop equals to zero, then we can rewrite the Eq.(5) as,

[TABLE]

Then what we are interested in is,

[TABLE]

So, one can find the zero of the Jones polynomial is equal to the zero of $Z\left(M;K\right)=\int DA\exp\left(iCS\right)\prod_{i=1}^{r}W\left(K_{i},R\right)$ . It equals to say that phase transition do not happen in the topological field theory in the previous two case.

The above has shown that there is no phase transition in the trivial knot. Then it is necessary to study some nontrivial knots which phase transition could happen. Now Let us study the zeros of Jones polynomial of torus knot $J(m,n)$ [19] which distributes on $x$ axis. one should note that the simplest nontrivial example of this knot is the $(2,3)$ torus knot which is also known as the trefoil knot. In physics, since real-zeros of an equation usually represent observable values, it is interested to investigate them in advanced. The Jones polynomial of torus knot write as,

[TABLE]

If taking the limit of $n\rightarrow\infty$ , this leads to $n$ for all $m$ , then one can find that all roots are distributed uniformly on the unit circle. In fact, for all Jones polynomials of torus $J(p,q)$ , there are two positive real-zeros with one is 1 and another is inside $1<r<2$ [20]. Then by Lee-Yang theorem, one can find that the two real zeros are phase transition points. In other word, phase transition happen in this type topological field theory.

Topological field theory has been important for establishing symmetries and dualities in field theory and string theory [21, 22, 23]. A generalization of electric-magnetic duality to Yang-Mills theory, known as $S$ -duality, acts naturally on Vafa-Witten theory [9]. This duality, proposed by Montonen and Olive [24], states that Yang-Mills theory with gauge group $G$ and complexified coupling constant

[TABLE]

has a dual description as Yang-Mills theory, whose gauge group is the Langlands dual group and with inverse coupling constant $-1/\tau$ . Together with the periodicity of the $\theta$ -angle, this generates the $SL(2,\mathbb{Z})$ $S$ -duality group,

[TABLE]

The Vafa-Witten twist of $N=4$ super symmetric Yang-Mills theory with gauge group $SU(N)$ contains a commuting BRST-like operator $Q$ . For a suitable $I$ , the topologically twisted action of Vafa-Witten theory can be expressed as a $Q$ -exact term $\{Q,I\}$ , plus a term multiplying the complexified coupling constant $\tau$ :

[TABLE]

where $n$ denotes the instanton number,

[TABLE]

In fact, the partition function of the topological twisted super Yang-Mills field theory depend only on $\tau$ and the gauge group $SU(N)$ chosen, which is a modular form [9, 25, 26],

[TABLE]

Where $\chi$ is Euler characteristics, $w$ is the modular weight. For any finite value of $N$ , the partition function $Z(\tau)$ is smooth as a function of $\tau$ , but for large $N$ limit, as $N\rightarrow\infty$ , the function becomes non-smooth [14]. Or, in other words, the function is not a analytic function now. The original idea of Lee and Yang is that although a system in finite volume can have no phase transition, its partition function, depending on the complexified thermodynamic variables, can have zeroes. Then, in the infinite volume limit, the zeroes become more numerous and may become dense. Then, a true phase transition can emerge. In our problem, the limit $N\rightarrow\infty$ , is analogous to a thermodynamic limit. Or, in other words, Lee-Yang type phase transition happens in the topological twisted super Yang-Mills field theory in the large $N$ limit.

In summary, we introduce the Jones polynomial and topological field theory. In fact, the expectation value of the Wilson loop is the Jones polynomial. Witten find that the Hawking-page phase transition in $Ads_{3}$ space is Lee-yang type. Then we study the zeros of the topological field theory. And we shown that the zero of hopf link obey the Lee-Yang unit circle theorem. It is suggested that there is no phase transition in this link. Since all the Jones polynomials of torus $J\left(m,n\right)$ there are at least two real zeros. Then we argue that phase transition can happen in the torus knot. Or, more accurately, phase transitions can happen in the topological field theory which the corresponding knot is torus knot. And we also show that the Lee-Yang type phase transition happens in the topological twisted super Yang-Mills field theory in the large N limit.

The $\frac{1}{N}$ expansion of the free energy of Chern-Simons theory [27, 28] takes the form $F=\sum_{g,h}C_{g,h}N^{h}\kappa^{2g-2+h}=\sum_{g,h}C_{g,h}N^{2-2g}\lambda^{2g-2+h}$ . However, in the large N limit, it is possible to have phase transitions even in finite volume. Further evidence was found in that the Wilson lines observable in the topological string theory gives the same knot invariants [29]. Then there is natural question to ask: Can phase transition really happen in the topological string theory?

Acknowledgements.

We thanks for discussing with Xun Chen. Part of the work was done when Jing Zhou visited the Yau Mathematical Science Center.

Bibliography29

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