3-manifolds, q-series, and topological strings

Overview

Introduction

This means that the BPS state vector space provides a natural categorization of the homology block of 3-manifoldness. In other words, the homology block ˆ𝑍𝔤is a searchable, categorizable invariant of 3-dimensional manifolds.

This thesis

The second identity is the inverse of the first identity and can be verified with qZeil. Similarly, two of the main features of B𝔤(𝑌) are that it is affine isomorphic to 𝐻2(𝑌;𝑄) and that it allows an action of the Weyl group 𝑊 (and thus an action of 𝐻2(𝑌;𝑄) ⋊𝑊 performs).

Background

Chern-Simons theory

It can be shown that this is independent of the choice of a braid representing the bond. Homology blocks, which we introduce in the next section, can be thought of as a generalization of these analytically continued WRT invariants.

Figure 2.1: Yang-Baxter equation encodes the braid relation.

Homological blocks

Setting 𝑞 = 𝑒ℏ, its ℏ expansion corresponds to the Melin-Morton-Rozansky expansion of the colored Jones polynomials. In the classical limit, the 𝑅 matrices ˇ𝑅(𝑥)and ˇ𝑅−1(𝑥) can be obtained by specializing the parameters of the parametrized 𝑅 matrix.

Figure 2.3: A plumbing graph and the corresponding surgery link.

Large-color 𝑅 -matrix

The large-color 𝑅 -matrix

It is easy to see that all arcs of the open thread must carry 𝑣. 2 are (the squares of) the eigengenes of the holonomy around the meridians of the two strings, in 𝑆 𝐿.

Figure 3.4: Pairing and copairing.

Inverted state sum

If exactly two of the four arcs are in 𝑆 for each intersection, connect the ends of the two arcs. When all four bows are in 𝑆, connect the ends of the bows of the same strand (i.e. the same as in the link).

Figure 3.9: An inversion datum for the figure-eight knot.

Proof of Theorem 3.2.2

The significance of the parameterized 𝑅 matrix arises from the fact that they induce an algebra morphism in the following way. It follows that the result of the state sum of this new model is of random walk of free bosons. The effect of the sign factor on the positive and negative stabilization movements can be summarized as in Figure 3.18.

In a similar way, we get candidates of the inverted Habiro coefficients for all mirror twirls. The 𝑞 holonomicity of these cyclotomic coefficients follows from the 𝑞 holonomicity of the colored HOMFLY-PT polynomials. At the end of the previous chapter, we presented a conjecture about the existence of a three-variable sequence𝐹𝐾(𝑥, 𝑎, 𝑞) ​​​​which interpolates the fore𝔰𝔩𝑁𝐹𝐾.

The existence of the quantum 𝐴 polynomial was independently conjectured in the context of the quantization of the Chern-Simons theory [Guk05]. For example, the 𝑏 degree of the 𝐵 polynomial is equal to the 𝑦 degree of the 𝐴 polynomial (since there is a one-to-one correlation between the branches). The universal 𝑅 matrix, Burau representation and the Melvin-Morton expansion of the colored Jones polynomial”.

Figure 3.12: Four possible types of crossings

Dehn surgery

Inverted Habiro series

The series on the right is what we will call an inverted Habiro series. A priori, the "reverse Habiro series" is a completely different object compared to the original Habiro series. For another thing, the inverse Habiro series can be extended to a power series in𝑥or𝑥−1, which is not possible for the regular Habiro series.

The distinction is clearer in the classical limit𝑞 → 1, where the usual Habiro sequences take values ​​in Z[[𝑥+𝑥−1−2]] (a completion in a finite place𝑥+𝑥−1 = 2), while the converse Habiro series take values ​​in Q[[𝑥+𝑥1−. For every node 𝐾 with Δ𝐾(𝑥) ≠ 1, it has an inverted Habiro sequence, and corresponds to the 𝐹𝐾 in the sense that . With the following identity we can transform a power series in 𝑥 into an inverted Habiro series and vice versa.

It follows from this proposition and Theorem 3.3.1 that for every nice knot 𝐾 (such as every homogeneous braid knot) that is not a non-knot, 𝐹𝐾 can be expressed as an inverted Habiro series. Proposition 4.1.3 states that the inverted Habir series and the v𝑥 power series contain the same amount of information. Nevertheless, the inverted Habiro series is a particularly nice way to represent 𝐹𝐾, since it is clearly Weyl symmetric.

Connection to indefinite theta functions

This three-variable series has a natural interpretation in terms of topological strings, which is the subject of the next chapter. The contribution of the vectorℓ = ( ®ℓ𝑙,0,ℓ®𝑟) for the upper graph to the theta function is the same as the contribution of the vectorℓ′=( ®ℓ𝑙,ℓ®𝑟) for the lower graph. 5.8) Therefore, ˆ𝑍𝔤 is invariant under the Type A Neumann shift. This is due to the reduction of the Weyl symmetry to Z2, as we specialize to symmetric representations.

According to the volume conjecture, it also captures the asymptotics of the colored Jones polynomials 𝐽𝐾 ,𝑟(𝑞) for large colors𝑟. The vacuum modulus space in 4dN =4 SYM is Sym𝑁(C3).) The gauge symmetry 𝑆𝑈(𝑁) of this theory appears as a global symmetry of the 3d limit theory𝑇[𝑀𝐾]. This brane configuration is essentially a variant of (6.2) with conormal Lagrangian notation).

For the SFT-stretch to work well, we should be able to move the Lagrangian completely off the zero section𝑆3. The variables 𝑥 and 𝑦 of the 𝐴-polynomial correspond to the eigenvalues ​​of the meridian holonomy and the knot length. Therefore, it is natural to expect that there are invariants analogous to 𝐹𝐾(𝑥 , 𝑎, 𝑞) ​​associated with the other branches of the 𝐴-polynomial.

We have seen that in the Abelian branch, the expectation value of the ˆ𝑏 operator gives a 𝑎 deformation of the inverse Alexander polynomial. On other branches 𝑦(𝛼)(𝑥 , 𝑎), the expectation value of the ˆ𝑏 operator will be some other function; let's define.

Figure 4.1: Double cone vs one-sided cone.

Higher rank

Higher rank ˆ 𝑍

Higher rank 𝐹 𝐾

Moreover, this series should be annihilated by (higher rank) quantum 𝐴- polynomial:. 5.23) Our main result in this section is an explicit expression for 𝐹𝔤. This can be derived either directly from (5.2) using plumbing description or by reverse construction using the higher-order operational formula that we discuss below. To calculate the integral (5.2) with 𝑥−𝑠𝑡 left unintegrated, we just replace the theta functionΘ−𝐵(𝑥−1, 𝑞) with.

Since 2−degree𝑣 is non-zero for only 3 vertices (central vertex𝑣− . 1 and 2 terminal vertices) it is very easy to calculate. This is a theorem for knots and 3-manifolds represented by negative-definite hydraulics, as a direct generalization of Theorem 1.2 of [GM21].

Figure 5.2: Plumbing diagram of the complement of 𝑇 𝑠,𝑡 . where 𝑃 𝑗 ( x ) ∈ Z [ 𝑥

Symmetric representations and large 𝑁

The 𝐴-polynomial 𝐴𝐾(𝑥 , 𝑦) is an invariant of the polynomial knot defining the algebraic curve {(𝑥 , 𝑦) ∈ (C∗)2 | 𝐴𝐾(𝑥 , 𝑦) = 0}, which is the projection of the characteristic variety of the complement of the knot onto the boundary torus [Coo+94]. Indeed, near the limit 𝑇2= Λ𝐾 =𝜕 𝑀𝐾 compactification of 𝑁 of five branes produces a 4dN =4 theory which has the space of vacuum modules Sym𝑁(C2×C∗) [Chu+20]. If 𝑦(𝛼)(𝑥) ∼𝑥𝑑is asymptotically close to𝑥 = 0, for some 𝑑 ∈Q, then we can use this as an initial condition and find a solution of the form.

Since there appears to be some correlation between the window of good surgery coefficients (Remark 4.0.2) and the boundary slope of the 𝐴 polynomial Newton polygon, it is not too far-fetched to speculate that this non-abelian branch 𝐹𝐾's may play a role in gaining a full understanding of ˆ𝑍. The DT invariants have two geometric interpretations, either as the intersection homology Betti numbers of the moduli space of all semisimple representations of 𝑄 of dimension vector, or as the Chow-Betti numbers of the moduli space of all simple representations of𝑄of dimension vectorized; see [MR19;FR18]. It is an interesting problem to understand, given a socket for one branch, how to get a socket for another branch of the same node.

Topological strings

Topological strings and 𝐹 𝐾

More generally, the colored HOMFLY-PT polynomials𝑃𝐾 , 𝑅(𝑎, 𝑞) ​​are polynomial knot invariants that generalize the HOMFLY-PT polynomial, which also depends on a representation (a Young diagram)𝑅. We will be mainly interested in the HOMFLY-PT polynomials colored by the totally symmetric representations. To simplify the notation, we will denote them by 𝑃𝐾 ,𝑟(𝑎, 𝑞) ​​​​and simply call them the HOMFLY-PT polynomials.

The 𝐴 polynomial can be further generalized for the colored HOMFLY-PT polynomials [AV12] and colored superpolynomials [Awa+12;FGS13], which we briefly introduced in (6.1). Equivalently, one can use the topological invariance along 𝑆3 to move the tubular neighborhood from 𝐾 ⊂ 𝑆3 to "infinity". This creates a long neck isomorphic to R×𝑇2, as in the discussion above. Anyway, we end up with a system of 𝑁 five branes on the node complement and no extra branes on 𝐿𝐾, so that the choice of 𝐺 𝐿(𝜌,C) flat connection on𝐿𝐾 is now encoded in the boundary condition for 𝑆 𝐿(𝑁 , C)connection2on𝑇2 =𝜕 𝑀𝐾.

To summarize, a system of 𝑁 pentabrane in a node complement (6.2) is equivalent to a brane configuration (6.4), with a suitable map connecting the boundary conditions in both cases. Both of these systems on the solved side compute HOMFLY-PT polynomials of𝐾 colored by Young diagrams with at most rows𝜌. Mathematically, what the physical picture above shows us is that 𝐹𝐾(𝑥 , 𝑎, 𝑞) ​​is the count of open topological sequences in the resolved conifold 𝑋, with the completion of the Lagrangian node𝑀𝐾 ⊂ ⊂ .

Branches

As mentioned briefly in the previous section, a branch choice corresponds to a vacuum choice in 3D theory𝑇[𝑀𝐾]. The abelian branch always corresponds to the slope∞(or equivalently𝑑 =0), which is why the two-variable series𝐹𝐾(𝑥 , 𝑞) we discussed in previous chapters do not have the exponential prefactor𝑒𝑑. One is the abelian branch, and there are two non-abelian branches with boundary slope ±1.

The abelian branch 𝐹𝐾(𝑥 , 𝑞), as we have considered at length in previous chapters, was part of a larger story involving closed 3-manifolds. On the other hand, it is currently not clear whether the non-abelian branch𝐹(𝛼).

Holomorphic Lagrangian subvarieties

Physically, this holomorphic Lagrangian corresponds to the Coulomb branch of a 3d-5d coupled system, which should have a quantization. This conjecture can be generalized even more by introducing the 𝑡 variable and its conjugate, which we denote by 𝑢.

Table 6.1: Classical 𝐵 -polynomials for some simple knots.

Knots-quivers correspondence

The similarity between knots and quivers can be generalized to knot complements, as proposed in [Kuc20] and extensively studied in [Ekh+22]. Then it is an assignment of a symmetric quiver𝑄, an integer 𝑛𝑖, and half integers,𝑎𝑖,𝑙𝑖,𝑖 ∈ 𝑄. 0 to a given node complement𝑀𝐾 =𝑆3\𝐾 in such a way that. 6.11). Ekh+] Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, and Piotr Sulkowski.𝑍bat large𝑁: From curve counts to quantum modularity.

Ekh+22] Tobias Ekholm, Angus Gruen, Sergej Gukov, Piotr Kucharski, Sunghyuk Park, Marko Stošić in Piotr Sułkowski.