Daniel Doro Ferrante

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Solution Space of Gauge Theories and Quantum Phases

Quantum Phases are defined to be the different quantum states at zero temperature. A Quantum Phase Transition (QPT) happens when a system undergoes a change from one quantum phase to another, at zero temperature. It describes an abrupt change in the ground-state of the system caused by its quantum fluctuations. A Quantum Critical Point (QCP) is a point in the phase diagram of a system (at zero temperature) that separates two quantum phases. In addition to this, QCPs distort the fabric of the phase diagram creating a phase of “quantum critical matter” fanning out to finite temperatures from the QCP. As expected, at the QCP, the system exhibits spacetime scale invariance, justifying the idea that it can be modeled via a Conformal Field Theory (CFT) — because of this, sometimes the QCP is referred only as a “conformal point”. The connection with High Energy Physics, namely with Gauge Theory, can be made almost instantly. There are two ways to best understand this phenomenon. One of them is via the analytic continuation of the partition function of a given system. However, this can be particularly tricky, since it involves $\mathbb{C}$omplex (Picard–Lefschetz) and infinite-dimensional Morse theory. On the other hand, this scenario can be addressed in a constructive manner, that reveals more clearly what is at stake. If we start from a $0$-dimensional gauge theory ($D0$-brane), for any field (scalar, fermion, vector, matrix, tensor, Lie or graded Lie algebra), and evaluate its Schwinger–Dyson equation (which is nothing but the differential version of the integral problem treated via a partition function), we clearly see that, because this differential equation has as many solutions as its degree, we are supposed to have one integral representation for each one of these solutions — and each one of these integral representations is what we call a partition function. Therefore, there will be more than one possible partition function, one for each available quantum phase. And the relation among them can be clearly seen through appropriate analytic continuations and asymptotic analysis (Stokes' phenomena, wall-crossing), which is how this approach ties in with the previously mentioned one. This maps out the Solution Space of a given Gauge Theory, revealing its “glassy” character, somewhat analogous to what happens in spin systems. Furthermore, understanding the Feynman Path Integral as a Linear Canonical Transform, we can make an analogy with the Penrose–Ward transform, establishing a correspondence between two spaces, where one contains the appropriate integration cycles, while the other contains a certain choice of [vector] parameters for a representation of the system's singularities in terms of Fox's $H$-function (which is a Mellin–Barnes transform, usually of a more general character than a polylogarithm). Also, with this understanding, we are also able to generalize the interpretation of Feynman's Path Integral as a sum of more general paths than just a Brownian one, such as Lévy flights, and so on. This brings to the foreground the nonlinear Fredholm theory of the Schwinger–Dyson differential operator, which in turn has something to say about the construction of Seiberg–Witten invariants. Finally, understanding the role of the matrix parameter of the Linear Canonical Transform helps to clarify the origin of some dualities.


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